Studies in Choice and Welfare Series Editors M. Salles (Editor-in-Chief) P. K. Pattanaik K. Suzumura
Jeffrey S. Banks 1958-2000
David Austen-Smith · John Duggan Editors
Social Choice and Strategic Decisions Essays in Honor of Jeffrey S. Banks
With 41 Figures and 17 Tables
^
Springer
David Austen-Smith Earl Dean Howard Distinguished Professor of Political Economy Department of Managerial Economics & Decision Sciences Kellogg School of Management Northwestern University 2001 Sheridan Road Evanston, IL 60208-2009 USA John Duggan Associate Professor of Political Science and Economics W. Allen Wallis Institute of Political Economy University of Rochester Rochester, NY 14627 USA
Library of Congress Control Number: 2004116749
ISBN 3-540-22053-4
Springer Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner Production: Helmut Petri Printing: betz-druck SPIN 11010081
Printed on acid-free paper - 43/3130 - 5 4 3 2 1 0
To Shannon, Bryan, and Danny
Preface
Jeffrey Scot Banks was born on December 15th 1958 in San Diego, California. He received his BA in Political Science from the University of California Los Angeles in 1982 and his PhD in Social Sciences from the California Institute of Technology in 1986. Jeff's first appointment was as an Assistant Professor of Political Science and Economics at the University of Rochester, New York. In 1988 the Political Science department asked if they could put him up for an early tenure decision in Political Science; Jeff denied the request but accepted a promotion to Associate Professor of Political Science and Economics without tenure. Deciding not to risk rejection yet again, the departments of Political Science and Economics at Rochester simply proceeded to promote Jeff to tenured Full Professor of Political Science and Economics in 1991. At least as of the time of writing, Jeff has been the only faculty member in the College of Arts and Science at University of Rochester to receive tenure simultaneously in two departments. Jeff remained at Rochester until 1997 when he joined the Division of Humanities and Social Sciences at the California Institute of Technology. During the intervening years he was at various times a visiting professor at the Universities of Arizona and Michigan and at the California Institute of Technology; he also spent a year as a Fellow at the Center for Advanced Study in the Behavioral Sciences. Jeff received many honors beyond a fairly regular flow of NSF Research Awards. These included an Alfred P. Sloan Foundation Research Fellowship (1989-91), a National Science Foundation Presidential Young Investigator Fellowship (1989-94), a National Academy of Sciences Award for Scientific Reviewing (1996) and being elected a Fellow of the Econometric Society (1996). Apart from generating a steady stream of important and influential papers, Jeff proved to be an outstanding teacher. His easy disposition and unbridled, infectious enthusiasm for research made his graduate game theory class at Rochester de rigueur for almost all of the economics and political science students who came through; indeed, one graduate student at Rochester was overheard to opine that "Jeff Banks could teach game theory to a stone."
VIII
Preface
In February of 1995, Jeff was diagnosed with leukemia. The situation worsened rapidly and he received a bone marrow transplant at Strong Memorial Hospital in Rochester in June of the same year. Although Jeff never complained and many never realized, the transplant proved increasingly difficult to manage and the associated complications became ever more severe. In October 2000 he was admitted into hospital in Pasadena, California. He died on December 21st 2000 in the City of Hope National Medical Center, Duarte, California, six days after his forty-second birthday. He is survived by his wife, Shannon, and sons Bryan and Danny. In the time between diagnosis of the cancer and his death, Jeff wrote eighteen papers, completed a book manuscript and advised several graduate students. He possessed a remarkable intellect and strength of will; but above all, Jeff expressed a joy in life that made it impossible not to like him. He is greatly missed. Acknowledgments The ten papers of this volume for Jeff are all invited contributions. The only requirement we asked of a contribution was that it be original and reflect some aspect of Jeff's own work and interests in economics and political science. Almost all of the contributors were either coauthors, faculty colleagues, sometime students, or long-time friends of Jeff, and our primary debt is to them for their unmitigated enthusiasm for the project. We are also grateful to Mark Fey, Ben Klemens, Tapas Kundu, and Piotr Kuszewski, all of whom helped at some stage or other in readying the manuscript for publication. Finally, we would like to thank Maurice Salles, managing editor of Social Choice and Welfare, and Martina Bihn of Springer-Verlag Publishers for their continued support, encouragement, and patience.
Rochester, NY December, 2004
David
Austen-Smith John Duggan
Preface
IX
Jeffrey S. Banks Publications and Papers Articles 1. Sophisticated voting outcomes and agenda control. Social Choice and Welfare, 1985, 1: 295-306. 2. Price-conveyed information vs. observed insider behavior: a note on rational expectations convergence. Journal of Political Economy, 1985, 93: 807-815. 3. Equilibrium selection in signaling games, (with Joel Sobel), Econometrica, 1987, 55: 647-661. 4. Endogenous agenda formation in three person committees, (with Farid Gasmi), Social Choice and Welfare, 1987, 4: 133-152. 5. An experimental analysis of unanimity in public goods provision mechanisms, (with Charles Plott and David Porter), Review of Economic Studies, 1988, 55: 301-322. 6. Elections, coalitions, and legislative outcomes, (with David Austen-Smith), American Political Science Review, 1988, 82: 405-422. 7. Voting games, indifference, and consistent sequential choice rules, (with George Bordes), Social Choice and Welfare, 1988, 5: 31-44. 8. Equilibrium outcomes in two-stage amendment procedures, Journal of Political Science, 1989, 33: 25-43.
American
9. Electoral accountability and incumbency, (with David Austen-Smith), in Models of Strategic Choice in Politics, Ordeshook, P. (ed) Ann Arbor: University of Michigan Press, 1989. 10. Allocating uncertain and unresponsive resources: an experimental approach, (with John Ledyard and David Porter), Rand Journal of Economics, 1989, 20: 1-25. 11. Agency budgets, cost information, and auditing, American Journal of Political Science, 1989, 33: 670-699. 12. Explaining patterns of candidate competition in congressional elections, (with Rod Kiewiet), American Journal of Political Science, 1989, 33: 9971015. 13. A model of electoral competition with incomplete information. Journal of Economic Theory, 1990, 50: 309-325. 14. Monopoly agenda control and asymmetric information. Quarterly Journal of Economics, 1990, 105: 445-464. 15. Repeated games, finite automata, and complexity, (with Rangarajan Sundaram). Games and Economic Behavior, 1990, 2: 97-117.
X
Preface
16. Equilibrium behavior in crisis bargaining games, American Journal of Political Science, 1990, 34: 599-614. 17. Stable governments and the allocation of policy portfolios, (with David Austen-Smith), American Political Science Review, 1990, 84: 891-906. 18. Monotonicity in electoral systems, (with David Austen-Smith), American Political Science Review, 1991, 85: 531-537. 19. The politics of commercial R&D programs, (with Linda Cohen and Roger Noll), in The Technology Pork Barrel, Cohen, L. and R. Noll (eds) Washington: Brookings, 1991. 20. The space shuttle program, in The Technology Pork Barrel, Cohen, L. and R. Noll (eds) Washington: Brookings, 1991. 21. Covering relations, closest ordering, and Hamiltonian bypaths in tournaments, (with Georges Bordes and Michel Le Breton), Social Choice and Welfare, 1991, 8: 355-363. 22. The political control of bureaucracies under asymmetric information, (with Barry Weingast), American Journal of Political Science, 1992, 36: 509-524. 23. A battle-of-the-sexes game with incomplete information, (with Randall Calvert), Games and Economic Behavior, 1992, 4: 347-372. 24. A class of bandit problems yielding myopic optimal strategies, (with Rangarajan Sundaram), Journal of Applied Probability, 1992, 29: 625-632. 25. Monopoly pricing and regulatory oversight. Journal of Economics and Management Strategy, 1992, 1: 203-233. 26. Denumerable-armed bandits, (with Rangarajan Sundaram), Econometrica, 1992, 60: 1071-1096. 27. Two-sided uncertainty in the monopoly agenda setter model, Journal of Public Economics, 1993, 50: 429-444. 28. Adverse selection and moral hazard in a repeated elections model, (with Rangarajan Sundaram), in Political Economy: Institutions, Information, Competition, and Representation, Barnett, W., M. Hinich and N. Schofield (eds) New York: Cambridge University Press, 1993. 29. An experimental analysis of Nash refinements in signaling games, (with Colin Camerer and David Porter), Games and Economic Behavior, 1994, 6: 1-31. 30. Switching costs and the Gittens index, (with Rangarajan Sundaram), Econometrica, 1994, 62: 687-694. 31. The design of institutions: an agency theory perspective, in Institutional Design, Weimer, D. (ed) Boston: Kluwer, 1995. 32. Acyclic social choice from finite sets. Social Choice and Welfare, 1995, 12: 293-310.
Preface
XI
33. Singularity theory and core existence in the spatial model. Journal of Mathematical Economics, 1995, 24: 523-536. 34. Information aggregation, rationality, and the Condorcet jury theorem, (with David Austen-Smith), American Political Science Review, 1996, 90: 34-45. 35. An experimental analysis of the two-armed bandit problem, (with Mark Olson and David Porter), Economic Theory, 1997, 10: 55-78. 36. Social choice theory, game theory, and positive political theory, (with David Austen-Smith), Annual Review of Political Science, 1998, 1: 259288. 37. Optimal retention in agency problems, (with Rangarajan Sundaram), Journal of Economic Theory, 1998, 82: 293-323. 38. Committee proposals and restrictive rules, Proceedings of the National Academy of Sciences, 1999, 96: 8295-8300. 39. A model of price promotions with consumer search, (with Sridhar Moorthy), International Journal of Industrial Organization, 1999, 17: 371-398. 40. Cycling of simple rules in the spatial model, (with David Austen-Smith), Social Choice and Welfare, 1999, 16: 663-672. 41. Cheap talk and burned money, (with David Austen-Smith), Journal of Economic Theory, 2000, 9: 1-16. 42. A bargaining model of collective choice, (with John Duggan), American Political Science Review, 2000, 94: 733-788. 43. Buying supermajorities in finite legislatures, American Political Science Review, 2000, 94: 677-681. 44. Costly signaling and cheap talk in models of political influence, (with David Austen-Smith), European Journal of Political Economy, 2002, 18: 263-280. 45. Bounds for mixed strategy equilibria and the spatial model of elections, (with John Duggan and Michel Le Breton), Journal of Economic Theory, 2002, 103: 88-105. 46. Strategic aspects of political systems, in Handbook of Game Theory III, Aumann, R. and Hart, S. (eds) Amsterdam: North Holland, 2002. 47. Probabilistic voting in the spatial model of elections: the theory of oflScemotivated candidates, (with John Duggan), in Social Choice and Strategic Decisions: Essays in Honor of Jeffrey S. Banks, Austen-Smith, D. and J. Duggan (eds) Berlin: Springer, 2004. 48. Social choice and electoral competition in the general spatial model, (with John Duggan and Michel Le Breton), Journal of Economic Theory, forthcoming.
XII
Preface
49. A social choice lemma on voting over lotteries with applications to a class of dynamic games, (with John Duggan), Social Choice and Welfare, forthcoming.
Books 1. Signaling Games in Political Science, Chur: Harwood Academic Publishers, 1991. 2. Modern Political Economy: Old Topics, New Directions, (co-edited with Eric Hanushek), New York: Cambridge University Press, 1995. 3. Positive Political Theory I: Collective Preference, (with David AustenSmith), Ann Arbor: University of Michigan Press, 1999. 4. Positive Political Theory II: Strategy and Structure, (with David AustenSmith), Ann Arbor: University of Michigan Press, 2005.
Comments and reviews 1. Comment on Jankowski's "Punishment in Iterated Chicken and Prisoners Dilemma Games" Rationality and Society, 1991, 3: 381-385. 2. Review of Cristina Bicchieri and Maria Luisa Dalla Chiara (eds) Knowledge, Belief, and Strategic Interaction, and Robert Koons, Paradoxes of Belief and Strategic Rationality, American Political Science Review, 1993, 87: 1000-1001. 3. Review of E. Roy Weintraub (ed) Toward a History of Game Theory, Journal of Interdisciplinary History, 1995, 25: 647-648. 4. Review of Melvin Hinich and Michael Munger, Analytical Politics, Journal of Economic Literature, 1998, 36: 1506-1507.
Unpublished manuscripts (as of October 2004) 1. A multidimensional model of repeated elections, (with John Duggan) 2. Existence of Nash equilibria on convex sets, (with John Duggan) 3. A bargaining model of legislative policy-making, (with John Duggan)
Honors and Activities Fellowships and aw^ards Clarence J. Hicks Memorial Foundation Fellowship, 1983-84
Preface
XIII
John and Dora Haynes Foundation Fellowship, 1984-85 Alfred P. Sloan Foundation Dissertation Fellowship, 1985-86 Alfred P. Sloan Foundation Research Fellowship, 1989-91 National Science Foundation Presidential Young Investigator, 1989-94 National Academy of Sciences Award for Scientific Reviewing, 1996 Elected Fellow of the Econometric Society, 1996 Fellow, Center for Advanced Study in the Behavioral Sciences, 1997-98 National Science Foundation Research Grants Co-PI with David Austen-Smith: 1988-1990, 1989-1991, 1995-1996 Co-PI with Randy Calvert: 1989-1991 Co-PI with John Duggan: 1999-2001 Editorial responsibilities Board member. Social Choice and Welfare, 1990-2000 Board member. Games and Economic Behavior, 1990-2000 Board member, American Journal of Political Science, 1997-2000 Associate editor. Journal of Economic Theory, 1991-2000 Associate editor. Review of Economic Design, 1993-2000 Associate editor. Journal of Public Economics, 1997-2000 Associate editor. Journal of Public Economic Theory, 1998-2000 Co-editor, Journal of Economics and Management Strategy, 1991-1998 Guest co-editor, (with Roger Myerson), Special Issue on Political Theory, Games and Economic Behavior, 1993, volume 5 Service Economics Panel member. National Science Foundation, 1992-94 Co-Organizer, (with Eric Hanushek), 1992 Wallis Conference on Political Economy Program Committee member, 2nd International Conference on Social Choice and Welfare, 1994 Program Committee member, 7th World Congress of the Econometric Society, 1995 Co-Organizer, (with David Austen-Smith), 1996 Wallis Conference on Political Economy Division Chair in Formal Political Theory, 1997 American Political Science Association meetings
XIV
Preface
Program Committee member, Econometric Society Summer 1997 meetings Program Committee member. International Conference on Logic, Game Theory and Social Choice, 1999 Co-Organizer, (with John Duggan and Mark Fey), 2000 Wallis Conference on Political Economy
Contents
Introduction and Overview David Austen-Smith,
John Duggan
1
Probabilistic Voting in the Spatial Model of Elections: The Theory of Office-motivated Candidates Jeffrey S. Banks, John Duggan
15
Local Political Equilibria Norman Schofield
57
Electoral Competition Between Two Candidates of Different Quality: The Effects of Candidate Ideology and Private Information Enriqueta Aragones, Thomas R. Palfrey
93
Party Objectives in the "Divide a Dollar" Electoral Competition Jean-Frangois Laslier
113
Generalized Bandit Problems Rangarajan K. Sundaram
131
The Banks Set and the Uncovered Set in Budget Allocation Problems Bhaskar Dutta, Matthew 0. Jackson, Michel Le Breton
163
Experiments in Majoritarian Bargaining Daniel Diermeier, Rebecca Morton
201
XVI
Contents
Legislative Coalitions in a Bargaining Model with Externalities Randall L. Calvert, Nathan Dietz
227
Testing Theories of Lawmaking Keith Krehbiel, Adam Meirowitz, Jonathan Woon
249
Deliberation and Voting Rules David Austen-Smith, Index
Timothy Feddersen
269 317
Introduction and Overview David Austen-Smith^ and John Duggan^ ^ Northwestern University dasm<9kellogg.northwestern.edu ^ University of Rochester duggStroi . cc. rochester. edu
Jeff Banks entered graduate school in the fall of 1982 and died in the winter of 2000, from complications following treatment for leukemia. Those years witnessed a rapid growth in non-coooperative game theory per se, in its application to political science and in the development of experimental methods with which to test such applications. Jeff made important contributions to each of these literatures. By the time he arrived at his first faculty position at the University of Rochester, Jeff (with Joel Sobel) had both proposed a useful and widely used equilibrium refinement, known as "divinity," for games of incomplete information [22] and (on his own) exploited this refinement to study the first model of Downsian electoral competition under incomplete information about candidate preferences [8]; he had characterized the set of (undominated subgame perfect Nash) equilibrium outcomes for committee voting over finite amendment agendas [5], subsequently called the "Banks set";^ and, with Charles Plott and David Porter, provided an experimental analysis of the unanimity rule in mechanisms for public good provision [21]. And, as even a cursory reading of his vita makes clear, he went on in subsequent years to contribute extensively to social choice, to positive political theory more generally, and to game theory. Jeff's influence in political science especially, both through his research and his teaching, was and remains considerable, and the ten original papers comprising this volume reflect this influence as well as his interests. We organize the contributions to this volume, with the relevant literatures in political science, by the scale of political decision-making: electoral competition among candidates (politics in the large) and legislative and committee behavior (politics in the small). This terminology for the Banks set made its creator feel a little awkward, and Jeff avoided it. In lighter conversation, however, he was known to refer to it as the "me set".
2
David Austen-Smith and John Duggan
1 Electoral Competition The theory of elections originated in locational models proposed by Hotelling [54], Downs [41], and Black [33], who focused on the policy stances taken by two candidates prior to an election. Emerging from that work, the median voter theorem dictates that the unique equilibrium is for both candidates to adopt the ideological location of the median voter as their campaign platforms. This result offers a precise and intuitively compelling prediction of electoral outcomes, but only under the key assumptions that political debate is restricted to a single policy issue and that voters preferences are single-peaked. The implication is that there exists a policy, the median ideal point, that beats or ties all others according to majority rule. Work in social choice theory (Mas-Colell and Sonnenschein [63], Nakamura [74], Banks [13], Schwartz [85]) has shown, however, that the existence of such a point can be problematic, due to the existence of "social preference" cycles. Later work by Plott [78] established the inherent mathematical intractability of more realistic models of elections in multidimensional environments: equilibrium points must satisfy conditions so stringent that they would be expected to exist only rarely. This generic non-existence result was formalized and extended to more general voting rules by Schofield [83, 84], Banks [12], and Saari [82]. Much of the recent development of the theory of elections can be seen as an attempt to generalize the basic model of elections to accommodate more realistic features of politics. DiflPerent lines of research have investigated incomplete information in elections, the use of mixed strategies by a candidate to prevent her opponent from adopting a winning policy, and the incorporation of dynamic aspects of elections. Incomplete information has been incorporated in a number of ways: in voters' knowledge of each other's costs of voting or other attributes (Ledyard [62], Palfrey and Rosenthal [76]); in voters' knowledge of candidate preferences (Banks [8]); and in candidates' knowledge of voter preferences. The latter vein of literature, known as "probabilistic voting," seeks to smooth out discontinuities in candidate payoffs by assuming the candidates are uncertain about voting behavior, thereby obtaining existence of equilibrium in multidimensional elections. Beginning with the work of Hinich et al. [52, 53], this literature has taken up issues of existence and characterization of equilibrium for different specifications of the model. In the second chapter of this volume. B a n k s and D u g g a n synthesize and generalize known results in this literature. The authors consider the class of models in which two candidates compete for votes and, as a special case, the ballot of each voter is determined by the voter's policy preferences and by an additive bias unobserved by the candidates. Banks and Duggan cover results on the existence of pure and mixed strategy equilibria, and they present several equilibrium characterization results in terms of social welfare optima. They show that, very generally, the candidates must adopt identical policy positions in equilibrium, a phenomenon they call "policy coincidence." They
Introduction
3
give a number of continuity results for electoral equilibria, and in particular they show that when bias is close to negligible, equilibria of the electoral game must be close to the core; and if there is no core point, then pure strategy equilibria will not exist in the probabilistic voting game. Thus, adding a small amount of uncertainty about voting behavior cannot overcome the equilibrium existence problem. Finally, the authors revisit the question of whether the median voter theorem is an "artifact," as claimed by Hinich [51], and they establish close connections to the class of binary Luce models studied by Coughlin and Nitzan [39] and Coughhn [38]. Schofield also contributes to the Uterature on probabilistic voting, considering local equilibria, in which parties cannot find small profitable moves from their policy platforms. In contrast to Chapter 2, and much of the literature on elections, Schofield allows for the possibility that multiple parties compete for votes. He finds a necessary and sufl&cient condition for the existence of a local equilibrium in multiple dimensions, one that involves the distribution of voter ideal points and the exogenous valences of the parties; he shows that this condition is satisfied generically, providing generic existence of a local equilibrium; and he examines a number of examples, including elections marked by "activist" valence, where some ideological voters condition their votes on a party's platform, irrespective of the platforms of other parties. Also in the probabilistic voting framework, Aragones and Palfrey analyze a diff"erent informational aspect of elections. As in Aragones and Palfrey [1], there are two candidates, a one-dimensional policy space, and one candidate has a valence advantage: other things equal, all voters would vote for her. In addition, the authors suppose the candidates have private information about their policy preferences and the weights they attach to holding oflSce. An advantage of this formulation is that there exist equilibria in pure strategies, whereas only mixed strategy equilibria obtain in the complete information model, and these strategies have a simple characterization in terms of cut-off values: in equilibrium, a candidate moderates if and only if the weight she attaches to holding office exceeds a certain threshold. The authors prove that there is a unique equilibrium in pure strategies. As the weight on holding office increases for both candidates, these pure strategy equilibria converge to the mixed strategy equilibrium found in Aragones and Palfrey [1], providing a purification argument for the mixing found in that model. Whereas the above papers modify the basic electoral framework by incorporating informational considerations, a different approach to the equilibrium existence problem is to retain the traditional framework but to relax the solution concept used: while results from social choice theory have shown that pure strategy electoral equilibria will rarely exist when the policy space is multidimensional, an alternative is to analyze mixed strategy equilibria of the electoral game. This approach has connections to the work of Borel [35] and Gross and Wagner [50] on the Colonel Blotto game, where two generals must allocate forces to a number of battle fields; at any battle field, the larger army
4
David Austen-Smith and John Duggan
is assumed to win; and each general is assumed to maximize the number of battles won. While this game has no pure strategy equilibrium, Borel [35], Gross and Wagner [50], and Laslier and Picard [61] give explicit constructions of mixed strategy equilibria. This model can be applied to elections for which distributive politics is the central issue: we interpret generals as candidates, who allocate pork barrel spending, and battle fields as voters, and we can then use the mixed strategy equilibria found by these authors to examine the distribution of pork barrel spending in elections. Mixed strategy equilibria in the context of the spatial model of elections were considered by Kramer [57], who proved existence of equilibrium in a model with a continuum of voters and vote-maximizing candidates. Banks et al. [18] take up the characterization issue. Assuming candidates seek to maximize probability of winning (in contrast to Kramer [57]), they prove that the support of mixed strategy equilibria must be contained in the uncovered set, a cooperative solution defined by Miller [69] and developed by McKelvey [64] and Banks et al. [19]. Laslier examines a special case of the Blotto game, in which the electorate is divided into three groups, which may be targeted by candidates in an election. Laslier compares two particular mixed strategy equilibria derived under alternative assumptions about the objectives of candidates: in one case, candidates care about their plurality (i.e., they maximize the number of votes), and in the other case, candidates care only about winning a majority of votes (so only the sign, rather than the magnitude, of the vote difference matters). The author finds that the two equilibria differ with respect to the degree of inequality in their treatment of voters. With plurality-maximizing candidates, all voters receive the same amount of spending by candidates; with candidates who maximize probability of victory, however, voters belonging to small groups receive a higher amount of pork. The literature on electoral accountability, beginning with Barro [32] and Ferejohn [46], explicitly models elections as repeated over time. Dispensing with the assumption that candidates can credibly commit to campaign promises, the literature examines the extent to which future elections can discipline the choices of current office holders. Austen-Smith and Banks [3] considered a two-period moral-hazard model of elections, allowing voters to condition on the discrepancy between an incumbent's previous campaign platform and her realized policy. Banks and Sundaram [25, 27] consider infinite-horizon environments in which a representative voter chooses between an incumbent and challenger in each period, with the goal of maximizing the effort levels of politicians. Duggan [42] considers a spatial setting with one policy dimension, and Banks and Duggan [16] provide existence and characterization results for a class of simple electoral equilibria. The contribution by Sundaram analyzes the class of dynamic decisionmaking problems known as "bandit" problems. Here, in each period over an
Introduction
5
infinite horizon, an agent must choose one of a number of "arms" to "pull." Each arm generates a stochastic payoff, but the payoff distributions of the arms are unknown to the agent. Rather, she must learn over time by experimentation. For the case of a finite number of possible arms, Gittens and Jones [49] show that the arms can be ranked by likely value, appropriately measured, in what is now called the "Gitten index," providing a characterization of a decision-maker's optimal strategy for choosing which arm to play at any stage of the process (see Gittins [48]). Banks and Sundaram [23, 24, 26] proved a variety of results on the structure of the Gitten index for the bandit problem with a countably infinite number of arms, capturing the mathematical structure of the accountability problem, where a representative voter must decide which from a set of candidates (i.e., arms) to elect (i.e., pull). In his contribution, Sundaram reviews and extends work on the subject, and he extends the Gittens characterization of the optimal strategy to problems in which the set of arms is uncountably infinite. The author also considers problems in which individuals may play several arms simultaneously and in which there are switching costs for changing arms, demonstrating difficulties for the Gittens index in these environments.
2 Legislative and C o m m i t t e e Behavior The formal theoretical literature concerned with understanding democratic legislative behavior essentially began with Black [33] and Farquharson [45], who explored some aspects of voting in committees, and Riker [79], who offered a cooperative game-theoretic theory of coalitions. Building on Farquharson and on Black and Newing [34], Kramer [56] and Shepsle [86] explore spatial models of legislative decision-making that explicitly focus on the institutional details of committee organization and the rules that govern agenda-formation and voting. Since then, the literature has grown rapidly with an important focus of the work on these issues among others. The canonical problem confronting a majoritarian committee is to select one alternative from a given finite set. There are many ways to do this but most of the literature on the problem assumes that the relevant scheme is a binary agenda with majority voting. Loosely speaking, a binary agenda is any process whereby a committee choice is determined by some prescribed sequence of binary votes (typically under majority rule) to eliminate all but one alternative (which may well be "none of the above"). The most studied of such choice processes are the amendment and the successive elimination agendas. Under an amendment agenda, the available alternatives are first ordered, say ( a o , a i , . . . ,am), and then voting proceeds by first choosing between ao and ai by majority vote, rejecting the loser and then choosing between the winner and a2 using majority vote, and so on until there remains only one alternative, the committee choice.
6
David Austen-Smith and John Duggan
A great deal is now known about this problem (see Miller [71]). In particular, McKelvey and Niemi [65] and Miller [68] showed that the set of alternatives achievable as undominated Nash equilibrium outcomes to some binary agenda was precisely the top cycle set for the underlying majority preference relation. Miller [69] defined a subset of the top cycle set, the uncovered set, showing that it necessarily contains the set of alternatives similarly achievable from amendment agendas. A full characterization of the set of alternatives achievable under amendment agendas, a subset of the uncovered set, was provided by Banks [5, 6]; subsequently, this set became known as the Banks set and has proven to be of wider interest in the theory of majority tournaments (Laslier [60]) and two-party spatial competition (Miller et al. [72]). So while agenda-setters constrained only to use a binary agenda can, with a suitable selection of the agenda, elicit any alternative at all from the top cycle set (which, in principle, can include all of the available alternatives), Banks's characterization result implies that agenda-setters constrained to using only amendment agendas possess less ability to manipulate committee decisions through agenda-selection. The first of the papers concerned with legislative decision-making in this volume, by Dutta, LeBreton, and Jackson, further explores the value of the Banks set and the uncovered set for committee choice. These authors consider the classical problem of a committee choosing allocations in a variety of distributive policy settings, including those with a continuum of available alternatives. Legislative choice of such distributive allocations is a terribly hard problem for which the typical result is that virtually every possible allocation can arise as an equilibrium outcome. Thus the focus of the Dutta et al. analysis is on the restrictions (if any) imposed in these settings by the uncovered set and the Banks set. Importantly, they show that both choice sets can limit the class of acceptable decisions, at times severely. Legislators and committee members do not simply vote over a given set of alternatives; inter alia they first decide upon the alternatives over which to vote. The first model of endogenous agenda-setting in committees is due to McKelvey [64], who allowed legislators in a spatial voting model each to ofi'er a proposal (or none at all) for consideration, following which an amendment agenda was chosen with which to make a final decision. McKelvey did not specify how the proposed alternatives might be ordered on the agenda, but assumed instead that individuals oflPered proposals under a "worst case" assumption. In a three person spatial voting model of endogenous agendaselection, Banks and Gasmi [20] assume that committee members offer proposals in a fixed order, say (ai, a2, as), and, once all proposals are determined, that an amendment agenda with the alternatives considered in the reverse order to the proposal sequence, a^ vs a2 with the winner against ai, is used to reach the final committee choice. More general versions of the Banks and Gasmi endogenous agenda-setting model are explored in subsequent work by Austen-Smith [2], Duggan [43] and Penn [77]. Unfortunately, although the
Introduction
7
Banks and Gasmi approach seems fairly natural, it is hard to analyze. An alternative approach to agenda-setting is through legislative bargaining theory. Rubinstein [81] is generally credited for introducing the two-person, alternating-offers model of bargaining to economics. Subsequently, Baron and Ferejohn [31] built on Rubinstein's insights to develop an n-person bargaining model of distributive politics. The basic Baron-Ferejohn problem is that of a majoritarian committee choosing how to allocate pork-barrel spending, viewed simply as allocations of a fixed and fully divisible "dollar". At the start of any period. Nature chooses one committee member to oflFer a proposed allocation which is then subjected to a majority vote for approval; if a majority votes in favor, then the process ends and the proposal is implemented; if a majority votes against the proposal, then the process moves to the start of a new period and repeats. The sequence continues until some proposal is accepted. Unlike unconstrained majority rule in this setting, the legislative bargaining model possesses many equilibria; restricting attention to stationary (history independent) equilibria, however, yields tight predictions. In particular, with complete information, the first proposal offered is accepted by a minimum winning coalition in which the agenda-setter receives the largest share of the dollar. Although Baron and Ferejohn constructed stationary equilibria for their particular model with risk-neutral legislators in the pure division problem, they offered no general equilibrium existence theorem. Such a result was provided by Banks and Duggan [15], who established the existence of stationary equilibria in the Baron-Ferejohn model, allowing for a very broad class of voting rules, risk-aversion and asymmetries between legislators. And, in Banks and Duggan [17], they extend the results and the model beyond the transferable utility case epitomized by divide-the-dollar to the general spatial policy model, showing some subtle connections between the stationary bargaining equilibrium set and the core of the relevant voting rule. Not surprisingly, the Baron-Ferejohn model has (and continues to be) widely applied and explored (e.g.. Baron [29], Winter [87], Merlo [67], Morelh [73], Banks [14], Eraslan [44], Kalandrakis [55]), and the contributions by Diermeier and Morton and by Calvert and Dietz are in this vein. Diermeier and Morton subject the Baron and Ferejohn model to experimental test. Their experimental design exploits a finite horizon, weighted majority bargaining model with a three person committee dividing a dollar. Although the design does not map exactly onto the basic Baron-Ferejohn model (in particular, the finite horizon precludes stationary strategies), the essential logic and qualitative properties of that model carry over. And in this setting, Diermeier and Morton find the Baron-Ferejohn model wanting: none of the theoretical predictions from the canonical bargaining model receive any support from their treatment. The authors observe that an equal sharing rule does best in explaining the observed final divisions of the resource, but argue that no strong inference can be drawn from the prevalence of this rule regarding any motivations of fairness on the part of the agents; equal sharing
8
David Austen-Smith and John Duggan
is a focal distribution and extremely easy to compute in what is a relatively complex experiment. Moreover, even the equal sharing rule accounts for only 50 to 75 percent of accepted proposals. Calvert and Dietz are interested in the formation of legislative parties. One implication of the canonical Baron-Ferejohn model is that while the winning proposal is supported by a minimum winning coalition, that coalition is chosen randomly by the agenda-setter. Such random coalition formation, both within and across periods, hardly reflects a legislative party. Baron [28, 30] attempts to introduce parties to the bargaining model, in the first case by looking at groups of legislators who can commit ex ante to offering the same (mutually profitable) proposal (a division of the dollar) conditional on being selected as agenda-setter and, in the second case, by considering bargaining in a spatial policy setting where parties have ideological preferences. Neither of these efforts is wholly satisfactory: the pure distribution model excludes a primary concern of parties with policy and the spatial policy model is sufficiently intractable that only a handful of special cases can be solved. Calvert and Dietz attempt to finesse both limitations by adopting the usual pure distribution setting (divide-the-dollar) but assuming externalities in consumption of the dollar. With this assumption, the authors derive "party-like behavior" that persists across periods and, while induced by preference-similarities, is derived from purely individually optimal strategic behavior. The paper by Krehbiel, Meirowitz and Moon gets away from legislative bargaining theory but is also concerned with the role of legislative parties in social choice. Their focus is on disentangling the two principal competing theories of the role of parties in Congress. The pivotal politics model (Krehbiel [58]) puts primacy on individual legislators' policy preferences and legislative rules, which induce two veto points, effectively attenuating the importance of party for legislative outcomes; in contrast, the party cartel model (Cox and McCubbins [40], Rohde [80]) highlights the significance of the party leadership and agenda-control over legislator preferences for legislative decision-making. Distinguishing these two models empirically is problematic, as many of their respective predictions coincide. Nevertheless, Krehbiel et al. manage to identify some observable distinctions between the two and also between each model separately and a benchmark, pure-preference, median voter model. Using data from the US Senate rather than the laboratory, their empirical results are not very encouraging for either the pivotal politics model or the party cartel model, at least under the one-dimensional politics assumption of the analysis. It has long been recognized that informational asymmetries and limitations play a role in political decision making, just as they do in more classical economic environments. The early formal work on informational limitations in politics largely concerned uncertainties in electoral competition (see Calvert [36] for a review of this literature). Efi'orts to consider the role of information in legislative settings are more recent and began in earnest with Gilligan and Krehbiel [47], who sought to explain the pattern of rules used under the
Introduction
9
committee system in the U.S. Congress. Jeff Banks' contributions to the informational literature in political science ranged from electoral competition (e.g., [8, 25]) to providing a foundation for the Niskanen model of bureaucracy ([7]), from the theory of monopoly agenda-setting ([9, 11]) to crisis bargaining in international relations ([10]). One strand of the hterature on information in politics, however, which has a peculiarly long history and was developed largely independently, concerns the Condorcet Jury Theorem. This theorem asserts that if committee members share identical full information preferences but have imperfect and distinct assessments about which of two alternatives (convict or acquit) is the "correct" committee choice (convict if guilty, acquit otherwise), then, subject to some regularity conditions, all individuals voting sincerely with their information leads to a majority decision being more likely to be "correct" than is any individual decision. Subsequent work has generalized the Jury Theorem both by weakening the original regularity assumptions and by extending its domain of application (e.g., Nitzan and Paroush [75], Miller [70], Ladha [59]). All of these contributions maintained the Condercetian assumption of sincere voting. Austen-Smith and Banks [4] relaxed this assumption and showed that, when there are no opportunities for communication in a committee, all individuals voting sincerely and informatively over a fixed binary agenda is almost never an undominated Nash equilibrium, despite it being common knowledge that all individuals have identical full information preferences. Moreover, while Condorcet's conclusion obtains at some equilibria (McLennan [66]), there are others where it fails. The assumption of no communication in the Condorcet setup is restrictive. In particular, if committee members are sure that they all share the same full information preferences, then it is intuitive that they would freely make public any decision-relevant information they might hold if given the opportunity prior to voting (Coughlan [37]). But when committee members are unsure of others' preferences, the extent to which individuals might be willing to share private information is not so clear. In the final contribution to the volume, Austen-Smith and Feddersen study committee decision-making with such communication, or deliberation, over a fixed binary agenda under incomplete information. They ask what happens when committee members may talk costlessly prior to voting. However, although identical full information preferences is one possible scenario, they allow for preference heterogeneity to the extent that the incomplete information not only concerns the character of the available alternatives but also a private preference parameter, a "bias" or predisposition in favor of one or other of the alternatives. In this context they argue that majority rule induces superior incentives for informative communication relative to unanimity rule and, inter alia, explore some of the qualitative properties of equilibrium speech-making.
10
David Austen-Smith and John Duggan
References [1] Aragones E. and Palfrey T. (2002) Mixed equilibrium in a Downsian model with a favored candidate. Journal of Economic Theory, 103: 131161. [2] Austen-Smith, D. (1987) Sophisticated sincerity: voting over endogenous agendas. American Political Science Review, 81: 1323-1329. [3] Austen-Smith, D. and J. S. Banks (1989) Electoral accountability and incumbency, in Models of Strategic Choice in Politics, Ordeshook, P. (ed) Ann Arbor: University of Michigan Press. [4] Austen-Smith, D. and J. S. Banks (1996) Information aggregation, rationality and the Condorcet jury theorem. American Political Science Review, 90: 34-45. [5] Banks, J. S. (1985) Sophisticated voting outcomes and agenda control. Social Choice and Welfare, 1: 295-306. [6] Banks, J. S. (1989a) Equilibrium outcomes in two-stage amendment procedures. American Journal of Political Science, 33: 25-43. [7] Banks, J. S. (1989b) Agency budgets, cost information, and auditing. American Journal of Political Science, 33: 670-699. [8] Banks, J. S. (1990) A model of electoral competition with incomplete information. Journal of Economic Theory, 50: 309-325. [9] Banks, J. S. (1990b) Monopoly agenda control and asymmetric information. Quarterly Journal of Economics, 105: 445-464. [10] Banks, J. S. (1990c) Equilibrium behavior in crisis bargaining games. American Journal of Political Science, 34: 599-614. [11] Banks, J. S. (1993) Two-sided uncertainty in the monopoly agenda setter model. Journal of Public Economics, 50: 429-444. [12] Banks, J. S. (1995a) Acyclic social choice from finite sets. Social Choice and Welfare 1995, 12: 293-310 [13] Banks, J. S. (1995b) Singularity theory and core existence in the spatial model. Journal of Mathematical Economics, 24: 523-536. [14] Banks, J. S. (1999) Committee proposals and restrictive rules. Proceedings of the National Academy of Sciences, 96: 8295-8300. [15] Banks, J. S. and J. Duggan (2000) A bargaining model of collective choice. American Political Science Review, 94: 733-788. [16] Banks, J. S. and J. Duggan (2002) A multidimensional model of repeated elections. Mimeo. University of Rochester. [17] Banks, J. S. and J. Duggan (2002) A bargaining model of legislative policy-making. Mimeo. University of Rochester. [18] Banks, J. S., J. Duggan, and M. Le Breton (2002a) Bounds on mixed strategy equilibria and the spatial model of elections. Journal of Economic Theory 103: 88-105. [19] Banks, J. S., J. Duggan, and M. Le Breton (2004) Social choice and electoral competition in the general spatial model. Journal of Economic Theory, forthcoming.
Introduction
11
[20] Banks, J. S. and F. Gasmi (1987) Endogenous agenda formation in three person committees. Social Choice and Welfare, 54: 133-152. [21] Banks, J. S., C. Plott and D. Porter (1988) An experimental analysis of unanimity in public goods provision mechanisms. Review of Economic Studies, 55: 301-322. [22] Banks, J. S. and J. Sobel (1987) Equilibrium selection in signaling games. Econometrica, 55: 647-661. [23] Banks, J. S. and R. Sundaram (1992a) A class of bandit problems yielding myopic optimal strategies. Journal of Applied Probability, 29: 625-632. [24] Banks, J. S. and R. Sundaram (1992b) Denumerable-armed bandits. Econometrica, 60: 1071-1096. [25] Banks, J. S. and R. Sundaram (1993) Adverse selection and moral hazard in a repeated elections model. In Political Economy: Institutions, Information, Competition, and Representation, Barnett, W., M. Hinich and N. Schofield (eds) New York: Cambridge University Press. [26] Banks, J. S. and R. Sundaram (1994) Switching costs and the Gittens index. Econometrica, 62: 687-694. [27] Banks, J. S. and R. Sundaram (1998) Optimal retention in agency problems. Journal of Economic Theory 82: 293-323 [28] Baron, D. P. (1989) A noncooperative theory of legislative coalitions. American Journal of Political Science, 33: 57-90. [29] Baron, D. P. (1991a) Majoritarian incentives, pork barrel programs, and procedural control. American Journal of Political Science, 34: 57-90. [30] Baron, D. P. (1991b) A spatial bargaining theory of government formation in parliamentary systems. American Political Science Review, 85: 137-164. [31] Baron, D. P. and J. A. Ferejohn (1989) Bargaining in legislatures. American Political Science Review, 83: 1181-1206. [32] Barro, R. (1973) The Control of Politicians: An Economic Model, Public Choice 14: 19-42. [33] Black, D. (1958) The Theory of Committees and Elections, Cambridge: Cambridge University Press. [34] Black, D. and R.A. Newing (1951) Committee Decisions with Complementary Valuations. London: William Hodge. [35] Borel, E. (1953) The theory of play and integral equations with skew symmetric kernels. Econometrica 21: 97-100. [36] Calvert, R. (1986) Models of Imperfect Information in Politics. Chur: Harwood Academic Publishers. [37] Coughlan, P. (2001) In defense of unanimous jury verdicts: mistrials, communication and strategic voting. American Political Science Review 94: 375-93. [38] Coughlin, P. (1992) Probabilistic Voting Theory, Cambridge: Cambridge University Press.
12
David Austen-Smith and John Duggan
[39] Coughlin, P. and S. Nitzan (1981) Electoral outcomes with probabilistic voting and Nash social welfare maxima. Journal of Public Economics, 15: 113-121. [40] Cox, G. W. and M. D. McCubbins (1993) Legislative Leviathan: Party Government in the House. Berkeley: Berkeley University Press. [41] Downs, A. (1957) An Economic Theory of Democracy. New York: Harper and Row. [42] Duggan, J. (2000) Repeated elections with asymmetric information. Economics and Politics 12: 109-136. [43] Duggan, J. (2003) Endogenous voting agendas. Social Choice and Welfare, forthcoming. [44] Eraslan, H. (2002) Uniqueness of stationary equilibrium payoffs in the Baron-Ferejohn model. Journal of Economic Theory, 103: 11-30. [45] Farqhuarson, R. (1969) Theory of Voting. New Haven: Yale University Press. [46] Ferejohn J. A. (1986) Incumbent performance and electoral control. Public Choice, 50: 5-25. [47] Gilligan, T. and K. Krehbiel (1987) Collective decision-making and standing committees: an informational rationale for restrictive amendment procedures. Journal of Law, Economics and Organization, 3: 287-335. [48] Gittins, J. (1989) Allocation Indices for Multi-Armed Bandits. London: Wiley. [49] Gittins, J. and D. Jones (1974) A dynamic allocation index for the sequential allocation of experiments. In J. Gani et al. (eds) Progress in Statistics Amsterdam: North Holland. [50] Gross, O. and R. Wagner (1951) A continuous Colonel Blotto game. Working Paper, RAND RM-408. [51] Hinich, M. (1977) Equilibrium in spatial voting: the median voter result is an artifact. Journal of Economic Theory, 16: 208-219. [52] Hinich, M., J. Ledyard, and P. Ordeshook (1972) Nonvoting and the existence of equilibrium under majority rule. Journal of Economic Theory, 4: 144-153. [53] Hinich, M., J. Ledyard, and P. Ordeshook (1973) A theory of electoral equilibrium: a spatial analysis based on the theory of games. Journal of Politics, 35: 154-193. [54] Hotelling, H. (1929) Stability in competition. Economic Journal, 39: 4 1 57. [55] Kalandrakis, T. (2004) A three player dynamic majoritarian bargaining game. Journal of Economic Theory, 116: 294-322. [56] Kramer, G. (1972) Sophisticated voting over multidimensional choice spaces. Journal of Mathematical Sociology, 2: 165-180. [57] Kramer, G. (1978) Existence of electoral equilibrium, in P. Ordeshook (ed) Game Theory and Political Science, NYU Press: New York. [58] Krehbiel, K. (1998) Pivotal Politics: A Theory of U.S. Lawmaking. Chicago: University of Chicago Press.
Introduction
13
[59] Ladha, K. (1992) The Condorcet jury theorem, free speech and correlated votes. American Journal of Political Science, 36: 617-634. [60] Laslier, J.-F. (1997) Tournament Solutions and Majority Voting. Berlin: Springer. [61] Laslier, J.-F. and N. Picard (2002) Distributive politics and electoral competition. Journal of Economic Theory, 103: 106-130. [62] Ledyard, J. (1984) The pure theory of large two-candidate elections. Public Choice, 44: 7-41. [63] Mas-Colell, A. and H. Sonnenschein. (1972) General possibility theorems for group decisions. Review of Economic Studies, 39: 185-192. [64] McKelvey, R. D. (1986) Covering, dominance and institution-free properties of social choice. American Journal of Political Science, 30: 283-314. [65] McKelvey, R. D. and R.G. Niemi (1978) A multistage game representation of sophisticated voting for binary procedures. Journal of Economic Theory, 18: 1-22. [66] McLennan, A. (1998) Consequences of the Condorcet jury theorem for beneficial information aggregation by rational players. American Political Science Review, 92: 413-418. [67] Merlo, A. (1997) Bargaining over governments in a stochastic environment. Journal of Political Economy, 105: 101-131. [68] Miller, N. R. (1977) Graph-theoretical approaches to the theory of voting. American Journal of Political Science, 21: 769-803. [69] Miller, N. R. (1980) A new solution set for tournaments and majority voting. American Journal of Political Science, 24: 68-96. [70] Miller, N. R. (1986) Information, electorates and democracy: some extensions and interpretations of the Condorcet jury theorem. In Grofman, B. and G. Owen (eds) Information Pooling and Group Decision-Making. Greenwich, Con.: JAI Press. [71] Miller, N. R. (1995) Committees, Agendas and Voting. Chur: Harwood Academic Publishers. [72] Miller, N. R., B. Grofman and S. Feld (1990) The structure of the Banks set. Public Choice, 66: 243-251. [73] Morelli, M. (1999) Demand competition and policy compromise in legislative bargaining. American Political Science Review, 93: 809-820. [74] Nakamura, K. (1979) The vetoers in a simple game with ordinal preferences. International Journal of Game Theory, 8: 55-61, [75] Nitzan, S. and J. Paroush (1985) Collective Decision-Making. Cambridge: Cambridge University Press. [76] Palfrey, T. and H. Rosenthal (1985) Voter participation and strategic uncertainty. American Political Science Review, 79:62-78. [77] Penn, E. (2003) A distributive N-amendment game with endogenous agenda-setting. Mimeo. Carnegie-Mellon University. [78] Plott, C. (1967) A notion of equilibrium and its possibility under majority rule. American Economic Review 57: 787-806.
14
David Austen-Smith and John Duggan
[79] Riker, W.H. (1962) The Theory of Political Coalitions. New Haven: Yale University Press. [80] Rohde, D. W. (1991) Parties and Leaders in the Postreform House. Chicago: University of Chicago Press. [81] Rubinstein, A. (1982) Perfect equilibrium in a bargaining model. Econometrica, 50: 97-109. [82] Saari, D. (1997) The generic existence of a core for g-rules. Economic Theory, 9: 219-260. [83] Schofield, N. (1980) Generic properties of simple Bergson-Samuelson welfare functions. Journal of Mathematical Economics, 7: 175-192. [84] Schofield, N. (1983) Generic instability of majority rule. Review of Economic Studies, 50: 695-705. [85] Schwartz, T. (2001) From Arrow to cycles, instability, and chaos by untying alternatives. Social Choice and Welfare, 18: 1-22. [86] Shepsle, K. A. (1979) Institutional arrangements and equilibrium in multidimensional voting models. American Journal of Political Science, 23: 27-59. [87] Winter, E. (1996) Voting and vetoing. American Political Science Review, 90: 813-823.
Probabilistic Voting in the Spatial Model of Elections: The Theory of Office-motivated Candidates'*^ Jeffrey S. Banks^ and John Duggan^ ^ California Institute of Technology ^ University of Rochester duggQtroi. cc. r o c h e s t e r . edu S u m m a r y . We unify and extend much of the literature on probabilistic voting in two-candidate elections. We give existence results for mixed and pure strategy equilibria of the electoral game. We prove general results on optimality of pure strategy equilibria vis-a-vis a weighted utilitarian social welfare function, and we derive the well-known "mean voter" result as a special case. We establish broad conditions under which pure strategy equilibria exhibit "policy coincidence," in the sense that candidates pick identical platforms. We establish the robustness of equilibria with respect to variations in demographic and informational parameters. We show that mixed and pure strategy equilibria of the game must be close to being in the majority rule core when the core is close to non-empty and voters are close to deterministic. This controverts the notion that the median (in a one-dimensional model) is a mere "artifact." Using an equivalence between a class of models including the binary Luce model and a class including additive utility shock models, we then derive a general result on optimality vis-a-vis the Nash social welfare function.
1 Introduction T h e spatial model of elections, originating in the work of Hotelling [19], Downs [12], and Black [8], is perhaps the simplest formalization of democratic poHtics: each of two candidates adopt policy platforms before an election, voters cast their votes, and the winner implements his or her platform. Theorists differ, of course, in their treatments of the details of the model. Common to most work on the spatial model is the assumption t h a t candidates can credibly commit to their campaign promises.'^ Beyond t h a t , more candidates may be * The second author gratefully acknowledges support from the National Science Foundation, grant number SES-0213738. ^ A branch of the literature considers citizen-candidates, who cannot commit to their platforms. See Osborne and Slivinski [27], Besley and Coate [7], and Banks and Duggan [5].
16
Jeffrey S. Banks and John Duggan
permitted; the policy space may be finite, one-dimensional, or even multidimensional; the set of voters may be finite or a continuum, with preferences satisfying assorted conditions; the candidates' objectives may be to maximize their probabilities of winning, or they may seek to maximize their pluralities, or they may be motivated by policy preferences, or their objectives may be some mix of these alternatives. In addition, the nature of the voting decision is at issue: in much work on the spatial model, the strategy of each voter is modelled as the solution of a known maximization problem depending on the platforms of the candidates, allowing the candidates to anticipate with certainty the voters' responses to diff'erent policy positions. Because political candidates are unlikely in reality to be able to predict election returns with certainty, let alone the vote of any one voter, and because models of that sort often suff'er from difficult technical problems, a theoretical literature has treated voting as a random variable, i.e., it is assumed that voters vote probabilistically from the perspective of candidates. This approach is consistent with the rational choice approach to voting and elections, if we regard this randomness as arising from unobserved parameters in the maximization problems of voters. As we discuss in Section 2, much of the theory developed in this paper is motivated by one such model in which randomness enters the decision of a voter through a shock, or "bias" term, added to the voter's payoff's from the candidates' policy platforms. Specifically, we might suppose that each candidate possesses non-policy characteristics, or there are some policy dimensions on which the candidates cannot commit, that are relevant to the voter. The voter's utility from having a candidate take office is then equal to the sum of the utility from the candidate's policy platform and a utility increment, unobserved by the candidates, due to these fixed factors. The voter simply votes for the candidate offering the highest utility. Though the literature on probabilistic voting has a long tradition in political science and political economy, the many models considered in that literature lack a unifying framework. As a consequence, the results of the models are difficult to compare, and superficial similarities and differences are difficult to interpret. Our objective in this paper is to supply such a framework, while generalizing many known results and providing a new, rigorous analysis of continuity of equilibria with respect to changes in the electoral environment. Our analysis is limited in two respects: we allow for only two candidates, and we assume that candidates maximize expected plurality. The first limitation is a widely considered benchmark. The second is common in the probabilistic voting literature, and is sometimes consistent with probability of winning maximization,^ but we limit ourselves to office-motivated candidates. Within the confines of those two qualifications, we cast our net quite widely: we allow for a multi-dimensional policy space, a finite set or continuum of voter types, and general concave voter utility functions. "^ See Patty [28] and Duggan [13] and references found there for results on the equivalence of expected plurality and probability of winning maximization.
Probabilistic Voting in the Spatial Model of Elections
17
Our analysis ranges over several traditional topics of interest: existence and uniqueness of electoral equilibria, the characterization of equilibrium outcomes as social welfare maxima, and the incentives for candidates to adopt identical platforms in equilibrium, a phenonomenon we refer to as "policy coincidence." In addition, we provide several results on the robustness of equilibrium platforms with respect to electoral parameters, such as the distribution of voter types. And we consider the nature of equilibrium platforms when the set of unbeatable policy positions, i.e., the core,^ is close to non-empty and voting behavior is close to deterministic. Throughout, we have attempted to disentangle these issues as much as possible. For example, some authors prove existence and uniqueness of pure strategy equilibria in the models they consider, simultaneously showing that the candidates choose the same platform in equilibrium, and also characterizing that platform as a social welfare optimum. In contrast, we first give general conditions under which mixed strategy equilibria exist, then stronger conditions under which pure strategy equilibria exist, and then stronger conditions guaranteeing uniqueness. Our results on existence and some on uniqueness of pure strategy equilibria involve continuity and concavity conditions on the voters' probabihty of voting functions and hark back to the work of Hinich et al. [17, 18]. We take up the issue of optimality of equilibria and find weak conditions under which equilibria correspond to utilitarian social welfare optima: in the Additive Bias Model, for example, if the voter' biases are identically and continuously distributed, and if voters have concave utility functions, then every pure strategy equilibrium exhibiting policy coincidence maximizes the sum of the voters' utility functions. Our general theorems unify results by Hinich [16], Lindbeck and WeibuU [23, 24], and Coughlin [10]. We then take up the issue of policy coincidence, found in numerous theoretical studies of elections, including Hotelling [19]. Combined with our existence results, a simple symmetry condition on probability of voting functions yields at least one equilibrium exhibiting policy coincidence. But the phenomenon of policy coincidence is much deeper than this. We drop the assumption of candidate symmetry and find that pure strategy equilibria necessarily exhibit policy coincidence under very weak assumptions: in the Additive Bias Model, for example, if voters have strictly concave utility functions, then every pure strategy equilibrium exhibits policy coincidence. We offer a general result on the robustness of equilibrium platforms with respect to variations in underlying electoral parameters, an issue broached by Calvert [9].^ The problem is formulated generally enough that we can consider variations in the distribution of voter types, or we can consider variations in the voters' probability of vote functions. We show, for example, that if voter types are distributed normally with some variance, say a, then a small change ^ Policy positions in the core are sometimes referred to ais "Condorcet winners." ^ Technically, we prove that the equilibrium correspondence of the game between the candidates has closed graph.
18
Jeffrey S. Banks and John Duggan
in a cannot lead to equilibrium outcomes far away from the equilibria of the original model — a desirable property when the parameters of a model are difficult to observe in applications. In the model with deterministic voters, the core, if non-empty, consists of exactly the platforms used in pure or mixed strategy equilibria. But emptiness of the core is typical when the space of policy positions is multi-dimensional, and this equivalence only obtains under deterministic voting in any case. Thus, by itself, this characterization is of limited use in understanding the location of equilibrium platforms in multi-dimensional models with probabihstic voters. We show, however, that if voter preferences are such that the core is "close" (in a precise topological sense) to being non-empty and voters are "close" to being deterministic, then pure and mixed strategy equilibrium platforms must be "close" to being in the core. These core convergence results shed light on an old debate, initiated by Hinich [15, 16] and taken up by Kramer [20], about the fragility of the median in the one-dimensional model with deterministic voters: an implication of our results is that, when a small amount of randomness is added to the choices of voters, pure strategy equilibria typically will not exist, hut mixed strategy equilibria will exist, and they must put probability close to one on platforms close to the median. The latter conclusion contradicts Hinich's [16] Theorem 2, which claims that equilibria would be at the mean (not the median), and corroborates Kramer's claim of median robustness. In much of the paper, we assume that voters respond to differences in utility levels, and accordingly we characterize equilibrium platforms as utilitarian social welfare optima. Finally, we consider a class of models in which voters respond to ratios of utility levels, including, for example, the Binary Luce Model used by Coughlin and Nitzan [11] and Coughlin [10]. Exploiting a tight connection between the two classes of models, we show how results for Utility Difference Models can be translated directly into results for Utility Ratio Models. For example, we derive the results of the latter authors on Nash social welfare optima from our results on utilitarian social welfare optima. Moreover, our result holds for the Additive Bias Model if the voter' biases are identically and continuously distributed and utility functions are log-concave — the assumption of the Luce functional form for probability of vote functions is unneeded. Two notable papers on probabilistic voting that are not covered by our analysis are Ledyard [22] and, recently, McKelvey and Patty [26]. In both papers, the probability a voter votes for one candidate or another, fixing their policy platforms, is determined in the equilibrium of a game of incomplete information among the voters. In Ledyard's paper, each voter has a private cost of voting which, along with the probability of being pivotal given others' strategies, determines the voter's decision to vote or abstain. This generates probability of vote functions for the candidates and defines the electoral game. Ledyard shows that there is a unique electoral equilibrium, and that the policy platforms adopted in equilibrium maximize the sum of voters' utilities.
Probabilistic Voting in the Spatial Model of Elections
19
Though these themes appear later in our paper, Ledyard's framework is quite different from ours: he assumes candidates maximize probability of winning, rather than expected plurality, and his probability of vote functions would be likely to fail many of our conditions. Thus, his results cannot be obtained from ours. McKelvey and Patty [26], unlike other work cited above, allow for multiple candidates. They suppose that each voter has a private value attached to voting for the different candidates, above and beyond the utility from their policy platforms, and they show that, for a large enough electorate, there exists a pure strategy equilibrium in which all candidates locate at the utilitarian social welfare optimum. In the two-candidate case with no abstention, their model appears to be quite similar to the Additive Bias Model, as the value of voting for a candidate looks like an additive utility shock. In the model of McKelvey and Patty, however, the utility increment for a candidate is realized if the voter merely votes for the candidate, regardless of whether the candidate wins: a voter must consider the sum of the utility from a candidate's policy platform, weighted by the probability of being pivotal, and the value of voting for that candidate. Because pivot probabilities are determined in equilibrium, their probability of vote functions will not typically satisfy our conditions, and their results cannot be obtained from ours.
2 T h e Framework 2.1 The Electoral Game Two candidates (or parties) A and B simultaneously choose points XA and XB in a multi-dimensional policy space X C R^. A measure space of voters then decide between the candidates in a majority rule election. We suppose the electorate is partitioned into a probability space (T, T, /j,) of types: the probability a voter's type lies in (measurable) 5 C T is fi{S). We take as given a family of functions Pf: X x X ^^ R, C e {A,B}, and where we impose the following minimal mathematical structure: •
Pf^ixA^^B)
is jointly measurable in t and XA for each fixed XB-
•
P^{XA',XB)
is jointly measurable in t and
•
—1 < Pf
{XA^XB)
XB
for each fixed
XA-
< 1 for each {t^XA^XB) and each C.
These functions are meant to capture uncertainty underlying the decisions of voters. It may be that, in contrast to the usual rational choice assumption, voters actually behave randomly; or the candidates may have subjective beliefs about how voters vote; or, the explanation we prefer, the candidates may face objective probabilities derived from an underlying environment of incomplete information. We give specific examples in the next subsection.
20
Jeffrey S. Banks and John Duggan
At this point, we have not given the functions Pf a specific interpretation. It could be, for example, t h a t P^{XA^XB) is the probability t h a t a type t voter or it could be the expected share would vote for C given platforms {XA,XB)\ of votes for C from type t voters; or it could be the expected plurality share for C from type t voters, where the share is calculated as a proportion of the total number of type t voters. We will select the latter interpretation by maintaining the following assumption. •
P^{XA,XB)
+ P^{XA,XB)
== 0 for each
{I^XA^XB)-
We then define the electoral game by identifying each candidate's payoff" with is a the candidate's expected plurality share in the election. Thus, {XA.XB) pure strategy electoral equilibrium if XA solves max: /
Pt'^{x,XB)Kdt)
xeX
and XB solves Pf{xA,x)ii{dt). max: / xex We say a platform pair {XA.XB) exhibits policy coincidence if XA — XB- We consider alternative objective functions t h a t have also been used to model oflfice-motivation at the end of this section. We will also consider equilibria in which each candidate C may use a mixed strategy, a Borel probably measure, denoted TTC on the policy space X. Formally, letting U denote the space of Borel probability measures on X , a pair (TTA^TTB) is a mixed strategy electoral equilibrium if TTA solves m a x / / / P^{xA,XB)7T{dxA)7rB{dxB) -^^n Jj, j ^ j ^
Kdt)
and TXB solves m a%X x ^ JT Jx
P^{xA,XB)T^{dXB)T^A{dXA)l^{dt). Jx
Equivalently, nc must p u t probability one on the platforms t h a t maximize C's expected plurality share. W h e n considering continuity properties of mixed strategy electoral equilibria, we endow the space of mixed strategy pairs with the weak* topology.^ ^ That is, a sequence {TT^} of probability measures converges to a probability measure TT if and only if, for every bounded and continuous f: X -^ R, we have //d7r--.//d7r.
Probabilistic Voting in the Spatial Model of Elections
21
2.2 Special Cases of the Model We would like foundations for the expected plurality share functions given above that connect candidates' payoffs to voting behavior based, at least in part, on voters' policy preferences. To this end, let the policy preferences of a type t voter be given by i^^: X —> R, where Ut{x) is bounded and jointly measurable. Let Ut = {ut{x) \ x G X} denote the range of the type t voters' utility functions. In the remainder of this section, we introduce some important special cases of the general model. We begin with one in which our results do not apply. The Random Utility Model. In its simplest form, the electorate is finite and voters' preferences are quadratic (i.e., Ut{x) = —\\x — x*|p), where the ideal point x^ of the type t voter is drawn from a continuous distribution function Ft on X. Then P^{XA-IXB) is the probability that the ideal point x^ is closer to candidate C's platform: in the one-dimensional model, for example, we have P^''{XA^XB)=
[Ft{^^^) liXA<XB I l-Ft{^^^)\iXB<XA \\ if ^A ^ XB'
It is a "folk theorem" that the unique pure strategy equilibrium is for both candidates to locate at the median of the distribution of the median voter's ideal point. Because the candidates' objective functions are clearly discontinuous here, and because continuity of the expected plurality share functions is critical for the results of this paper, our analysis does not cover the Random Utility Model. We will focus largely on the next model and several of its special cases. The Fixed Utility Model The policy preferences of voters are now fixed and expected plurality shares depend on policy positions only through utilities: there exist functions P^: R^ ^ R such that P^{XA^XB)
=
Pf{Ut{xA),Ut{XB)),
C G {A,B}. In this case, the expected plurality share functions are weakly monotonic if, for all u,v,u',v' ^Ut, u' > u and v' > v imply
pf'{uy)>pf{u,v). They are strictly monotonic if the above inequalities hold strictly. Note that, because they determine expected plurality shares, voter utility functions are used for more than the purpose of representing ordinal preferences over policies. We will often suppose the voters' utility functions are concave, and we sometimes wish to make use of strict concavity properties. Rather than assume
22
Jeffrey S. Banks and John Duggan
all (or even any) voters have strictly concave utility functions, we can assume aggregate strict concavity, meaning t h a t , for all distinct x,y G X and for all a G (0,1), t h e set of voter types t such t h a t Ut{ax-\-{l — cx)y) > aut{x) + (1 — a)ut{y) has positive measure. This weaker assumption allows us to capture economic models in which policies include allocations of private goods and voters' preferences are strictly concave in their own consumption b u t constant in t h e consumption of others.^ In this special case of t h e Fixed Utility The Utility Difference Model Model, expected plurality shares depend only on the difference in utilities from t h e candidates' platforms: there exist functions P^: M —> R such t h a t P^{XA,XB)
= P^{Ut{xA)
-Ut{xB)),
C e {A,B}. In the Utility Difference Model, weak monotonicity means t h a t P/^ is non-decreasing and P^ is non-increasing for all t, and strict monotonicity means P/^ is strictly increasing and Pf is strictly decreasing for all t. Note the implicit assumption t h a t Utility Differences are meaningful: scaling Ut by a positive constant for some types can affect the objective functions of the candidates. It may be, for example, t h a t Ut{x) — Ut{y) measures t h e type t voter's willingness to pay for x over y in terms of some (unmodelled) private good. The Additive Bias Model. In a further special case of the Utility Difference Model — and one we rely on to motivate many of our assumptions — each type t voter has a random utility bias /3 in favor of candidate B given by some marginal distribution function G^, where Gt{P) is jointly measurable in (t,/?).^ T h e n t h e type t voter votes for A if Ut{xA) > Uti^B) + /?, votes for B if the inequality is reversed, and "flips a fair coin" in t h e event of equality. In general, t h e candidates' expected vote shares are Vt^in)
= 2 Gtiu) + lira Gt{u')
V,^{u)=^l-\
Gtiu) + lim Gt{u')
and their expected plurality share functions are given by Pf = Vf — Vf. Thus, weak monotonicity is automatic. Discontinuities in Gt will lead to discontinuities in Vf and P^. If Gt is continuous, however, then V/^ = Gt and implying V^^ = l-Gt, P,^ = 2G, - 1 P,^ - 1 - 2G,. ^ The assumption is also fulfilled if voters have concave and strictly quasi-concave utility functions, and if, for all x^y G X and all a G (0,1), the set of voter types t such that ut{ax + (1 — a)y) > max{ut(x),ut{y)} has positive measure. For example, types may be continuously distributed on T = X and ut{x) = —\\t — x\\. ^ Biases need not be independently distributed across voter types.
Probabilistic Voting in the Spatial Model of Elections
23
Because they are positive affine transformations of the bias distributions, the assumptions we later impose on expected plurality share functions (e.g., strict monotonicity, differentiability, or concavity) have a clear interpretation in terms of the primitives of the Additive Bias Model. The Utility Ratio Model. In this alternative to the Utility Difference Model, it is assumed that Ut{x) > 0 for all x e X and Pf^iXA.XB) =
Pf{Ut{xA)/UtixB))^
An example would be the Multiplicative Bias Model, in which each type t voter draws bias /? and votes for A if Wt(x^) > /3ut{xB)^ votes for B if the inequality is reversed, and flips a coin in the event of equality. When Gt is continuous, expected plurality share functions are again positive affine transformations of bias distributions, and assumptions on the former translate directly to assumptions on the latter. In much of the paper, we focus on the Utility Difference Model rather than the Utility Ratio Model, but, in Section 9, we will see that the two models are essentially interchangeable: any result for one can be translated into a corresponding result for the other. 2.3 Alternative Objective Functions Under the expected plurality share interpretation of our model, there exist expected vote share functions Vf^ such that P^{XA,XB) — Vf^{XA.XB) — V^^{XA',XB) for all t. Naturally, these functions satisfy • •
Vf{xA.XB)>^ V,^{XA.XB)-^V,^{XA.XB)<1
for all t. An alternative would be to interpret P^ as expected vote share, equating P^ — Vf, in which case our maintained assumption that P^-\-Pf — 0 is violated. We stick with the expected plurality share interpretation for two reasons. First, when voters cannot abstain from voting, i.e., V/^ + V^^ = 1 for all /;, the two formulations are equivalent: expected vote share and expected plurality share are positive affine transformations of each other. Second, when abstention is possible, expected vote share maximization can lead to paradoxical conclusions. For example, suppose three voters have single-peaked preferences on the real line, with ideal points 0, 1, and 2; if one candidate's platform is preferred by a voter to the other's, the voter votes for the former with probability one; if indifferent, a voter votes for A with probability 1/4, for B with probability 1/4, and abstains with probability 1/2. Then, assuming expected vote maximization, locating at the median is not an equilibrium of the electoral game: candidate A's expected vote share is increased from 1/4 to 1/3 by moving away from the median, though this move guarantees a loss for A.
24
Jeffrey S. Banks and John Duggan
Of perhaps greater concern, our assumption that payoffs are additive across voter types implies that we cannot generally capture probability of winning maximization directly as a special case of our model. If there is a continuum of voters of each type, however, then we interpret P^{XA^XB) as the plurality share actually received by candidate A, so the probability A wins is deterministic. Assuming a plurality rule election, it is one if / P^dii > 0, zero if the reverse inequality holds, and one half otherwise. Clearly, every pure strategy equilibrium under expected plurality maximization is an equilibrium under probability of winning maximization. The converse may not hold because, under the latter hypothesis, it is possible that a candidate who cannot win picks a platform that does not maximize his/her expected plurality. This disparity is precluded if a minimal symmetry condition is imposed: given any platform for one candidate, the other can choose some platform that at least equalizes their expected plurality shares. This assumption holds, for example, if the voters "flip fair coins" when indifferent between the two candidates. It follows that if the electorate is a continuum, if the symmetry condition holds, and if voters cannot abstain, then the three objectives coincide, at least in regard to pure strategy equilibria. It appears the electoral game with expected plurality maximizing candidates may have different mixed strategy equilibria than the game with probability of winning candidates in a plurality rule election. Moreover, if the electorate is finite, then our additive formulation of payoffs may lead to different pure strategy equilibria than the probability of winning formulation.
3 Existence and Uniqueness In this section, we take up the issues of existence and uniqueness of equilibria in the electoral game. Hinich et al. [17, 18] first noted that, when uncertainty is added to the canonical spatial model of elections, the electoral game can have pure strategy equilibria, unique under certain conditions. Subsequent results have typically strengthened their original assumptions, yielding sharper characterizations of equilibrium, a topic to which we return in later sections. In this section, we first establish the existence of mixed strategy equilibria, an issue often overlooked, under minimal conditions of compactness and continuity, We then take up the existence of pure strategy equilibria and find that concavity of the candidates' objective functions is a key condition. Strengthening this to strict concavity, we derive uniqueness as well. While our main results are stated for the general model, we close the section with some implications for the Fixed Utility and Utility Difference Models. Theorem 1. Assume (i) X is compact and (ii) for each t and C, P^{XA,XB) is jointly continuous in {XA.XB)- Then there exists a mixed strategy electoral equilibrium of the electoral game,
Probabilistic Voting in the Spatial Model of Elections
25
Proof. T h e space of mixed strategies for the candidates is convex and is compact and metrizable in t h e weak* topology. We can apply Glicksberg's [14] fixed point theorem to deduce t h e existence of an equilibrium if t h e candidates' objective functions are jointly continuous: t h a t is, as long as (X^.X'Q)
-^
(XA.XB)
implies
/ Pf{x^,x'S)Kdt) JT
-. /
Pf{XA,XB)Kdt).
JT
Assumption (ii) implies that, for each t, Pf{x'^,x^) converges to Pf^XA^xs)By our background assumptions on expected plurality share functions, we know the sequence {PR{X'A , x^)} is bounded. Therefore, by Lebesgue's dominated convergence theorem, the integrals converge as desired. = An application to t h e Fixed Utility Model is clear: there, P^{XA,XB) Pf{ut{xA), Ut{xB)), SO continuity of i^^ and P^ yields continuity of Pf; with compactness of X , a mixed strategy equilibrium exists. Hinich et al. allow for "non-rectangular" sets of strategy profiles, incorporating restrictions such as one candidate cannot locate to the left of the other on a particular dimension.^^ Moreover, Hinich et al. prove uniqueness under the expected vote share interpretation of the model, which creates a nonconstant sum game when abstention is allowed. We do not take up these issues here, and our analysis is dramatically simplified as a consequence: whereas Hinich et al. rely on an existence and uniqueness result for "concave games" due to Rosen [31], our arguments employ elementary results of game theory. ^^ Before examining uniqueness, we first verify well-known sufficient conditions for existence of pure strategy equilibrium. T h e o r e m 2. Assume (i) X is compact and convex, (ii) for each t and C, P^{XA-IXB) is jointly continuous in {XA'>XB), (Hi) for each XA CL'nd XB, f P^{xA,XB)dfi is quasi-concave inxA and J Pf{xA,XB)dfi is quasi-concave equilibrium. in XB ' Then there exists a pure strategy electoral Proof. Continuity of t h e candidates' objective functions follows as in t h e proof of Theorem 1. Since the objective functions are quasi-concave, Ghckberg's [14] theorem can be applied in the space of platform pairs to deliver a pure strategy equilibrium. This kind of restriction may be viewed as a reduced form of a more complex model in which candidates can feasibly locate themselves independently but, due to the structure of payoffs, would never find it profitable to do so. In Theorem 2 we could have allowed for any compact, convex set of feasible strategy profiles, thereby allowing for non-rectangularities, by appealing to the theorem of Banks and Duggan [4].
26
Jeffrey S. Banks and John Duggan
Of course, since the integral of concave functions is concave, a sufficient condition for (iii) in Theorem 2 is that, for each t and XB, P^{XA,XB) is concave in XA' In fact, this condition is sufficient for concavity, rather than merely quasi-concavity, of the candidates' objective functions in their own strategies. A notable implication of this stronger condition is that, since expected plurality shares lie between zero and one, the policy space is either bounded or the candidates' objective functions are actually constant in their own strategies. We do impose compactness in Theorem 2, however, so the assumption of concavity would not imply the objective functions are constant. Strengthening conditions (iii) and (iv) of Theorem 2, we can establish uniqueness as well. The next result is closely related to the existence and uniqueness part of Lindbeck and Weibull's [24] Theorem 1, though we strengthen their strict quasi-concavity to strict concavity in order to capture mixed strategy electoral equilibria. We give further uniqueness results in Section 4. Theorem 3. Assume (i) X is compact and convex, (ii) for each t and C, Pf{xA,XB) is jointly continuous in {XA.XB), (Hi) for each XA cind XB, j P^{xA,XB)diJi is strictly concave in XA and f Pf{xA,XB)dfi is strictly concave in XB' Then there is exactly one electoral equilibrium, and it is in pure strategies, Proof, Existence of equilibrium follows from Theorems 1 or 2. By (iii), candidate A's expected payoff given a mixed strategy for B, / / Jx JT
P^{XA,XB)
fJ^{dt)7TBidxB),
is strictly concave in XA, and likewise for B. Thus, all best responses to mixed strategies, and therefore all electoral equilibria, are in pure strategies. If there are distinct equilibria, say (X'A^^B) ^^^ (^A^^B)^ ^^ ^^^ suppose without loss of generality that x^ ¥" ^A- '^^^ electoral game is constant sum, and therefore its equilibria are interchangeable. Thus, (a:^,x^) is an equilibrium, implying that x^ and x^ are distinct best responses to x'^. By strict concavity, however, ^^A + h^A yields a higher payoff to candidate A, a contradiction. The application of Theorems 2 and 3 to the Fixed Utility Model is straightforward, after establishing the following simple lemma. This gives us essentially Hinich et al.'s [18] Theorem 1. Note that we can assume only aggregate strict concavity, rather than assuming some types of voters have strictly concave utility functions. Lemma 1. In the Fixed Utility Model, assume (i) for each t and C, P^ is weakly monotonic, (ii) for each t and v, P^{u,v) is concave in u onUt, (Hi) for each t and u, Pf{u^v) is concave in v on Ut, and (iv) for each t, Ut is
Probabilistic Voting in the Spatial Model of Elections concave. Then for each XA cind XB, jPf'{xA,XB)dii f Pf{xA,XB)d/i is concave in XB-
27
is concave in XA ciTid
Assume, moreover, (v) for each t and C, Pf is strictly monotonic, and (vi) aggregate strict concavity holds. Then for each XA dnd XB, f P^{xA',XB)dii is strictly concave in XA cind JPf{xA,XB)d/j, is strictly concave in XBProof. We first check that, for all v^ J P^{ut{xA)^v)dfi is concave in XA- Take any distinct x^,:r^ and any a G (0,1). Then, for all t, P^'^iutiax'^ + (1 - a)x^X)^y) > Pt^{cxut{x'A) + (1 - a)ut{x'^),v) > aPt^{ut{x'A).v) + (1 - a)P,^[ut{x':^).v), where the first equality follows from (i) and (iv), the second from (ii). Since the integral of concave functions is concave, A's objective function is concave in XA' By (v) and (vi), the first equality holds strictly for a set of types with positive measure, implying that A's objective function is strictly concave in XA' A symmetric argument applies for B. We use Lemma 1 to provide two applications of the above results to the Utility Difference Model.^^ Corollary 1. In the Utility Difference Model, assume (i) X is compact and convex, (iij for each t and C, P^ is continuous and weakly monotonic, (Hi) for all t, P^{u) is concave on Ut —Ut, (iv) for all t, PP{—u) is concave on Ut —Ut, and (v) for each t, ut is continuous and concave. Then there exists a pure strategy electoral equilibrium. Assume, moreover, (vi) for each t and C, Pf is strictly monotonic, and (vii) aggregate strict concavity holds. Then there is exactly one electoral equilibrium, and it is in pure strategies. If nothing more than concavity is known of voter utility functions, then concavity of P^{u) and Pf{—u) is needed to guarantee concavity of the candidates' objective functions in Corollary 1. But this has the strong implication that P^ and Pf are afRne linear on Uf—Ut- Thus, Corollary 1 applies to the Additive Bias Model as long as the bias term is uniformly distributed over a wide enough range, namely, a range including the set Ut —Ut of possible utility differences. As we show in Section 4, however, we can prove much more than the existence and uniqueness result of Corollary 1 when we know the expected plurality share functions of the candidates are afffne linear. Namely, we can show that pure strategy equilibrium platforms must maximize a utilitarian ^^ In fact, Lindbeck and Weibull's [24] argument in the proof of their Theorem 1 establishes that the unique electoral equilibrium of the second part of Corollaries 1 and 2 must exhibit policy coincidence. We give more general results on policy coincidence in Section 5.
28
Jeffrey S. Banks and John Duggan
welfare function, and that these platforms are actually dominant strategies for the candidates. If we strengthen concavity of the voters' utility functions, however, then we can weaken concavity of the expected plurality share functions, allowing for different distributions of the bias term in the Additive Bias Model. The next corollary does just this, and it anticipates our discussion, in Section 9, on the connections between the Utility Difference and Ratio Models. Corollary 2. In the Utility Difference Model, assume (i) X is compact and convex, (ii) for each t and C, Pf{\u{u)) is continuous and weakly monotonic in u, (Hi) for each t, Pf^iln^u)) is concave on hit — hit, ('^v) for each t, P^{— ln{u)) is concave on hit — hit, and (v) for each t, Ut is continuous and e^* is concave. Then there exists a pure strategy electoral equilibrium. Assume, moreover, (vi) for each t and C, Pf is strictly monotonic, and (vii) aggregate strict concavity holds with respect to the exponentials of voter utility functions. Then there is exactly one electoral equilibrium, and it is in pure strategies. The proof follows easily upon verifying the assumptions of Theorems 2 and 3. For an example of expected plurality share functions in the Utility Difference Model satisfying (ii)-(iv), (vi), and (vii) from Corohary 2 but not concavity, consider
^ti^)
= v-zz^i ^nd Pf' (u) 1 + e-^ ' '
1 + e-^
Ultimately, of course, the minimal condition on expected plurality shares and utility functions needed for existence of pure strategy equilibrium is just: for all XB, Pf{ut{xA)—Ut{xB)) concave in XA (and similarly for B). Corollaries 1 and 2 just serve to break this into separate conditions on voter utility functions and the expected plurality share functions P^ in the Utility Difference Model. See Lindbeck and Weibull's [23] Theorem 2 for a joint sufficient condition on the two primitives in their distributive setting.
4 Social Welfare O p t i m a In this section, we establish a correspondence between pure strategy equilibria of the electoral game and optima of a utilitarian social welfare function: under very general conditions, pure strategy equilibrium platforms must maximize welfare. Indeed, our most general necessity result is stated in the Fixed Utility Model and assumes only differentiability of expected plurality share functions and concavity of voter utility functions. This result is in the same spirit as Ledyard's [22] utilitarian result, proved in a different framework, and extends
Probabilistic Voting in the Spatial Model of Elections
29
results by Lindbeck and WeibuH [23, 24] and Coughlin [10]. In fact, we can drop concavity of voter utility functions and, when the expected plurality share functions are linear, obtain a strong equivalence between utilitarian optima and pure strategy equilibria. Finally, using our utilitarian theorems, we derive Hinich's [15, 16] result on the mean voter, a topic we return to in Section 8. Regarding their implications for social welfare, the theorems of this section do yield Pareto optimality of the candidates' equilibrium platforms. Because they are stated in terms of utilitarian social welfare functions, the results may seem to suggest even more: the selection of a normatively superior Pareto optimal policy. It is well-known, however, that such a selection requires interpersonal comparisons of utility and raises a prior problem: the selection of "normatively superior" welfare weights and utility representations of voter preferences. The welfare weights defined in the theorems below are given by each voter type's marginal expected pluralities in favor of one candidate or the other, and these would appear to have no special normative basis. Utilities can be compared across voters in the Fixed Utility Model because the voter utility functions selected there contain information about probabilities of voting for one candidate or the other — and these probabilities can be compared. Though that selection of utility representations is technically convenient and even natural, however, there is no presumption of its normative priority. 4.1 Characterization Results We first use strong restrictions on expected plurality share functions, namely that they are linear, to show that maximizing social welfare is necessary and sufficient for platforms to be used in equilibrium. Because we do not need to assume concavity of the candidates' objective functions (or, specifically, the voters' utility functions), this gives alternative conditions to those of Theorem 2 for existence, and it gives alternative conditions to those of Theorem 3 for uniqueness. Theorem 4. In the Fixed Utility Model, assume that for each t and all u,v G hit, P^{u^v) is affine linear in u with coefficient at > 0 (independent of v), and Pf{u,v) is affine linear in v with coefficient bt > 0 (independent of u). Then (7r^,7r^) is a mixed strategy electoral equilibrium if and only if T^\ and TT^ put probability one on maximizers of the weighted utilitarian social welfare functions I atUt{x) fi{dt) and / btUt{x) ii{dt), JT JT respectively. If at and bt are independent oft, then TT\ and TTQ both put probability one on the maximizers of the utilitarian social welfare function / Ut{x)/j.{dt).
30
Jeffrey S. Banks and John Duggan
If the welfare functions have unique maximizers, say XA dnd XB, then {XA^ XB) is the unique electoral equilibrium. Proof. Note that XA is a best response to TT^ for A if and only if x\ maximizes / P^{ut{x),ut{x%)) JT
ii{dt) ^ /
[atut{x)^ct]ix{dt),
JT
which is a positive affine transformation of A's welfare function. A similar argument for B shows that maximizing the welfare function is necessary and sufficient for an electoral equilibrium. The last claims of the theorem are selfevident. As is clear from the proof, under the conditions of Theorem 4, the welfare maximizing platforms are actually dominant strategies for the candidates, in the sense that those platforms are best responses for a candidate regardless of the other candidate's platforms. Furthermore, in the special case of the Utility Difference Model with linear P^, we have
Pf{u,v) = P^{u-v) = at{u-v)-\-Ct = Pfiu-v) = PP{u,v), = -bt{u-v)+dt so that the coefficient at is automatically independent of v (and bt is independent of u)^ and at = bt for all t. Thus, in any electoral equilibrium of the Additive Bias Model with a bias term uniformly distributed over a wide enough range, the candidates maximize the same weighted utilitarian welfare function. We can derive much more general necessary conditions for pure strategy equilibria to solve social welfare maximization problems. The next theorem generalizes the first order analysis of Theorem 1 of Lindbeck and Weibull [23], who consider the distributive setting. With Theorem 3, it extends Lindbeck and Weibull's [24] Theorem 1, which includes sufficient conditions for existence, to mixed strategy equilibria in the multidimensional setting. It also generalizes half of Coughlin's [10] Corollary 4.4, proved in connection to his results on Nash social welfare, a connection we develop in Section 9. In contrast to our theorem, Coughlin's utilitarian result is stated as a necessary and sufficient condition for pure strategy equilibrium, because he imposes the Binary Luce functional form, fulfilling the sufficient conditions for existence in Theorem 2. Theorem 5. In the Fixed Utility Model, assume (i) for each t and C, Pf{u, v) is continuously differentiable in {u,v), (ii) for each C, the partial derivatives of Pf{u,v) are bounded in {t,u,v), and (Hi) for each t, Ut{x) is continuously differentiable on the interior of X with derivatives bounded in (t,x). / / (x^,x^) is an interior pure strategy electoral equilibrium, then x\ is a critical point of the weighted utilitarian social welfare function
Probabilistic Voting in the Spatial Model of Elections
L
atUt{x)iJ.{dt),
31 (1)
where at = Du=ut.{x* )Pt^{^'> '^t{^%)) /^^ ^^^ ^; ^^^ ^% ^-5 a critical point of the weighted utilitarian social welfare function
L
Ptut{x)fi{dt),
(2)
where A = Dy=utXx%)Pf M^A)^'^) /^^ ^^^ ^• Assume, moreover, (iv) at and Pt ^^^ positive and independent oft. x\ and x'^Q are critical points of
I ut{x)^{dt).
Then
(3)
Assume, moreover, (v) X is convex and (vi) aggregate strict concavity holds. Then x\ = x*^ is the unique maximizer of (3). Proof. If
{X^.X'Q)
is an interior equilibrium, then x\ solves max /
P^{ut{xA),Ut{x%))^x{dt).
By (i) and (ii), and since (x^, x^) is interior, the above integral is differentiable at ( x ^ , x ^ ) , and the necessary first order condition for a maximum holds: DxA=x\
/ Pt^{Ut{xA).Ut{x%))^{dt)
= ^.
Using (i)-(iii), we interchange differentiation and integration and use the chain rule to obtain / Du^^^^^*^)P^{u,Ut{x'JQ))D^=^*^Ut{x)^i{dt) = 0 , which implies atD^=:,*^Ut{x) ^{dt) = 0.
(4)
/.•
Therefore, critical point of jatUt{x)dii, with a similar argument applying for B. Assuming (iv), let a == a^ > 0 for all t. Dividing both sides of (4) by a, we see that x\ is a critical point of (3), and similarly for B. Adding (v) and (vi), f Ut{x)d/j. is strictly concave, and the last claim of the theorem is self-evident.
32
Jeffrey S. Banks and John Duggan
As is clear from the proof, the necessary conditions given in the first parts of Theorem 5 actually hold for all interior local electoral equilibria, which requires merely that each candidate's platform maximize the candidate's expected plurality share over some open set containing the platform. The result can be proved without imposing differentiability conditions on voter utility functions, and without assuming interiority of equilibrium, if we add some convex structure to the problem. Next, we impose concavity of voter utility functions at the outset to prove the utilitarian result. Theorem 6. In the Utility Difference Model, assume (i) X is convex, (ii) for each t and C, Pf{u,v) is strictly monotonic and has partial derivatives with respect to the Cth component bounded in (t^u^v), and (Hi) for each t, ut is concave. If {X%X''Q) is a pure strategy electoral equilibrium, then x\ maximizes (1) and x^ maximizes (2). Assume, moreover, (iv) at and Pt o.re positive and independent oft. x\ and x^ maximize (3).
Then
Assume, moreover, (v) aggregate strict concavity holds. Then x\ — x^ is the unique maximizer of (3). Proof. We first prove the result assuming T is finite. Suppose (x^,x^) is an equilibrium but x\ does not maximize (1), so there exists x' £ X such that
T
T
Let P : R^ ^ M be the function defined by P(^) = ^ P , ^ ( z „ ^ , ( x ^ ) ) / i ( W ) teT
for aU z — {zt)teT ^ ^'^, and let
U = {zeR^ \3xe x,\/t eT,zt< ut{x)}, a convex set by (i) and (hi). By the assumption that {x\,x*^) is an equilibrium, z* = {ut{x*j^))teT solves max^GM^ P(^) s.t. z eU. By (ii), P is differentiable at z*, and D^==^*P{z) = {at,f^{{ti}),...
,at^iJ.{{tK})),
where at is defined as in Theorem 5, and where T = { t i , . . . ,^i^}. Now let z' = (ut{x'))t£T^ and note that D,^,,P{z)-{z'-z*)
= Y.at[ut{x') teT
- ut{x'')\ij^{{t})
> 0.
Probabilistic Voting in the Spatial Model of Elections
33
Thus, Dz=z*P{z) ' a{z' - z*) > 0 for all a > 0. Taking a > 0 small enough, we therefore have z* + a{z' - z"") £ U and P(^* + a{z' - ^*)) > P(z*), a contradiction. Therefore, x\ maximizes (1). When T is infinite, the proof proceeds exactly as above, but some concepts must be extended to the general case. Integrals replace summation signs, and P may be defined on the subspace of R^ consisting of the bounded, measurable functions oft, endowed with the sup norm. Thus, P is a "functional." If we use the concept of Prechet derivative (see Luenberger [25]), then P is differentiable at 2;* under our assumptions, and D,=,^P{z){z'-z'')
=j
at[ut{x')-ut{x^)]^ji{dt)
> 0,
as above, again resulting in a contradiction. Of course, the same argument applies to x^. The remainder of the theorem is self-evident. Theorem 6 has an easy implication, stated next without proof, for pure strategy electoral equilibria exhibiting policy coincidence in the Utility Difference Model. We have already established conditions under which such equilibria exist. In the next section, we give several sets of conditions under which pure strategy equilibria of the electoral game necessarily exhibit policy coincidence. Corollary 3. In the Utility Difference Model, assume (i) X is convex, (ii) for each t and C, Pf{u) is strictly monotonic and differentiable with derivatives hounded in {t,u) and with DU^QP^ {u) independent oft, and (Hi) for each t, Ut{x) is concave. //(x*,x*) is an interior pure strategy electoral equilibrium, then x* maximizes (3). Assume, moreover, (iv) aggregate strict concavity holds. Then (a:*,x*) is the unique interior pure strategy electoral equilibrium exhibiting policy coincidence. The assumptions in Corollary 3 isolate one case in which the welfare weights of the different voter types are equal. The key condition used there is that the marginal expected plurality shares of the candidates are equal across voter types when all voter types are indifferent between the two candidates, a condition satisfied if expected plurality share functions are symmetric across voter types. The theorem has two interesting connotations. First, if we view voter utility functions as valuations in terms of a quasi-linear private good, then it is well-known that Pareto optimality is equivalent to maximizing the unweighted sum of voter utilities. Under the assumptions of the theorem, therefore, pure strategy equilibria exhibiting policy coincidence are Pareto optimal. But, if voter preferences are quasi-linear and the marginal expected plurality shares of the candidates are not equal across voter types when indifferent toward the platforms of the candidates, equilibrium outcomes may not be
34
Jeffrey S. Banks and John Duggan
Pareto optimal. Second, under the conditions of the theorem, the pure strategy equilibria exhibiting policy coincidence, if they exist, are independent of the candidates' expected plurality share functions and, therefore, independent of the structure of incomplete information in the model. 4.2 The Mean Voter Theorem We say preferences are generalized Euclidean if (a) for each t, there exists x^ e X and a positive n x n matrix At = (a^^) such that ut{x) = -{x - x^)At{x - x^), for all a: G X, (b) x^ and At are bounded and measurable with respect to t, and (c) letting A't denote the transpose of At, J [At + At]dfi is invertible. A special case is quadratic preferences, where At =^ / , x^ is the ideal point of type t voters, and Ut(x) — — ||x — x^|p. We next derive half of Hinich's [16] Theorem 1 on the "mean voter" from our utilitarian results. His theorem is stronger than our result in one respect: whereas we assume policy coincidence for now, Hinich derives it. In the next section, we take up this issue in full detail. Corollary 4. In the Utility Difference Model, assume (i) for each t and C, Pf{u) is strictly monotonic and differentiable with derivatives bounded in {t,u) and with Du=oPf (u) independent oft, and (ii) preferences are generalized Euclidean, //(x*,x*) is an interior pure strategy electoral equilibrium, then
'JjAt + A',] n{dt)^
(^l^[At + A'M' t,{dt)^ .
(5)
Assume, moreover, (iv) At is symmetric and independent oft, then x* = / x^ ^{dt). JT
Proof. From Corollary 3, we know x* is a critical point of JUt{x)dfi. Thus, j Dx=x*ut{x)dii — 0. Note that, for each t, n
n
ut{x) = = - ^ Y ^ 1=1
Dut{x) =
alji^i - ^Diocj - x]) j=i
-[At+A[]{x-x*).
Probabilistic Voting in the Spatial Model of Elections
35
Thus, the first order condition reduces to
Ij[At-^ A[]{x* - x^) fi{dt)
-0,
which gives us (5). Assuming (iv), x* is the mean ideal point, as claimed. The mean voter result also appeared in Hinich's [15] 'Three Voter Example,' and earlier in Hinich et al.'s [18] Theorems 5 and 6. Their Theorem 5 differs from our Corollary 4 in that it restricts the distribution of voter ideal points (imposing symmetry) rather than any single voter's utility function; they do, however, also assume that the utility functions of voters with diametrically opposed ideal points are "mirror images" of each other. Though we do not formalize this claim here, their result can also be derived from Theorem 5: under their conditions, the mean (also mode) of the voters' ideal points maximizes the utilitarian social welfare function. Hinich et al.'s Theorem 6 assumes the candidates' expected plurality share functions are linear in platforms adopted: in the Fixed Utility Model, this amounts to assuming linear expected plurality share functions (as functions of utility levels) and generalized Euclidean voter utility functions. 4.3 A Further Uniqueness Result Corollary 3, and Theorems 5 and 6 more generally, give conditions under which there is at most one interior pure strategy electoral equilibrium exhibiting policy coincidence. We end this section by adding a uniqueness result based on the familiar property of interchangeability of equilibria for two-player zerosum games and holds in the general electoral model. Theorem 7. Assume (i) for each t, C, and {x,y), P^{x,y) = Pf{y,x), and (ii) (a:*,x*) is the unique (interior) pure strategy electoral equilibrium exhibiting policy coincidence. If (XA.XB) is an (interior) pure strategy electoral equilibrium, then XA — XB=X*. Proof. By (i), the. electoral game is a symmetric, two-player, zero-sum game. If {XA.XB) is an (interior) pure strategy electoral equilibrium, then so is {XB.XA)' By interchangeability, [XA.XA) is an (interior) pure strategy electoral equilibrium, and by (ii) we have XA — X"" , Similarly, XB = X*. Thus, under the assumption of symmetry, we can extend the above uniqueness results for electoral equihbria exhibiting policy coincidence to all pure strategy electoral equilibria. Of note, Lindbeck and Weibull's [23] uniqueness argument (in their Section 4) does not impose symmetry and does not follow from our results: their argument uses the special structure of the distributive model.
36
Jeffrey S. Banks and John Duggan
5 Policy Coincidence In this section, we consider more deeply the incentives for candidates to match each other's policy platform, giving voters essentially just one choice of policy. If the expected plurality share functions of the candidates are symmetric, in the sense of condition (i) from Theorem 7, then, under the conditions of Theorem 2, the existence of symmetric equilibria follows from standard arguments. While we therefore have conditions under which some equilibria exhibit policy coincidence, we now examine whether all equilibria must exhibit policy coincidence, a feature of elections first noted by Hotelling [19] in the Deterministic Model. Theorem 7, with Corollary 3, currently delivers policy coincidence in the Utility Difference Model under rather weak conditions: chief among them are that the marginal expected plurality shares of the candidates are independent of voter type when voters are indifferent, and that the electoral game is symmetric. In the existing literature, Hinich et al. [17] give conditions necessitating policy coincidence, ones involving strict concavity of expected plurality share functions. Calvert [9] weakens strict concavity to the existence of an "estimated median." Coughlin [10] weakens strict concavity of voter utility functions to strict "log concavity," but he imposes the Binary Luce functional form on P^}^ Furthermore, all of these authors assume symmetry of the electoral game. In fact, all of these results on policy coincidence are proved under assumptions guaranteeing a unique pure strategy electoral equilibrium. But if (x^,x^) is the unique pure strategy equilibrium of the symmetric electoral game, then it is immediate that x\ = ^^, for otherwise, (x^,a:^) would be a distinct equilibrium. Is policy coincidence necessitated by the logic of electoral competition under weaker assumptions? Hinich [16] gives one such set of conditions: he shows that, in the Additive Bias Model with generalized Euclidean voter utility functions, every pure strategy equilibrium has the candidates adopting the generalized mean and, therefore, exhibits policy coincidence. He assumes equal marginal expected plurality shares across voter types, but he does not assume symmetry (in his notation, F^ need not be symmetric) or concavity of the candidates' objective functions, and indeed there may not be a pure strategy equilibrium. Lindbeck and Weibull's [23] Theorem 1 establishes that policy coincidence holds generally in the distributive politics model without assuming symmetry or strict concavity of expected plurality share functions, whereas Lindbeck and Weibull [24] add strict quasi-concavity of expected plurality share functions to obtain this result in a one-dimensional model. In a different framework, Ledyard [22] uses symmetry to establish policy coincidence, but his result does not rely on concavity of the candidates' objective functions. ^^ Here, we add strictness to log concavity in Coughlin's [10] Corollary 4.4 to get policy coincidence.
Probabilistic Voting in the Spatial Model of Elections
37
We provide general sufficient conditions for policy coincidence in the Utility Difference Model, obtaining the above results on policy coincidence as special cases. We drop the assumption of equal marginal expected plurality shares across voter types; we do not assume candidate symmetry; and we do not impose concavity conditions on expected plurality share functions. We find that, under natural concavity conditions on voter utilities and monotonicity conditions on expected plurality share functions, every pure strategy electoral equilibrium exhibits policy coincidence. Theorem 8. In the Utility Difference Model, assume (i) X is convex, (ii) for each t and C, Pf is strictly monotonic, (Hi) for each t, ut is concave, and (v) aggregate strict concavity holds. If (x^,a:^) is a pure strategy electoral equilibrium, then Proof Suppose x\ ^ x^, and let y = ^x\ + \^^B' Because {X\,X*Q) is an equilibrium, we know that the net gains to A from deviating to y are nonpositive: / P,^{ut{y)-ut{x*B))^x{dt)
< [ P,''{utix*^)-ut{x%))f,{dt).
IT JT
(6)
JT
Define T+ = {t€T\ TX = {teT\
ut{y) > ut{x*^)} utiy) < utix*^)}
as the sets of types on which A gains and loses, respectively. Since P^ is strictly increasing, we have teT+4^
P,^{ut{y) - ut{x*B)) - P^^{u,{x\) - ut{x%)) > 0
t&T^^
Pt'^iutiy) - utix%)) -
PHM^I)
- nt{x*B)) < 0.
Thus, we can decompose A's net gains from deviating to y into gains, gA=
f
[Pt'^iUtiy] - Utix*s)) - Pt'^iUtix*^) - Ut{x*s))]
flidt),
JT+
and losses CA = f
_[P^{ut{x*A)-Ut{x%))-P,\u,{y)-Ut{x%))\ii{dt).
By construction, the integrands of QA and CA are positive. Rewriting (6), we have QA < CABy the same logic, the net gains to B from deviating to y are non-positive:
X
-P^^{ut{x*^)-ut{y))\^x{dt)
<
j[-P^{ut{x'A)-u,{xl))iJi{dt),
38
Jeffrey S. Banks and John Duggan
where we use P^ = —P^- Define T+ = {teT\ TB ={teT\
ut{y) > utixi)} utiy) < utix%)}
as the sets of types on which B gains and loses, respectively. Since P / ' is strictly increasing, we have tGT+^
P^{ut{x\)
- utix^e)) - Pt^{ut{x\)
- ut{y)) > 0
teT^^
Pt{ut{x\)
- utixD)
- ut{y)) < 0.
- Pt^{ut{x\)
We can therefore decompose B's net gains from deviating to y into gains GB^
I
[P^{ut{x\)-Ut{x^B))-Pt^{ut{x\)~Ut{y))]iJi{dt),
^B = f
_[Pt''iu,{x*A)-ut(y))-P,''iutix\)-Utix*s))]pi{dt).
and losses
By construction, the integrands of QB and CB are positive. Rewriting (7), we have GB < CBIn the remainder of the proof, we show JCA ^ GB and CB < SA? at least one inequality strict. With (6) and (7), we then have GA
< CA
at least one inequality strict, a contradiction. We first show CA ^GB^^ similar argument delivering CB ^ GA- TO see that T^ C T^", note that t G T^ if and only if Ut{y) < Ut{x\)] by concavity Ut{y) > llutix'X) + Ut{x%)]; thus, Ut{y) > Ui(x^), or equivalently t G T^. Now take any t G T^ and compare the integrands of GB and CAP,^{u,{x\)
- ut{x*s)) - Pt{ut{x\)
> P^{ut{x\)
- ut{x%)) - Ptiutiy)
- ut{y))
(7)
- wt(x^))
if and only if Pt^{ut{x\)
- ut{y)) < Pt^iutiv) - ut(x*s)).
Since P^ is strictly increasing, this holds if
My) - M^B) ^ M^A) - My)^ or Ut{y) > ^[ut{x\) + Utix"^)], which indeed follows from concavity. This, together with the fact that the integrand in the definition of GB is non-negative on T^\T^, gives us CA < GB-
Probabilistic Voting in the Spatial Model of Elections
39
To prove t h a t either £ ^ < QB or CB < GA^ ^^^ S = {t e T \ Ut{y) > |'Ut(x^)-f ^ ? i t ( x ^ ) } . By concavity, S C T^UT^. By aggregate strict concavity, S has positive measure. Thus, without loss of generality, suppose S fi T^ has positive measure. Since T^ C T ^ , we then have either fj.{S Ci T^) > 0 or /j.{Sr\{T^ \T^)) > 0. In the first case, for all t € SOT^, the inequality in (7) holds strictly and CA < GB follows. In the second case, for all t € TQ \ T ^ , the integrand of GB is positive, and the desired inequality follows.
6 Robustness We next take up the issue of robustness of equilibria broached by Calvert [9]. To do so, we formulate candidate preferences more broadly. Namely, let A be an arbitrary metric space of parameters with generic element A, and let U^{XA', XB, A) represent ^ ' s payoff. Similarly, let t / ^ ( x ^ , X B , A) represent B's payoff. Let -^(A) represent the set of mixed strategy equilibria of the electoral game corresponding to A, where we give the space of mixed strategies the weak* topology. Let A* be such t h a t [/^(xA,xs,A*) -
/
P^{xA.XB)^Ji{dt),
C G {A, B]. T h a t is, A* is the probabilistic voting model introduced in Section 2. Because A* lies in a metric space, we can examine the robustness of equilibria when the parameters of our model are subject to small perturbations; t h a t is, we can examine the continuity properties of E. It may be, for example, t h a t A = [0,1], t h a t candidates have preferences over platforms adopted given by continuous functions uc '• X ^^ R, t h a t U^{XA,XBA)
- Xucixc)
+ (1 - A) /
P^{xA.XB)lJi{dt),
and hence t h a t A* = 0. Thus, we can examine the effects of small departures from our assumption of expected plurality share maximization.^^ T h e next result states the well-known implication of upper hemicontinuity of the mixed strategy equilibrium correspondence t h a t E has closed graph.^^ ^^ In this example, candidates have preferences over their respective platforms. Because we have formulated the candidates' objective functions in terms of expected plurality share, rather than probability of winning, we have no way of naturally introducing preferences over policies implemented by the winner, as Calvert does. Because he assumes probability of winning functions are continuous, his Theorem 6 follows from known continuity results, as Theorem 9 does. ^^ We say E has closed graph if, for every sequence {A"^} and every sequence (7rX,7rS^) such that (7rX,7r^) e E{X^) for all m, if A^ -> A and (7r:?,7r^) -> (7rA,7rB), then (7rA,7rB) G ^(A).
40
Jeffrey S. Banks and John Duggan
Theorem 9. Assume (i) X is compact and (ii) for each C, jointly continuous in ( X ^ , X B , A). Then E has closed graph.
U^{XA,XB,X)
is
Because A is an arbitrary metric space, A may index distributions of voter types, and we can consider robustness with respect to changes in demographic variables. Or we can let A index families of expected plurality share functions, and we can examine the effects of small changes in the underlying structure of incomplete information in the electoral game. We first apply Theorem 9 to establish robustness of equilibria of the electoral game with respect to demographic variables captured by the distribution of types, //. Corollary 5. Assume (i) X is compact, (ii) T is a compact metric space, (Hi) for each C, PfixA^ocs) is jointly continuous in {t.XA.^B), (iv) /^^ -^ M weak"^, and (iv) for each m, (7r^,7r^) is a mixed strategy equilibrium of the electoral game parameterized by fi'^. If (7r^,7r^) —^ (T^A^^B), ihen {TTA.TTB) is a mixed strategy equilibrium of the electoral game parameterized by //. Proof The result follows from Theorem 9, where A is the space of probability measures on (T, T) endowed with the weak* topology. Define U^{XA^XB,I^)
= /
Pf{xA^XB)Kdt),
C e {A,B}. To establish continuity of [/^, let {(x^,x^,/i'^)} converge to {XA.XB, 1^)- Given XA and XB, define (p: yl —^ R by ^{y,z)
= max\Pf{y,z)
-Pf
(XA,XS)|.
This function is continuous by the Theorem of the Maximum, and 0(XA, XB) = 0. Therefore, 0 ( x ^ , x g ) -^ 0. That is, P'^,^{xJ,x'g) - P^^{XA,XB) ^ 0 uniformly, and / Pf{x^,X^)n"'{dt) JT
^
/
PfiXA,XB)Kdt),
JT
follows from Aliprantis and Border's [1] Corollary 12.6. We now apply Theorem 9 to establish that equilibria of the electoral game are robust with respect to variations in the underlying structure of incomplete information in the model: given one specification of expected plurality share functions, "nearby" functions will not admit equilibria far away from those of the first. Much hinges on the meaning of "nearby" in reference to expected plurality share functions. In our next theorem, we use the notion of uniform convergence, and hence "nearby" has a very strong meaning. Corollary 6. Assume (i) X is compact, (ii) for each t and C, {P^ '^} is a sequence of continuous functions converging uniformly to Pf and (Hi) for
Probabilistic Voting in the Spatial Model of Elections
41
each m, {TT^.TT'^) is a mixed strategy equilibrium of the electoral game parameterized by {Pi '^}t,c- ^f i^A^^^) ~^ {^A^TTB), then (TTA^TTB) is a mixed strategy equilibrium of the game parameterized by {Pf}t,C' Proof The result follows from Theorem 9, where A = {HtV) x {IltV) and V is the space of continuous expected plurality share functions. We endow V with the topology of uniform convergence and A with the product topology. Define W^ixA.XBAPfKc)
= I
P^{XA.XB)lJi{dt),
C e {A,B}. To establish continuity of U^, let {(x^,:r^, {if^'^^j^c)} converge to {XA,XB, {Pf}t,c)- Take any t, and note that, because Pf is continuous and Pf'"^ -> Pf uniformly, P^'"^{x"^,x'g) -^ P^{XA,XB)^ Therefore, [/^(x:?,x^,{Pf'"^kc) -
U^{XA.XBAP?KC)
follows from Lebesgue's dominated convergence theorem. We can weaken uniform convergence to pointwise convergence if we go to the Utility Difference Model and make the weak assumption that expected plurality share functions of the candidates are weakly monotonic. This gives us a much weaker notion of "nearby" and a stronger robustness result. Corollary 7. In the Utility Difference Model, assume (i) X is compact, (ii) for each t, Ut is continuous, (Hi) for each t and C, Pf is continuous, (iv) for each t and C, [P^ '"^} is a sequence of weakly monotonic functions converging pointwise to Pf, and (v) for each m, (7r2^,'7r^) is a mixed strategy equilibrium of the electoral game parameterized by {P^'^}t,C' If (TT^^'TT^) ~^ {^A^'^B), then {TTA^T^B) is a mixed strategy equilibrium of the game parameterized by {P?]t,cProof. The result follows from Theorem 9, where A = {{P^ '^}t,c \ ^ = 1, 2,...} U {{Pf}t,c} is endowed with a simple topology: the only non-trivial convergent sequences are {{P^ '^}t,c} and its subseqences. Define U^{xA.XBAPf}t,c)
= J
P^{Ut{xA)-Ut{xB))^i{dt),
C e {A,B}. To establish continuity of U^, let {(a:^,^^, {Pf'^^j^c)} converge to {XA.XB, {Pt^}t,c)' Take any t and C, and note that {P^ '"^} is a sequence of weakly monotonic functions converging pointwise to Pf on the interval [ut{xA) — Ut{xB) — l,'Ut(x^) — Ut{xB) + !]• Therefore, {P^ '"^} converges uniformly on this interval. (See Aliprantis and Burkinshaw, [2], Sec.7, ex.14.) The proof then proceeds as in the proof of Corollary 6.
42
Jeffrey S. Banks and John Duggan
7 Core Convergence In the context of the Utility Difference Model, we now take up a different kind of robustness issue. Define the Deterministic Model by the expected plurality j\
Q
share functions P^ and P^ , where 1
Pf{u) = {0
if w > 0
i{u = 0 1 if u < 0,
and P^ = —P^ . We parameterize voter preferences by A, as in Ut{x^ A), where A again lies in an arbitrary metric space. We define the core at A as the set of policies X G X for which there does not exist y E X such that fi{{teT\ut{yA)>ut{x,X)})>
i.
It is clear that the pure strategy equilibrium platforms of the electoral game, when the Deterministic Model obtains, must lie within the core. Thus, if the core is empty, then pure strategy equilibria will not exist. Suppose that the core at A is strongly externally stable, i.e., for every x in the core at A and for every other y, we have fj.{{t G T \ Ut{x,X) > Ut{y,X)]) > ^. If the core is non-empty and strongly externally stable, then we can say more: the core consists of one point, x*, and {XA,XB) is a pure strategy equilibrium if and only if XA = XB = X*. Further, the equilibrium mixed strategies are then exactly those placing probability one on x*. It is well known that core points must meet stringent conditions and that, even when the core is non-empty, small perturbations of voter preferences may annihilate it (see Plott [30]). In this section, we show that small perturbations of preferences and small departures from the Deterministic Model cannot produce electoral equilibria "far away" from the core, if non-empty. More than that, if pure strategy equilibria exist for models arbitrarily close to the Deterministic Model, then the core must be non-empty. Because of discontinuities inherent in the Deterministic Model, a difficult technical issue not addressed in the previous section, Theorem 9 and Corollary 7 do not yield the desired core convergence results, so that separate arguments are required. Our first theorem on core convergence establishes a robustness result using minimal background assumptions: models nearby the Deterministic Model cannot possess pure strategy electoral equilibria far away from the core. We then extend this result to mixed strategies by assuming that the core at a parametrization A is weakly externally stable, i.e., for every x in the core at A and for every other y, we have //({t G T | ut{x^X) > Ut{y, A)}) > ^, replacing the strict inequality above with a weak one. This holds, for example, if X is convex and each Ut is strictly quasi-concave. We find that, if there exist mixed strategy electoral equilibria of models close to deterministic that converge to
Probabilistic Voting in the Spatial Model of Elections
43
a pure strategy pair, then those pure strategy platforms must be in the core. In the following, let 5x denote the probability measure with point mass on x. Theorem 10. In the Utility Difference Model, assume (i) for each t, Ut{x, X) is jointly continuous in {x^X), (ii) {A^} is a sequence converging to X, and (Hi) for each t and C, {P^ ' ^ } is a sequence of weakly monotonic func(J
tions converging pointwise to P^ . For each m, let ( x ^ , x ^ ) be a pure strategy equilibrium of the electoral game parameterized by X'^ and {P^ '^}t,c- If ( x ^ , x ^ ) -^ {XAIXB), then XA ci'^d XB CLTC in the core at X. Assume, moreover, (iv) the core at X is weakly externally stable. For each m, let (7r^,7r^) be a mixed strategy equilibrium of the electoral game parameterized by A^ and {P^'^]t,c- U {^^•>^^) -^ {^XA^^XB)^ ^^^^ ^A ^^^ ^B o^re in the core at A. Proof, For the first part of the theorem, suppose XA is not in the core at A, so that /i({t G T I Ut{y,X) > ut{xA,X)}) > \ for some y G X. Let S ^ {teT \ ut{y, A) > ut{xA^X)}, so that / P^ (utixA, A) - ut{y, A)) ij.{dt) = /i(5), and let a < 1 satisfy a > 2fi{T \ S). Take any t G 5, and note that from (i), (ii), and x^ -^ XA, there exists e > 0 and M such that, for m > M, Ut{y, A^) — ut{x^, A^) > e. By (in), we have P^ '"^(—e) > a for high enough m. And since P^ '^ is weakly decreasing, P^ '^{ut{x^,X^) — Ut{y,X^)) > a for high enough m. By Fatuou's Lemma,
L /
Pf'^^iutix^X, A-) - utiy, A™)) ^iidt) > a/i(5)
(8)
Pf^^iutix^,
(9)
A-) - ut{y. A-)) f,{dt) > - ( 2 - a)f,{T \ S)
/T\S
for high enough m, and therefore
I Pf •'"(wt W , A-) - Utiy, A'")) Kdt) >a- 2/i(r \ S)
(10)
for high enough m. For such m, since x ^ is a best response to x ^ , it must be that /.
P^^'^{ut{x'2.X'^)-ut{x'g,X'^))lJi{dt)>a-2^{T\S)
> 0.
It follows that candidate A's payoff is less than /i(T \ 5) — | < 0 for high enough m. Since P^ '^(0) —> P^ (0) = 0 for each t, Lebesgue's dominated convergence theorem yields
44
Jeffrey S. Banks and John Duggan
i
P,'''"'iO)n{dt)>n{T\S)
for high enough m. But x^ is then a better response to x'^ for A than x ^ , and (x^, x^) cannot be an electoral equilibrium. A symmetric argument addresses the case in which XB is not in the core. For the second part, suppose XA is not in the core at A, so that again li{{t e T I Ut{y,\) > Ut{xA,X)}) > \ for some y G X. By Proposition 12 in Banks et al. [6], there exist a measurable set 5 C T with ii{S) > ^ and an open set y C R^ such that y ^ Y and, for all t G 5 and all z G X OY, Ut{y, A) > Ut{z, A). Let a < 1 satisfy a > 2/j,{T \ S). Prom TT^ -^ S^^^, we have 7r^{X\Y)<[a-2i^{T\S)]n^{Y), for high enough m. Take any any z e Y and any t G 5, and note that, by (i) and (ii), there exists e > 0 and M such that, for m > M, Utiy^X^) — Ut{z,X^) > 6. By (iii), we have P^ '^(—e) > a for high enough m. And since P^ '^ is weakly decreasing, we have P^ '^{ut{x'^, X'^) — Ut{y,X^)) > a for high enough m. Por any z e Y, we therefore again have (8)-(10), with x ^ replaced by z. Therefore,
JY
JT
+ f
f Pf'^'iutiz,
JX\Y
A-) - utiy, A-)) fiidt) TT^idz)
JT
>[a-2KT\S)]ir^iY)-n^iX\Y) >0, for high enough m. Por such m, since TT^ is a best response to TT^, it must be that candidate B's expected payoff from using TT^ against TT^ is at least a - 2/i{T \S) - n'XiX \Y) > 0. This implies that candidate A's expected payoff from using n]^ against TT^ is less than - f + fi{T \ 5) -f ""^"'^f^^^ < 0 for high enough m. To derive a contradiction, note that either XB is in the core at A or not. In the latter case, there is some y G X such that fi{{t G T \ Ut{y,X) > ut{xB,X)}) > | , and the above argument shows that ^ ' s expected payoff must be positive for high enough m, a contradiction. In the former case, let y = XB' Using (iv), we can argue as above that liminf / J yi
f Pf''^{ut{yAn-^t{xB.Xn)Kdt)T^'S{dxB)
> 0,
J 1
again a contradiction. A symmetric argument addresses the case in which XB is not in the core, completing the proof.
Probabilistic Voting in the Spatial Model of Elections
45
Note that condition (iv) in the second part of Theorem 10 could be replaced by the symmetry condition in Theorem 7, which reduces to P^{u) = Pf{—u) in the Utility Difference Model. Condition (iv) is used in the proof only to argue that candidate A could guarantee a (close to) non-negative expected payoff against TT^ for high enough m. In the proof, this was accomplished by letting A choose a platform in the core, and under the symmetry condition this could be done by letting A adopt B^s mixed strategy. Theorem 10 does not require the assumption that the core at A is nonempty. As a consquence, when the core is empty, the implications of Theorem 10 are definitive: pure strategy equilibria are guaranteed to not exist in models close enough to deterministic. Though Theorem 2 gives conditions under which pure strategy equilibria electoral equilibria would exist, they cannot exist in or "near" the Deterministic Model when the core is empty. The problem is that, because of non-concavities inherent in the Deterministic Model, the concavity condition needed in Theorem 2 is necessarily violated in models close enough to the Deterministic Model. Of course, there is no guarantee that pure strategy will exist after perturbing the Deterministic Model, even when the core is non-empty. We return to a related issue in the next section. Corollary 8. In the Utility Difference Model, assume (i) X is compact, (ii) for each t, Ut{x,X) is jointly continuous in {x,X), (Hi) [X^] is a sequence converging to X, and (iv) for each t and C, {P^ '^} is a sequence of weakly (J
monotonic functions converging pointwise to P^ . If the core at X is empty, then there exists M such that, for all m > M, the electoral game parameterized by A"^ and {Pf'"^1^,0 has no pure strategy electoral equilibria. Proof If the conclusion of the corollary did not hold, we could extract a subsequence {{x^'^.x^^)} of pure strategy equilibria converging to some {XA.XB)By the first part of Theorem 10, XA and XB would be in the core at A, a contradiction. Finally, we extend the second part of Theorem 10 by addressing the possibility of mixed strategy electoral equilibria in models that do not converge to pure strategies. We assume that the core at A is non-empty and strongly externally stable, the latter holding if X is convex, each ut is strictly quasiconcave, and majority rule is strong.-^^ We find that in models close to the Deterministic Model, mixed strategy electoral equilibria must be close to the unique core point. Theorem 11, In the Utility Difference Model, assume (i) for each t, Ut{x, A) is jointly continuous in (x^X), (ii) {A^} is a sequence converging to X, (Hi) for ^^ We say majority rule is strong if, for all S G E, either /x(»S) > 1/2 or /x(5') < 1/2. Banks et al. [6] show that majority rule need not be strong for this claim when T is a continuum if voter preferences are sufficiently heterogeneous.
46
Jeffrey S. Banks and John Duggan
eacht andC, {Pf'"^} is a sequence of weakly monotonic functions converging Q
pointwise to P^ , and (iv) the core at A is strongly externally stable and consists of the policy x*. For each m, let (7r^,7r^) be a mixed strategy equilibrium of the electoral game parameterized by X^ and {P^ ''^}t,c- Then (7r^,7r^) -^ Proof. We claim that 0: X —^ M defined by 4^{z)^^{{t^T\ut{x\X)>Ut{zA)}) is lower semicontinuous. Let Sz — {t £ T \ Ut{x*, X) > Ut{z,X)}. Given a sequence z^ -^ z, by (ii) we have Sz Q hminf S';2rn. Then the claim follows from liminf/i(52m) > fi{\[miniSz^) > /J^{Sz), an implication of Fatou's Lemma. Now suppose {TT^} does not converge weakly to Sx*. Then there is some b > 0 and some open set Y containing X* such that, for all M, there exists m > M with 7r^(X \Y) > b. Going to a subsequence if needed, we can suppose this is true of all m. From (i) and lower semicontinuity oi (f), (j) achieves it's minimum, say i/, on X \Y. From (iv), we have v > ^. Let a < 1 satisfy a > 2(1 — i/). Take any z e X \Y and any t E Sz- By (ii) and (iii), there exists e;^ > 0 and Mz such that, for m > Mz, we have ut(x*, A^) — Ut{z, A^) > e^. By (iii), we have A '^{—^z) > o, for high enough m. And then since P^ '"^ is weakly decreasing, Pf''^{ut{z,X'^)-Ut{x\X'^)) > a for high enough m. For emy z e X\Y, Fatou's Lemma implies
i-
Pf'"^(u,(z, A-) - ^,(x*, A-)) fi{dt) > afi{Sz) - fi{T \ Sz) >
a-2(1-1^)
for high enough m. Therefore, / f Pf^'^{ut{z,X'^)-Ut{x\X'^))fi{dt)7T'^{dz)>{a-2{l-iy))b Jx\Y JT
> 0,
for high enough m. Taking m high enough, a similar argument shows that
LL
Pf'"'iut{z,X^)-Ut{x*,X^))n{dt)Tr^idz)
Y JT
can be bounded below by negative numbers arbitrarily close to zero. Therefore, there exists c > 0 such that 5's expected payoff using x* against TT^ is at least c for high enough m. Since TT^ is a best response to TT^^, ^ ' S expected payoff from using TT^ against TT^ is at least c for high enough m, and therefore A's expected payoff is less than or equal to —c. Finally, an argument similar to that above yields a lower bound on A's expected payoff using x* against
Probabilistic Voting in the Spatial Model of Elections
47
TT^ that goes to zero. But then, for high enough m, x* is a better response to TT^ for A than TT^, a contradiction. A symmetric argument addresses the case in which {TT^} does not converge to Sx*It is important to note a difference between Theorem 10 and Theorem 11: non-emptiness of the core at A is a conclusion in the former, allowing us to derive Corollary 8, whereas in the latter it is an assumption. If we drop the assumption of non-emptiness and have a convergent sequence of mixed strategy equilibria, can we deduce non-emptiness of the core? Evidently not: even when the core is empty, the Deterministic Model can be approximated by continuous models generating a convergent sequence of mixed strategy equilibria. From the second part of Theorem 10, however, we can say that such a sequence of mixed strategy electoral equilibria in nearby models could not converge to a pure strategy electoral equilibrium in the Deterministic Model.
8 Is the Median an Artifact? In two papers, Hinich [15, 16] has suggested that the median voter result of Black [8] and Downs [12] is an "artifact" of their assumption of deterministic voting behavior. In the earlier of his papers, Hinich assumes three voters, quadratic utility functions with median ideal point distinct from the mean, and he approximates the one-dimensional Deterministic Model with a sequence of continuous expected plurality share functions converging pointwise to the Deterministic Model. He finds that they do not possess pure strategy equilibria close to the median and that, if the continuous models do have pure strategy equilibria, then they must be at the mean rather than the median. Hinich gives an example, 'Convergence to the Mean,' suggesting that, arbitrarily close to the Deterministic Model, the mean is indeed an equilibrium. Thus, the median appears fragile: adding the slightest uncertainty, the equilibrium jumps to the mean. In the later of the papers, Hinich generalizes his earlier observations, allowing for a continuum of voters, multiple policy dimensions, and generalized Euclidean preferences. His Theorem 1 (see our Corollary 4) establishes that, in the presence of uncertainty about voters' choices, if pure strategy equilibria of the electoral game exist, then the candidates must locate at the mean ideal point. His Theorem 2, proved in the context of the Additive Bias Model with normally distributed bias and a one-dimensional policy space, states that, for small enough variance, the mean is a pure strategy equilibrium of the electoral game. If this were true, and if the mean were distinct from the median, then the equilibrium would jump from the median to the mean in the presence of the slightest uncertainty. Hinich does not consider mixed strategies.
48
Jeffrey S. Banks and John Duggan
Our Theorems 5 and 10 can be used to formalize and clarify the suggested fragility of the median. Suppose, for example, that the median and mean are distinct, and that voters have quadratic utility functions. Consider a sequence of normally distributed Additive Bias Models with variance converging to zero (generating a sequence of continuous expected plurality share functions converging pointwise to the Deterministic Model), and suppose that, indeed, pure strategy equilibria exist for small enough variance. By Corollary 4, we know that the equilibrium platforms of the candidates converge to the mean as the variance gets close to zero. By the first part of Theorem 10, however, this sequence of equilibria must converge to the core, i.e., the median. But the median is distinct from the mean, an impossibility. Given Corollary 4 and Theorem 10, pure strategy equilibria must fail to exist beyond some point in this sequence, i.e., there is some variance level such that any Additive Bias Models with smaller variance fail to admit pure strategy equilibria, contradicting Hinich's [16] Theorem 2.^^ Thus, though the median must cease to be an equilibrium when arbitrarily small amounts of uncertainty are added to the Deterministic Model, the equilibrium cannot jump to the mean. A further observation arises from Theorems 1 and 10. Namely, we know from Theorem 1 that mixed strategy equilibria do exist as the variance of the bias term goes to zero. And we know from the third part of Theorem 10 that these equilibrium mixed strategies must converge weak* to the median. In other words, given an arbitrarily small open interval around the median, the probability that the equilibrium mixed strategies of the candidates determine policies within that open interval goes to one. Thus, we may expect outcomes close to the median, even when small amounts of uncertainty are added to the Deterministic Model. The next theorem restates these observations formally, echoing Kramer's [20] criticism of Hinich's earlier analysis. Kramer also argues that Hinich's examples fail to illustrate the fragility of the median, and, assuming candidates are motivated by probability of winning, he proves that mixed strategy equilibria of models close to the Deterministic Model must converge to the median.-^^ We offer similar conclusions in the expected plurality maximizing framework of Hinich, and in the more general context of a multidimensional policy space and general concave voter utility functions. To facilitate further discussion of Hinich's examples, we also allow the voters' preferences to vary. Theorem 12, In the Utility Difference Model, assume (i) X is compact and convex, (ii) for each t, Ut{x^ A) is continuous in (x. A); (Hi) {\^} is a sequence converging to X, (iv) for each t and m, ut{x, A^) is concave in x, (v) for each m, aggregate strict concavity holds, (vi) for eacht andC, {P^ '"^} is a sequence (J
of strictly monotonic functions converging pointwise to P^ , and (vii) for each This observation Wcis made independently by Laussel and Le Breton [21]. ^^ In fact, he proves something stronger: each non-median platform is strictly dominated in models close enough to the Deterministic Model.
Probabilistic Voting in the Spatial Model of Elections
49
t, C, and m, P^ '^{u) is differentiable with derivatives bounded in {t,u) and with Du=oPt '^('^) independent oft. If the core at A is disjoint from argmax / ut{x^ X) ii{dt), then there exists M such that, for all m > M, the electoral game parameterized by A"^ and {Pt^''^}t,c has no pure strategy electoral equilibria. Assume, moreover, (viii) the core is strongly externally stable and consists of just one policy x*, For each m, there exists a mixed strategy equilibrium (7r^,7r^) of the electoral game parameterized by X'^ and {P^ '^}t,c- For all sequences of such mixed strategy equilibria, (7r^,7r^) -^ (Sx^.^x*)Proof. Suppose the first conclusion of the theorem is false. Then we can extract a subsequence {{x^'^,x^'^)} of pure strategy equilibria. By Theorem 8, x'^'"' = x^^ for ah k. Then, by Corollary 3, we have x^^' G argmax / Ut{x,y^^')
^{dt)
for all k. By (i), {{x'^^, x^^)] has a subsequence converging to some By (i), (ii), and the Theorem of the Maximum, XA—XB^
argmax / uAx.X)
^ J
{XA.XB)-
iiidt).
But by the first part of Theorem 10, XA and XB are in the core at A, a contradiction. By the third part of Theorem 1, mixed strategy equilibria exist for each m. Since X is compact, so is the space of mixed strategy pairs, and so every sequence of mixed strategy pairs has a limit point. Under (viii), the third part of Theorem 10 yields x* as the only limit point. There are two points where Theorem 12 must be reconciled with Hinich's work. The first point is Hinich's [15] 'Convergence to the Mean' example. Here, we have voters with quadratic utility functions and a sequence of continuous expected plurality share functions, parameterized by 5, converging to the Deterministic Model. But given a particular (5, the candidates' objective functions are concave only within an interval of the mean, creating a problem for existence of pure strategy equilibria. This concavity problem forces Hinich to arrange voter ideal points within this interval, which itself depends on 5 and collapses to the mean as 6 goes to zero. Thus, we cannot interpret this example as adding an arbitrarily small amount of uncertainty to the Deterministic Model. Furthermore, Hinich's argument does not simply require "small" movements of ideal points, for the voters' ideal points will have to be collapsed to within an arbitrarily small neighborhood of the mean as 5 goes to zero. Theorem 12, because it allows the voters' utihty functions to
50
Jeffrey S. Banks and John Duggan
vary with the expected plurahty share functions, formalizes the point that, as expected plurality share functions converge to the Deterministic Model, the mean cannot be made an equilibrium by arbitrarily small changes in voter preferences. The second point is Hinich's [16] Theorem 2, where he claims existence of a pure strategy equilibrium when bias terms are distributed with small enough variance. Hinich does prove that, for a fixed platform XB for candidate J5, A's expected plurality maximizing response converges to XB as variance goes to zero, an intuitive result: as voters become more deterministic, candidate A is better able to "capture" the voters to one side of B and can then move toward B to increase his/her plurality. Hinich continues the argument by fixing XB at the mean and proving that there is a certain open interval around XB within which the best response to XB is the mean itself (his Lemma). Letting the variance go to zero, Hinich reasons that the best response x^'s must eventually reach that open interval and must then be the mean itself. Thus, the mean would seem an equilibrium. But the open interval itself depends on the variance, and it collapses to the mean as the variance goes to zero. Hinich's proof breaks down because the best response XA does not reach the open interval. In fact, the reverse of Hinich's claim is true: if the variance of the bias term is large enough, pure strategy equilibria will exist and they will be at the mean. From Theorem 2, we know that the key sufficient condition for existence of pure strategy equilibria is the concavity of the expected plurality shares as functions of policy platforms. Assume voter utility functions are quadratic and therefore concave (with second derivatives bounded below zero). When Gt is the distribution of a normally distributed random variable, it will never be concave, but it will be arbitrarily close to concave as the variance of the distribution goes to infinity. Composed with the voters' utility functions, the candidates' expected plurality share functions will eventually be concave, as needed. Then Corollary 4 implies the equilibrium platforms are at the mean.
9 The Utility Ratio Model Some of our results (namely. Theorems 8, 10-12 and Corollaries 1-4, 7, 8) are stated and proved for the special case of the Utility Difference Model, rather than the Utility Ratio Model. In this section, we will show that the two models are interchangeable, in a sense, and that corresponding results hold for the latter model. Given utility functions Ut for voters and expected plurality share functions Pf^ in the Utility Ratio Model, define the following Utility Difference Model: Ut{x) = ln{ut{x)) P^{u)=P^{e^)
Probabilistic Voting in the Spatial Model of Elections
51
for all t. The candidates' objective functions in the two models are clearly identical: Pf(xA,XB)
=
Pf{UtixA)/ut{XB))
= P^{Ut{XA)
-M^B))-
Furthermore, many properties of utility and expected plurality share functions in the Utility Ratio Model translate directly to properties in the associated Utility Difference Model. Continuity, differentiability, and monotonicity properties are preserved by the translation. Concavity of voter utility functions is clearly preserved, as well. The core is invariant under this transformation. And if {P^ '^} converges pointwise to P^^, then {P^ '^} converges pointwise
to Pf. As a consequence, Theorems 8 and 10-12 and Corollaries 7 and 8 carry over to the Utility Ratio Model directly: we need only replace "Utility Difference Model" with "Utility Ratio Model." To make this clear, consider the assumptions of the first part of Theorem 10 in the context of the Utility Ratio Model, i.e., Ut{x,X) jointly continuous, P^ ' ^ weakly monotonic and converging pointwise to Pf, pure strategy equilibria (x^, x ^ ) converging to (XA^XB), etc. Now go to the associated Utility Difference Model: Ut, {pf^'"^}, Pf- All of the assumptions of the first part of Theorem 10 hold, and the theorem yields XA and XB in the core defined using iit. Since the core is invariant under our transformation, we have XA and XB in the core of the original model. The same logic holds for the rest of Theorem 10. In fact. Corollaries 1 and 2 carry over as well, in a way that highlights the connection between the Utility Difference and Utility Ratio Models. The next result follows directly from Corollary 1 by going to the associated Utility Difference Model. Let Ut/Ut — {ut{x)/ut{y) \ x,y e X} denote the set of possible Utility Ratios for type t voters. Corollary 1' In the Utility Ratio Model, assume (i) X is compact and convex, (a) for each t and C, Pf is continuous and weakly monotonic, (Hi) for each t, P/^(e^) is concave on Ut/Ut, (iv) for each t, Pf{e~'^) is concave on Ut/Ut, and (v) for each t, Ut is continuous and \n{ut) concave. Then there exists a pure strategy electoral equilibrium. Assume, moreover, (vi) for each t and C, Pf is strictly monotonic, and (vii) aggregate strict concavity holds with respect to logs of voter utility functions. Then there is exactly one electoral equilibrium, and it is in pure strategies. The proof follows easily from Corollary 1: iit is continuous and concave, and Pf{u) is weakly monotonic, continuous, and concave, so there is a pure strategy equilibrium of the associated Utility Difference Model. Since candidates' payoffs are invariant under the transformation, we have an equilibrium
52
Jeffrey S. Banks and John Duggan
of the original model. The second part also follows directly from Corollary 1. The next result follows directly from Corollary 2. Corollary 2' In the Utility Ratio Model, assume (i) X is compact and convex, (ii) for each t and C, Pf{u) is continuous and weakly monotonic on Ut/Ut, (Hi) for each t, P^{u) is concave on Ut/Ut, (iv) for each t, Pf{-^) is concave on Ut/Ut, and (v) for each t, Ut is continuous and concave. Then there exists a pure strategy electoral equilibrium. Assume, moreover, (vi) for each t and C, Pf is strictly monotonic, and (vii) aggregate strict concavity holds. There is exactly one electoral equilibrium, and it is in pure strategies. For an example of expected plurality share functions in the Utility Ratio Model satisfying (ii)-(iv) and (vi) from Corollary 2^ consider ^A^ ^
U - l
PC{u) ^ ——
,
AR.
ix + l Note that these functional forms give us Pf{XA,XB) ?B/ ^_^ _ Pf{XA,XB) =
.
and P , ^ H = ' ' '
1
-U
u^ 1
Ut{xA) + Utixs)
Ut{xA) + Ut{XB)
M^B) Ut{XA) + Ut{xB)
Ut{XA) Ut{xA) + Ut{xB) '
which is just the Binary Luce Model used by Coughlin and Nitzan [11] and Coughlin [10]. Thus, we have derived their existence and uniqueness result from results for the Utility Difference Model, under minimal assumptions — in particular, their assumption of the specific Binary Luce functional form is unneeded. We can also derive their characterization in terms of Nash, rather than utilitarian, social welfare optima. Again, their result follows quite generally, relying only on the structure of the Utility Ratio Model — the Binary Luce functional form is unneeded. Corollary 3' In the Utility Ratio Model, assume (i) X is convex, (ii) for each t and C, Pf{u) is strictly monotonic and differentiable with derivatives bounded in (t^u) and with Du=iPf{u) independent oft, and (Hi) for each t, ln{ut) is concave. //(a:*,a:*) is a pure strategy equilibrium of the electoral game, then x* maximizes / ln{ut(x)) fi{dt). Assume, moreover, (iv) aggregate strict concavity holds with respect to logs of voter utility functions. Then (x*,a:*) is the unique interior pure strategy electoral equilibrium exhibiting policy coincidence.
Probabilistic Voting in the Spatial Model of Elections
53
The result for the Utility Ratio Model follows by transforming the model into its associated Utility Difference Model. Doing so, Corollary 3 implies that X* maximizes / ut{x) fj.{dt) = / JT
\n{ut{x))id{dt),
JT
and it implies the remainder of the theorem. We started with the Utility Difference Model because we appreciated the interpretation of the Additive Bias Model, but from a technical perspective we could equally well have begun with the Utility Ratio Model: any result for one model translates into a result for the other. More importantly, the apparent conflict between theories that predict utilitarian optima and those that predict Nash optima is resolved as a matter of modelling preference.
10 S u m m a r y We have presented numerous results predicated on numerous assumptions, those results often stated in a general form and then specialized to the case of the Utility Difference Model (or the Utility Ratio Model), which has the clear interpretation in terms of a random Additive Bias term for each voter. We end with a quick summary of our results for the Utility Difference Model, under the following minimal background assumptions. When they hold, our results for the Utility Difference Model are quite sharp. •
X is compact and convex.
•
For each t and C, Pf is strictly monotonic and differentiable with derivatives bounded in (t^u) and with Du=oPf{u) independent oft.
•
For each t, Ut is continuous and concave.
•
Aggregate strict concavity holds.
The above conditions on expected plurahty share functions hold, for example, in the Additive Bias Model whenever voter biases are identically distributed according to a continuous, strictly increasing distribution function. Under these assumptions, Theorem 1 yields the existence of a mixed strategy electoral equilibrium. If a pure strategy electoral equilibrium exists, then, by Theorem 8, it exhibits policy coincidence. By Corollary 3, the policy platform adopted by the candidates in any such equilibrium maximizes the utilitarian social welfare function. When do pure strategy equilibria exist? If nothing more than concavity of the voters' utility functions is known, we must assume the expected plurality share functions of the candidates are affine linear, as when the Additive Bias term for each voter type is uniformly distributed. In that case. Corollary 1
54
Jeffrey S. Banks and John Duggan
gives us existence and uniqueness of a pure strategy equilibrium. Moreover, as discussed after Theorem 4, that equilibrium platform is actually a dominant strategy for the candidates, in the sense that it is a best response to every platform one's opponent might adopt. Otherwise, if it is known that voter utilities possess stronger concavity properties, then weaker conditions may be imposed on expected plurality share functions, as illustrated in Corollary 2. Pure strategy electoral equilibria will not exist, by Theorem 12, if the core is non-empty, it does not include the utilitarian optimum, and the expected plurality share functions are close enough to the Deterministic Model. In the one-dimensional model with an odd number of voters and quadratic utilities, this simplifies to the following: if the mean ideal point is distinct from the median, then in models close enough to deterministic, pure strategy equilibria will not exist. From Theorem 12, however, mixed strategy electoral equilibria will exist, and the equilibrium mixed strategies of the candidates must put arbitrarily high probability near the median.
References [1] Aliprantis, C. and K. Border (1994) Infinite Dimensional Analysis: A Hitchhiker^s Guide New York: Springer-Verlag. [2] Aliprantis, C. and O. Burkinshaw (1990) Principles of Real Analysis New York: Academic Press. [3] Ball, R. (1999) Discontinuity and non-existence of equilibrium in the probabilistic spatial voting model. Social Choice and Welfare, 16: 533556. [4] Banks, J. S. and J. Duggan (2004) Existence of Nash equilibria on convex sets. Mimeo. University of Rochester. [5] Banks, J. S. and J. Duggan (2002) A multi-dimensional model of repeated elections. Mimeo. University of Rochester. [6] Banks, J. S., J. Duggan, and M. Le Breton (2004) Social choice and electoral competition in the general spatial model. Journal of Economic Theory, forthcoming. [7] Besley, T. and S. Coate (1997) An economic model of representative democracy. Quarterly Journal of Economics, 112: 85-114. [8] Black, D. (1958) The Theory of Committees and Elections Cambridge: Cambridge University Press. [9] Calvert, R. (1985) Robustness of the multidimensional voting model: candidate motivations, uncertainty, and convergence. American Journal of Political Science, 29: 69-95. [10] Coughlin, P. (1992) Probabilistic Voting Theory Cambridge: Cambridge University Press.
Probabilistic Voting in the Spatial Model of Elections
55
[11] Coughlin, P. and S. Nitzan (1981) Electoral outcomes with probabilistic voting and Nash social welfare maxima. Journal of Public Economics, 15: 113-121. [12] Downs, A. (1957) An Economic Theory of Democracy New York: Harper and Row. [13] Duggan, J. (2000) Equilibrium equivalence under expected plurality and probability of winning maximization. Mimeo. University of Rochester. [14] Glicksberg, I. (1952) A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium. Proceedings of the American Mathematical Society, 3: 170-174. [15] Hinich, M. (1977) Equilibrium in spatial voting: the median voter result is an artifact. Journal of Economic Theory, 16: 208-219. [16] Hinich, M. (1978) The mean versus the median in spatial voting games. In P. Ordeshook (ed) Game Theory and Political Science New York: NYU Press. [17] Hinich, M., J. Ledyard, and P. Ordeshook (1972) Nonvoting and the existence of equilibrium under majority rule. Journal of Economic Theory, 4: 144-153. [18] Hinich, M., J. Ledyard, and P. Ordeshook (1973) A theory of electoral equilibrium: a spatial analysis based on the theory of games. Journal of Politics, 35: 154-193. [19] Hotelling, H. (1929) Stability in competition. Economic Journal, 39: 4 1 57. [20] Kramer, G. (1978) Robustness of the median voter result. Journal of Economic Theory, 19: 565-567. [21] Laussel, D. and M. Le Breton (2002) Unidimensional Downsian politics: median, utilitarian or what else? Economics Letters, 76: 351-356. [22] Ledyard, J. (1984) The pure theory of large two-candidate elections. Public Choice, 44: 7-41. [23] Lindbeck, A. and J. WeibuU (1987) Balanced-budget redistribution as the outcome of political competition. Public Choice, 52: 273-297. [24] Lindbeck, A. and J. Weibull (1993) A model of political equilibrium in a representative democracy. Journal of Public Economics, 51: 195-209. [25] Luenberger, D. (1969) Optimization by Vector Space Methods New York: Wiley. [26] McKelvey, R. and J. Patty (2003) A theory of voting in large elections. Mimeo. Carnegie Mellon University. [27] Osborne, M. and A. Slivinski (1996) A model of political competition with citizen-Candidates. Quarterly Journal of Economics, 111: 65-96. [28] Patty, J. (2002) Equivalence of objectives in two candidate elections. Public Choice, 112: 151-166. [29] Patty, J. (2003) Local equilibrium equivalence in probabilistic voting models. Mimeo. Carnegie Mellon University. [30] Plott, C. (1967) A notion of equilibrium and its possibility under majority rule. American Economic Review, 57: 787-806.
56
Jeffrey S. Banks and John Duggan
[31] Rosen, J. (1965) Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33: 520-534.
Local Political Equilibria ^ Norman Schofield Washington University at St. Louis schofldOwueconc.wustl.edu
S u m m a r y . This article uses the notion of a "Local Nash Equilibrium" (LNE) to model a vote maximizing political game that incorporates valence (the electorally perceived quality of the political leaders). Formal stochastic voting models without valence typically conclude that all political agents (parties or candidates) will converge towards the electoral mean (the origin of the policy space). The main theorem presented here obtains the necessary and sufficient conditions for the validity of the "mean voter theorem" when valence is involved. Since a pure strategy Nash equilibrium (PNE), if it exists, must be a LNE the failure of the necessary condition for an LNE at the origin also implies that PNE cannot be at the origin. To further account for the non-convergent location of parties, the model is extended to include activist valence (the effect on party popularity due to the efforts of activist groups). These results suggest that it is very unlikely that Local Nash equilibria will be located at the electoral center. The theoretical conclusions appear to be borne out by empirical evidence from a number of countries. Genericity arguments demonstrate that LNE will exist for almost all parameters, when the policy space is compact, convex, without any restriction on the variance of the voter ideal points or on the party valence functions.
1 Introduction Many of the decisions of a society depend on choices of elected representatives, over social and property rights, taxation, government regulation, etc. These * This article is based on research supported by NSF grant SES 0241732. I would also like to acknowledge the debt that I owe to Jeff Banks, for inspiration and motivation in my attempt to outline an equilibrium theory of politics. I am very grateful to Lexi Shankster for preparing the figures, and to Ben Klemens for his help in dealing with Scientific Word. John Duggan very kindly went through earlier versions of the paper and pointed out problems with the proof of the Theorem. Gaetano Antinolfi showed me how to use Mathematica to check some calculations.
58
Norman Schofield
representatives, or political agents, are comparable to the entrepreneurs of an economic system. How these political decisions are made, and the relationship between the decisions and the nature of economic equilibrium, are part of the discipline of political economy. To understand this relationship it is necessary to have an equilibrium theory of political choice that is comparable to the equilibrium theory of economics (Austen-Smith and Banks [8, 9]). In principle, it would then be possible to create a more general theory of political economy. It has, however, proved difficult to construct such a political theory. The general perspective I adopt is that it worthwhile attempting to use the very abstract ideas of topology that proved so useful in the earlier demonstration of the existence of economic equilibria by Dierker [18] and Smale [48]. These studies did not use the apparatus of convex analysis and fixed point theory as usual in economics, but rather the notions of critical and local equilibria taken from the qualitative theory of dynamical systems. While these equilibrium concepts are weaker than the usual "global" concepts, it is possible to prove the "generic" existence of local economic equilibria under very weak assumptions. In the last main section of this article, I shall offer a way of using the same tools to demonstrate generic existence of local political equilibria. A development of this technique would then provide a way of studying the political economy in a qualitative fashion. However, before discussing political equilibria, it is necessary to have a good idea about what is entailed in such a notion. The usual concept of equihbrium is the "global" one of Nash equilibrium (see Nash [31]), but proof of existence is extremely difficult because of the necessity of requiring "convexity", in some form, of the preferences of political agents (see Schofield [38]). For example, even in a very simple model, where it is assumed that representatives adopt positions to maximize votes, there is no obvious reason why the convexity assumption should be satisfied (see Schofield and Parks [42]). As an illustration of the applicability of the concept of local political equilibrium, this article will examine this model in detail, for the general case in which the number of political agents, and the dimension of the space of political decisions, are not arbitrarily restricted. More specifically, we shall model the choices of the leaders of political parties as equilibria to a vote maximizing electoral game. Since the work of Downs [19], such models have tended to lead to the inference that representatives will adopt a position at the electoral mean (see Banks and Duggan [11] and Hinich [23]) or near the electoral median (Banks et al. [13, 14] and McKelvey [27]). However, this conclusion seems to contradict extensive evidence that parties do not converge in this way. In electoral systems based on proportional representation (PR), there may be a large number of parties, located at very different non-centrist positions (see Schofield and Sened [43, 44]). In electoral systems based on plurality (or first past the post), either there is no party near the center, as in the U.S. (Miller [29], Schofield et al. [41], Poole and Rosenthal [32]), or, as in Britain, the centrist party is not a likely candidate for government (Schofield [36]).
Local Political Equilibria
59
The model presented here attempts to account for non-centrist choices of parties by extending the stochastic model of voting due to Lin et al. [26] to include " valence" (Stokes [49]). Valence here is assumed to have two different sources. It may either be an exogeneous effect due to the differences in the average perceived quality of the party leaders (see Ansolabehere and Snyder [6], Aragones and Palfrey [7], Groseclose [22], Schofield [36]), or more generally, valence can be endogeneous, due to the indirect effect of party leader choices on the willingness of activists to support the party by fund raising, et cetera. Such activist support allows a party leader to increase the electoral perception of the valence of the party, through the use of the media and by other means (see Aldrich [1, 2] and Aldrich and McGinnis [3]). The chapter first models the situation where each agent (whether a candidate or party leader), j , is characterized by an average level of (exogenous) perceived competence (or quality), A^. That is to say, a typical voter, i, when making comparisons between agents j and k will compare their policies, Zj and Zk, and also their valences A^ and A/c. We assume that each agent adopts a policy position Zj so as to maximize the agent's own share of the popular vote. Because the agents do not know precisely what weight each voter gives to the perceived quality of the agent, the agent uses the "expected" vote share as a utility function. This version is an extension of the standard "probabilistic" or stochastic vote model (Banks and Duggan [11] and Coughhn [17]). The addition of exogenous valence changes the result known as the "mean voter theorem." Although the first-order condition for vote maximization is satisfied when all agents adopt the mean voter position, it may be the case that one of the agents with a low valence will, in fact, minimize vote share at the mean. Consequently, the second-order (or Hessian) condition has to be considered. A standard condition, or assumption, that is usually made for proof of existence of pure strategy Nash equilibrium (PNE) is "concavity" of the payoff function (in this case the vote share). Concavity is equivalent to the requirement that the Hessian, for the vote share function of each agent, be negative semi-definite on the entire strategy domain. Because this condition is unlikely to be met, I consider a weaker equilibrium concept, that of "local strict pure strategy Nash equilibrium" (LSNE). A vector of strategies is a LSNE whenever the appropriate Hessian for each agent is negative definite at that strategy vector. I shall refer to this condition as strict local concavity. Since it is generically the case that smooth functions do not exhibit degenerate critical points (with at least one zero eigenvalue of the Hessian) we may assert that, generically, a PNE, if it exists, is a LSNE. The converse, of course, is not true, since a LSNE may not be a PNE. Theorem 1 gives the necessary and sufficient conditions for strict local concavity at the joint electoral mean position. These conditions can be expressed as upper bounds for a single convergence coefficient which is defined
60
Norman Schofield
in terms of the parameters of the model, including the number of parties, the maximum valence difference, the dimension of the policy space, the stochastic variance a^, together with the variance of the set of voter ideal points. Empirical studies of elections in the Netherlands (Schofield et al. [40]) and in Israel (Schofield and Sened [44]) have obtained values for these parameters such that the necessary condition is violated. Consequently, there is no reason to expect convergence by parties to the electoral mean in these polities. In the case when the policy space is unidimensional, then a corollary of Theorem 1 gives a readily computed necessary and suflacient condition for existence of a LSNE at the voter mean. This chapter uses related empirical work by Schofield [36], based on a unidimensional model, to show that this condition was satisfied in elections in Britain in 1992 and 1997. It may not prove surprising that the mean voter theorem is invalid for multiparty competition. In the U.S. for example, candidates must win primaries in order to compete in presidential elections, and it may be the case that a policy position which wins a primary is much more radical than a position which wins the presidential election. However, political parties in Britain do not have to face primaries. Moreover, the fact that there are two principal parties in Britain, competing under an electoral system based on plurality rule, implies that maximizing vote share is almost identical to maximizing the probability of winning the election (Duggan [20]). According to the empirical analysis, under the assumption of exogenous valence, all parties would have maximized their vote share by adopting an identical position at the voter mean in these two elections. The evidence offered in Schofield [36] and supported by Alvarez [5] is that the British parties were perceived by the electorate to have very different positions on the single economic policy axis. To account for this commonly perceived divergence of party position, the model of British elections was developed in two directions. Firstly, the empirical analysis was extended to include a second dimension involving nationalism, in particular attitudes towards the European Union. The increase in the number of dimensions, and in the electoral variance in this policy space, implied that the necessary condition for strict local concavity at the electoral mean was violated. Although this explains the non-centrist position of the low valence Liberal Democrat Party, the analysis does not explain the location of the high valence Conservative Party, perceived to be far from the center on both axes. In an attempt to account for the extreme divergence of this party, the simple voting model with exogenous valence was extended to include activist support (Aldrich and McGinnis [3]). The activist model assumes that by contributing time and support to a preferred candidate, activists enhance overall voter support. Such activist support for a candidate is a function of the party or candidate position. Moreover, activist contributions to a party can be expected to exhibit decreasing marginal returns. It is plausible, therefore, that each party's activist valence function will be concave in the party strategy.
Local Political Equilibria
61
The most general model that I consider is one where both exogenous valence and activist valence are included. It is shown that the first-order condition for the optimal party position involves a trade-ofi" or balance between what I shall call the marginal "electoral pull" and the marginal "activist pull." In other words, the electoral gradient vector and the activist gradient vector must be equal in magnitude, but in opposite directions, at an equilibrium position for each party. Another way of interpreting this result is that when exogenous valence falls, for whatever reason, then activist valence becomes more significant in determining the optimal position of the party. Moreover, if we assume that activist valence is indeed a concave function of strategy, then it is possible to determine conditions on the eigenvalues of the Hessian of each of the parties' activist vote-share functions, sufficient to guarantee that PNE do indeed exist. The purpose of this article is to present the argument that, in the general spatial electoral model considered here, existence of PNE need not be guaranteed. However, transversality arguments, based on differentiabilty, can be used to show that LSNE will typically exist. Since the set of LSNE generically includes the set of PNE, it is possible to use simulation techniques in actual empirical situations to determine the set of LSNE, and by further analysis determine whether any of these LSNE are PNE. To determine whether PNE do indeed exist, it appears necessary to model, in a more complete fashion, the eff'ect of activists on political support. The notion of LSNE has not generally been used in modeling political phenomena. It is much more common when existence of PNE cannot be guaranteed, to use the notion of mixed strategy Nash equilibria (MNE). There seem to be two reasons to reject the applicability of the notion of MNE. First of all, as Banks et al. [13, 14] demonstrate, in two-party competition, the support of MNE will belong to a subset of the uncovered set (McKelvey [27]). It is conjectured that the uncovered set will tend to be small and centrally located with regard to the electoral distribution; see for example Schofield [35]. Consequently, models based on the MNE concept suggest that two party competition may lead to party locations that are "close" to the electoral center. In general, there is no empirical evidence for such a conclusion. A second objection is that party leaders are usually obliged to make policy pronouncements in the form of manifestos, etc. Empirical analyses with current techniques require that parties be precisely located and the stochastic models of electoral behavior, based on specific estimates of party position, typically give statistically significant empirical models of voter behavior (see Alvarez and Nagler [5], Poole and Rosenthal [32], Schofield [36] and Schofield et al. [41]). It would appear, therefore, that formal models, using the theoretically appealing notion of MNE, cannot be readily tested by empirical analysis. The next section of the article introduces the stochastic model, and proves Theorem 1, and its two corollaries, on existence of electoral equilibrium at the electoral mean. Section 3 off'ers some examples based on the theorem. Section
62
Norman Schofield
4 extends the model to include activist valence and presents a preliminary result on optimal party strategy in this more complex game. The empirical example from Britain is given to illustrate the balancing of marginal "electoral pull" and "activist pull". A short Section 5 attempts to address possible shortcomings of the model, but argues for the usefulness of the notion of local Nash equilibrium. Since there are many possible alternatives to the assumption of vote share maximization. Section 6 considers the more general question of generic existence of local Nash equilibria when agent payoffs are differentiable. Section 7 concludes.
2 Nash Equilibria in t h e Voter Model The situation we consider is that of a collection P = { 1 , . . . , j , . . . ,p} of political agents (whether candidates or parties). Each agent, j , chooses a policy Zj in a set X. Let z = {zi,... ,Zp) € X^ denote a strategy vector for the set of agents. The game form h : X'P -^ W maps from the set of strategy vectors to a space, W, of outcomes, on which the j t h agent has a utility function Uj : W -^ R. The game is {U^ : X^ -^ R}jeP or U^ : XP -^ W, where U^{z) = U,{h{z)). Definition 1. A pure strategy Nash equilibrium (PNE) for the game {UJ']P is a vector z* E X^ with the property that, for each j G P, there exists no Zj G X such that U j yZi^ . . . ^Zj_i^Zj^
^ j 4 - l 5 • • • , ^p j > ^ j \Zi^
. . . ^ Zj_i^
Zj , ^j-\-l 5 • • • 5 ^ p J •
A more general notion, that of mixed strategy Nash equilibrium (MNE) is similar, but considers strategies for each agent in a space of lotteries, or mixtures, defined over X. In the case where X is a compact, convex subset of a topological vector space, there are well known properties of U^ sufficient to gurantee existence (Austen-Smith and Banks [8]). Some of these focus on the properties of the underlying preferences induced by U^ on X. These are based on the Fan [21] Theorem. For example, quasi concavity and continuity of U^ are sufficient for existence of PNE. A function U is quasi-concave if U{ax + (1 - a)y) > min[U{x), U{y)] for all x,y eW
and a G [0,1].
Concavity is a stronger property that also suflSces. The function U is concave if U{ax + (1 — a)y) > aU{x) + (1 — a)U{y)^ for every real a. In the topological category where X is a topological "weaker"equilibrium concept is local Nash equilibrium.
space,
a
Local Political Equilibria
63
Definition 2. (i) A local pure strategy Nash equilibrium (LNE) for the game {^^}p is a vector z* G X'^ with the property that, for each j G P, there exists an open neighborhood Wj of z^ in X, such that, for no Zj in Wj is it the case that
(a) An LNE, z*, is locally isolated if there exists a neighborhood W of T^"" in X^ such that z* is the unique LNE in W. In the diflFerentiable category, where X has a differentiable structure and {Uj' : XP -^ R} are smooth functions, then it is natural to use the weaker equilibrium concept of critical Nash equilibrium (CNE). We give the definition when X is a vector space. However, the definition is also applicable to the general case when X has a differential structure; that is, when X is a smooth manifold (see Hirsch [24]). Definition 3. Suppose that X is a compact topological vector space of dimension w with smooth boundary. Let U^ : X^ —^ R^ be C^ -differentiable. Then z* G X'P is a critical Nash equilibrium (CNE) iff the first order vector equation -^j^(z*) = 0 Z5 satisfied for all j E P. In this category, if all /7^ are C^-differentiable, then analysis of the second order Hessian conditions at z* can be used to determine if the CNE z* is a LNE. Excluding boundary situations, every LNE must be a CNE, and the Hessian of each Uj' : X^ —^ R must be negative semi-definite (with all eigenvalues non-positive) at z*, with respect to Zj. It is therefore natural to say that a strategy vector z* is local strict Nash equilibrium, or LSNE, if and only if it is a CNE the and the Hessian of each Uj" at z is negative definite. Consequently, if the eigenvalues of all Hessians are negative at a CNE, z*, then this vector will be a LSNE and therefore a LNE. Since every PNE must be a LNE, this can be used to determine existence of PNE. Since we focus on LSNE, we essentially ignore the case where one of the eigenvalues is zero.. In this situation, the determinant of the Hessian is zero, so that the CNE is degenerate. As we shall see, this corresponds to a knife edge property of the parameters of the model. More abstractly, the degeneracy of a critical point is a non-generic property in the space of C^differentiable utility profiles, when this is endowed with the C"^-topology (see Hirsch [24]). It is obvious that a PNE can be characterized by a locally "flat" domain in utility space, so that such a PNE is not a LSNE. However, the generic situation is when a PNE is strict, in the sense that the payoff for each agent j at z* is strictly greater than at any different ^^.Such a strict PNE is clearly also a LSNE.Thus the necessary condition for a LSNE is generically a necessary condition for existence of a PNE.
64
Norman Schofield
To examine LSNE, we first seek the condition for z* to be a CNE, and then fix z!.^ = ( z ^ , . . . , Zj_-^,z*^i,...). We then determine the conditions under which the Hessian of the j ' s vote share function is negative definite at Zj and then examine Zj in the j t h strategy set. If the induced utility function for j is concave on this strategy set (with z*_j fixed) then the Nash equilibrium property holds for j at z^. Reiteration for each j gives a method of determining, at least generically, whether PNE exist. I shall use the term " strict local concavity" for the Hessian condition for z* to be a LSNE. This technique allows us to determine those constraints on the parameters of the model which are necessary and sufficient for strict local concavity.This allows us to obtain a necessary condition for LNE, and thus for PNE. The stochastic electoral model was originally developed for two-agent competition. In this case, it is natural to suppose that, for each agent j , Uj'{z) = Vj{z) — Vk{z) where k is the opposing agent to j , and Vj is j ' s "expected vote share function". Implicitly, such a model assumes a game form h : X'^ -^ W = [0,1]^, with h{z) = {Vj{z),Vk{z)) where the prize for each agent j is the plurality over j ' s opponent. Banks and Duggan [11] give an extensive discussion of this symmetric zero-sum case. In addition to compactness and convexity of X, and joint continuity of {Vj, Vk) they assumed a further property, "aggregate strict concavity"(a property on voter utility functions), and showed that then there would exist a unique symmetric PNE (2;*, z^) with Zj^z^ equal to the mean of the voter ideal points. One purpose of this article is to present an extension of the multiparty stochastic model of Lin et al. [26] by inducing asymmetries due to differing valences between the parties. As we shall see, this changes the "mean voter theorem", so as to bring its conclusions more into line with the empirical evidence as regards non-convergence. There are a number of possible choices for the appropriate game form for multiparty competition (see Schofield and Sened [43]). The simplest one, which is used here, is based on the assumption that the prize for agent j is proportional to Vj. With this assumption, we can examine the conditions on the parameters of the stochastic model which are necessary and sufficient for strict local concavity and thus for existence of strict LSNE. From this we can infer the necessary condition for existence of PNE. The key idea underlying the formal model is that party leaders attempt to estimate the electoral effects of party declarations, or manifestos, and choose their own positions as best responses to other party declarations, in order to maximize their own vote share. The stochastic model essentially assumes that party leaders cannot predict vote response precisely. In the model with "exogenous" valence, the stochastic element is associated with the weight given by each voter, z, to the average perceived quality or valence of the party leader.
Local Political Equilibria
65
The data of the model is a set {xi € X}i^]sf of "voter ideal points" for the members of the electorate, N, of size n. As usual we assume that X is a compact convex subset of Euclidean space, R^, with w finite. Each party in the set P = { 1 , . . . , j , . . . ,p} chooses a policy, Zj G X, to declare. Let z = {zi,... ^Zp) G X^ be a typical vector of party policy positions. Given z, each voter, i, is described by a vector u^(x^,z) = (un^Xi^zi)^... ^ Uip{xi,Zp)), where Uij{xi, Zj) = Xj + fij{zj) - f3\\xi - ZjW^ + Cj. Here, Xj is the "exogenous" valence of party j , lJij{zj) is the "activist" valence, /? is a positive constant and || • || is the usual Euclidean norm on X. The terms {cj] are the stochastic components, assumed to be independently and identically normally distributed (iind), with zero expectation, each with standard deviation cr. The independence assumption is used in the estimations discussed in Section 4, and in the proof of Theorem 1. However, we show that analogous results can be obtained, if the more general assumption is made that the stochastic errors are multivariate normal with general variance/covariance matrix, 9. Because of the stochastic assumption, voter behavior is modeled by a probability vector. The probability that a voter i chooses party j is Pij{7,) = Vx[[uij{xi,Zj) > Uii{xi,zi)], for ah / 7^ j]. Here Pr stands for the probability operator. The expected vote share of party j is
In the vote models it is assumed that each agent j chooses Zj to maximize Vj, conditional on z_j = {zi,..., Zj-\,Zj^i^..., Zp). The first result will focus on "exogenous" valence and assume that JJLJ = 0 and that the exogeneous valences are ranked Xp > Xp-i > • • • > A2 > Ai. I denote this model by M. In this model it is natural to regard Xj as the "average" weight given by a member of the electorate to the perceived competence or quality of candidate j . The "weight" will in fact vary throughout the electorate, in a way which is described by the normal distribution. Because of the differentiability of the cumulative normal distribution, the individual probability functions {pij} are C^-differentiable in the strategies {zj}. Thus, the vote share functions will also be C^-differentiable. Let x* = {l/n)UiXi. Then the mean voter theorem for the stochastic model asserts that the "joint mean vector" ZQ = ( x * , . . . , x*) is a PNE. Lin et al. [26] used C^-differentiability of the expected vote share functions, in the situation with zero valence, to show that the validity of the theorem depended on the condition that a'^ was "sufl&ciently large."
66
Norman Schofield
To determine whether this result can be applied to empirical cases, where the parameters of the model can be estimated, I shall obtain the precise necessary and sufficient conditions for existence of a LSNE of the form (a;*,..., x*). Of course, even if the sufficient condition for ZQ = (x*,..., x*) to be an LSNE is satisfied, this does not imply that it is a PNE. However, the necessary condition for ZQ to be a LSNE immediately gives a necessary condition for ZQ to be a PNE. I shall argue that, in general, this necessary condition will be violated. First, we may transform coordinates so that in the new coordinates, x* = 0, and I shall often refer to ZQ as the joint origin. The voter variance covariance matrix: To characterize the variation in voter preferences, we must represent in a simple form the variance covariance matrix (or data matrix, V) of the voter ideal points. Let X be endowed with a system of coordinate axes ( l , . . . , t , 5 , . . . , i t ; ) . For each coordinate axis let ^t = i^it^ ^2t5 • • • 5 Xnt) be the vector of the t^^ coordinates of the set of n voter ideal points. We use (^s,^t) to denote scalar product. The symmetric w x w voter variance/covariance data matrix V is then defined to be
/(6,6
(6,6) (6,6)
\
(6,6) (6,6) (6,6) (6,6)
v=
(6,6) (6,6)
(6,6) (6,6) (6,6) (6,6)
We next define the convergence coefficient of the model M. Definition 4. (i) For each agent j , define Xay(j) = ^
E/CGP-O)
'^^•
(a) Define the coefficient Aj for the contest of agent j against the competing agents to be A,=
-2
pa^
['^avU) ^
•^j)-
(Hi) The Hessian matrix Cj associated with agent j is defined to be Cj = {2Aj/n)V — I, where I is the w x w identity matrix. (iv) The "convergence coefficient'^ of the model M is given by c{M) = 2A\v^ where v"^ = (^) ^^^li^t^^t) 'is the total variance of the voter ideal points. We now state the main result on the model M, with arbitrary p and w.
Local Political Equilibria
67
Theorem 1. The necessary and sufficient condition for ZQ to he a LSNE is that the eigenvalues of the Hessian matrix Ci be all negative. The proof of Theorem 1, below, depends on considering the first and second order conditions at ZQ for each vote share function. The first order condition is obtained by setting dVj/dZj — 0 (where we use this notation for full differentiation, keeping ^^i,..., Zj-i^ ^j-^i^ - - -1 ^p constant). Because the voter probabilities involve correlated errors, we first transform the problem so that these terms are uncorrelated. This allows us to show that ZQ satisfies the first order condition. The second order condition is that the Hessian d^Vj/dz'^ be negative definite at the joint origin. If this holds for all j at ZQ, then ZQ is a LSNE. However, we need only examine this condition for the vote function V\ for the lowest valence party. As we shall show, this condition on the Hessian of Vi is equivalent to the condition on C\, and if it holds for Vi, then the Hessians ior V2,... ,Vp are all negative definite at ZQ . As usual conditions on Ci for the eigenvalues to be negative depend on the trace, trace(Ci), and determinant, det(Ci), of Ci. These turn on the value of Ai and on the electoral variance/covariance matrix, V. Using the determinant of Ci, we can show that 1 > 2Aiv'^ is a sufficient condition for the eigenvalues to be negative. In terms of the "convergence coefficient" this is 1 > c{M). In a policy space of dimension w, the necessary condition on Ci, therefore, on the Hessian of Vi is that w > 2Aiv'^. This condition is obtained from examining the trace of Ci and gives w > c{M), Conversely, if the necessary condition for Vi fails, then ZQ can be a neither a LNE nor a LSNE. Ceteris paribus, a LNE at the joint origin is "less likely" the greater are the parameters p, /?, Xav{i) ~ '^i^'^^? ^^^ is "more likely" the greater is the stochastic variance. To illustrate, in the case with p = 2, the expression for Ai is just 2^2-(A2 — Ai). Thus, in the very simplest case, with two parties and one dimension the convergence coefficient of the model M is
c(M) = 4 (A2-Ai)t;2. As Corollary 1 shows, the single eigenvalue is c{M) — 1, so the necessary condition for an LNE is c{M) < 1. Note that the case Xp — Ai was studied by Lin et al. [26]. In that case, the convergence coefficient c{M) is zero so the joint origin, ZQ, is an LSNE. However the examples presented below to illustrate the theorem suggest that even when the joint origin is an LSNE, the concavity condition will not be satisfied. This suggests that PNE are unlikely to exist at the origin. The proof of the theorem is presented in two parts. In Part A we show that the multivariate integral problem can be reduced to a univariate problem by suitable choice of a matrix transformation. In Part B, we examine the first and second order conditions.
68
Norman Schofield
Proof of Theorem 1, Part A: Choice of orthonormal variates. Because the vote functions involve correlated variates, we show first how to construct a transformation of the variates so as to make them orthonormal. This allows us to show that the competition between an agent j and the other agents can be represented as a contest between j and a composite competitor. We then use the "convergence coefficient", c{M), as given in Definition 4, to show that c{M) determines the eigenvalues of the Hessians, and thus classifies the model M in a crucial sense. From the above definitions, the probability, pii{z), that voter i chooses agent 1 is given by Pr [[Ai - p\\xi - zi||2 + 61 > A, - /3||x, - Zj\f + e,-], for all j ^ l] = P r [[Ai - f3\\xi - zi\\^ - Xj + p\\xi - Zj\\^ > Cj - ei], for all j ^ l] .
Now let ei = (e2 — ei, €3 — e i , . . . , Cp — ei) be the (p—l) dimensional variate. It is obvious that ei has the mutivariate normal distribution with covariance matrix U. Unfortunately the components of ei are correlated, so that U has off-diagonal terms. Indeed it is easy to see, in the case p — 4:, for example, that E^a^\
121 1 1 2^
To show this, note that ei == F(e) where e is the error vector, and
F =
Thus Z" = (j^F • F-^, where T denotes transpose. Because the components of ei are correlated, the expression for Pii{z) cannot be readily differentiated. However we may make a transformation to new orthogonal variates. Consider a transformation matrix Bi of rank {p — 1) with yi = Bi{ei). A standard result is that the random vector yi has the multivariate normal distribution with covariance matrix {Bi)E{Bi)^. Now consider the solution to the matrix equation where / is the {p — 1) x {p — 1) identity matrix. It can be readily shown that a solution /
^1,1 ^1,2 hj
B, = \^p-i,i
. bk,j • bp-ij
h,p-i
bp-i^p-i
\
Local Political Equilibria
69
can be found such that X]?=i ^kj = 0 for each /c = 1 , . . . ,p — 2, and such that bp-ij = bp-i^p-i for j = 1 , . . . ,p — 2. These additional 2(p — 2) restrictions can be satisfied because the symmetry of the covariance matrix, E, gives (p- ){p- ) (jegj-ees of freedom on the choice of entries of Bi. We have shown that this transformation exists in the simple case that the original components of e were independent and identically normally distributed (iind). Suppose more generally that these errors were mutivariate normal with an arbitrary covariance matrix 9. Then it is easy to show that the required matrix Bi satisfies the matrix equation {BiF)9{BiF)^ = G where G is an appropriate diagonal matrix. In principle there is no difficulty in finding ^ i in this more general case. For the purpose of exposition, this chapter will focus on the simpler,iind case. Consider the transformed (p — 1) dimensional variate p
yi =^hj{ej
-ei).
Because the components of yi are independent, we may write p^l{z)
= ^ i ( ^ n , i ( z ) ) , . . . ,^/e(^^i,/e(z)) • (^p_i ( ^ a , p - l ( z ) ) ,
where ^k{d) stands for the univariate cumulative normal distribution (end) with appropriate variance, up to the value d. The values {gn^ki'^)} are the upper bounds of these univariate integrals. It follows directly from the definition of Bi that p-i
9ii,k{z) = "^bkj
[Ai - P\\xi - zi\\^ - Aj-+i + P\\xi - Zj^iW'^] .
j=i
Since Xl^Zi bkj — 0 for /c = 1 , . . . ,p — 2, we see that this term is independent of zi. However it is a function of p-i 3=1
Now consider gii,p-i{z) = ^ 6 p _ i j [ A i - p\\xi - ziW'^ - >^j-\-i + f3\\xi - Zj^i\\'^]. It is easy to see that the variate Yl0=2(^0 ~ ^1) ^^^ variance
[{p-l) +
{p-lf]a^=p{p-l)a^.
70
Norman Schofield
Thus the coefficients {bu-i A must be
. ^
Without loss of generality we
may define the (p — 1) dimensional variate yi,p-i by
r) —
I
j=2
= -ei
Note that this new variate has variance -^cr^. The relevant upper bound on the integral is then /^zi,p-i(z) - - ^
Y.[Xi - f5\\x, - z , f -
Xj + p\\x, - ZjW'].
Now for each k = 1 , . . . , p define Xav{k) == — J
2^
A^-.
jeP-{k}
It then follows that 1 ^ /lil,p_l(z) = [Ai - Xavil) - P\\Xi - ^llP] + - — Y ^ / ? | | X i - Zj\\^. As a consequence we may write Pil{z)
= ^ i ( ^ a , l ( z ) ) , . . . , ^ A ; t e l , / c ( z ) ) • ^p-2{gil,p-2{z))
' ^p_i(/ln,p-l(z)).
The terms ^k{9ii,k{z))^ for /c = 1 , . . . ,p — 2, are independent of zi. The (p — l)th term is given by the univariate end for the variate yi,p-i with variance -3J
Again, the first (p — 2) terms are independent of Zj, while the expression ^p-i{hij^p-i{z))
involves the term [Xj — Xav{j) ~/?ll^2 ~^jlP]- ^^ essence, the electoral competition of agent j against the other agents can be regarded as a contest between j and a composite opponent with the average exogeneous valence XavU)- The P-r
Local Political Equilibria
71
Thus, the coefficient Aj = "^2 {^av(j) ~ ^j)^ previously defined, characterizes the contest of party j against the opposing parties. Having shown Part A we may now proceed to the second part of the proof of the theorem. Part B: Equilibrium conditions. We first seek the first order condition. Consider the lowest valence party, 1, and any voter i. Prom Part A, we have seen that the probability that voter i picks agent 1 over the other agents, at the joint strategy z, is given by pn(z)
= ^ i ( ^ n , i ( z - i ) ) , . • • ,^/ctei,/e(z-i)) • ^p_i(/ia,p-i(z))-
Now Vi(z) = ^EiPii{z). To obtain the condition for zi to be a local strict best response to z_i = ( 0 , . . . , 0), observe that we may write Pii{z) = anixi,z-i)
'^p-i{hii^p-i{xi,z)).
Because of the definition of Bi, the first term in this expression is independent of zi, and we may treat it as a constant. Notice however that this term is a function of Xi and z_i. Taking the first differential gives the first order condition on Vi(z) as I^^=iaii{xi,z_i)(t){hii^p-.i{z))dhii^p-i/dzi
= 0,
where <j){hii^p-i) is the normal probability density function for the variate yi,p-i at the value ^a,p-i- Now, dhii^p-i
.
——^^— - -2f3{zi - Xi). dzi Consider the vector zl^ = ( 0 , . . . ,0), where Y^Xi = 0. By the definition of ^ 1 , the term aii{xi,z'i-^) is now independent of Xi, and can be written as ai. Note that ai is positive. To test whether 2:1 = 0 satisfies the first order condition, let ZQ = ( 0 , . . . , 0). Then, for each i,
is independent of Xi, and thus of i. Hence, the first order condition for a CNE with respect to agent 1 is
i
The solution to this equation is zi = ^Z^^z- But Yl^i — 0, by the choice of coordinates. Thus the solution is ^1 = 0 . Obviously, an identical argument can be carried out for each agent j , by noting that QJ = Xj — Xavij)- Thus zj = ( 0 , . . . , 0) solves the first order condition.
72
Norman Schofield
The second order condition on the Hessian Hi — (PVi/dzf is that this be negative definite. With zl^ — ( 0 , . . . , 0 ) we may write hii^p-i{z) = hii,p-i{z_i^Zi) = gii{zi). Computation of Hi shows that it is given by i7i(z_i,zi)
(1) (2)
i
p-l
-9ii{zi)['^ii{zi)] cr"
+
(fignizi)
dzl
• (3)
Without loss of generality we may normalize this expression by dividing by ai . The term [Vii('^i)] is a -u; x it; symmetric matrix involving the differentials dgn/dzi, whose entries are 4/3^(x^s — zis){xit — zu) in the (5, t) position. The term [d'^gn/dzi] is the negative definite Hessian of gup (namely, —2/3/, where / is the identity matrix). Thus, X]2GAr^^i(^) — 4/?^V, where V is the voter covariance matrix, given above. As we have just seen, gii{0) • gi = ' ^ l - ^ a t ; ( l ) is independent of Xi. We thus obtain Hi{zl)
2/3
{[*;^][A..,.,-A.iv-„/} 2Ai
V
-I
n Obviously, the first term involving (f){\i — )^av{i)) ^^ this expression is positive. So the condition that zi == 0 constitutes a local best response to z_i = ( 0 , . . . ,0) depends on the eigenvalues of Ci — [^^V — / ] . If Ci has negative eigenvalues, then the same argument can be repeated for each agent 2 — 2 , . . . ,p. To see this note that the above argument immediately shows that the Hessian for j is
if,(zS)==2^0(A,-A,,(,))
2A,
V-/
Thus the Hessian for j has negative eigenvalues if and only if Cj is negative definite. Because the valences are ranked \p > Xp-i > • • • > A2 > Ai, it follows that Ai > A2 > •'' > Ap. This implies that trace(Ci) > trace(C2) > • • • ^ trace(Cp) and det(Ci) > det(C2) > ••• > det(Cp). Thus if Ci has negative eigenvalues then so do C 2 , . . . , Cp, and this implies that Zi = Z2 = - - - = Zp = 0 will all be mutual local best responses. This shows that the stated condition is sufficient for ZQ = ( 0 , 0 , . . . , 0) to be an LSNE. Obviously, if Ci does not have negative eigenvalues, then ZQ cannot be a LSNE. This completes the proof of the theorem. Note also that if the valences are not all identical then 0 > Aj^ for some jo > 1, thus guaranteeing that Cj has negative eigenvalues whenever j > JQ.
Local Political Equilibria
73
On the other hand, if all valences are identical then Ci must have negative eigenvalues. Although Theorem 1 is stated in terms of LSNE, it clearly gives analogous conditions for ZQ = ( 0 , . . . , 0) to be an LNE. Thus if ZQ is an LNE then the eigenvalues of Ci must be non-positive, while negative eigenvalues imply that ZQ must be an LNE. It is also worth remarking that with a general covariance matrix, 6, for the errors, we can use the above procedure, based on the matrix transformation {BiF)9{BiF)^ = G, to obtain a Hessian matrix Ci{9) whose eigenvalues give conditions for a LSNE at the joint origin. To illustrate this theorem, we can give corollaries for the one and two dimensional cases. Corollary 1, If X is one dimensional, and there are two parties, then the necessary condition for ZQ to be a LNE is that ^(A2—Ai)t;^ < 1. The sufficient condition for the joint origin to be a LSNE is that this inequality holds strictly. Proof Since we assume p = 2, then Xavii) = ^2- Just as in the above, ^^12(0) = Ai — A2 is the same, for all i. Moreover, V^^ =
2o2 {dg^l/dz^y=4xip•
Thus, (A2-Ai)(x2/?)
(4)
Now '^^ — \ S j ^? is the total electoral variance. Clearly, B.\(0) is nonpositive if and only if cr^ > /3(A2 — X\)v^. If this condition fails, then z\ — ^ is a local strict minimum of the vote share function, Vi, given Z2 — 0, so that ZQ cannot be an LNE. This gives the necessary condition. On the other hand, if G^ > /3{X2 — Ai)i;^, then zi = 0 gives a local strict maximum of Vi. The sufficient condition for V2 is a'^ > /3{Xi — X2)v'^. Since A2 > Ai, this condition must hold. Consequently the strict inequality gives a sufficient condition for z5 to be a LSNE. Notice that as cr^ ^ 0 , then it becomes impossible for ZQ to be an LSNE if the valences differ. (See also Groseclose [22] for a one-dimensional model involving valence.) Corollary 2. Assume X is two dimensional. Then ZQ is a LSNE if c{M) is strictly less than one, and is a LNE only if c{M) < 2. Proof. The condition that both eigenvalues be negative is equivalent to the condition that det(Ci) is positive and trace(Ci) is negative. Now det(Ci) = {2A^/nf
[(6,6) • (6,6) - (6,6)']
+l-(2/li/n)[(6,6) + (6,6)]-
74
Norman Schofield
The first bracket [(^i,Ci) * (^25^2) — (Ci5&)^] is an "inverse correlation coeflacient" associated with the covariance matrix of the voter distribution. By the triangle inequality, this term must be non-negative. Thus det(Ci) is positive if 1 > ( 2 ^ 1 / n ) [ ( 6 , 6 ) + ( 6 , 6 ) ] , or 1 >
fc^/?(A,,(i)
- Ai).;^
(5)
This gives the sufficient condition that 1 > c{M) for a LSNE at the joint origin, zj. The necessary condition for ZQ to be an LNE is that the eigenvalues be non-positive. Since trace(Ci) equals the sum of the eigenvalues we can use the fact that trace(Ci) = (2Ai/n)[(^i,^i) + ( 6 , 6 ) ] ~ 2, to obtain the necessary condition ^^^^/3(Aa.(i)-Ai)t;2<2.
(6)
Thus c{M) < 2 gives the necessary condition, completing the proof. If the first inequality (given in (5)) fails, but the second (given in (6)) is satisfied, then the eigenvalues may still be non-positive, and can be explicitly computed in terms of the model parameters and data. If the second condition fails then obviously at least one of the eigenvalues must be strictly positive, and so ZQ cannot be an LNE. The condition c{M) = 2 includes the non-generic case where both eigenvalues are zero. In the two dimensional case there is one situation where computation of eigenvalues is particularly easy. If the covariance ( 6 , 6 ) of the electoral data on the two axes is (close to) zero, then the voter covariance matrix is (approximately) diagonal and the two policy dimensions can be treated separately. In this case we obtain two separate necessary conditions ~2 {^p — Ai)t'^ < 1, for both t = 1,2, for ZQ to be a LNE. If v^ < v'^ then we obtain essentially identical necessary and sufficient conditions in terms of v"^. It is obvious that trace(Ci) involves the dimension, w, so we obtain the necessary condition c{M) < w for the joint origin to be a LNE in this more general case. In the examples below we deal with the one dimensional case first, and then give a two dimensional illustration of the result.
3 Examples Example 1, Consider the model, M, but with zero valence, in one dimension, with two voters {1,2} at Xi = —1 and X2 = + 1 , and two parties {1,2} at zi^ Z2. As observed above, with p = 2, it is not necessary to perform the transformation to orthogonalise the variables. The probability voter i picks party 1 over 2 is
Local Political Equilibria Pr [-P{x^ - zif
75
+ ei + /3(x, - Z2f - es > O] .
We can write gi = —0{xi — zi)'^ -{- P{xi — z^)^ for the "comparison function" for voter i in comparing party 1 with party 2 when the parties are at {z\,Z2). This probability is ^(^i), where ^ is the cumulative normal distribution for the variate e2 — 6i, with variance 2cr^, and expectation 0. Suppose now that 2:2 = 0 (the mean of the voter ideal points). We can proceed just as above to test whether zi — 0 comprises a best response to Z2 — 0, thus making up a component of a Nash equilibrium.. Now V\ = \[^{gi) -f ^(^2)], so dVi/dzi = \\(t){gi)dgi/dzi + 4){g2)dg2/dzi], where 0(^i) is the value of the normal probability density function, (j)^ at gi. Since zi = Z2, we see gi = ^2 and the first order condition is dgi/dzi + dg2/dzi = 0 or zl = (l/2)(a:i + X2) = 0. As before, z^ -= x\ Prom (l)-(3),
"'"'»•>-«».)
d^gi _ _gi_ f dgi_^ ^^ dzl 2(72 y^^^
Clearly, if zi = 0, then p^ = 0 for i = 1,2. Moreover, d^gi/dzl — —2(3 is negative definite. Consequently, Vi has a negative definite Hessian at zi ==0. Although Vi has a local maximum at zi — 0, this does not imply that (0,0) is a Nash equilibrium. (Obviously, the vector (0,0) is what we have termed a LSNE.) The sufficient condition for 2:1 == 0 to be a best response to Z2 = 0, when zi can lie in the domain [—1, +1], is that Vi is a concave function in zi (for Z2 fixed at 0), in the domain zi G [—1,4-1]. We use the standard result in convex analysis that a necessary and sufficient condition for concavity of a differentiable function on some domain is that its Hessian be negative semidefinite on the domain. To see whether this is so, we evaluate the Hessian of ^(92) at zi = - 1 . We obtain ^2 = -P{2)^ + ^(1)^ - -3/?. So, d'^{g2)/dzf
= (/)(-3/3) [-2/?+ [3/3/2^2] (4^)2]
Similarly d^^{gi)/dz\ ==0(/?)[—2/3], which is clearly negative. The first expression is negative only if cr > 12/?^. If cr < /3\/l2, then the Hessian of ^(^2) at zi = —1 will be positive. Thus, for a "sufficiently" small, the Hessian of Vi at zi — —1 will also be positive. In this example, therefore, for some value of a, with a < 12/32, the expected vote function, Vi will not be concave on [—1, +1]. Although concavity may fail, the weaker condition of quasi-concavity appears to be satisfied, so that the origin is a local "attractor" on the domain. As the example should make clear, the requirement of concavity imposes a relationship between the variance of the voter ideal points and the stochastic variance. As expressed in Theorem 1, if the stochastic variance is given, then the constraint is imposed on the variance of the set of bliss points. Alternatively, as in the example, if the bliss points are given then the constraint is imposed on the stochastic variance.
76
Norman Schofield
Example 2. We now modify Example 1, by introducing valence terms A2,Ai with A2 > Ai- We first use Corollary 1 to determine the conditions for (0,0) to be a LNE. Now dgi/dzi == 2[3xi = ±2/? depending on whether i = 1 or 2. By (4), iJi(O) = \(t){\i - Ap) Y.i [((Ap - Ai)/2a2)(2/?Xi)2 - 2/3], so the necessary and sufficient condition is (Ap — Ai)/3 < cr^, as stated in Corollary 1. (In this case the electoral variance is 1.) Suppose this condition is satisfied, so that (0,0) is an LNE. Then we can obtain a condition for concavity so as to ensure that (0,0) is a PNE. Let Z2 = 0 . A sufficient condition for 2:1 == 0 to be a best response in the domain [—1,4-1] is that the Hessian of Vi be non-positive on this domain. Consider zi — —1. The comparison functions ^2 and gi are now 92 = Xi- P{X2 - zif = Ai - A2 - 3/3
- A2 -f I3{X2 - Z2f
Pi - Ai - /3(xi - zif = Ai-A2+/3.
- A2 + /3(xi - Z2f
The Hessian at zi = — 1 is then 0(Ai - A2 - 3/3) {-2/3 + [(3^ + A2 - Ai)/2(72] [4/3]'} + (/)(Ai - A2 +/3){-2/3}. While the second term is clearly negative, the first term may be positive. A sufficient condition for the first term to be negative at 2:1 = —1 is then cj^ > 4(3/3 + A2 — Ai)/3. Since A2 > Ai this condition is more severe than the one obtained in Example 1. Obviously a sufficient condition for the Hessian of Vi to be negative at 2:1 — —1 is much more restrictive than the one obtained in Example 1. Moreover, even when the condition for (0,0) to be a LNE is satisfied, there is no formal reason to expect (0,0) to be a PNE. Results in Groseclose [22] suggest that even this simple model it is difficult to determine even if PNE exist. Symmetry suggests however that if the origin is not an LSNE, there will be two symmetrically located LSNE, with the high valence party nearer the origin. Example 3. To illustrate the computation of eigenvalues in two dimensions, consider a situation where the valences for 1, 2, 3 are ranked with Ai < A2 < A3, and three voters, labeled {1,2,3}, have the ideal points in M^ given by (0,1), ( - Y ^ s T i , - ^ ) and ( ^ ^ 4 , - ^ ) . With all agents at the origin, the comparison function gi for each voter is given by Ai — ^(A2 + A3), in comparing agent 1 against agent 2 and 3. So the Hessian for agent 1 involves the matrix Hi = Ei(l){gi
^^^^[v.1/9-2/ pa^
Here Vi is the 2 x 2 matrix generated by the gradients of gi at Xi. Summing the terms [V^] gives 4V, where V is the voter variance/covariance data matrix
Local Political Equilibria V=(
77
So 2 3 0§
As before we let ^i = ( ~ v f^O^ v/1) ^^^ ^2 = ( - ^ , 1, - | ) be the vectors of voter ideal points on the first and second coordinates. The the diagonal terms in V are given by the quadratic forms (^i,^i) = (^2,^2) = | and the off diagonal terms are given by the scalar product ((^1,^2) = 0- Thus the covariance between ^1 and ^2 is zero. The Hessian for agent 1 is given by
A necessary condition for this matrix to have negative eigenvalues is that its trace be negative. If the trace is zero or positive, then one of the eigenvalues must be positive (or both eigenvalues are zero). Now the variance of the voter ideal points on the two axes is given by f ^ = |[(Ci, ^i)-f (^2, C2)] — 3 bi+'^2] — 1. Because of the zero covariance, the two eigenvalues of the Hessian must be equal. Thus the necessary condition that the the eigenvalue be negative gives the same condition on the trace /?(p-l)[A2 + A3-2Ai] pcr^
-2.
Because of the symmetry, this condition is identical to the one obtained from the determinant. If the trace is positive, then each eigenvalue of the Hessian will be positive at the origin. Thus, if cr^ is sufficiently small then z^ = 0 cannot be a local best response and so the origin cannot be an LSNE. Note that due to the symmetry of the voter ideal points, if the above condition fails, then agent 1 can increase vote share by moving in any direction away from the origin. In the non symmetric case, where V is non-diagonal, when the trace condition fails then one eigenvalue will exceed the other, and agent 1 should move away from the origin in the "major" eigenspace associated with this eigenvalue. This eigenspace will depend on V, and in particular, on the variances vf and t'f. Again, symmetry suggests that there are two different cases, depending on the degree of covariance, and on the variances Vi and f I . In the case where there is a major axis, and one eigenvalue is positive, while the other is negative, then, in equilibrium, all three agents will be positioned on this major axis. In the second case, where both eigenvalues are positive, then the agents will move away from the origin in these eigenspaces. In both cases, the highest valence agent will be positioned nearest the origin. Obviously, by symmetry, there will be many LSNE.
4 Activist Valence We now consider a more general model, M(/i), involving both exogenous valence {Xj} and activist valence {/Xj}. To keep the analysis relatively simple
78
Norman Schofield
we shall focus below on the competition between two parties, called j and p. We can then apply the results to the case of Britain, where there are two principal parties. As in the proof of Theorem 1, the first order condition for agent j is
-^
i
However, now gij{z) = Xj -\- fij{zj) - P\\xi -ZjW'^ - Xp - /^p{zp) -}- P\\xi - ZpW^, Hence the first order solution for agent j is
In this equation, the coefficients {aij} involve z and the exogenous valence terms {Xj}. Moreover, each aij is strictly increasing in Xj and decreasing in Xp. Let us denote the vector ^ ^ aijXi by dV*/dzj and call it the "(marginal) electoral pull" due to exogenous valence. Then the first order condition can be written dV;
1 dfij
,
^
,^,
Say the electoral pull and activist pull are "balanced" if this equation is satisfied. The first term in this expression (the "marginal or gradient electoral pull") is a gradient vector pointing towards the "weighted electoral mean." (This weighted electoral mean is simply that point where the electoral pull is zero.) As Xj is exogenously increased, this vector increases in magnitude. The vector dfjLj/dzj "points towards" the position at which the total of activist "contributions" is maximized. We may term this vector the "(marginal or gradient) activist pull." Moreover, if the activist function is "sufficiently concave" (with negative eigenvalues of large modulus), then the second order condition (the negative definiteness of the Hessian of the "activist pull") will guarantee that the vector z* given by the solution of the system of equations given by (7), for all j , will be an LSNE. This can be seen by examining the Hessian as in (l)-(3). The following theorem states these conclusions (see Schofield [37] for a proof). Theorem 2. Consider a vote maximization model, M{/j,), with both exogenous popularity valences {Xj} and activist valences {/J^J}. The first order condition for z* to be an equilibrium is that, for each j , the electoral and activist pulls must be balanced. Other things being equal, the position Zj will be closer
Local Political Equilibria
79
Pro-Britain
Pro-Europe
Fig. 1. Estimated party positions in the British Parliament in 1997 (based on MP survey data and a National Election Survey), with the highest density plots of the voter sample distribution at 95%, 75%, 50% and 10% levels.
to a weighted versely, if the willingness of to the activist
electoral mean the greater is the exogenous valence, Xj. Conactivist valence function (ij is increased (due to the greater activists to contribute to the party) then the nearer will z* be preferred position.
If all activist valence functions are sufficiently concave (in the sense of having negative eigenvalues of sufficiently great magnitude) then the solution given by (7) will be a PNE. Example 4- Fig. 1 presents d a t a on the voter distribution, in a two-dimensional policy space derived from an empirical model for Britain (see Schofield [36] for further details). In this figure, the positions of the Conservative P a r t y ( C O N S ) , Labour ( L A B ) , Liberal Democrats ( L I B ) , Plaid Cymru ( P C ) , Ulster Unionists (UU) and Scottish Nationalists ( S N P ) were estimated from M P surveys. T h e two dimensional policy space was estimated from factor analysis of an electoral sample survey. T h e left right axis is the usual economic axis, while the north south axis represents attitudes to the European Union (with south being pro-Europe, and north pro-Britain.) Fig. 1 shows the estimated density function of the distribution of voter ideal points. T h e origin in this
80
Norman Schofield
Economic leftist indifference curve
•ptimal Conservative Position Economic conservative .ndifference curve
ECONOMIC DIMENSION Capital
Contract curve between economic leftists and proEurope activist
Pro-E Indifference curve
Pro-Europe
Fig. 2. Illustration of vote maximizing positions of Conservative and Labour party leaders in a two-dimensional policy space.
figure gives the mean of the voter distribution. It is obvious from the figure t h a t the covariance (^1,^2) term in the voter matrix V is close to zero. We can therefore estimate the eigenvalues of the Hessians on these two axes separately. Using the single economic axis alone, the empirical estimate of the exogenous valence of the Conservative P a r t y dropped from 1.58 at the election of 1992 to 1.24 at the 1997 election, while the Labour valence increased from 0.58 to 0.97 over the same period. T h e Liberal Democrat P a r t y was given the label 1, and its valence was set at zero for both elections. Because the scale of the figure is indeterminate, we are at liberty to use the standard deviation as the unit of measurement. For 1997, the spatial coeflScient, /?, was estimated to be 0.50, while the average valence difference Xav{i) was estimated to be 1.105. 2
T h e ratio ^ of the electoral variance (on the economic axis) to the stochastic variance was 2 / 3 . Using the equation 2(p-l)/3
Local Political Equilibria
81
and considering only competition between three parties {p = 3) we find Ai = 0.25, so the estimated convergence coeflScient of the model is 0.5. Therefore the necessary and sufficient conditions for an LSNE at the joint origin were satisfied, at least when only one axis of policy is relevant. Indeed, computation of the vote share functions indicated that, in one dimension, all the gradients of the party vote share functions pointed towards the origin. In other words, the origin (in one dimension) was an attractor of the electoral model. We can infer that all parties should have converged to this electoral mean. This conclusion changes, however, when the second policy axis, involving attitudes to Europe, is included. As Fig. 1 indicates, the electoral variance v'^ increases to approximately 2, and we can estimate that the suflficient condition for a LSNE at the origin is not satisfied. When the eigenvalues for the Liberal Democratic Party are computed, this party's eigenvalue at the origin on the European axis is positive. As Fig. 1 suggests, the Liberal Democratic party did indeed vacate the electoral origin, and adopted a pro-European position. We may infer that such a position was also more acceptable to its party activists than the electoral mean. Once the Liberal Democrat Party is no longer positioned at the origin, the results presented in Theorem 1 cannot be used. However, the high relative valences of the Conservative and Labour parties would suggest that their local equilibrium positions would be nearer to the origin than the Liberal Democrat Party. Simulation of vote maximizing models by Schofield and Sened [44] suggests that parties with high exogeneous valence tend to adopt positions nearer the origin than parties with low valence. I conjecture that the Conservative and Labour Parties did adopt positions close to those marked CONS and LAB, respectively, in Fig. 1. A basis for this conjecture can be found in Theorem 2, which develops the idea that the high valences for these two parties were due partly to exogenous valence and partly to valence derived from activist support. Analysis of survey data suggests that the Labour exogenous valence {XLAB due to Blair) rose in this period (see Clarke et al. [16]). Conversely, the relative exogenous term, XCONI for the Conservatives fell. Since the coefl5cients (in the equation for the electoral puh) for the Conservative party depend on (XCON — XLAB), these must all fall in this period. This has the effect of increasing the marginal effect of activism. Indeed, it is possible to include the effect of two potential activist groups for the Conservative Party — one "pro-British" and one "pro-Capital." The optimal Conservative position will be determined by a version of (7) which equates the "electoral pull" against the two "activist pulls." Since the electoral pull fell between the elections, the optimal position ZQQ^^ will be one where ^CON ^^ "closer" to the locus of points where the marginal activist pull is zero (i.e., where djicoN/dzcoN = 0). This locus of points I shall refer to as the "activist contract curve" for the Conservative party. Note that in Fig. 2, the indifference curves of representative activists for the parties are described by ellipses. This is meant to indicate that preferences of
82
Norman Schofield
different activists on the two dimensions may accord different saliences to the policy axes. The "activist contract curve" given in the figure, for Labour say, is the locus of points satisfying the equation d/j,LAB/d^LAB = 0- This curve represents the balance of power between Labour supporters most interested in economic issues and those more interested in Europe. The optimal positions for the two parties will be at appropriate points on the locus between the respective "activist contract curves" and a point "near" the origin (where the electoral pull is zero). As relative exogenous valence for a party falls, then the optimal party position will approach the activist contract curve. Moreover, the optimal position on this contract curve will depend on the relative intensity of political preferences of the activists of each party. For example, if grass roots "pro-British" Conservative party activists have intense preferences on this dimension, then this feature will be reflected in the activist contract curve and thus in the optimal Conservative position. For the Labour party, it seems clear that two effects are apparent. Blair's high exogenous valence gave an optimal position closer to the electoral center than the optimal position of the Conservative party. Moreover, this affected the balance between "old left" activists in the party, and "new Labour" activists, concerned to modernize the party through a European style "social democratic" perspective. This conclusion is compatible with Blair's successful attempts to bring "New Labour" members into the party (see Seyd and Whiteley [46]).
5 Comments on the Electoral Model A number of objections can be raised against the model presented here. I believe it worthwhile addressing these. Voter rationality. Why is it appropriate to build a stochastic element into the model? Does the model suppose that voters randomize across their choices? The purpose of the model is to determine how political agents choose their positions, given their information about electoral response. Opinion poll data will give approximate vote shares for each agent. Each agent can build a rough model of electorate response to party declarations. This is proxied by the econometric models of electoral response. These models are stochastic, and have proven quite successful at estimating voter response. Various assumptions are made about the distribution of the error terms, but the model based on the multivariate normal distribution is the most general (see Alvarez et al. [4], [5], Poole and Rosenthal [32], Quinn et al. [33], Schofield et al. [40, 44, 45]). Agent rationality. Why should political agents (party leaders or candidates) attempt to maximize expected vote share?
Local Political Equilibria
83
In two party competition, maximizing expected plurality or probability of winning is an obvious alternative (see Duggan [20]). In this case, empirical estimates of the variance of the vote share functions are generally low, so probability of winning is a close proxy to expected vote share. In multiparty competition, the objections are more serious. A party on the electoral periphery with a large vote share is unlikely to be asked to join a government coalition. If party leaders care about policy, then vote share alone is not an accurate measure of payoflF. Modeling a party's effect on final policy outcome has proved very difficult, since it is necessary to combine a model of post-election bargaining (Banks and Duggan [12]) with pre-election maneuvering (Schofield and Sened [43]). Although the simple model of vote maximization presented here is likely inadequate for multiparty situations, it is offered as a first step at classification of electoral systems. It suggests that convergence to the electoral mean is unlikely, and that the notion of LSNE is compatible with the very different poUtical configurations that have been found to occur. Activists. An early model of activists by Aldrich [1, 2] and Aldrich and McGinnis [3] assumed that activists controlled candidate policies. In contrast, the activist model outlined here is based on the assumption that different potential activist groups compete with one another to influence candidates or parties. Moreover, activists are likely to prefer policy positions far from the electoral origin. Theorem 2 asserts that this poses a complex optimization problem for party leaders. If a leader's exogenous valence falls, then activists become more important for the party. Thus a party led by a low valence politician may be forced away from the electoral origin. The effect of activists on valence. It seems most natural in modeling plurality electoral systems to assume activists affect valence. In such systems, a small advantage in vote shares results in a large seat advantage. Activist effort and contribution then becomes crutial in obtaing votes. I have not attempted to model the activist calculus, since there are many possible ways to do this. It is plausible that activist motivations play a predominant role in the two-party system of the U.S. (see Miller and Schofield [29] and Schofield et al. [41]). Local Nash equilibria. In empirical studies the LSNE can be determined by simulation. These local equilibria are not identical to actual policy positions of parties, but they can be used to distinguish between parties that principally interested in maximizing vote shares, and those that are policy motivated (see Schofield and Sened [44]). It is clear from the simulation that exogenous valence terms have considerable impact on equilibrium positions in multiparty systems based on proportional representation. Since these valence terms fluctuate because of exogeneous shocks, such shocks will induce small changes in local equilibria. Indeed, the local equilibrium concept is compatible with small
84
Norman Schofield
policy changes by parties as they attempt to adapt to the changing electoral environment. A possible conjecture is that such shocks will induce only small changes in party configurations, when the electoral system is based on proportional representation. In contrast, under plurality rule, exogeneous shocks may increase the importance of activists for some parties, thus inducing large policy changes.
6 Extensions of t h e Model and Generic Existence of LSNE One feature of the "mean voter theorem" is that each agent's optimum strategy is independent of the strategies of other agents, and is determined only by the distribution of voter ideal points. This has been regarded as an attractive feature of the spatial model. However, as Theorem 1 indicates, there are conditions under which the joint electoral origin will not be a LSNE, and definitely not be a PNE. As Example 4 indicates, a key element of the necessary condition for convergence is the ratio of the variance of the voter ideal points to the stochastic variance. When the electoral variance is high, and there are many parties with differing exogeneous valences, as in Israel, then simulation of the vote maximizing model has found multiple non-convergent LSNE (see Schofield and Sened [44]). For these non-convergent LSNE, agent equilibrium strategies are "strongly interdependent", in the sense that each party's equilibrium position is sensitive to the positions of all other parties. Simulation, under the vote maximizing assumption, shows that higher valence parties adopt positions nearer the electoral center than low valence parties. Determining the nature of these LSNE by analysis, rather than by simulation, is extremely difficult. Because convexity properties, like quasi-concavity of the utility functions, are not be satisfied, it is diflScult, if not impossible to determine if PNE exist. For example, failure of the second order, Hessian condition at the origin immediately implies that the usual suflBcient conditions for existence of PNE will not be satisfied. For these reasons we now consider the question of existence of LSNE under more general conditions that those discussed above. Firstly, in the proof of Theorem 1, it was assumed that the errors were independent and identically normally distributed (iind). However, estimation of such voter models has found it necessary to adopt the more general hypothesis that the error structure is multivariate normal (that is, allowing for non-zero covariance terms in the error covariance matrix 6). See Alvarez and Nagler [4], Quinn et al. [33] and Schofield et al. [40]. The proof procedure of Theorem 1 does carry through in this case. In particular, it will still be possible to find a solution to the orthonormal matrix equation {B\F)9{BiF)^ — G, but the solution will depend on the error covariances. The solution will then lead to the determination of a convergence coeflScient, c(^). Because of the symmetry
Local Political Equilibria
85
induced at the origin, the computation of c{0) is relatively simple. The problem arises when the induced necessary condition for an LSNE at the origin is not satisfied. In this case, the analytical computation of the first and second order equations for equilibria away from the origin will be extremely diflScult. The equilibria can generally be found by simulation. The first order equations are matrix equations involving n smooth functions {giJN- Since there are p agents, each choosing a strategy in X (of dimension w) we obtain wp equations. In the case of a general covariance matrix, these wp equations will involve interaction terms induced by the error covariance. Nonetheless, transversality theory (Austen-Smith and Banks [9], Banks [10] and Saari [34]) can be used to show that these wp equations are "generically" independent. Since wp is the dimension of the joint strategy space, the solution is generically of dimension 0. Consequently, the first order equations can be solved, even for general valence functions, as long as these functions, and the multivariate cumulative probability distribution of the errors are different iable. Thus, CNE will generically exist. Indeed, it can be shown that one of these CNE will be a LSNE. A similar argument can be carried out even when the payoff functions of the agents are defined not simply by their expected vote shares, but also by agent policy preferences. To outline a proof of this assertion, we need to introduce the idea of a tangent bundle (see Hirsch [24]). At z e X^, the tangent space is TZ{XP) and the tangent bundle is defined by T{XP) = U^r^(XP). We assume all utility profiles in the set E = {U^ : X^ -^ W} are C^ different iable. The differential dUj'{z) at z can be regarded as a linear map from R^^ to R so dUj' : X^ —> Lin(R^^,R). Since dUj' is C^ different iable, this map is continuous. Moreover, there is a C^-topology on the set, J5, under which two profiles are close, if their components are close as linear maps, and all their Hessians are close as bilinear maps, at every z G X^ (see Smale [48]). The differential dUj' can also be projected onto T^iX). Then this projection DU^ : XP -^ Lin(R^,R) can be identified with the gradient of Uj' in X, when Zk (for k ^ j) are held fixed. That is to say, DU^{z) can be regarded as an element of a subtangent space Tz{X) C Tz{X''P)^ where X corresponds to the j t h strategy space. This uses the fact that Tzix^) =Tz{X)X'--x Tz{X). We use the idea of a conic field generated by U^. Let D{z) = {ConU^){z) be the convex huh in T^iX^) of the set of vectors {DU^{z) : j e P}. Then ConU^ : X^ -> TXP is a generalized vector field over X^. Its image in TzX^ is convex and the field is continuous as a correspondence. The Selection Theorem of Michael [28] can be deployed to construct a continuous selection m : X^ —^ TX^ of D. That is to say, at every point zeX^, m{z) e D{z) C TzX^ (see Schofield [35]). Note that by this construction each DU^{z) lies in a different tangent space. Thus, 0 G ConU^{z) if and only if DU^{z) = 0 for every j G P. We shall now use standard transversality theory to show CNE generally exist, and are "locally isolated." We say a property of points in X^ is locally
86
Norman Schofield
isolated if it holds for 0-dimensional submanifolds of X^. This is simply a more general version of the definition given with regard to local equilibria. Say a property K is C^-generic in E if the set E^ = {U^ e E : U^ has property K} is open dense in the C^-topology on E. Definition 5. Let Ki he the property that there exists a locally isolated LSNE for the profile U^ and let K2 he the property that a CNE exists and is locally isolated. We seek to show Ki is C^—generic. We shall prove this by two lemmas. L e m m a 1. K2 is C^-generic. Proof. Given U^, let Tj{U^) be the set in the j t h strategy space such that -j^ = 0. By the inverse function theorem, Tj(U^) is generically a smooth submanifold of X'^ of dimension {p — l)dim(X), i.e., of codimension dim(X). But then r]jTj{U^) is of codimension p • dim(X). This is precisely that the property K2 is C^— generic. Definition 6. A profile U^ satisfies the houndary condition if for any z in the houndary, dX^, the differential {DU^{z)^... ,DU^{z)) points inward. This boundary condition simply means that if 5{z) is the normal to the boundary at z, then DU^{z){5{z)) > 0, for all j . L e m m a 2. IfU'^ satisfies the houndary condition and ifU^ exhihits a locally isolated CNE, then there exists a LSNE. Proof We have defined the conic field D = Con{U^)XP -^ TXP in the preliminary to this section. The conic field D admits a selection m : X^ —> TX^, that is a continuous function such that m{z) G Con{dU^(z)). Moreover, m(z*) = 0 if and only if DU^iz"") = 0, for all j G P. Clearly, 0 G Con{dU^{z)) iff DU^{z) = 0 for all j G P. The selection is a vector field on X^. Moreover, m can be selected to be a gradient vector field on X^ which satisfies the boundary condition, i.e., m{x){5{x)) > 0 for all x on the boundary of X^. Clearly, 771(2:*) = 0 iff z* is a CNE. Because of the Morse inequalities (Milnor [30]), m must exhibit a stable equilibrium, 2:*, say. Let mj{z) be the projection of 777 at 2; onto the jth tangent space. Because 2:* is a stable equilibrium, there exists a neighborhood Y of z* with the following property: if 2; G Y" then for each j G P , mj{z) "points" towards ^*. In the vector space context this means that m{z){z* — z) > 0. Let Yj be the projection of Y onto the j t h strategy space. Then DUj{z) points towards 2;* for all Zj G Yj. The Cartesian product of {1^} gives a neighborhood of 2;* satisfying all second order conditions. Clearly, ^* is an attractor and therefore a LSNE.
Local Political Equilibria
87
(Note that this lemma is proved here for X compact and convex, but it is valid for a gradient field on a space with non-zero Euler characteristic; see Brown [15].) Theorem 3. Existence of LSNE is generic in the topological suhspace E' = {U^ £ E : U^ satisfies the boundary condition]. Proof. By Lemma 1, there exists an open dense subset E" on which K2 is satisfied. But then E' D E" gives an open dense subspace of E. For every profile in this set, the procedure of Lemma 2 gives a LSNE. Now let ^ * = {u : X ^ + P -^ R^^} denote the set of all electoral systems with n voters and p agents. In this formulation the valence functions are treated as parameters. We can also regard the covariance matrix ^, and therefore the error variances {al,..., (7^}, as fixed, since these are essentially scale factors. We may think of a specific u as the political institution. Once voter ideal points in X^, and agent strategies in X^, are specified, then the n X p array of voter utilities is defined by u. Then u e E* and the set of voter ideal points x € X'^ leads to the political game U^ : X^ —^ R^.The vector X G X'^ may then be thought of an electoral map x : E* ^ E. The electoral models that we have considered here possess the property that this electoral map is continuous. By Theorem 3, existence of LSNE is generic in E' within the space E. Assuming the electoral map x is continuous, the inverse image of an open set in E' is open in the co-image of x. Suppose x is also proper: that is, if Y is open in £"*, then its image is open in E\ In this case, the inverse image of a dense set is dense. This suggests the following conjecture. Conjecture 1. Existence of LSNE is a generic property in the space of political institutions. The results on transversality theory used in Theorem 3 can be found in Hirsch [24]. Dierker [18] and Smale [47, 48] discuss the Morse inequalities. A review of this material can be found in Schofield [38]. A minor point concerns the nature of the topology. Theorem 3 uses topologies on utilities. Equilibrium concepts in both politics and economics should be based on preferences. Schofield [35] has proposed a C^-topology on preferences, which can be used when the utility representation of the preferences are diff'erentiable. This suggests that the Conjecture can be expressed as a genericity result for preferences.
7 Conclusion In this exposition, I have assumed that the policy space, X, is fixed. In fact, a possible extension is where there is a map ifj : X —^ Z., where Z is the
88
Norman Schofield
full economic commodity space. Voters have economic preferences on Z, and these can, in theory, induce preferences on '0~^(Z) C X. These preferences then define voter ideal points in X. There are certain subtleties of such a model which were explored by Konishi [25]. The principal difficulty with Konishi's model was in locating appropriate political equilibria in X, This paper has presented one way of inferring existence of LSNE in X^, given electoral data about voter ideal points. The results of Banks and Duggan [11], for example, can then be used to infer the political outcome in X. Note, however, that there are a number of further theoretical difficulties. Firstly, the map il) may be multi-valued. Given the structure of the economy, there will be many possible Pareto incomparable local economic equilibria. Each of these can arise from a single political decision in X. However, it is likely that ij; will be locally single-valued. That is, if the economy is initially at a particular state, TT, in Z, then voters may compute back to X to determine how changes in political decisions will affect economic outcomes in a neighborhood of TT within Z. Similarly, political decision-making will be locally single-valued. That is, for a given economic and political situation, the local political equilibrium will be determined by the particular basin of attraction within which the status quo is located. Although this article has focused on party leaders who attempt to maximize vote share, the general model can, in principle, be extended to include more general candidate or leader motivations, as long as these can be assumed to be C^-diff'erentiable functions.
References [1] Aldrich, J. (1983) A spatial model with party activists: implications for electoral dynamics. Public Choice^ 41: 63-100. [2] Aldrich, J. (1983) A Downsian spatial model with party activists. American Political Science Review, 77: 974-990. [3] Aldrich, J. and M. McGinnis (1989) A model of party constraints on optimal candidate positions. Mathematical and Computer Modelling, 42: 437-450. [4] Alvarez, M. R. and J. Nagler (1998) When politics and models collide: estimating models of multicandidate elections. American Journal of Political Science, 42: 55-96. [5] Alvarez, M. R., J. Nagler and S. Bowler (2000) Issues, economics and the dynamics of multiparty elections: the British 1987 general election. American Political Science Review, 94: 131-150. [6] Ansolabehere, S. and J. M. Snyder (2000) Valence politics and equilibrium in spatial election models. Public Choice, 103: 327-336.
Local Political Equilibria
89
[7] Aragones, E and T. Palfrey (2004) Spatial competition between two candidates of different quality: the effects of candidate ideology and private information. This volume. [8] Austen-Smith, D. and J. S. Banks (1998) Social choice theory, game theory and positive political theory. In N. Polsby (ed), Annual Review of Political Science vol I: 259-287. [9] Austen-Smith, D. and J. S. Banks (1999) Positive Political Theory I: Collective Preference Ann Arbor: The University of Michigan Press. [10] Banks, J. S. (1995) Singularity theory and core existence in the spatial model. Journal of Mathematical Economics^ 24: 523-536. [11] Banks, J. S. and J. Duggan (2004) The theory of probabilistic voting in the spatial model of elections: the theory of office-motivated candidates. This volume. [12] Banks, J. S. and J. Duggan (2000) A bargaining model of collective choice. American Political Science Review, 94: 73-88. [13] Banks, J. S., J. Duggan, and M. Le Breton (2002) Bounds for mixed strategy equilibria and the spatial model of elections. Journal of Economic Theory, 103: 88-105. [14] Banks, J. S., J. Duggan, and M. Le Breton (2004) Social choice and electoral competition in the general spatial model. Journal of Economic Theory, forthcoming. [15] Brown, R. F. (1970) The Lefshetz Fixed Point Theorem Glenview IL: Scott and Foreman. [16] Clarke, H., M. Stewart and P. Whiteley (1998) New models for new labour: the political economy of labour support: January 1992 - April 1997. American Political Science Review, 92: 559-575. [17] Coughlin, P. (1992) Probabilistic Voting Theory Cambridge: Cambridge University Press. [18] Dierker, E. (1974) Topological Methods in Walrasian Economics Lecture Notes on Economics and Mathematical Sciences, 92, Heidelberg: Springer. [19] Downs, A. (1957) An Economic Theory of Democracy New York: Harper and Row. [20] Duggan, J. (2000) Equilibrium equivalence under expected plurality and probability of winning maximization. Mimeo. University of Rochester. [21] Fan, K. (1961) A generalization of Tychonoff's fixed point theorem. Mathematische Annalen, 42: 305-310. [22] Groseclose, T. (2001) A model of candidate location when one candidate has a valence advantage. American Journal of Political Science, 45: 862886. [23] Hinich, M. (1977) Equilibrium in spatial voting: the median voter result is an artifact. Journal of Economic Theory, 16: 208-219. [24] Hirsch, M. W. (1976) Differential Topology Heidelberg: Springer.
90
Norman Schofield
[25] Konishi, H. (1996) Equilibrium in abstract political economies: with an application to a public good economy with voting. Social Choice and Welfare, 13: 43-50. [26] Lin, T. M., J. Enelow and H. Dorussen (1999) Equilibrium in multicandidate probabilistic spatial voting. Public Choice, 98: 59-82. [27] McKelvey, R. (1986) Covering, dominance, and institution-free properties of social choice. American Journal of Political Science, 30: 283-314. [28] Michael, E. (1956) Continuous selection I. Annals of Mathematics, 63: 361-382. [29] Miller, G. and N. Schofield (2003) Activists and partisan realignment in the United States. American Political Science Review, 97: 245-260. [30] Milnor, J. (1963) Morse Theory Annals of Mathematics Studies, Series 51 Princeton: Princeton University Press. [31] Nash, J. (1951) Non-cooperative games. Annals of Mathematics, 54: 289295. [32] Poole, K. and H. Rosenthal (1984) Presidential elections 1968-1980: a spatial analysis. American Journal of Political Science, 28: 283-312. [33] Quinn, K., A. Martin and A. Whitford (1999) Voter choice in multiparty democracies. American Journal of Political Science, 43: 1231-1247. [34] Saari, D. (1997) The generic existence of a core for q-rules. Economic Theory, 9: 219-260. [35] Schofield, N. (1999) The C^ topology on the space of smooth preference profile. Social Choice and Welfare, 16: 445-470. [36] Schofield, N. (2003a) A valence model of political competition in Britain, 1992-1997. Electoral Studies, forthcoming. [37] Schofield, N. (2003b) Valence competition in the spatial stochastic model. Journal of Theoretical Politics, 15: 371-383. [38] Schofield, N. (2003c) Mathematical Methods in Economics and Social Choice Heidelberg: Springer. [39] Schofield, N. (2004) Equilibrium in the spatial valence model of politics. Journal of Theoretical Politics, 16: 447-481. [40] Schofield, N., A. Martin, K. Quinn and A. Whitford (1998) Multiparty electoral competition in the Netherlands and Germany: a model based on multinomial probit. Public Choice, 97: 257-293. [41] Schofield, N., G. Miller and A. Martin (2003) Critical elections and political realignment in the US: 1860-2000. Political Studies, 51: 217-240. [42] Schofield, N. and R. Parks (2000) Nash equilibrium in a spatial model of coahtion bargaining. Mathematical Social Sciences, 39: 133-174. [43] Schofield, N. and I. Sened (2002) Local Nash equilibrium in multiparty politics. Annals of Operations Research, 109: 193-210. [44] Schofield, N. and I. Sened (2003) Multiparty competition in Israel. British Journal of Political Science, forthcoming. [45] Schofield, N., I. Sened and D. Nixon (1998) Nash equilibrium in multiparty competition with 'stochastic' voters. Annals of Operations Research, 84: 3-27.
Local Political Equilibria
91
[46] Seyd, P. and P. Whitely (2002) New Labour's Grassroots Basingstoke UK: Macmillan. [47] Smale, S. (1960) Morse inequalities for a dynamical system. Bulletin of the American Mathematical Society^ 66: 43-49. [48] Smale, S. (1974) Global analysis and economics I: Pareto optimum and a generalization of Morse theory. In M. Peixoto (ed) Dynamical Systems New York: Academic Press. [49] Stokes, D. (1992) Valence politics. In D. Kavanagh (ed) Electoral Politics Oxford: Clarendon Press.
Electoral Competition Between Two Candidates of Different Quality: The Effects of Candidate Ideology and Private Information* Enriqueta Aragones^ and Thomas R. Palfrey^ ^ Universitat Autonoma enriqueta.aragonesOuab.es ^ Princeton University tpalfreySprinceton.edu S u m m a r y . This paper examines competition in a spatial model of two-candidate elections, where one candidate enjoys a quality advantage over the other candidate. The candidates care about winning and also have policy preferences. There is twodimensional private information. Candidate ideal points as well as their tradeoffs between policy preferences and winning are private information. The distribution of this two-dimensional type is common knowledge. The location of the median voter's ideal point is uncertain, with a distribution that is commonly known by both candidates. Pure strategy equilibria always exist in this model. We characterize the effects of increased uncertainty about the median voter, the effect of candidate policy preferences, and the effects of changes in the distribution of private information. We prove that the distribution of candidate policies approaches the mixed equilibrium of Aragones and Palfrey [2J, when both candidates' weights on policy preferences go to zero.
1 Introduction Several recent papers have used a framework for studying the effect of candidate quality on political competition,'^ based on the standard Downsian model of competition between two candidates who maximize the probability of winning, b u t with an important twist: one candidate has a quality advantage. T h a t is, any voter will strictly prefer the "higher quality" candidate (Candidate A) to the "lower quality" candidate (Candidate D) if the candidates Aragones acknowledges financial support from the Spanish Ministry of Science and Technology, grant number SEC2000-1186. Palfrey acknowledges financial support from the National Science Foundation, grant number SES-0079301, and from the Institut d'Analisi Economica. He is also grateful for the hospitality of lAE in May 2003. We thank Clara Ponsati and participants of several seminars for helpful comments. See, e.g., Aragones and Palfrey [2, 3] and Groseclose [7].
94
Enriqueta Aragones and Thomas R. Palfrey
locate so t h a t the voter is indifferent between the two candidates on the policy dimension. In Aragones and Palfrey [2], we showed t h a t candidates diverge, and t h a t this divergence occurs in predictable ways. In equilibrium the higher quality candidate ends up reinforcing her advantage by adopting relatively more centrist platforms, in a probabilistic sense. Three limitations of t h a t simple model are (1) candidates may have policy preferences, but the model assumes they only care about holding office; (2) the equilibrium is in mixed strategies;^ and (3) candidates have perfect information about each other's objective function, which is unrealistic. This paper extends the model in a natural way t h a t relaxes all three limitations, and leads to new insights about candidate competition when there are quality differences between the two candidates. A key insight comes from Harsanyi's [8] paper on purification of mixed strategies. T h a t paper shows t h a t for games like the one considered in Aragones and Palfrey [2] one can almost always approximate a mixed strategy equilibrium by a pure strategy equilibrium of a game in which the players have private information. T h a t is, if we consider the model with complete information to be only a first approximation to the real world, where the "correct" model would be one with private information, then indeed the mixed strategy equilibrium is reasonable since it is close to an equilibrium of a more complicated and realistic game. Our approach is to introduce incomplete and asymmetric information about candidate policy preferences. We consider two-dimensional private information. It is common to assume t h a t the candidates care not only about the probability of winning, but also about the policy t h a t is implemented by the winning candidate.^ In our model, the weight each candidate places on winning is private information and is independently drawn for each of the two candidates. T h e second component of private information is t h a t neither candidate is certain of the other candidate's exact ideal point. Both of these generalizations capture important and realistic aspects of political competition. While candidates may have some information about each other's ideal point, based on past records, and candidates may know a little bit about how much the other candidate trades off policy preferences and the value of holding office, both are arguments of a utility function, and neither can be observed directly. Moreover, much of what a candidate says is rhetorical, ^ It is hard to imagine how candidates would actually implement mixed strategies in a location game. ^ In a related paper, Groseclose [7] examines a model of asymmetric candidates where candidates have a mixture of policy preferences and preferences for holding office. However, in that paper the exact weights between the two objectives are the same for both candidates and are common knowlegde. As a result, pure strategies equilibrium often fails to exist in that model. Other recent theoretical papers on candidate competition with quality asymmetry are Ansolabehere and Snyder [1] and Berger et al. [5].
Electoral Competition and Quality
95
which makes it difficult to take campaign platforms of candidates as straightforward representations of their ideal points. In fact, we know from results by Wittman ([10, 11]), Calvert [6], and others, that pohcy motivated candidates will generally not adopt their ideal point as a platform. Furthermore, the actual policies adopted by the elected candidate may not necessarily reflect her ideal point, since it may simply be done to fulfill campaign promises or to satisfy her constituency or party. In this two-dimensional asymmetric information model, we characterize the best response functions of the two candidates and use the properties of these best response functions to fully characterize the equilibrium. Best responses of each candidate depend on five variables: the candidate's quality, the amount of uncertainty, the probability the other candidate locates at the center, the candidate's ideal point, and the candidate's own value of holding office. First, we show that locating at an extreme position other than one's own ideal point is never a best response for either candidate. Next, we show that this implies, that best responses are fully characterized by cutoff rules, which means that it is optimal for a candidate to locate in the center if and only if his or her value of holding office is sufficiently great. Third, we show that, for the advantaged candidate, best responses are upward sloping, in the sense that her cutoff value increases in the cutoff value of the disadvantaged candidate. That is, candidate A is more likely to locate in the center if she thinks candidate D is more likely to locate at the center. The opposite is true for candidate D, who is less likely to locate in the center, the more likely he thinks A will locate at the center. Fourth, we show that an increase in uncertainty about the median voter leads both candidates to be less likely to adopt the moderate platform. An alternative interpretation is that as the electorate becomes more polarized (i.e., the probabihty the median voter is moderate decreases), the candidates also become more polarized. Fifth, putting these results together we can show how the equilibrium distributions of candidate locations vary with the polarization parameter. Here we find that the equilibrium platform of A becomes more polarized when the electorate becomes more polarized, but that is not the case for candidate D. In fact, for D the effect can go either way because of conflicting forces. On the one hand, locating at his ideal point is more attractive for D because the probability the median voter has the same ideal point as D has increased. On the other hand, since that is A's equilibrium response, it is less attractive. The sum of these two effects can be either positive or negative. We then look at the effect of decreasing the asymmetric information between the two candidates. When both candidates' office-holding weights collapse to 1 (it becomes common knowledge between the candidates that both only care about holding office), we recover all of the results of the symmetric
96
Enriqueta Aragones and Thomas R. Palfrey
information model. However, the direction of convergence is interesting. The equilibrium probability that D locates in the center converges from above, and the equilibrium probability that A locates in the center converges from below. Thus, one surprising effect of asymmetric information is that it leads D to moderate. This occurs even though the expected value of holding office is decreasing. In contrast, however, asymmetric information leads A to adopt more extreme policies on average. Finally, we characterize the boundary case of complete information about A, which provides insights into the intuition for the general case. First, we show that only mixed strategy equilibria exist when the value of holding office is high enough. If this occurs, then we obtain comparative statics similar to Aragones and Palfrey [2]. Increased uncertainty leads the advantaged candidate to adopt more extreme positions and the disadvantaged candidate to be more moderate. However, in contrast to the earlier paper, each candidate simply mixes between its ideal point and the central policy rather than mixing over all three policies. Thus, a new interpretation of this result is that the effect of increased uncertainty is for the advantaged candidate to move closer to her ideal point (in expectation) and for the disadvantaged candidate to move away from his ideal point. Results of previous work on competition with policy preferences suggest that more uncertainty would lead both candidates to move toward their ideal points. This points to an interesting interaction effect between candidate quality, uncertainty, and policy preferences, which can lead to non-intuitive results. In this boundary case we also analyze the effect of the value of holding oflSce on equilibrium location choices. We again find an opposite effect for the two candidates. Candidate A adopts more central locations when the value of holding office increases, but Candidate D adopts more extreme locations when the value of holding office increases, another counterintuitive effect, driven by the fact that candidate D needs to differentiate his position from A in order to win. The rest of the paper proceeds as follows. The next section describes the formal model. Section 3 presents the derivation of the unique equilibrium. The properties of the equilibrium are analyzed in Section 4. Finally, Section 5 contains some concluding remarks.
2 T h e Model The policy space, p consists of 3 points on the real line, {0, .5,1}, which we will refer to as L (left) C (center), and R (right). There are two candidates, A and D, who are referred to as the advantaged candidate and the disadvantaged candidate, respectively. Each voter has a utility function, with two
Electoral Competition and Quality
97
components, a policy component, and a candidate image component.^ The policy component is characterized by an ideal point in the policy space p, with utility of alternatives in the policy space a strictly decreasing function of the Euclidean distance between the ideal point and the location of the policy, symmetric around the ideal point. We assume there exists a unique median voter ideal point, denoted by Xm- Candidates do not know x ^ , but share a common prior belief about it, which is symmetric around C. We denote by a E [0,1/2] the probability that Xm = L^ which also equals the probability that Xm = R. Hence the probability that Xm = C equals 1 — 2a. The quality advantage of A is captured by an additive constant to the utility a voter obtains if A wins the election. That is, the utility to a voter i with ideal point xi if A wins the election is Ui {XA) = 5 — \xi — XA\ and the utility to i if candidate D wins is Ui {XD) = — |xi — XD|, where candidates' policy positions are denoted by XA and XD and the magnitude of ^ ' s advantage is 0 < J < 1/2.'^ 2.1 Candidates' Objective Functions Candidates have ideal points, just like voters. The ideal point of candidate j is denoted yj. Candidates know their own ideal point. They do not know the ideal point of the other candidate, but do know that the other candidate's ideal point is equally likely to be L or R. The game takes place in two stages. In the first stage, candidates simultaneously choose positions in p. As in the standard Downsian model, candidates implement their announced positions if they win the election. In the second stage, each voter votes for the preferred candidate (taking account of the quality advantage). In case of indifference, a voter is assumed to vote for each candidate with probability equal to 1/2. Since the behavior of the voters is unambiguous in this model, we define an equilibrium of the game only in terms of the location strategies of the two candidates in the first round. Given a pair of candidate locations, {XA^^D) we denote by TTA{^A^XD) and ITD{^A^XD) the probability of winning for candidate A and for candidate D, respectively, as a function of (x^,xjr>), where 7^A{XA,XD)
+ ^D{XA,XD)
= 1.
Each candidate maximizes an objective function that is a linear combination of the probability of winning and a second component corresponding to the candidate's privately known policy preferences. Formally, the objective function of candidate A and D are given, respectively, by: ^ There could be either a finite number of voters or a continuum. ^ Two further generalizations of this model would be: (1) to allow different candidates to have different beliefs about x; or (2) to allow different voters to have different image terms.
98
Enriqueta Aragones and Thomas R. Palfrey UA{xA,XD\yAAA)
= >^A7TA{00A^XD)
- (1 - XA){^A{XA^
+7TDixA,XD)\yA UDixA^XDlyD^D)
= ^D7TD{XA^XD)
OCD) IVA - XA\
- XD\} " {I - >
+7rD{xA^XD)\yD
XD) {VD - XA\
-XD\}'
Thus, Xj is the weight candidate j places on holding office. This weight is private information , That is, candidate j knows Xj but does not know the other candidate's value of holding office. Each Xj is independently drawn from a commonly known distribution, with cdf Fj over [0,1]. We assume, for each j , Fj{0) = 0, Fj{l) = 1, and Fj is continuously increasing on [0,1], and refer to this as the regularity assumption. To summarize, each candidate has a two-dimensional type (yj^Xj) which is private information . The types are drawn independently and the distribution of types is common knowledge.
3 Derivation of Unique Equilibrium The first thing to notice is that if candidate D's ideal point is 0, then locating at 1 is never a weak best response for all XD ^ [0,1). Similarly, if candidate D's ideal point is 1, then locating at 0 is never a weak best response for all XD ^ [0,1). Therefore, in equilibrium, the probability that D locates at 0 is bounded above by .5 and the probability that D locates at 1 is also bounded above by .5. Iterating this never a weak best response elimination for A implies that if candidate A's ideal point is 0, then locating at 1 is never a weak best response for all A^ G [0,1). Similarly, if candidate A's ideal point is 1, then locating at 0 is never a weak best response for all A^ G [0,1). Therefore, given candidate j ' s ideal point, and given any strategy of the other player, we only need to consider two possibilities for j ' s best response. Either j ' s best response is to locate at his ideal point, or to locate at .5. Which is optimal will depend not only on the opponent's strategy, but also on Xj. Specifically, there will exist a outpoint, A* G [0,1] such that locating at .5 is strictly optimal for j if and only if Xj > A*. Hence equilibrium strategies take a very simple form, where candidate j chooses to moderate or not, depending only on the value of Xj. Thus, an equilibrium will consist of a pair, (A^, A^) such X\ is an optimal response to A^, and A^^ is an optimal response to A^. Given (A^,A^), this determines the probabihty candidate j locates at .5, which is simply prob{Xj G [A*,l]}. We denote PAi = prob{XA e [A:^, 1]} = 1 - FAiX*^) qxi, = prob{XD e [X*D, 1]} = 1 -
FoiXh)
and, dropping the dependence on A, we refer to p (or q) as the induced mixed strategy of candidate A (or D).
Electoral Competition and Quality
99
Finally, by symmetry, this implies that the induced mixed strategy for A is ( ^ ^ , p , ^-^) and the induced mixed strategy for D is {^^,q, ^ ^ ) . Given any symmetric induced mixed strategy for A, (—^,p, —^), we can derive the optimal A-cutpoint for D, from which we can derive the induced mixed strategy for D, from which we can derive the optimal A-cutpoint for A, from which we can derive the induced mixed strategy for A. A Bayesian Nash equilibrium is a fixed point of this composed mapping. Formally, one calculates the equilibrium by the equality conditions that must hold at each of the cutpoints. That is, at a candidate's (interior) cutpoint,^ the candidates are exactly indifferent between locating at their ideal point or locating at .5. 3.1 Candidate A's Best Responses Without loss of generality, assume that A's ideal point is L.^ We derive best responses for A, by identifying conditions on a, A^, and q, such that choosing C is a best response. With this in mind, fix a and XA and suppose that D is using some type-contingent (possibly mixed) strategy that implies an induced mixed strategy oi q G (0,1). Then the expected payoff to A for locating at L when his office holding weight is equal to A^ is given by: ^/ = " f^ A ^ +
+
^ X A
1-A^
^ f^-^^ +a i^-^XA = ^[2-q
QXA
- g ^ -
+ qa]-^[{l-2a)q
f^-An
\^
- ^ ^ - j (1 - A^) + a].
Similarly, the expected payoff to A for locating at C when his office-holding weight is equal to A^ is given by:
= ^[2qa-2a
+
3)]-\.
If the outpoint is at A = 0 or A = 1 then we have an inequality condition. By symmetry, the payoffs and strategy calculations are the same when candidate A^s ideal point is R.
100
Enriqueta Aragones and Thomas R. Palfrey Comparing payoffs for A:
V^{a,\A,q)
_
If 0 < g < 2 ^ ^ then X\{q) > 1, so the best response is to locate at her ideal point. Thus, p = 0 for all values of A^i < 1, for this range of q. If 2 3 ^ < Q' < 1 then \\{q) G (0,1). In fact, over this range, we get — ^ ^ < 0. T h a t is, A's A-cutoff is strictly decreasing in q over this range of g, from a maximum of X\{i^r^) == 1 to a minimum of A ^ ( l ) = 2 ^ Similarly, suppressing the dependence of the reaction function of a , we can write P ( ^ ) == l-FA[\\{q)], and we have ^ ^ ^ > 0 when q e (23^51]^ ranging from a minimum of P ( 2^^) = 0 to a maximum of P ( l ) — 1 — -^yi[23^]- Thus, the reaction function of candidate A is
This is illustrated by the solid upward sloping curve in Fig. 1.-^*^
3.2 C a n d i d a t e D ' s B e s t R e s p o n s e s Next consider candidate D. W i t h o u t loss of point is R,^^ Fix a and Xo- Suppose A is induced mixed strategy p e (0,1). T h e n the at R when his office holding weight is equal
generality, assume t h a t D's ideal using a strategy t h a t implies an expected payoff to D for locating to XD is given by:
+ (1-2a) f - f i ^ V l - A . ) - , 1 : 1 ^ + 1^0 +a
(^-^XD+PXD
I -\- pa = XD—;:;
+
^^^0
I —a —.
^° The curve represents P{q) as a concave function. This is done because in a sense this is the typical case when the distributions of A converge to 1. A necessary and sufficient condition for P to be a concave function of g is: F4 > ^^ By symmetry, the payoff's and strategy calculations are the same when candidate D's ideal point is L.
Electoral Competition and Quality
101
l-FA(a/(2-a))
(I
\
1
/
/
4
/
/
/
/
l-Fo(a/(2-2a))
/
(p\q)a'>a
/
'V"''
^— \ a/(2-a)
\ \Q(P)
2-3a 2-a
Fig. 1. Unique equilibrium and comparative statics in a Similarly, the expected payoff to D for locating at C when his office-holding weight is equal to A^ is given by:
VS{a,XD,p)
),_.,, .,1.^,(1^) -M"^:
(3^))
M(^)-^H^)(^
+ (l-2a)(^( - ( ( ^ ) = Ac
-1
(^)_,i^.i^„)
2pa — 2p-- 2a + 3
1 2'
To compute best replies for D, we simply compare V^{a^X]j^p)
V^ia,XD,p):
and
102
Enriqueta Aragones and Thomas R. Palfrey
XD{l+pa)-l-a>XD
(2pa - 2p - 2 a + 3) - 1 <^ a>
(2(l-a)-p(2-a))Ai^.
If ^ ^ < p < 1 then a > (2(1 — a ) — p(2 — a ) ) , so the best response for D is to locate at his ideal point. Thus, g' = 0 for all values of A D , for this range of p. If 0 < p < | E ^ then 2ii-a)-p(2-a) range, we get —f^
= ^hiP)
^ (0. !)• In fact, over this
> 0. T h a t is, D ' s A-cutoff is strictly increasing in p
over this range of p, from a minimum of A^(0) — 2-2a ^^ ^ maximum of A ^ ( ^ ^ ) = 1. Similarly, we can write Q(p) = 1 — FD[X'J^{P)],
and we have
^ ^ ^ < 0 when p G [0, ^ E ^ ) , ranging from a maximum of Q(0) = j : ^ to a minimum of Q ( ^ i ^ ) = 0. Thus, the reaction function of candidate D is
This is ihustrated by the solid downward sloping curve in Fig. 1."^^ It is evident from the figure t h a t there is a unique equilibrium in pure strategies, which we state and prove formally below.
T h e o r e m 1. There is a unique equilibrium in pure strategies for all values of a, and for all FD dnd FA satisfying the regularity assumption. Proof
There are two cases.
Case 1: i^^ < 1 — ^ D [ 2 I ^ ] - This case is illustrated in the Fig. 1. At p == 0, candidate D ' s cutoff value, X'p{p) equals 2-2a ^ ^ ^ increases continuously to 1, which occurs when p = ^ir^ < 1- Hence, D ' s induced mixed strategy if p = 0, and decreases continuously response, Q ( p ) , is equal to 1 — FD[2Z^] (by the regularity assumption) to 0 for ^ E ^ ^ P- Foi" candidate A, A^(Q') = 1 for all values of g G [0, 2^]- Then A^(g) is strictly and continuously decreasing until g = 1, at which point, A^(g') = 2^- Therefore A's induced mixed strategy response, P{q), is equal to 0 if g G [0, 2^]^ ^ ^ ^ increases continuously (by the regularity assumption) to 2^ when q = I. Since j ; — ^ < l - ^ i : > [ 2 i ^ ] there is exactly one intersection between Q{p) and P ( g ) . This intersection point is in the interior of [0,1]^ and takes on values (7* G (2^^^ 1"~ ^ ^ [ 2 ^ ^ ] ) andp*G(0,|E^). ^^ The curve represents Q(p) as a concave function. This is in some sense a typical case, particularly when the distributions of A converge to 1. A necessary and sufficient condition for Q to be a concave function of p is: F^^, > —2Fj:,/X*p{p).
Electoral Competition and Quality
103
Case 2: 2Z^ ^ 1 — Fp [ 2-2a 1 • There is again a single intersection, but it is not interior, since the intersection occurs at p* = 0, g* = ^~^D[2-2oi\ — 2 ^ ' completing the proof.
4 Properties of t h e Equilibrium Mapping Here we study several properties of the equilibrium mapping. First, we look at how the equilibrium changes when a, the index of voter polarization (or uncertainty about the median voter), changes. Then, we study the effects of changing the distribution of weights that candidates place on their policy preferences. 4.1 The Effects of Changing a It is straightforward to show that P(g) is weakly decreasing in a (strictly decreasing for q > 2^)- This is illustrated in Fig. 1, with the dotted upward sloping curve to the upper left of the solid P(^) curve. As a increases the ^-intercept of P(g'), which equals 2^^ increases and the p-intercept of P(g), which equals 1 — ^ A (23^)5 decreases. Similarly, Q(p) is also weakly decreasing in a (strictly decreasing for p < ^'2^a ) • ^^^^ is shown in Fig. 1, by the dotted downward sloping curve to the lower left of the solid Q{p) curve. As a increases the g-intercept of Q(p) which equals 1 — -^L>( 21^)5 decreases as does the p -intercept of Q(p), which equals ^2"!^^ • These two results are stated and proved below in Proposition 1. Proposition 1. The comparative statics with respect to a are:
a)^<0
for dip
and^
b)^<0
for all q and ^
^ .
Proof. An informal argument is given in the paragraph above. The formal argument simply requires partial differentiation of P (q) and Q (p) with respect a. For Q (p) when p < ^ E ^ , we get
dQ (p) ^ da -
dFp dXJ^ dX da ^^^
dX
^2{l-a)%{2-a)
da
_ dFp 2(1 - g) - p(2 - g) + a(2 - p) '~~ dX [2(l-g)-p(2-g)]2 dFp 2(1-p) dX [ 2 ( l - g ) - p ( 2 - g ) ] 2 <0
104
Enriqueta Aragones and Thomas R. Palfrey
since ^
> 0 and ^d-ai-pS-a)]^ > 0- When ^
that ^
= 0.
< p < 1 we always have
Similarly, for P {q) when i^^ < g, we get
dv {q)
OFA dx\ dX da
da
dX
da ( - 1 + 2q){l - 2a + q{l - a)) + (2 - q){l - a - q{l - 2a)) dX [ l - 2 a + g ( l + a)]2
DFA
dFA 2g2 4- (g - 1)' dX [ l - 2 a + g ( l + a)]2 <0 since ^ ^ > 0 and 2q^ + (^ — 1)^ > 0. When q < ^^
we always have that
—Q^ = 0, completing the proof. Both of these effects, which lead candidates to adopt less moderate positions when a increases, are intuitive, since they are direct effects. As a increases, the median voter's ideal point is more likely to be at one of the two extremes, either L or R. Therefore all types of both candidates find it less advantageous to locate in the center, holding constant the strategy of the other player. Hence either player's cutoff value increases, given any induced mixed strategy of the other player. The equilibrium effect of this shift reflects the same intuition as discussed in Aragones and Palfrey [2]. In order to increase the chance of winning, candidate A wants to locate close to the median voter, and also wants to locate close to D. Since the direct effect on D is to move in the direction of the median voter (i.e., X}y{p) decreases when a increases), both of these effects on A go in the same direction. Hence ^ < 0. The effect on D is more complicated. While the direct effect on D is to follow the median voter (suggesting that q* should decrease), the indirect effect on D goes in the opposite direction, since D wants to distance himself from A. Since these effects go in opposite directions, we cannot sign ^ . The sign can be either positive or negative. Fig. 1 shows a case in which ^ > 0, but it could easily go the other way. Proposition 2. The equilibrium comparative statics with respect to a are: b)^<0^fandonly^f^|^<^. Proof. An informal argument is given in the paragraph above.
Electoral Competition and Quality
105
(a) The formal argument that ^ < 0 is straightforward. Consider (p* (a) ,^* (a)) and (p* (a') ,g* (a')) and suppose that ot < a'. We will show thatp*(aO < p * ( a ) . If g* (aO > 9* (Q^), we have that g* (aO = Q (a',P* (aO) < Q (a,p* (a^) since a < a' and Q^' < 0. Therefore, since q* (a') > g* (a), we have that Q*(a) - Q ( a , p * ( a ) ) < Q(a,p*(aO)- Since ^ § ^ < 0, this implies that If g* (aO < g* (a), we have that p* (a') = P (a^g* (a^) < P (<^'.^* (o^)) because ^* (a^ < ^* (c^) and ^ | i ^ > 0. And P (a^g* (a)) < P (a,g* (a)) = p* (a) since - ^ ^ < 0 and a < a^ Therefore, we have that p* (a') < p* (a). (b) To prove that ^
< 0 iff "^'glg^ < ^
notice that
c/g* _ d ( l - F z , ( A | , ( p * ( a ) ) ) ) da da 'aFz3(AB(p*(a)))\ / d ( A l , ( a , p * ( a ) ) )
dX
da
I \
which is equivalent to da since ^ ^
~
> 0.
Prom above, d{Xh {a,P* (g))) ^ 2 (1 - p*) + g (2 - g) (^g
[2(l-a)-p*(2-a)]^
and hence d(AB(g,p*(«))) >0, dg or equivalently,
2(l-p*)+g(2-a)^5ElM>0. Therefore, ^
< 0, i.e.,
' aa — '
'
- 2 ( l - p * ) ^ 6/p* a (2 — a) ~ da' completing the proof.
^
106
Enriqueta Aragones and Thomas R. Palfrey
4.2 The Effects of Changing the Distribution of OfRce-holding Weights, FA and FD Converging to Complete Information about A In this subsection we study the effects of changing the distribution of A^ and XD' When either distribution function shifts to the right, the value of the corresponding A is more likely to be higher, in the sense of stochastic dominance. ^"^ This implies that the reaction function of the candidate whose distribution function has shifted will also shift in the same direction. That is, the candidate's best response is more likely to locate in the center since the candidate is more likely to place a higher weight on winning. This in turn implies unambiguous comparative statics results for p* and g*, which are summarized in the next proposition. Proposition 3. Let FQ {\D) < Go (Aj^) for all XD, and FA (XA) < GA {XA) for all XD, where FD, GD, FA, a'^d GA each satisfy the regularity assumption. Then P^FA^FD)
q^FA,FD)q''{GA.FD)Proof Since FD {XD) < GD (AD) for all AD we obtain P (FA, q) >P {GA,q) for all q. Similarly, if FD (XD) < GD{XD) for ah AD, then Q ( F D , P ) > Q{GDIP) for cill p. This implies that the equilibrium values for p* will be larger when either distribution function shifts to the right. That is, if Fj first order stochastically dominates Gj (j = A, D) we will have that p* (FA^FD) < p* (GA.FD) and p* (FA^FD) < p* (FA^GD), because Q (p) is decreasing and P {q) is increasing. The equilibrium values for q* will be greater when FD shifts to the right, that is, if FD first order stochastically dominates GD we will have that q* {FA, FD) < q* {FA, GD), because P {q) is increasing. Finally, q* decreases when FA shifts to the right. That is, if FA first order stochastically dominates GA we will have that q* {FA, FD) > q* {GA,FD), because Q (p) is decreasing. Therefore, we have that on the equilibrium path as both distribution functions shift to the right p* increases and q* could either increase or decrease. This completes the proof. As we continue to shift these distributions to the right (keeping the support at [0,1]) in the limit the distributions become concentrated at AA = AD = 1. This is illustrated in Fig. 2. The solid curves show the same reaction functions ^^ Formally, given two distribution functions F and G defined on [0,1], F stochastically dominates G if F (A) < G (A) for all A G [0,1].
Electoral Competition and Quality
107
as in Fig. 1. The dotted curves show the reaction functions when the distributions are very close to degenerate on A^ = AD = 1. We have also marked the limit equilibrium, for A^ = AD = 1. 2-3a P ^ 2-a a 2-a which is the same equilibrium point as in Aragones and Palfrey [2]. Thus, the mixed strategy equilibrium in that paper can be approximated arbitrarily closely as a pure strategy equilibrium when players have private information about policy preferences. That is, this limiting case gives identical mixed strategies as in Aragones and Palfrey [2],^^ except here the candidates have policy preferences that are private information . It is also worth remarking on the direction of convergence as the distributions approach A^ = AD = 1. Candidate A converges to p* == ^ ^ from below while candidate D converges to g* = ^^ from above. That is, for any distributions FA and Fo that satisfy the regularity condition, the effect of policy preferences on the two candidates is for A to be more extreme that she would be without policy preferences, while D is more moderate than the case of no policy preferences. Recall that when candidates only care about holding office, then D tends to hold extremist views (even though he does not prefer them) and A tends toward the moderate location (even though she does not prefer a moderate policy). The effect of incomplete information and policy preferences is to dampen this extremist/moderate distinction between D and A, The effect is especially interesting for D, since (stochastically) increased preferences by D for extreme policies lead him to adopt equilibrium strategies that are actually less extreme. The Boundary Case of Complete Information about A We next examine the properties of the equilibrium correspondence in the boundary case where FA and Fjj converge to any degenerate pair of weights for holding office, {XA, AD) G [0,1]^. This is illustrated in Fig. 3, which shows the equilibrium limit points for all values in the unit square. First consider the diagonal of this figure, corresponding to limiting distributions where at the limit XA = XD = A. As a reference point, the point of the upper left, W, corresponds to both candidates only caring about winning, where we know from above that the unique equilibrium has mixed strategies, p* = gr* m ^ ^ . For almost all values of A the equilibrium is unique. ^^ However, the players actually mix only at the limit. For any distributions of XA and AD satisfying the regularity assumption, no matter how concentrated around XA = I and AD = 1, there is a unique pure strategy equilibrium in type-contingent strategies.
108
Enriqueta Aragones and Thomas R. Palfrey
I-F^(a/(2-a))
^ / \
P(ci)/
J-Fo(a/(2-2a))
Q(P)^
/
/
j
1 1
1 a/(2-a)
*(pWh=i
2-3 a 2-a
Fig. 2. Comparative statics as FA —> 1 If A < 2-2a' ^^^ unique equilibrium is pure, with p* = 0 and q* = 0. That is, if the candidates place enough weight on policy preferences, they locate at their ideal points and never in the center. If A >
2-2a
there is a unique equilibrium in mixed strategies with: , _
2{l-a)X-a A(2 -a) - a - {l-2a)X A(l + a) + l - 2 a
If A = 2-2a' th^^^ is ^ continuum of equilibria. In all of these equilibria, A plays p* == 0. When p* = 0 and A — 2-2a^ ^ ^^ indifferent between locating at the center and at his ideal point. As long as D chooses C with probability no greater than 2-ba-{-5a'^' ^'^ ^^^^ response is her ideal point, so the set of equilibria are p* - 0, g* G [0, ^ ^ ^ ^ ] . The comparative statics of (p*, q*) when A is increased along the diagonal is qualitatively the same as the comparative statics of stochastically increasing
Electoral Competition and Quality
109
W
AD
0
p=0 q-1
/ /
a/(2-2a)
/
p=0 q-0
a/(2-a)
Fig. 3. Equilbrium strategies for A^ and AD \A and Xo. That is, ^ > 0 and ^ This is formally proved below.
< 0. The intuition is exactly the same.
Proposition 4. The comparative statics with respect to \, when A is common knowledge, are: - ^ > 0 and ^ < 0. Proof. The formal argument simply requires partial differentiation of p* and g'* with respect A. For g*, we get Qg* _ dX ~
-(l-2a)^-(l-a^) <0 [A(l + a) + l - 2 a ] 2
For p*, we get dp^ a >0, dX " A 2 ( 2 - a ) completing the proof.
110
Enriqueta Aragones and Thomas R. Palfrey
Next, we consider the case where A^ and X^ are common knowledge, but XA y^ XD- These correspond to the off-diagonal points in Fig. 3. There are three regions to consider. First, if XD < 2-2a' ^^^^ ^ cares enough about pohcy that there is a unique pure strategy equilibrium with p* = q* = 0. That is, both candidates locate at their ideal points. If AD > 2-2a ^^^ '^^ ^ 2 ^ ' ^^^^^ is a unique pure strategy equilibrium with p* = 0 and g* = 1. In this region, policy matters much more to A than to D. If Xjj > 2-20L ^^^ ^^ ^ 2 ^ ' Then both care enough about winning that a pure strategy equilibrium cannot exist, and we are in the region with a unique mixed strategy equilibrium. On the boundaries between the mixed and pure strategy regions, multiple equilibria typically exist, with one player indifferent (with a continuum of possible equilibrium mixing strategies) and the other player adopting a pure strategy. Finally, we consider the comparative statics results with respect to a, in the mixing region.^^ Straightforward derivations give: a^* da
- 2A + 1 >0 [A(l + a) + l - 2 a ] ^ 3A2
and dp^ ^ - 2 ( 1 + A)a da A(2-a)2 ^''• These comparative statics are qualitatively the same as the case studied in Aragones and Palfrey [2], with A^ = AD — 1.
5 Conclusions This paper examined an equilibrium model of candidate competition, combining the effects of five variables that are important factors shaping voter and candidate behavior in competitive elections: candidate quality, candidate policy preferences, the value of holding office, asymmetric information between candidates, and the uncertainty that candidates face about the distribution of voter preferences. It extends in a significant way the results of earlier models of candidate quality by Aragones and Palfrey [2] and Groseclose [7], and shows how results in those papers are special cases in the framework of this paper. Asymmetric information arises naturally because candidates do no know the other candidate's value of holding office and do not know precisely the policy preferences of the other candidate. This asymmetric information not only makes the model more realistic, but actually simplifies the analysis as ^^ The comparative statics with respect to a are flat in the other regions. However, the boundaries between regions will change as a function of a.
Electoral Competition and Quality
111
well. In particular, we show that even if candidates have very little private information , a unique pure strategy equilibrium always exists. Furthermore, due to the approximation result of Harsanyi [8], this implies that the mixed strategy equilibria identified in Aragones and Palfrey [2] are limit points of the pure strategy equilibria in this paper. In other words, the mixed equilibria, which are difficult to interpret empirically, can be viewed as an artifact of the complete information in the basic model. Even a tiny amount of asymmetry will convert these mixed equilibria into pure equilibria that share similar qualitative properties. With asymmetric information, we show that an increase in uncertainty about the median voter leads both candidates to be less likely to adopt the moderate platform. An alternative interpretation is that as the electorate becomes more polarized (i.e., the probability the median voter is moderate decreases) the candidates also become more polarized. In equilibrium we find that ^ ' s platform becomes more polarized when the electorate becomes more polarized (a increases), but that is not the case for candidate D. In fact, for D there are two effects that go in opposite directions, so the total effect is ambiguous. With complete information about A, we show that there is a unique mixed strategy equilibrium if and only if the value of holding office is high enough for both candidates. In this case, we obtain the same main comparative static results of Aragones and Palfrey [2]. The case of complete information also allows comparisons to the model of Groseclose [7], although he considers a continuous policy space with known candidate ideal points and does not look at mixed equilibria. The two similar findings are that A moves to the center as A increases, and that only mixed equilibria exist if the value of holding office is sufficiently high. Our theoretical findings complement the wealth of empirical evidence about the importance of candidate quality in competitive elections, evidence that has for the most part been gathered and studied without the guidance of formal theoretical models.-^^ Dating back at least to the seminal work of Stokes [9] on the "valence dimension" of politics, numerous studies have identified a wide variety of effects of quality and other valence factor. This paper combines several essential features of candidate competition in a simple model that has clear and interesting implications about the nature of equilibrium platforms. Among the most interesting is the interactive effects of candidate quality, the degree of polarization (or uncertainty) in the electorate, and the information candidates have about each other. There is a strong interaction between quality and these information variables. That is, the effects of polarization on candidate behavior go in opposite directions depending on ^^ A notable exception is the work of Banks and Kiewiet [4] which investigates the effect of candidate quality and asymmetric information on entry decisions by challengers in congressional elections.
112
Enriqueta Aragones and Thomas R. Palfrey
candidate quality. This suggests a role for empirical studies to explore these theoretical hypotheses. Experimental research (Aragones and Palfrey [3]) has verified all of the qualitative implications of the model, but it would be very useful to obtain field data and see if the conjectures also hold up in mass elections.
References
[2] [3] [4] [5] [6] [7]
Ansolabehere, S. and J. M. Snyder (2000) Valence pohtics and equilibrium in spatial election models. Public Choice, 103: 327-336. Aragones, E. and T. Palfrey (2002) Mixed equilibrium in a Downsian model with a favored candidate. Journal of Economic Theory, 103: 131161. Aragones, E. and T. Palfrey (2004) The eff'ect of candidate quality on electoral equilibrium: an experimental study. American Political Science Review, 98: 77-90. Banks, J. S. and D. R. Kiewiet (1989) Explaining patterns of candidate competition in congressional elections. American Journal of Political Science, 33: 997-1015. Berger, M. M., M. C. Munger and R. F. Potthoff (2000) With uncertainty, the Downsian model predicts divergence. Journal of Theoretical Politics, 12: 262-268. Calvert, R. (1985) Robustness of the multidimensional voting model: candidate motivations, uncertainty and convergence. American Journal of Political Science, 28: 127-147. Groseclose, T. (2001) A model of candidate location when one candidate has a valence advantage. American Journal of Political Science, 45: 862-
[8] Harsanyi, J. (1973) Games with randomly disturbed payoffs: a new rationale for mixed strategy equilibrium points. International Journal of Game Theory, 2: 1-23. [9] Stokes, D. E. (1963) Spatial models of party competition. American Political Science Review, 57: 368-377. [101 Wittman, D. (1977) Candidates with policy preferences. Journal of Economic Theory, 14: 180-189. [11] Wittman, D. (1983) Candidates motivation: a synthesis of alternatives. American Political Science Review, 77: 142-157.
Party Objectives in the ^^Divide a Dollar" Electoral Competition Jean-Frangois Laslier Ecole Polytechnique [email protected]
S u m m a r y . In the "divide a dollar" framework of distributive politics among three pivotal groups of unequal size, the paper compares two variants of two-party competition, the objective of a party being the probability of winning ("majority tournament" game) or the expected number of votes ("plurality" game). At a mixed equilibrium, all individuals are, on expectation, treated alike in the plurality game while the tournament game favors individuals in small groups.
1 Introduction In basic models of two-party Downsian political competition, maximizing the number of votes (the "plurality") or trying to win the election by whatever margin (the "majority tournament") are equivalent objectives for the parties in the sense t h a t they lead to the same equilibria. But the equivalence result is often lost in richer models t h a t take into account abstention, uncertainty, or a non-trivial decision structure for parties. T h e hypothesis t h a t parties maximise the expected number of votes rather t h a n the probability of winning usually (but not always) gives rise to more mathematically tractable models; for t h a t reason, as noticed by Coughlin [11], "In most of the public choice literature, it is assumed t h a t each candidate wants to maximize his expected plurality". A notable exception is Roemer [29] for whom the "probability of winning" hypothesis is important. It may be the case t h a t specific models can be interpreted in one way or another, like for instance, t h e model of Aragones and Palfrey [1]. T h e equivalence result is very sensitive to various kinds of uncertainty. Uncertainty in electoral competition may arise from exogenous reasons such as parties having imperfect information about the voters' preferences or a technical impossibility of communicating clearly with voters. Uncertainty may also arise for endogenous strategic reasons.
114
Jean-Frangois Laslier
Models of imperfect information in politics are surveyed by Banks [5], where instances can be found in which payoffs are uncertain, be it for exogenous or endogenous reasons. According to the Condorcetian philosophy, gathering information in an uncertain world is one reason of being for democratic institutions, as expressed in the Condorcet jury theorem. The "jury" framework is used by Austen-Smith and Banks [2] to discuss in a modern fashion information aggregation by direct voting. With the same framework, Laslier and Van der Straeten [22] discuss information aggregation by electoral competition and the equivalence problem. The equivalence problem is studied with great details by Patty [27] and Duggan [12] in "probabilistic voting" models. Here the uncertainty about the number of votes to be received by a party derives from a random noise added to each voter's utility function and the probability distribution of this noise is exogenous. The question of endogenous strategic uncertainty is of particular importance because electoral competition games usually have no pure strategy equihbria (the book Austen-Smith and Banks [3], contains a modern survey of the "chaos" theorems of McKelvey, Schofield, and others). The question is thus raised of the mixed strategy equilibria of these games. Laffond et al. [15] give a simple example with a finite set of alternatives in which the majority tournament game and the plurality game each have unique equilibria and the two equilibria are completely different. Dutta and Laslier [13] provide a more detailed comparison of the social choice correspondences based on the majority tournament and plurality scores. Laslier [18] proposes an unusual interpretation of mixed strategies in these specific games. There are two standard interpretations for the mixed strategy probability p{x) associated to a proposed alternative x. (i) Random choice: p{x) is the probability that the party chooses x at random, (ii) Belief: p{x) is the belief of the other party as to the choice oi x. In electoral competition games, one can furthermore interpret p{x) as the fraction of the electorate who judges the party according to proposition x, or probability that a voter understands that the party will implement x. The equilibrium of the electoral competition under this third interpretation can curiously be the equilibrium of the payoff matrix of the plurality game even if the parties maximise the probability of winning.^ This is a theoretical point in favor of the "plurality" model and against the "majority tournament" model. Another point in favor of the plurality model is that it has a natural extension to the case of more than two parties, under the "Borda electoral competition", even in the presence of a mixed equilibrium (Laffond et al. [16]). Laslier [20] studies the robustness of mixed strategy equilibria in electoral ^ The key point is here: do the various voters judge a party fuzzy proposal independently the ones from the others?
The "Divide a Dollar" Electoral Competition
115
competition with respect to the voters disliking ambiguity and the parties prudent behavior. On economic domains, mixed strategy equilibria are usually hard to compute. Technical difficulties abound, no general existence theorem is available, and the inclusion of the support of optimal strategies in the uncovered set, which is easy to prove in the finite setting, is no longer a trivial matter (see McKelvey [23], Banks et al. [6]). One important economic model, which serves as a benchmark in social choice theory and formal political science, is the "divide a dollar" model. When voting is not mediated by parties. Banks and Gasmi [7] and Penn [28] have studied the three-voter divide a dollar game, following the ideas developed in the "sophisticated voting" literature (see the survey by Miller [24]) which led to the definition of the "Banks set" (Banks [4]). When voting is mediated by parties, the "divide a dollar" model is equivalent to a problem of strategic resource allocation that has been studied by game theorists (Borel [8], Gross and Wagner [14], Owen [26]). It is used in political science as a model of pure redistributive politics in order to study the question of the treatment of minorities by democratic rules (Myerson [25], Laslier and Picard [21], Laslier [19]). The present paper considers a "Divide a dollar" framework, in which a fixed amount of money has to be divided among three groups of individuals of unequal size. I exhibit a mixed strategy equilibrium for the associated tournament game and compare it with the mixed strategy equilibrium for the associated plurality game studied by Laslier [19]. The optimal strategy for the plurality game always treats individuals in the same way, whatever group they belong to; but the optimal strategy for the tournament game treats better the individuals belonging to small groups. To highlight the difference, I further consider the situation with two large groups of equal size and one small "pivotal" group. By computing expected values for two indices of inequality (variance and Gini index) I show that the optimal strategy of the tournament game generates more inequality than the one of the plurality game. The paper is organized as follows: after this introduction. Section 2 presents the model. Section 3 describes optimal strategies and contains the statement (Proposition 6) about the shares received by individuals in different groups. Section 4 is devoted to the analysis of inequality in the case of one small pivotal group. Section 5 is a short conclusion.
2 T h e Model The population N is formed of n individuals and is partitioned in 3 groups Ni, i = 1, 2, 3. There are ui individuals in group Nf.
116
Jean-Frangois Laslier 3
3
2=1
i=l
The partition {Ni,N2,Ns) contains a majority if there exists an i such that Hi > n/2; the majority is strict if the previous inequality is strict. In the "divide the dollar" model, a quantity Q G 1R++ of money must be divided among the A^ individuals, but individuals belonging to the same group must receive the same amount. Let Xi be the amount received by an individual in group i. The set of possible outcomes is: 3 i=l
For two outcomes x, y G Z\^^^ the net plurality for x against y can be written: g{x,y) = Y^niSgn{xi
-yi),
i=\
where, for any number i G M, sgn(t) equals — 1 if t < 0, 0 if t == 0 and +1 \i t > 0. Call the plurality game the two-player, symmetric, zero-sum game defined by g. The net plurality g{x,y) is the gain for the strategy x against strategy y. A mixed strategy is a regular probability distribution on Z\^^^; if p and q are two mixed strategies the payoff to p against q is the expected margin of votes: Jxex
Jvex
For two outcomes x,y e A^^^ the net majority for x against y is by definition +1 if a majority of individuals prefer a: to y, —1 in the opposite case, and 0 in case of a tie. It can be written:
m{x,y) = sgn[g{x,y)] = sgn Y^niSgn{xi
- yi)
li=l
The two-player, symmetric, zero-sum game defined by m is called the majority tournament game or simply the tournament game. The net majority m{x,y) G { —1,0,+1} is the payoff" for the strategy x against the strategy y. If p and q are two mixed strategies the payoff to p against q is: m{p,q) =
/ m{x,y) dp{x) dq{y) Jxex Jyex = Fi[g{x,y)>0]Pi [g{x,y) < 0],
If ties can be neglected, that is if p (g) ^ [^'(a:, ?/) = 0] = 0, then m(p, q) is, up to constants, equal to the probability of winning.
The "Divide a Dollar" Electoral Competition
117
3 Optimal Strategies The two games considered in this paper are zero-sum, therefore the solution concept is min-max equilibrium. Recall that, in a zero-sum game with payoff function g, one defines the value for player 1, or "gain-floor" as: vi = maxmin^(p, g). p
Q
The symmetric definition for player 2 is f 2 = max^ miup {—g{p^q)) and it is easy to see that: V2 — — minmaxpfp, g). q
p
The values for the two players are well-defined numbers, provided for instance that the payoff function is bounded, and are such that f 1 + t'2 < 0. If vi -\- V2 = 0 or, equivalently, maxp miug g{p^ q) = min^ maxp g{p, g), one simply says that v\ is the value of the game. According to von Neumann's min-max theorem, finite games in mixed strategies have a value, but infinite games may have no value, even in mixed strategies. A strategy p* for player 1 is optimal if min^(p*,g) = maxmin^(p, g) == minmax^(p, g); q
V
q
Q
P
that is: p* is optimal if the game has a value and p* guarantees this value to player 1. The definition is symmetric for the other player. A min-max equilibrium is a pair of optimal strategies.^ When the game is symmetric, that is if both players have the same strategy set and g{x^y) = —g{y,x), optimal strategies, when they exist, are obviously the same for both players and the value of the game can only be 0. (All games in the present paper are symmetric.) 3.1 Cases with Pure Strategy Equilibrium It is a standard result that, in pure strategies, the two objectives for parties (plurality or majority) are equivalent, both parties proposing the Condorcet winner policy. Proposition 1. The plurality and tournament games have the same optimal pure strategies. But a pure equilibrium appears only when the partition contains a majority; more precisely, Laslier [19] notices: ^ See Owen [26] for an introduction to the theory of zero-sum games. For these games, a strategy profile is a Nash equilibrium if and only if each player's strategy is optimal.
118
Jean-Prangois Laslier
Proposition 2. (i) If the partition of the population contains a strict majority then each party has a unique optimal strategy. This strategy is pure, it consists in giving all the money to the largest group. (a) The same result holds if one group contains exactly half of the population and there are at least two other groups. (Hi) If the partition is (A^i, A^2) 'with m = n2 = n/2 then any strategy is optimal. Unfortunately, the existence of pure strategy equilibria is not a general phenomenon. Proposition 3. / / the partition of the population contains no majority then no pure strategy is optimal, and the plurality game and the tournament game have no pure strategy equilibrium. 3.2 Cases with N o Pure Strategy Equilibrium We now turn to the cases where no pure strategy equilibrium exists. None of the three groups is a majority: n>o = m -\-n2-\-n3 0 < Hi < n2 < Us < n/2. The set A^^^ of possible divisions can then be represented as a triangle (^1,^2,^3) with sides A1A2 — ns, A2A3 = ni, AsAi = 712- For a point M inside the triangle, let Hi, i = 1,2, 3, be the projections of M on the sides, as in Fig. 1. Then for any M, J2i=i '^liMHi = 25, where 5 is the surface of (^1,^25^3)- Changing the unit of money so that Q = 25, we can represent any division of Q by a point M. The height MHi is the amount of money that each individual in group i gets.
H3 Fig. 1. Representing divisions by heights in a triangle
^'
The "Divide a Dollar" Electoral Competition
119
We look for optimal strategies in the tournament game and in the plurality game. We now define a strategy that will prove to be optimal for the tournament game: Definition 1 (An optimal strategy for the tournament game). Let ip and 6 be two independent random variables such that (p has density (1/2) simp on [0, TT] and 9 is uniform on [0, 27r]. For i = 1,2, 3^ let: a^ = (1/3) (1 - sin(/Pcos(<9 - 2i7r/3)). Let p'^^ be the probability distribution of the variable (xi,X2,X3) with: ciiQ Note that the ai sum to one, with the marginal on each ai being uniformly distributed on [0,2/3]. Proposition 4. With three groups, pj^^^ is an optimal strategy for the tournament game. Proof. With probability one, x € A^^\
m{pl,^,x')^E
Consider a pure strategy x' G
Y^sgn{xi-x[) Li=l
The events such that {xi — x[} have probability 0, thus the only case to be distinguished are of the form "two groups against one". Therefore the payoff of pj^a against x' in the tournament game is:^
— 2_. P^ [^i < ^'i ^^d Xj < Xj and x^ > xj^] . where the sum ^ . • ^ goes over suitable distinct indices i,j,/c = 1,2, 3. The random variable a = (ai,a2,a3) has the property that Ylii^i^i — ^ and, for i — 1,2,3, a^ is uniform on [0,2/3]. From this, one can easily infer that: [3 "I \/a' G Z\3, K X ! sgn(a^ - a'M > 0, where Ea denotes expectation, here with respect to the probability distribution of a (see Laslier and Picard [21]). Write a^ = ^ ^ 5 ^^^ notice that Xi > x[ is equivalent to a^ > a^, it follows that This is the point where the proof given here is only valid for three groups.
120
Jean-Prangois Laslier
i,j,k
— Y J P r [a^ < a[ and aj < aj and a^ > a[^] 3
= E X^sgn(ai-a-)
]
>0
Li=l
Thus we proved t h a t the mixed strategy p ^ ^ has non-negative payoff against any pure strategy. In a symmetric and zero-sum game, this is sufficient to prove t h a t p^^ is optimal, and the proof is complete.
Al
A2
Fig. 2. Draws from an optimal strategy for the tournament game To illustrate, Figs. 2 shows 1000 draws according to the probability distribution p ^ ^ . T h e support is an ellipse, with more weight on the border t h a n in the center region. In the case of the plurality game, we can define an optimal strategy t h a t can be compared with pj^^ and can be described geometrically. D e f i n i t i o n 2 ( A n o p t i m a l s t a t e g y for t h e p l u r a l i t y g a m e ) . In the triangle {Ai,A2,As), consider the inscribed circle and a hemisphere erected over that circle (see Fig. 3), Choose a point R at random uniformly on the hemisphere and denote by M the projection of R on the plane that contains ^ 1 , ^ 2 , ^ 3 (see Fig, 4)- ^^^ Pp\ be the probability distribution of the variable (2/1,^2, ?/3) with y^ = MHi. Laslier [19] proves optimahty of p*j. P r o p o s i t i o n 5. With three groups, p*j is an optimal strategy for the game.
plurality
From the properties of the strategies pj^^ ^^<^ Ppi' ^^^ immediately deduces the following proposition.
The "Divide a Dollar" Electoral Competition
121
Proposition 6. (i) In the tournament game, according to the optimal strategy p^a^ ^^^ individual share in group i is uniform on [0, 3^]. On average, all groups receive the same amount and individual belonging to smaller groups receive more. (a) In the plurality game, according to the optimal strategy p*^, the individual share in group i is uniform on [0, ^ ] . On average, all individuals receive the same amount. Fig. 4, to be compared with Fig. 2, shows 1000 draws according to the probability distribution p*i. The support is now a circle, again with more weight on the border than in the center region.
Fig. 3. Hemisphere erected over the inscribed triangle
Al
A2 Fig. 4. Draws from an optimal strategy for the plurality game
4 A Small Pivotal Group In order to underline the difference of between the two games, consider the following example: there are two groups of large, identical, size and one small
122
Jean-Prangois Laslier
"pivotal" group. For an overall population of size 1 and for a pivotal group of size e G]0, 1/3], m = n2 = (1 - e)/2 ns = e. Figs. 2 and 4 suggest that, as e gets close to zero, the support for the tournament case looks like a long and fiat ellipse while the support for the plurality case looks like a small circle. The two cases are thus obviously very different for the small group. It seems that, because the individuals in the small group occasionally get a large share, the tournament case generates more inequality than the plurality case. This section shows that the intuition is right, despite the fact that the number of favored individuals vanishes as e gets small. To do so, two indices of inequality are considered: the variance and the Gini index. 4.1 Inequality According to the Variance To make comparisons for different values of c, we assume the amount of money to be divided is 1. Therefore the amount that an individual gets in group i is ^i — hi/{2S), where hi is the height and S is the surface of the triangle:
S - (1/4) e %/r=^. The variance at point M is: 3
var(M) = ^riix'^
- 1.
i=l
The next result identifies the expected value of this variance for the two games. Proposition 7. (i) In the tournament game, according to the optimal strategy p^^: r.*
r
1
4
16
•^ma 1
It thus tends to -]-oo when the size e of the pivotal group tends to 0. (a) In the plurality game, according to the optimal strategy p*i; E;,
[var] = 1/3.
The "Divide a Dollar" Electoral Competition
123
Proof. According to Proposition 6, for pj^^, the individual share is uniform between 0 and ^ == 3 ^ , therefore:
27ni 27(l-e)/2 27(l-e)/2 16 4 2 7 ( l - e ) H-:;;^ 27e - 1 .
+ 27e 4--1
Likewise, for Ppj, the individual share is uniform between 0 and ^
= 2. Hence:
^pi[var] = ^ n i - - l - - , 1
^
completing the proof. 4.2 Inequality According to the Gini Index When the shares have been ordered X(i) < X(2) < ^(3), the Gini index is given by the following formula: gini(M) = - 1 - 2
^(1) I _^^
2
; +^(2) (^
2
)
/2n(i)X(i)+2n(2)X(2)+n(3)X(3) \
which can be written in a more convenient way: gini(M) = 1 - (^n^i) + 2n(i)n(2) + 2n(i)n(3) j a:(i) - (^71^2) + 2n(2)n(3)j :r(2) - n^3)^(3)Proposition 8 gives the expected value of the Gini index when e is small. Proposition 8. (i) In the tournament game, let E^^ [gini] denote the expected value, according to p"^^, of the Gini index,
lim£;jgim] = i ± ^ ^ . 4 8 . (a) In the plurality game, let E*^ [gini] denote the expected value, according to Ppp of the Gini index, hm^p*! [gini] - .25.
124
Jean-Prangois Laslier
Proof. Computing E^^: I will prove the stronger result that, when e is small, the expected value, according to p^^, of the Gini index is approximately: ^* r . ..
4 + v / 3 - ( 1 2 + y3)e
Following the definition (1) of pj^^, one writes: 2
^1 "" TT, \ (^ ~ sin(^cos((9 - 27r/3)) 3(l-e) ^2 = T7^ ^ (1 - sin (f cos{9 + 27r/3)) 3(l-e) xs = — (1 — sin(/?cos0) oc
(1)
The Gini index is a piecewise affine function of Xi,X2,X3, depending on how they are ordered. First notice that, because of the symmetry between groups 1 and 2, one can compute expected values conditionally on xi < X2E:^^ [gini] = 2 /
gini(M) dp^^.
-'{xi
From (1), xi < X2 if and only if cos(^ - 27r/3) > cos(^ + 27r/3), which is equivalent to 0 < ^ < TT. I break up the remainder of the proof for tournaments into three steps. A Claim I now claim that, according to pj^^^, the probability of the event Xs > xi,X2 tends to one when e tends to zero. To check that point, one studies the inequality x^ > X2, which writes equivalently: 1 2 - - (1 — sin(pcos 0) > -— r (1 — simpcos(9 + 27r/3)) 3e 3 (1 — e) 1 - 3e > ((1 - e) cos ^ - 2e cos((9 -h 27r/3)) sin (f 1 — 3e > Ml — e)cos^ — e — cos6> — \/3sin^ j sin(p l - 3 e > ('cose^ + eVSsine^') simp. Since 0 < (^ < TT, one can see that the inequality holds as soon as 1 - 3e > cos6> + e\/3sin6>. Studying the function cos6> + e^/SsinO for 0 e [0,7r], one finds that (4.2) is equivalent to: l-3e-3eV2e(l-e) 0 > 0{e) = arc cos —^ .
The "Divide a Dollar" Electoral Competition
125
Recall that the random variable 9 is uniform on [0,7r]; since 9{e) tends to 0 with e, the probability of the event 9 > 9{e) tends to 1 when e tends to 0. The claim follows. An affine approximation As a consequence, the expected value of the bounded function gini on the event xi < X2 is approximately equal to the expected value of the function that coincides with gini on the event Xi < X2 < x^^ Let gini(M) 3= 1 - (rii + 2nin2 + 2nin3) xi - (n^ + 2n2n3) X2 -
nlxs,
then
lira /
[gini(M) - ihTiCM)] dp*^, = 0.
One can write:
2f
^i{M) dp*^, = 2 r
^{xi
r
iiiTiCM) /(^, e) d^ d9
Je=OJ(p=0
where / is the density:
The expression gini can be simplified given the expressions of n^ : -^/.^N 1 ( l - e ) ( 3 + e) ( l - e ) ( l + 3e) 2 gmi(M) - 1 - ^ ^ '-xi - ^ '-^ -X2 - e^X3 Final computations The integral to be computed is thus:
2/
ihTiCM) dp^,
^{xi
2r
pTT
r ^liM) ^-^ d
=0 JifiO J(fO Je=o
/7 ^
47r
'\^d^dO
Jo( lJo -e)(3
4-e) r r sin^ ^^ 4 Jo Jo ^^ p7T ( l - e ) ( l + 3e) [^ rr7\ sinc^ ' ' X2 —— 4 ./n ./n 27r dip d9 Jo Jo
-if Substituting x:
Jo
2:3 - r — dip d9. ZTT Jo
126
Jean-Prangois Laslier 27r
Jo Jo
^
/ f {l-simp cos(6> 12 7o Jo (2 + 6e) r r , . ,. 1-sin(^cos(^ + / ./n / ^ 12 Jn Jo Jo {1 — sin if COS 6) 3 70 Jo 27r Integrating with respect to 9: pTT
IT
PIT
27r/3)) ^ ^ ^ d(^ de* 27r ^ ,^^^ sin(^ 27r/3)) —-^ dap dO 27r
p-K
dipdO.
27r (6 + 2e) r (
fi •
\ sin^ ,
(2 + 6e) r
/- .
\ sirup _,
e f^ 3 7o
/
sin 09 ^ 27r
Integrating with respect to if :
(3 + e) / _ V3 \
12 ^^- 2J
and the result follows. Computing £^*j: The disk solution (definition 2) Ppj in that case can be defined by the equations: xi — r (1 + sin (f cos{9 + a)) X2 = r {1-\-sin (f cos{0 — a)) xs = r (1 — sin (/9 cos ^ ) , where r is the radius of the inscribed circle, a is the angle at ^1 (or A2), 1 r = - e V l - 2e, cos a
(2)
The "Divide a Dollar" Electoral Competition
127
and if and 6 are two independent random variables such that if has density (1/2) sin(^ on [0, TT] and 0 is uniform on [0,27r]. This is easily seen graphically (Fig. 5): C the center of the inscribed circle, H — (^i + A2)/2 is the foot of A3, ^ is the angle between the vectors CH and CM^ and r sin ^p is the distance CM. Then, if H\ is the projection of M on the side A2AS, with xi = MHi, one gets by considering the projection P of C on MHi: llCP=
i:-ot,
HCM = e,
CMHi - ^ + a, MHi = PHi + CM cos (9 + a) = r + rsm(f cos{6 + a). There is no simpler way to write down equations (2), but the regions of integration, for computing the expected Gini index are very simple: for 0 < ^ < 7r/2 - a , x^ < Xi < X2 for 7r/2 - a < ^ < 7r/2 + a , xi < xs < X2 , xi < 0:2 < X3 for 7r/2 -i- a < 6 < TT and so on... On each of these regions, integrating the Gini index with respect to the density ^ sin (p is a long but straightforward computation, suitable for a symbolic computation software. Using Mathematica, one obtains that the expected value, according to p*j, of the Gini index is:
El, [var] =
VT^^ l - 2 e
21
/4 +
1 - 2e + 2e2 ^/2~-2e
1-e
and therefore: limg^o E*, [var] = .25, completing the proof.
Fig. 5. Construction for the proof of Propopsition 8
/4,
128
Jean-Frangois Laslier
5 Conclusion In a redistributive problem among three groups of unequal size, this paper showed that at a mixed equilibrium of the two-party electoral competition, all voters are, on expectation, treated symmetrically in the plurality game while the tournament game favors individuals in small groups. In this model, Downsian competition is predicted to generate more or less inequality depending on an assumption on parties' objectives, more inequality arising if parties do not care about their margin of victory. One may wonder about the possibility that the "party objective" question be settled by observation. To that end, one would need differentiated testable predictions based on the kind of models proposed in the present paper. In the 70's, several scholars, following Brams and Davis [9], tried to relate the amount spent during their campaign by presidential candidates in the different States in the USA, to the size of the States. Brams and Davis, used a "plurality" objective and derive the conclusion that resources are allocated proportionally to the power 3/2 of the electoral vote. Lake [17] used a "probability of winning" objective and found that candidates should spend a disproportionately large amount of their funds in large states. These analyses use a "local equilibrium" approach which is not totally satisfying, and the conformity of observations with prediction has been disputed (see Colantoni et al. [10], and the exchanges that followed; see also Young [30]). My opinion is that there is little hope, when trying to figure out what is the true objective of a party, to find a simple answer of the form "parties maximise the expected plurality and not the probability of winning", and this for two reasons. One obvious reason is that a party is not a single decision-maker but a complex organization whose members have different goals: winning the next election, but also winning other elections, promoting ideas, representing some part of the electorate, etc. Another reason is the following. When we simplify things to a point where parties are "Downsian" and the party objective is one of the two objectives under scrutiny, we are often confronted with mixed strategy equilibria, whose predictions are difficult to test. I therefore submit that the main thing we need is to better understand the nature of political competition in the absence of a Condorcet winner policy, which includes understanding how a party behaves when it has no pure optimal strategy. I suspect that a key point here is to deepen our theoretical knowledge of the communication process that takes place between a candidate and voters. Thus the "party objective problem" is probably a case in point where we are in need of theory before observations.
The "Divide a Dollar" Electoral Competition
129
References [1] Aragones, E. and T. R. Palfrey (2002) Mixed equilibrium in a Downsian model with a favored candidate. Journal of Economic Theory, 103: 131161. [2] Austen-Smith, D. and J. S. Banks (1996) Information aggregation, rationality and the Condorcet jury theorem. American Political Science Review, 90: 34-45. [3] Austen-Smith, D. and J. S. Banks (1999) Positive Political Theory I: Collective Preference Ann Arbor: The University of Michigan Press. [4] Banks, J. S. (1985) Sophisticated voting outcome and agenda control. Social Choice and Welfare, 1: 295-306. [5] Banks, J. S. (1991) Signalling Games in Politics Chur: Harwood Academic Publisher. [6] Banks, J. S., J. Duggan and M. Le Breton (2002) Bounds for mixed strategy equilibria and the spatial model of elections. Journal of Economic Theory 103: 88-105. [7] Banks, J. S. and F. Gasmi (1987) Endogenous agenda formation in threeperson committees. Social Choice and Welfare, 4: 133-152. [8] Borel, E. (1938) Application de la Theorie des Probabilites aux jeux de Hasard Paris: Gauthier-Villars. [9] Brams, S. J. and M. D. Davis (1974) The 3/2's rule for presidential campaigning. American Political Science Review, 68: 113-134. [10] Colantoni, C , T. J. Levesque and P. C. Ordeshook (1975) Campaign resource allocation under the electoral college. American Political Science Review, 69: 141-154. [11] Coughlin, P. J. (1992) Probabilistic Voting Theory Cambridge: Cambridge University Press. [12] Duggan, J. (2000) Equihbrium equivalence under expected plurality and probability of winning maximization. Mimeo. University of Rochester. [13] Dutta, B. and J. F. Laslier (1999) Comparison functions and choice correspondences. Social Choice and Welfare, 16: 513-532. [14] Gross, O. and R. Wagner (1950) A continuous Blotto game. Mimeo. The Rand Corporation. [15] Laffond, G., J. F. Laslier and M. Le Breton (1994) Social choice mediators. American Economic Review (proc), 84: 448-543. [16] Laffond, G., J. F. Laslier and M. Le Breton (2000) A:-player additive extension of two-player games, with an application to the Borda electoral competition game. Theory and Decision, 48: 129-137. [17] Lake, M. A new campaign resource allocation model. In S. J. Brams, A. Schotter and G. Schwodiauer (eds) Applied Game Theory Wiirzburg: Physica-Verlag. [18] Laslier, J. F. (2000) Interpretation of electoral mixed strategies. Social Choice and Welfare, 17: 283-292.
130
Jean-FYangois Laslier
[19 Laslier, J. F. (2002) How two-party competition treats minorities. Review of Economic Design, 7: 297-307. [2o; Laslier, J. F. (2003) Ambiguity in electoral competition. Mimeo. Laboratoire d'econometrie de I'Ecole polytechnique. [21 Laslier, J. F. and N. Picard (2002) Distributive politics and electoral competition. Journal of Economic Theory, 103: 106-130. [22; Laslier, J. F. and K. Van der Straeten (2004) Electoral competition under imperfect information. Economic Theory, 24: 419-446. [23; McKelvey, R. (1986) Covering, dominance, and institution-free properties of social choice. American Journal of Political Science, 30: 283-314. [24 Miller, N. R. (1995) Committees, Agendas and Voting Chur: Harwood Academic Publishers. [25; Myerson, R. B. (1993) Incentives to cultivate favored minorities under alternative electoral systems. American Political Science Review, 87: 856869. [26 Owen, G. (1982) Game Theory New York: Academic Press. [27; Patty, J. W. (1999) Equilibrium equivalence with J candidates and A^ voters. Mimeo. California Institute of Technology. [28 Penn, E. M. (2002) A distributive A'-amendment game with endogenous agenda formation. Mimeo. California Institute of Technology. [29 Roemer, J. E. (2001) Political Competition Cambridge: Harvard University Press. [30 Young, H. P. (1978) The allocation of funds in lobbying and campaigning. Behavioral Science, 23: 31-31.
Generalized Bandit Problems'* Rangarajan K. Sundaram New York University rsundaraOstern.nyu.edu
Summary. This chapter examines a number of extensions of the multi-armed bandit framework. We consider the possibility of an infinite number of available arms, we give conditions under which the Gittins index strategy is well-defined, and we examine the optimality of that strategy. We then consider some difficulties arising from "parallel search," in which a decision-maker may pull more than one arm per period, and from the introduction of a cost of switching between arms.
1 Introduction T h e framework of multi-armed bandit problems is among the most widelyused models for the study of optimal information acquisition and "learning" by economic agents. An important factor in this popularity is the considerable degree of analytical tractability possessed by even very general formulations of problems in this framework. In large part, this tractability derives from the remarkable Theorem of Gittins and Jones [11], which asserts t h a t in any independent-armed bandit problems with geometric discounting over an infinite horizon, it is possible to associate with each arm an index, known as the Gittins index^ with the property t h a t a strategy in the bandit problem is an optimal strategy if, and only if, it (almost) always involves playing an arm with the highest value of the Gittins index at t h a t point. (Such a strategy is called a Gittins index strategy.) T h e feature t h a t gives this result potency is t h a t the Gittins index on an arm depends solely on characteristics of t h a t arm (and on the rate of discounting), but not on any other feature of the problem under study. The questions addressed in this paper grew out of my work with Jeff Banks on bandit problems and their applications (Banks and Sundaram [3, 4, 5]) and owe much to many discussions I had with him on this subject. I also had the benefit of several discussions with Andy McLennan, especially regarding the material in Sections 4 and 6 of this paper.
132
Rangarajan K. Sundaram
The fact that indices of the arms can be computed separately and then put together to generate the optimal strategy makes the Gittins-Jones Theorem a powerful tool in the characterization of optimal plans. For instance, Banks and Sundaram [3] show that the Gittins index function possesses certain curvature and continuity properties. Exploiting these properties, they show that in a very large class of bandit problems, it is the case that with non-zero probability, the optimal strategy will involve the play of just a single arm of the bandit forever. Moreover, this need not be the "best" type of arm (i.e., one that would be optimal under complete information); indeed, it could even be the case that arms that are optimal under complete information will be discarded in finite time with probability one, under the optimal strategy. Similarly, Banks and Sundaram [4] demonstrate — once again by identifying and exploiting properties of the Gittins index function — the existence of a class of bandit problems in which optimal strategies are myopic, i.e., in which regardless of the rate of discounting, the optimal strategies remain the same as those that are optimal at a discount factor of zero. Since the latter are trivial to compute, characterizing optimal behavior is made simple. Given the importance of the Gittins-Jones Theorem in characterizing optimal plans, a natural question that arises concerns the extent to which the assumptions of the theorem may be relaxed, without doing violence to its conclusions. A number of papers in the literature have addressed various aspects of this question. On the positive side, it has been shown that an index theorem continues to hold when new projects are continually appearing (Whittle [21]); when project stages are variable and the bandit problem is a semi-Markovian decision problem (Gittins [9]); for the scheduling of priorities in a queue (see Whittle [22, Ch. 14.12]); and in problems that resemble the bandit framework such as the "Pandora's Box" (or "treasure-hunt") problem (Weitzman [20]; Whittle [22, Ch.14.10]). On the downside, the stationarity of the underlying decision problem is very important: Berry and Fristedt [2] show that for the Gittins index to be able to identify optimal strategies in all bandit problems, it is not only sufficient that discounting be geometric, it is also necessary. This paper summarizes and generalizes some recent work in this area, and presents some new results on this issue. Three questions are studied, all motivated by economic considerations. In a number of economic applications, notably in the field of job-search and matching, it is natural to posit the presence of an infinite number of available arms. In Section 4, we examine the extension of the Gittins-Jones result to such a setting, from its original assumption of a finite number of available arms. When the number of arms is not finite, the Gittins index strategy may itself cease to be well-defined, i.e., there may exist a set of histories which occur with positive probability, such that after any of these histories, it is no longer possible to identify an arm that attains the supremum of the Gittins indices following that history. We begin, therefore, with the identification of a set of conditions that are both necessary and suflficient for the Gittins index
Generalized Bandit Problems
133
strategy to be well-defined from a given initial state (Theorem 2). Then, we show (Theorem 3) that regardless of the cardinality of the set of arms, as long as the Gittins index strategy is well-defined from a given initial state, a strategy is optimal from that state if, and only if, it is a Gittins index strategy. To complete the analysis, we examine the converse question: if the Gittins index strategy is not well-defined, is it the case that an optimal strategy does not exist from that initial state? It appears a strong conjecture that the answer to this question is in the afl&rmative, but this paper provides only a partial answer (Theorem 4). Two conjectures, which are of independent interest, but will also help provide a more complete answer answer are stated. The standard formulation of the bandit problem limits the decision-maker to only choosing one arm in each period. This precludes the possibility of "parallel search" by the decision-maker, in which several options may be learnt about simultaneously. In Section 5, we examine a simple generalization of the bandit framework, which allows the decision-maker to play a fixed number k > 2 arms per period, and examine the validity of a conjecture that arises naturally: whether it is the case that optimal strategies involve the decisionmaker playing the arms with the k highest values of the Gittins index in each period. Somewhat unfortunately, this conjecture turns out to be false. We describe a three-arm problem in which k = 2. At the given initial state in this example, arm 3 has the highest index, arm 2 the next highest, and arm 1 the lowest, but we show that beginning with arms 1 and 3 strictly dominates beginning with arms 2 and 3. Finally, in Section 6, we consider a generalization of the bandit framework that is of considerable interest for economic analysis: the introduction of a cost for switching between arms.^ Indeed, it is difficult to imagine a relevant decision problem in which the decision-maker may move between alternatives in a costless manner. Since switching costs do not affect the model's stationarity, it appears a strong conjecture that a suitably generalized version of the Gittins index will continue to identify all optimal strategies in this problem. Moreover, as Weitzman [20] has shown, optimal index strategies do exist in the related problem of "Pandora's Box" which involves switching costs. Unfortunately, this conjecture also turns out to be false: Banks and Sundaram [5] show that in the presence of switching costs, it is no longer possible to define any index on the arms which will unfailingly identify optimal strategies. In Section 6, we provide an alternative proof of their theorem. The rest of this paper is organized as follows. Section 2 describes the framework of bandit problems and discusses some applications. Section 3 describes the construction of the Gittins index, and reviews the Gittins-Jones Theorem on the optimality of Gittins index strategies. Then, as stated above. Section 4 ^ Despite their obvious appeal, bandit problems with switching costs have not been studied very widely. The literature consists only of a few papers including Kolonko and Benzing [12] and Agarwal, et al [1].
134
Rangarajan K. Sundaram
focusses on infinite-armed bandits; Section 5 on bandits with multiple plays allowed in each period; and Section 6 on bandit problems with switching costs. The Appendix contains proofs of results omitted in the main text.
2 Bandit Problems This section describes a typical bandit problem and identifies some useful classes of strategies in these problems; it also provides a brief description of some of the principal applications of the bandit framework to economic analysis. 2.1 Description of the Framework An independent-armed bandit problem with geometric discounting (hereafter, simply bandit problem) is specified by the following objects: 1. A finite set I = { 1 , . . . , n} of arms of the bandit, with generic element z. 2. A tuple Ci = (Xi^ri^Qi) for each arm i, where a) Xi, a, Borel subset of some Polish (i.e., complete, separable, metric) space, describes the set of possible states of arm i] b) ri : Xi —^ R is a, bounded measurable function describing the instantaneous reward from arm i; and c) Qi represents a family of transition probabilities on Xi, i.e., for each Xi G Xi, qi{.\xi) is a probability measure on Xi, and for each fixed Borel subset G of Xi, qi{G\.) is a measurable function from Xi to [0,1]. 3. A discount factor S G [0,1). We will denote such a problem by (/, (Ci)i^/,(5), or, suppressing dependence on (5, sometimes simply by (/, (Ci)^^/)- We also define X = Xi^jXi, and denote by x = ( x i , . . . , x^) G X a typical state of the bandit problem. There is a single decision-maker in the problem, who must decide, in each period t — 0 , 1 , 2 , . . . , of an infinite horizon, the arm z G / of the bandit to be activated that period. This decision is made with full knowledge of the history of play up to that point, including the state Xt = {xu,..., Xnt) at the beginning of period t. If the decision-maker chooses arm i in period t, two things happen; first, he receives an instantaneous reward of ri{xit)\ second, the state of arm i moves to its period-(t + 1) value xa^i which is realized according to the distribution qi{.\xit). The state of all unused arms remains frozen, so we have Xjt^i = Xjt for all j y^ i. The decision-maker discounts future rewards by S G [0,1) and aims to maximize total discounted expected reward over the infinite horizon.
Generalized Bandit Problems
135
Formally, a t-history, denoted /i^, is a list of the states of all the arms in each period up to t, the arm chosen in each of those periods, the consequent reward witnessed, and the period-t state. A strategy a for the decision-maker is a sequence {o-^}, where for each t, a^ specifies the action to be taken (i.e., the arm to be activated) in period-t as a measurable function of the history up to t. Let H denote the set of all strategies for the bandit problem. Each strategy a and each initial state x G X define, in the obvious way, a t-th period expected reward, denoted rt[cr, x], for the decision-maker for each t. The worth of the strategy a from the initial state x, denoted W{a){x)^ is defined as X^^o ^*^^[^'^]- ^ strategy cr* is optimal from an initial state x if its worth from x is maximal amongst all possible strategies, i.e., we have W{o-''){x) > W{a){x) for all a e U. U a. strategy is optimal from all x e X, then we simply say it is an optimal strategy. The value function V: X -^ R of the problem is defined by V(x) = sup^^^ W{a){x). Note that, as the supremum of the set of attainable rewards, the value function is well-defined whether or not an optimal strategy exists since rewards are bounded and there is strict discounting (i.e., 5 < 1.) On the other hand, it is obvious that if an optimal strategy does exist, then V must be equal to the worth H^(cr*)(.) of any optimal strategy. 2.2 The "Classical" Framework The framework we have described above treats each arm of the bandit as an abstract Markov process whose "state" remains frozen when the arm is not in use. In what we shall call the "classical framework" of bandit problems, these states of an arm correspond to the belief of a decision-maker regarding the "true" distribution of rewards from that arm. The independence of arms implies that a decision maker's beliefs about an arm change only when that arm is used. We provide a brief description of this framework here. For omitted details, we refer the reader to the excellent monograph of Berry and Fristedt [2]. Let D denote the set of all probability distributions on the real line, and let A represent the set of all probability distributions on D. Both D and A are given the topology of convergence in distribution. The space A represents the state space Xi for any arm z, and is interpreted as the space of possible beliefs regarding the true distribution of rewards from arm i. F G A then denotes the decision-maker's belief (or prior) regarding the likelihood of these different distributions. Thus, the set A'^^ defined as the n-fold Cartesian product of A and endowed with the product topology, arises as the state space of the bandit problem. Let F = ( F i , . . . ,Fn) denote a typical element of Z\^. The reward functions r^: X^ ^^ R are defined in the obvious manner as
136
Rangarajan K. Sundaram
ri{F,)= [ [ rp,{dr)Fi{dpi). J A JR
Finally, there remains the definition of the transition probabilities. For any Fi e A, let Fi[r] represent a version of the conditional probability distribution (i.e., the posterior belief) that arises when arm i is used and the reward r is witnessed. We assume that for each Fi and r, some version has been chosen and is held fixed. (Berry and Fristedt [2] show that Fi[r] can be chosen to depend measurably on i and r.) The posteriors Fi[r] then implicitly define the transition probabilities qi on the space Xi(= A): given any Borel subset S of ZA, the probability qi{B\Fi) of the posterior belief being in B given the prior belief Fi, is simply the probability under Fi of observing a reward r such that Fi[r] G B. 2.3 Markovian Strategies and Index Strategies We close this section with a brief discussion of two subclasses of strategies of special interest in the sequel: stationary Markovian strategies (hereafter, simply Markovian strategies), and strategies defined through an "index." Given any t-history ht, let x{ht) = {xi{ht),... ,Xn{ht)) denote the period-/; vector of states under ht. Markovian Strategies A stationary Markovian strategy is a strategy a in which at each t, at depends on ht only through x{ht), but not through t or any other details of the history up to t. Such a strategy can evidently be represented completely by a measurable function g: X —^ / , with the interpretation that g{x{ht)) G / is the arm to be in played if the t-history ht has occurred — and indeed, every such function gives rise to a stationary Markovian strategy. Abusing notation, we will refer to such a strategy by simply the fuction g. Index Strategies An index for the bandit problem is with each arm i and each state Xi speaking, the number X{xi,Ci) may i whose characteristics are given by will not push this interpretation.
a measurable function A that associates G Xi, a real number X{xi,Ci). Loosely be interpreted as the "worth" of an arm Ci and whose current state is Xj, but we
A strategy a is said to be an index strategy with respect to the index A if at each t and each t-history ht up to t, we have (Tt{ht) G {i G / | \{xi{ht), Ci) > X{xj{ht),Cj) Mj G / } , i.e., if a always selects one of the arms whose index, calculated according to A is maximal at that point.
Generalized Bandit Problems
137
Intuitively speaking, the notions of Markovian strategies and index strategies appeal to different aspects of the framework we have defined. The former owe their interest to the fact that the optimization problem facing the decisionmaker in our model is itself a stationary Markovian one, i.e., the current state of the arms encapsulates all relevant information concerning current and future payoff possibilities. Thus, stationary Markovian strategies constitute a natural starting point for analysis in these problems. The idea of using an "index" to define strategies, on the other hand, has its genesis in the independence assumption that when any arm i is in use, the states of all remaining arms are frozen. To wit, suppose that in some independent-armed bandit problem, it is optimal to discard the current incumbent arm i in favor of some arm j if the history h transpires when i is in use; and that it is optimal to replace i with a third arm /c, if the history that is witnessed from i is instead /l^ Since no information is being obtained on j (or k or any other arm) when arm i is in use, it seems very plausible — indeed, almost "obvious" — that there must also exist an optimal strategy which chooses arm j after the history h\ Put differently, the converse seems very improbable: that it is optimal to continue with j after the history /i, but not after the history h\ The conjecture that an index exists which gives rise to an optimal strategy is a stronger version of this statement: it asserts that this continuation arm j can be chosen by calculating its "worth" in some manner, independent of the other available arms. 2.4 Applications of the Bandit Framework The bandit framework has found a vast degree of applicability in economics and allied disciplines. We present a brief description of simple formulations of a few of these in this section, in part to motivate some of the generahzations considered later in the paper. Market Pricing: Learning the Demand Curve It is common in most of economic theory to assume that firms and managers act under perfect knowledge of market conditions when choosing price and output decisions. One of the first papers to move away from this perfect information formulation was Rothschild [17] who introduced the bandit framework to economic theory. Rothschild considers the problem of a seller who does not know the parameters of the "true" demand curve he faces. Specifically, there are a finite number of possible prices ( p i , . . . ,Pn) the seller can charge. Associated with each price Pi is a "probability of purchase" TT^ (the probability that a consumer faced with the price pi will actually buy the good). If the vector (TTI, . . . , TT^) is known, then the seller maximizes expected profits simply by setting the
138
Rangarajan K. Sundaram
price equal to that i at which piiTi is maximized. When the TT^'S are unknown, however, the seller faces a bandit problem in the classical framework where the n prices act as the n arms. The reward from choosing arm i is either Pi (if a sale occurs) or 0 (if one does not)."^ Each time the seller chooses a price Pi, he learns something about the probability of purchase at that price. Rothschild shows that when n = 2 (the case he studies throughout), there is a non-zero probability under the optimal strategy that the seller will choose one arm and stick to it forever. He further shows that this arm could be the inferior Sim. (i.e., the arm that would not be optimal under full information). Banks and Sundaram [3] generalize these findings substantially. They prove that both results remain true under very general conditions in independentarmed bandit problems in the classical framework, i.e., there is a non-zero probability that an arm that becomes optimal at some point will remain optimal forever, and as in Rothschild's case this need not not be the "best" type of arm, that is the arm that would be optimal under full information. Indeed, a particularly strong version of this last result is true. Banks and Sundaram provide an example of a bandit with a countable infinity of arms where each arm is one of the same three possible types, and show that in any optimal strategy in this example, the best type of arm (the one with the highest expected reward) will be rejected in finite time with probability one, while the arm that would be second-best under complete information survives forever with probability one. Job Search and Matching The framework of bandit problems has also proved a popular one for the analysis of decision making in labor markets, notably in the theory of job-search and matching (see Mortensen [14] for references). A typical "matching" model has the following structure: there is a single decision-maker (the "worker") who faces a infinite number of a priori identical potential employers. The productivity of the worker is match-specific: it depends on the firm she opts to work for. However, the productivity levels of some or all of the matches are a priori unknown. Instead, both firm and worker begin with (the same) belief regarding her productivity, which is updated as information comes in. In the meanwhile, the worker receives as compensation the expected value of her match as given by the current belief on this score. It is easy to see that, treating the firms as arms of the bandit, and the productivity of a match as the "true" distribution of rewards from that arm, the optimization problem facing the worker is precisely a bandit problem in the classical framework, the only difference being the assumption of an infinite number of available arms (firms). However, it is shown in Banks and ^ The situation is, thus, analogous to one where the arms generate rewards according to Bernoulli distributions with unknown probabilities.
Generalized Bandit Problems
139
Sundaram [3] that the independent-armed bandit framework we have outlined above is easily generalized to accomodate a countable infinity of arms, and existence of optimal strategies may be guaranteed under very general conditions that includes the case where all arms are a priori identical. In addition to existence of optimal strategies. Banks and Sundaram also show that the assumption of an infinite number of a priori identical arms has the following implications. First, there is always at least one optimal "no recall" strategy which never re-uses a once-tried and discarded arm. Second, the expected number of arms used in any optimal strategy is finite, so that, with probability one, the worker uses only a finite number of arms. And finally, when the true distribution of rewards from any match is one of only two possible types, myopic strategies (strategies which select the arm to be played purely on the basis of current expected rewards from each arm) are fully optimal. Technology Choice and Learning-By-Doing Consider the problem faced by a firm which has available to it two possible modes of production. The first (the "old" technology) has associated with it a constant marginal cost of production, denoted say c. The second (the "new" technology) has an initial marginal cost that is higher than c, but this marginal cost declines over time as the technology is used owing to a learningby-doing eff'ect. For definiteness, assume that the marginal cost of production using the new technology is given by g{do.dud2.r'X)=do
+ Cdie-'''^
(1)
where do,di,d2 > 0; do < c < do ^- di\ r is the number of periods the second technology has been used; and C is a noise term with known distribution Z and mean 1. Note that if r == 0 (i.e., the new technology has not yet been used), then the (expected) marginal cost of production using this technology is do + d\ > c, while as r ^ oo, this marginal cost approaches do < c. Moreover, d2 parametrizes the rate of decline of this cost. The firm produces one unit of output per period, and wishes to minimize the sum of discounted expected costs over an infinite horizon. For notational ease, let 0 represent the parameter vector {do,di,(^2). When the vector 0 is known with certainty, the resulting optimization problem facing the firm is a simple one: since costs are declining under the new technology and are constant under the old, if ever it becomes optimal to switch to the new technology remaining with this choice forever must also be optimal. Thus, the firm only needs to calculate the discounted expected cost of using the old technology forever, which is c/(l — J), and compare it with the cost of switching to the new technology forever, which is Ylu=o /^(^5^'C)<^^(C)- The
140
Rangarajan K. Sundaram
smaller of the two then is the firm's optimal choice. The choice problem becomes non-trivial, however, if some or all of these parameters are not known with certainty. Moreover, the problem cannot be solved by putting it into the classical bandit framework, since the firm's belief (denoted, say F) regarding the vector 9 is not suflScient to describe the state of the problem at any point. Rather, it is also important to know the number of periods r the new technology has been used so far. Indeed, even the current marginal cost of production fjg{0, r, ()dF{6)dZ{Q cannot be calculated without this piece of information. Nonetheless, the problem can be cast as a bandit problem in the framework of subsection 2.1. To do so, let arm 1 be defined by Xi = {c}, ri(c) = —c, and gi({c}|c) = 1. This denotes the fact that arm 1 is in a constant state c, from which the rewards are — c. As regards arm 2, let ^ denote the space of all possible beliefs regarding the parameter vector 6, and let X2 = T x {/, 00, G,...} be the state space of arm 2. The expected reward r2(F, r) from using arm 2 at the state (F, r) can be written as r2{F, r)^-J
J g{0, r, OdF{e)dZ{C). / /
Finally, suppose F{w} represents the posterior belief regarding the value of 0 given the prior F and the observation w = —9{0,r]Q. Then, given the current state (F, r ) , if the reward w is witnessed from this arm, the new state of the arm is given by {F{w},r 4-1). Since the distribution of w is completely specified by knowledge of F and r, this implicitly defines the transition probabilities q2 on arm 2. The vectors (Xi,ri,gi) and (X2,r2,g2) complete the specification of this problem as a bandit problem. Agency Problems The framework of bandit problems can also be thought of as representing a problem of pure agency, in which each arm of the bandit represents an agent and the difi'erent distributions in the support of that arm represent the different types the agent can be. The problem facing the decision-maker is then to select an agent whose type is "best" (from the decision-maker's point of view) subject to the implicit search costs involved. On interpretation arises from treating the decision-maker as the median voter in some constituency, the arms as the various candidates offering themselves for election, and the payoffs from an arm as the government-controlled benefits that become available to the constituency. Thus, a candidate's type is an index of that candidate's ability to divert funds to his constituency. More generally, by expanding the state space to include variables apart from beliefs
Generalized Bandit Problems
141
(as in the subsection above), the impact of features such as term-limits and incumbency effects could also be studied. Alternatively, a different labor-market interpretation of the bandit problem from that presented above is obtained by treating the decision-maker as a firm or employer searching over workers (arms). Here, once again, the generalized bandit framework is particulary useful, since the state of an arm (i.e., the characteristics of the worker) could be used to incorporate a variety of detail apart from the worker's expected productivity.
3 Optimal Strategies and the Gittins Index The existence of optimal strategies in the bandit problem — in fact, the existence of Markovian strategies that are optimal — is easily ascertained using standard arguments from dynamic programming. These arguments are briefly outlined in below. Following this, we describe the construction of the Dynamic Allocation Index (better known today as the Gittins Index) of Gittins and Jones [11], and state their powerful theorem that a strategy in a bandit problem is an optimal strategy if, and only if, it is an index strategy with respect to the Gittins Index. 3.1 Markovian Optimal Strategies The description of the bandit problem as a dynamic programming problem, and the obtaining of the value function and an optimal strategy using a contraction mapping argument, are wholly routine. Our exposition in this section is correspondingly brief. As a first step, recall that since rewards are bounded, the value V{x) = sup^^j; W{a){x) is well defined from any x. A standard argument establishes that at any x e X, V must satisfy the time-consistency principle embodied in the "Bellman Equation" for this problem: V{x) = \/UV{x),
(2)
iei
where LiV{x) = ri{xi) + 5
V{x-i,Xi)qi{dxi\xi).
(3)
(Here, in obvious notation, {x-i,Xi) refers to the vector x, but with Xi replaced by Xi.)
To recover the function F , we use the following procedure. Let M{X) be the space of all bounded measurable functions on X, endowed with the sup-norm. Define the map T: M{X) -^ M{X) as follows: for w G M{X), let
142
Rangarajan K. Sundaram Tw{x) = V Liw{x)
(4)
iei
where Liw{x) = ri{xi) 4- S
w{x-i,Xi)dqi{xi\xi).
(5)
Let v,w e M{X) and a G R. If t; > to, then Liv > LiW for all z, so certainly Tv > Tw. Moreover, it is evident that T{v -\- a) = Tv -{- 5a. Since (5 < 1, it follows from Blackwell [7] that T is a contraction on the complete metric space Mj(X), and hence has a unique fixed point it;*. Since V also meets this equation, we must have ic'* = V. Finally, let g denote any measurable selection from the correspondence of maximizers of (2)-(3). (The existence of such a selection can be shown by appealing to standard selection theorems.) It follows easily from the recursive nature of these equations that the total worth of the stationary Markovian strategy defined using g (and denoted simply by g) is, in fact, V. Thus, any measurable selection defined in this manner is a Markovian optimal strategy, establishing simultaneously the existence of optimal strategies, and the existence of Markovian optimal strategies. 3.2 The Gittins Index The Theorem of Gittins and Jones [12] is perhaps the single most powerful result in the literature on bandit problems. It asserts that in all independentarmed bandit problems under geometric discounting, the set of all optimal strategies may be recovered by solving a family of optimal stopping problems which associate with each arm an index known as the Dynamic Allocation Index, or, more popularly, as the Gittins Index. The feature which gives this result power is that the index on an arm depends solely on the characteristics of that arm, and not on any other feature of the problem. The Gittins index is constructed as follows. Let a bandit problem (/, (Ci)i^/, S) be given, where, of course, Ci = (Xi.ri^qi). Select any arm i. Consider the following optimal stopping problem involving arm i. In each period (conditional on the termination option not having been accepted yet) the decision-maker is given the choice between (a) terminating the problem and accepting a fixed terminal reward of m € M, or (b) continuing with playing arm i for one more period and facing the same choice next period. Standard techniques from the theory of dynamic programming show that this problem is well defined from any initial state x^ G X^, and that the value of the problem V(xj, Ci, m) from the initial state Xi is a measurable function on Xi that satisfies at each xf.
Generalized Bandit Problems
V{xi,Ci,m)
= max{m, ri{xi)-\-5
Vi{xi,Ci,m)dqi{xi\xi)}.
143
(6)
The Gittins Index fi{xi,Ci), on an arm i, whose characteristics are given by Ci and which is currently in the state x^, is then defined as /j.{xi, Ci) = inf{m | V{xi, Ci,m) = m}.
(7)
Since r^ is a bounded function by hypothesis, it is easy to see that for m sufliciently large we must have V(xj,Ci,m) = m for all Xi G Xi] while, for —m sufiicently large it must be the case that V(xi,Ci,m) > m at all Xi. It follows that the Gittins Index is well defined. The following result establishes the importance of this index: Theorem 1 (Gittins and Jones [11]). The optimal selections in the bandit problem {I,5,{Ci)i^i) o.t the state {xi,... ,Xn) are those arms i which are Gittins index maximal at that state, i.e., which are such that li{xi,Ci) = \J ii{xj,Cj).
(8)
Equivalently, a strategy a for a bandit problem {I,5,{Ci)i^i) is an optimal strategy for that problem if and only if, the set of histories on which its recommendations differ from the set of Gittins Index maximal arms following that history has probability zero.
4 Infinite-Armed Bandits and the Gittins Index In many economic applications (such as the literature on matching models mentioned above), it is natural to allow for an infinite number of available arms, to leave open the possibility that there is at least one untried arm available each period. It would, therefore, be interesting to know if the GittinsJones Theorem can be extended to cover this case also. Increased applicability is only one reason why one might wish to allow for an infinite number of arms in the bandit problem. Perhaps a deeper one is the intuitive feeling that, as an expression of the independence between arms, the "correct" statement of the Gittins-Jones Theorem should not depend on the cardinality of the set of available arms. More precisely, consider the statement made in Section 2 that in an independent-armed bandit problem, if is is optimal to switch from arm i to arm j after the history h (on arm i), and to switch to arm k after the history h\ it "should" be optimal to switch to j after h^ also. The Gittins-Jones Theorem shows that this statement is indeed
144
Rangarajan K. Sundaram
correct when the number of arms is finite; but it is unclear why finiteness of the set of available arms should m a t t e r for the t r u t h of this statement."^ We examine the optimality of Gittins index strategies in bandits with a (possibly) infinite set of arms in this section. As before, we denote a bandit problem by (/, (Cj)^^/), but, for the purposes of this section alone, we will no longer assume t h a t / is a finite set. Rather, / will be allowed to be of arbitrary cardinality. Notation is otherwise largely unchanged. T h e index i will continue to denote a generic member of / , and Ci = {Xi^Vi^qi) will denote the characteristics of a generic arm i. In addition, the following assumption will be maintained throughout this section to ensure t h a t total discounted rewards under an arbitrary strategy are finite: Assumption 1
There is ^ G M such t h a t s u p ^ . ^ ^ . ^^7 k i ( ^ i ) | ^
^•
For expositional reasons, we break up the discussion in this section into two parts. We first present a simple example to show t h a t when the cardinality of the set of available arms is allowed to be infinite, the Gittins index strategy need not be well-defined: there could exist a set of histories of positive probability under any of which there is no longer an arm attaining the supremum of the indices. We then identify a set of conditions t h a t are necessary and sufficient for the index strategy to be well defined from a given initial state. Finally, we examine the optimahty of Gittins index strategies when there are an infinite number of arms.
4.1 T h e G i t t i n s I n d e x S t r a t e g y Any strategy in the bandit problem must prescribe the continuation arm to be picked after any history. Thus, if we are to examine the optimality of the Gittins index strategy, we must first ensure t h a t it is really a "strategy," i.e., t h a t after any history, there is at least one arm t h a t attains the supremum of the set of indices. It is a simple matter to construct seemingly well-behaved bandit problems with a compact set of arms in which this is not the case. Consider the following: E x a m p l e Let I — [0,1]. For each i e I, let Xi — [0,1], and let '^ii^i) = ^i foi* ^11 ^i ^ ^i' Finally, define the transition probabilities by qi{{l}\xi) = 1 — qi{{0}\xi) = Xi/2. Since the characteristics of all arms are the same, we denote the index on arm i at the state Xi by 4
In Theorem 4.1 of Banks and Sundaram [3], it is shown that the Gittins index strategies continue to identify all the optimal selections at every point, when the number of available arms is allowed to be countably infinite. While sufficient for most applications, this result still begs the question of whether the intuition given here has a validity that is independent of the cardinality of the set of available
Generalized Bandit Problems
145
simply ii{xi). Suppose now that the initial state is given by Xi — i/2 for alH. A straightforward calculation shows that
(l-(5)(2-2(5 +Ji) so that arm 1 has the highest index at the initial state. However, after the 1-history in which the state of arm 1 moves to 0 (a 1-history which occurs with probability 1/2), the index on arm 1 drops to zero; it is a trivial matter to see that there is no longer an arm that attains the supremum of the indices at this new state. In the rest of this subsection, we focus on identifying a set of conditions which guarantee that the Gittins index strategy will be well-defined from a given fixed initial state x = {xi)i^j. The Gittins index of arm i at the fixed initial state Xi is denoted fti. Consider the following strategy in the optimal stopping problem used to define the Gittins index on arm i, when m is the terminal reward: Select arm i initially. In each subsequent period, stay with arm i if the index fii on arm i at the beginning of the period satisfies i^i >m. At the first point where fj^i < m, accept the terminal reward m. Denote this strategy by <Ji[?Ti]. For t = 1, 2 , . . . , let iJ^(t, m) denote the set of thistories in this stopping problem (from the fixed initial state Xi) under which ai[m] accepts the terminal reward in period t. Also, denote by Pi{t,m) the probability of the set Hi{t,m), and by Ri{t,m) the total expected discounted reward from arm i over this period. Theorem 2. The Gittins index strategy is well defined from the initial state ^ — {^i)iei if, cii^d only if either (a) there are infinitely many arms i ^ I such that pi > m*, or (b) there is i E I such that fti > m* and pi{t^ m*) = 0 for all t, where m* == sup{m \jli>m
for infinitely many i e I}.
Remark Under the convention that the supremum of the empty set is — oo, this result applies when / is finite also. Proof We begin by proving that under (a) or (b), the Gittins index strategy is well defined. First suppose (a) holds. Define the sequences m^. It and Jt inductively as follows. Let mi = sup^^j /i^, and set /i = {z G I\fii = mi} and Ji = I - / i . For t > 1, let mt+i = sup{fli\i G Jt}, /t+i = {i e Jt\fli = m^+i}, and Jt+i = Jt - / t + i .
146
Rangarajan K. Sundaram
It is obvious from the definition of m* and the hypothesis that (a) holds that /i is non-empty. If /i contains an infinite number of elements, the Gittins index strategy is evidently well defined. Suppose /i contains only a finite number of elements. Then, we claim, I2 must be non-empty, and, in fact, that 7^+1 must be non-empty whenever U^^i IT contains only a finite number of elements. For, suppose the contrary. Clearly, we must then have m* < m^+i, for if m* > rut^i and ^i such that Jxi = rrit^i, (a) would be violated. On the other hand, since there is no i such that /2i — m^+i, it also follows that for any e > 0, there are infinitely many i such that fti G (mt+i — e, mt+i]. For e suflSciently small, m* is not in this interval, implying that there exists m > m'' such that fti > 7% for infinitely many i. This violates the definition of m*, and establishes the claim. It trivially follows from this claim that the Gittins index strategy is well defined from the initial state x. On the other hand, suppose (b) holds; assume without loss that (a) does not. The set /* = {j G /|/ij > m*} contains at least i, and since (a) is violated contains at most a finite number of elements. Moreover, after any history, the index on arm i will always be at least as large as m* with probability one, so in following the Gittins index strategy, there will never be a call to play an arm not in /*. It follows that, since /* is finite, the Gittins index strategy is well defined. Now suppose both (a) and (b) are false. If the set /* of the previous paragraph is empty, then it follows from the definition of m that there is also no j such that ftj = sup^^//2^. So suppose that /* is non-empty. Since (a) is violated, /* can contain at most a finite number of elements. By (b) being false, for each arm j in this set, there is a positive probability that when j is played, the index on j falls strictly below m* in finite time. Thus, with nonzero probability, the indices on all these arms will fall below m* in finite time. From the definition of m*, it follows that there is no longer any i attaining the supremum of the indices. 4.2 Optiraality of the Gittins Index Strategy The first result of this subsection, proved in the appendix, shows that whenever the Gittins index strategy is well-defined, a strategy in the bandit problem is an optimal strategy if, and only if, it always recommends picking an arm that is Gittins index maximal (except, possibly, after a set of histories of collective probability zero). As usual, W{a)(x) will denote the total discounted reward to the decision-maker under the strategy a from the initial state x. Observe that since Assumption 1 holds, the value function V{x) = sup^ W{a){x) is well-defined from every initial state x, even if an optimal strategy does not exist from that state. Given an initial state x, and the history ht from x, let x{ht) = {xi{ht))iei represent the resulting new vector of states.
Generalized Bandit Problems
147
Theorem 3. Suppose that the Gittins index strategy is well defined from the initial state x € X, i.e., that after any t-history ht from x, there exists i G I such that iii{xi{ht)) = \/j^jl^jixj{ht)). Then, a strategy a satisfies W{a){x) = V{x) if, and only if, (Jt{ht) e{iel\
iii{xi{ht)) = y
fij{xj{ht))}
except possibly after a set of histories of probability zero. Remark The proof of this theorem actually follows from a close repetition of arguments in the original Gittins-Jones proof. It appears a plausible conjecture that the reason this has gone unnoticed is that much of the recent literature (including Berry and Fristedt [2]) uses the shorter and more elegant proof of the Gittins-Jones Theorem due to Whittle [22]. However, unlike the GittinsJones proof, Whittle's arguments rely in an essential way on the finiteness of the set of available arms. Theorem 3 establishes that provided the Gittins index strategy is welldefined, only strategies coinciding with it almost surely can be optimal. However, this result leaves unanswered the question of what happens if the Gittins index strategy is not well defined. In particular, is it the case that if the Gittins index strategy is not well defined, optimal strategies no longer exist? The following theorem establishes that in a large class of problems, this is indeed the case: Theorem 4. Suppose that for each i G I and each Xi G Xi it is the case that q{'\xi) has countable support, i.e., there is a countable subsetYi of Xi (possibly depending on Xi) such that q{Yi\xi) = 1. Then, a strategy is optimal in the bandit problem if, and only if, it is a Gittins index strategy. In particular, optimal strategies fail to exist whenever the Gittins index strategy is not well defined. Proof. Under the stated hypothesis, any strategy in the bandit problem will involve only the use of (at most) a countable number of arms, since with probability one, there are only a countable number of distinct t-histories for any t. Given any countable subset J of / , a strategy aj is optimal in the bandit problem (J, {Cj)j^j) if, and only if, it is a Gittins index strategy (Banks and Sundaram [3]). The theorem follows. On the other hand, the assumption of countable support seems restrictive, and it appears a strong conjecture that Theorem 4 is valid even without this restriction. I have not been able to prove this conjecture or to provide a counterexample. However, the following related — but simpler — conjectures,
148
Rangarajan K. Sundaram
which pertain to e-optimal strategies rather than fully optimal ones, would help provide an answer if true. (Recall that a strategy cr* is e-optimal from the initial state x if it is the case that W{(J''){X) > W{a){x) — e for all a e U.) Conjecture 1. For all e > 0, there is ry > 0 such that if a strategy a always picks an arm whose index is within 77 of the supremum of the indices, then a is e-optimal. Conjecture 2. For all e > 0, there is 77 > 0 such that if a is e-optimal, then it always picks an arm whose index is within rj of the supremum of the indices at that point.
5 Multiple Plays We return now to the framework of Section 2 where the set of arms / is finite, / = { l , . . . , n } . As discussed in the Introduction, one somewhat restrictive feature of this framework is that only one arm may be activated by the decision-maker at any point in time. In this section we consider a situation where the decision-maker is allowed to play k arms each period, where 1 < A: < n is a given, fixed number, and examine whether it is the case (as intuition would suggest) that the optimal strategy consists of playing the arms with the k highest Gittins indices in each period. Thus, the action space for the decision-maker in this set-up consists of the set of all possible combinations of k of the n arms. Let C denote this set. A typical action will be denoted C = ( c i , . . . ,c/e) where Ci G I for each i (and, of course, Ci ^ Cj if z 7^ j . Given an action (7 G C, the reward R{x, C) from taking the action C at the state x = ( x i , . . . ,Xn) ^ X — x^^iXi is given by: i?(x,C)-^r,(x,). When the action C is taken at a state x, the state of all untried arms (i.e., arms j ^ C) remain frozen, while the state of arm i e C moves to a new state according to the distribution qi{.\xi). More formally, let qi{.\xi,C) be the measure qi{.\xi) if i G C, and let qi{.\xi,C) be the measure Xii^i) that places point-mass on Xi ii i ^ C. Then, the transition probability Q{.\x,C) from taking the action C at the state x in this modified bandit problem is simply given by the product measure Q(.|x,C) =qi{.\xi,C)
X ...
xqn{.\xn,C).
The tuple {X, C,i?, Q} then represents the decision-maker's optimization problem as a Markovian dynamic programming problem with state space X, action space C, reward function i?, and transition probability Q.
Generalized Bandit Problems
149
We first establish the existence of Markovian optimal strategies in this problem. Then we turn to the central question of this section: is it the case that at all points it is optimal to select the k arms with the highest Gittins indices at that point? 5.1 Markovian Optimal Strategies As in previous sections, the existence of Markovian optimal strategies follows from a straightforward application of dynamic programming results: Theorem 5. The dynamic programming problem {X,C, R, Q} is well defined. The value function V: X ^^ R is a measurable function that satisfies the Bellman Principle of Optimality at each x G X : V{x) = V LcV{x), cec
(9)
where LcV{x)
= r{x, Q+d = ^^riixi)
fv{x)dQ{x\x, +S
V{xi,...
C) ,Xn)dqi{xi\xi,C)
" • dqn{xn\xn^C).
Any Markovian strategy defined through a measurable selection from the correspondence of optimizers of equation (9) defines an optimal strategy. 5.2 Optimality of Gittins-Index Strategies When /c = 1, the Gittins-Jones theorem shows that any optimal strategy involves (almost) always picking an arm with the highest current value of the Gittins index. In this section we provide a counter-example which shows that it is not necessarily optimal to play the arms with the k highest values of the Gittins index when k> 1. The example has 3 arms: I = {1,2,3}, and C - {(1,2), (1,3), (2,3)}. The state space Xi of each arm i is the unit interval [0,1]. The reward functions ri'. Xi —^ R are given by ri{xi) = ixi, Xi G [0,1], i — 1,2,3. Finally, the transition probabilities qi{.\xi) are given by qi{{l}\xi) = 1 ~ qi{{0}\xi) — x^, XiG [0,1], z = 1,2,3. Some simple calculation shows that the indices on the three arms are given by the following expressions:
150
Rangarajan K. Sundaram
^^(^^^ = {I - S)iZ\ + x,S)
^''^
Let (xi, X2, X3) € [0,1]^ denote the initial state of this bandit problem. We begin by identifying a set of conditions under which, conditional on optimal continuations in either case, it is strictly preferable to choose arms 1 and 3 in the first period, rather than arms 2 and 3. Then we will show that it is possible to choose the initial state to satisfy these conditions and simultaneously also have pi < ft2 < fis^ where pi = fJ^i{xi) is the Gittins index of arm i at this initial state. Consider first the case where arms 2 and 3 are used in the first period at the initial state. Then, there are four possible continuation states that could result at the beginning of the second period: ( x i , l , l ) , ( x i , 0 , l ) , (xi,0,1), and (xi,0,0), with respective probabilities X2X3, (1 — X2)x3, X2(1 — X3), and (1—X2)(l —X3). Since all arms yield a reward of zero when they are in the state 0, and strictly positive rewards when they are not, the optimal continuation strategy is obvious in each case: continue with arms 2 and 3 forever if the new state is (xi, 1,1); switch to arms 1 and 3 forever if the new state is (xi,0,1); switch to arms 1 and 2 forever if the new state is (x — 1,1,0); and play either 1 and 2, or 1 and 3, if the new state is (xi,0,0). The expected continuation payoffs are: y(xi,l,l)-5/(l-J) F ( x i , 0 , l ) = (3 + x i ) / ( l - 5 ) y(xi,l,0)-(2 + xi)/(l-5) y(xi,0,0)-xi/(l-(5). Thus, the total value Z/(2,3)V(xi,X2,X3) of starting with arms 2 and 3 is: ^(2.3)V'(xi,X2,X3) = 3X3 + 2X2 + (5[X2X3F(X1, 1, 1) + X3(l - X 2 ) F ( x i , 0, 1) +X2(1 - X3)F(X1, 1,0) + (1 - X2)(l - X3)F(X1,0,0)] = 3X3 + 2X2 + ^(1 - (5)"^[5X2X3 + (3 + Xi)x3(l - X2) + (2 + Xi)x2(l - X3) + Xi(l - X2)(l - X3)].
Now suppose instead that arms 1 and 3 were used in the first period. The the four possible second period states are (1,X2,0), (0,X2,1), (0,X2,0), and (1,X2,1), which occur with the respective probabilities x i ( l — X3),(l — Xi)x3, (1 — xi)(l — X3), and X1X3. The optimal continuation in three of these cases is obvious: if (1,X2,0) occurs, the optimal continuation is to play arms 1 and 2 forever; if (0, X2,1) occurs, the optimal action is to play 2 and 3 forever; while
Generalized Bandit Problems
151
if (0,^2,0) occurs, any continuation is optimal that involves playing arm 2 forever. The case where the state at the beginning of the second period is (1, X2,1) is a little more complicated. There are two possibilities here. (There are more, but these are the only two relevant possibilities.) One option is to play arms 1 and 3 at this state; if this is an optimal choice, it must remain optimal forever, since all the states remain frozen, and the continuation reward from employing this choice forever is 4/(1 — 5). A second option is to play arms 2 and 3, and to switch to arms 1 and 3 forever if, and only if, the state on arm 2 moves to 0; the continuation reward from this option is {3 + 2x2 + 5(1 — X2))/{I — S), It is easily seen that the first option is strictly preferable as long as X2 < (1 — 6)/{2 — 5). Assuming this to be the case (we will ensure it later), the four continuation values that result are F(0,X2,1) = (3 + 2 X 2 ) / ( 1 - J ) ] / ( l , X 2 , 0 ) - ( l + 2x2)/(l-(5) V(0,X2,0) = 2 x 2 / ( 1 - ( 5 ) y(l,X2,l) = 4 / ( 1 - 5 ) . So, the value of beginning with arms 1 and 3 is: ^(l,3)V'(xi,X2,X3)
= 3x3 + xi + S{1 - (5)"^[4xiX3 + X3(l - xi)(3 + 2x2) + X i ( l - X3)(l + 2x2) + 2X2(1 - Xi)(l - X3)].
Subtracting L(i^3)V from 1/(2,3) "i^ ^^^ cancelling common terms results in the following: ^(2,3)V'(xi,X2,X3) - L ( i , 3 ) V ( x i , X 2 , X 3 ) = 2 X 2 " ^^1 + ^ 3 7 ^ 2 ^ 3 .
(13)
This difference is negative provided:
xi > X2 h+Y^^y
'
^^^^
Thus, as long as (14) holds (and X2 < (1 — S)/{2 — 5)), it is the case that beginning with arms 2 and 3 is dominated by the situation where the decisionmaker begins with arms 1 and 3. Now consider the following parametrization. Let 5 = 1/2, xi = 7/12, X2 = 1/4, and X3 = 1/6. At these values, using the expressions (10)-(12) for the indices on the three arms, it is readily calculated that pi = 28/19, fl2 = 8/5, and fts — 12/7, so we have fti < /22 < ^3- In particular, the two arms with the highest Git tins indices are arms 2 and 3.
152
Rangarajan K. Sundaram
Note that at the given values of the parameters, we have {1 — 5)/{2 — 5) = 1/3 > 1/4 = X2. Thus, (13) applies and we have _ _ _ 1 7 1 1 ^(2,3)V(xi,X2,X3)-L(i,3)V(xi,a:2,X3) "" 3 ~ 12 "^ 24 " ~ 2 4
^^
so that beginning with arms 1 and 3 and continuing optimally is strictly preferable to beginning with arms 2 and 3 and continuing optimally.
6 Switching Costs in t h e Bandit Framework Another variant of the basic bandit framework that is of considerable interest concerns the introduction of costs for switching between arms. Indeed, it is difficult to imagine a relevant economic decision problem in which the decisionmaker may costlessly move between alternatives. Unfortunately, Banks and Sundaram [5] show that it is not possible, in the presence of switching costs, to define an index on the arms such that the resulting strategy is invariably optimal. We provide in this section an alternative proof of their non-existence result. Recall that in our current framework, the tuple Ci = {Xi, Vi.qi) completely describes arm i. We introduce switching costs by including in this tuple two real numbers Q and di with the interpretation that Ci is the cost of switching to arm i (from any other arm), and di is the cost of switching away from arm i (to any other arm). Thus, the characteristics of arm i are given by the quintuple {Xi^ri^qi^Ci^di), To distinguish this from the case of no switching costs, we will denote this quintuple by C*. Note that in this scenario, the total cost of a switch from arm i to arm j is given by di -\- Cj.^ When switching costs are allowed to be non-zero, the attractiveness of an arm evidently changes depending on whether or not that arm is the one "currently in use" (i.e., whether or not it is the arm that was used in the previous period by the decision-maker). For it is obvious that in comparing two otherwise identical arms, of which one was used in the previous period, the one that was in use must be more attractive than the one that was idle. This motivates the following modification in our definition of an "index" on the arms: Definition An index in the presence of switching costs is any function A which specifies for a generic arm i a value X{xi^ C*, Si), where C* denotes the ^ It is apparent that the framework described here is the most general framework of switching costs in which the existence of an optimal index may reasonably be expected. Under more inclusive situations (such as, say, where the cost of switching from arm i to arm j is given by c^j), it is clearly not possible to have an optimal index strategy if the index on arm i is to depend solely on the characteristics of arm i.
Generalized Bandit Problems
153
characteristics of arm i, xi is the current state of arm i, and si G {0,1} is a variable that denotes whether (si = 1) or not (si = 0) i is the arm currently in use. An index A will be said to be optimal in the presence of switching costs if the strategy it induces is optimal in every bandit problem {/, (C*)ie/,(5}, in which switching costs are possibly non-zero. 6.1 Markovian Optimal Strategies Let a bandit problem {/, (C*)^^/, J} be given. As in the previous sections, the existence of Markovian strategies that are optimal in the bandit problem can be ascertained using standard arguments from the theory of stationary dynamic programming. Indeed, the only modification in the arguments that is required from those used in Section 4 lies in the definition of the state space to be used. When switching costs are allowed to be non-zero, the state of the bandit problem cannot be adequately described by just the vector of current states (xi)i^i of the individual arms (i.e., optimal continuations cannot be calculated with just this knowledge); rather it is also important to know the arm that was in use in the previous period. Thus, in the representation of the bandit problem as a dynamic programming exercise, the state space is given by Z\ = X x 7, where as always X = Xi^jXi. Routine arguments now yield the following result. Theorem 6. The value function V: A ^^ R of the problem {/, (C*)^^/} satisfies the Bellman optimality equation at all (x, j) G A: V{XJ)^\JL,V{XJ),
(15)
where, for i y^ j , we have LiV{xJ)
=^ ri{xi) - Ci - dj -{- 6
V{{x-i,Xi),i)dqi{xi\xi),
(16)
while LjV{xJ)
= rj{xj) -{- 5 I V{x-j,Xj)dqj{xj\xj).
(17)
The Markovian strategy defined through any measurable selection from the correspondence of maximizers of (15) is an optimal strategy. 6.2 Optimal Index Strategies The following result, the main result of this section, shows that optimal index strategies no longer exist when switching costs are allowed to be non-zero:
154
Rangarajan K. Sundaram
T h e o r e m 7. There is no index A such that the strategy induced by A is optimal in every bandit problem {/, (C*)^^/}The proof of this theorem occupies the rest of this section, and comes in two stages. In the first stage a simple argument is used to show that any bandit problem in which there are both costs of switching "to" and costs of switching "from" is equivalent to another bandit problem in which there are only costs of switching "to." In the second stage, we consider bandit problems with only costs of switching "to," and use a reductio ad absurdum method to show that an optimal index strategy cannot exist on this class of bandit problems. Let a bandit B = {/, (C*)^^/} be given. Define the bandit B = {/, ((7*)ie/} from B as follows. For each i e I, let Xi = X^; ri{xi) = ri{xi) + (1 - S)di for all Xi G Xi\ Qi = qi; Ci = Ci -\- di\ and, finally, di =^ 0. Note that B only has costs of switching "to." The only difference between the bandits B and B is that in the bandit 5 , a cost of di is paid every time a switch away from arm i occurs, while in bandit B, di is paid "in advance" when the switch to i occurs, but an additional reward oi {1 — S)di is received in each period that arm i is in use. It follows that if arm i is used for t continguous periods, the present value of the total switching cost paid in the bandit B is Ci-\- 5^di, In bandit B, this total cost is (cj -}- di); however, an additional reward of {1 — S)di is received in each of these t periods, so that the net cost paid under B is Ci -\- di{l — X^^ZQ S^{^ — ^)) = Ci -\- 5^di^ which is exactly the same cost as in the bandit B. The equivalence of B and B is immediate now, completing the first stage of the proof. We proceed to the second stage of our proof. Suppose that an optimal index did exist on the class of bandit problems with only costs of switching "to." Denote this index by A. We wih show the presence of a contradiction. Some new notation will help simplify exposition since all the arms we consider here involve similar characteristics. For a, 6 G M, x € [0,1], and c > 0, let {xxia) + (1 — x)x{b), c} denote the arm with state space [0,1] and initial state x; reward function r{x) — xa + {1 — x)b; transition probabilities ^({1}|2:) = 1 — ^({O}!^;) = z for all z G [0,1]; and switching cost c.^ As a further simplification, let {x(<^),c} denote an arm with switching cost c that pays a reward of a in each period with certainty. In this notation, X{{xx{ci) + (1 — x)x{b), c}; s) will denote the value of the hypothetical optimal index on the arm {xx(a) -f- (1 — x)x{b),c} at the state s G {0,1}. (Recall that s — 1 signifies that the arm is currently in use.) We proceed in a series of claims that establish properties that any optimal index A must satisfy. Then we show that these properties are not mututally That is, (xxia) + (1 — x)x{b),c) is an arm which pays (xa -f (1 — x)b) in the first period of its use, and thereafter either pays a forever (which happens with probability x) of b forever (which happens with probability (1 — x)).
Generalized Bandit Problems
155
consistent. The intuition underlying this construction is quite straightforward: if there is any possibility of switching back to the arm currently in use after a switch away from it (say, the switch back depends on the realizations from the other arms), then the index on the incumbent arm must be increasing in the cost of switching to it, since a higher cost of switching back ought to make the decision-maker more reluctant to leave the arm. On the other hand, if switching back is a zero-probabiUty event (say, because the worst observations on the other arms would still dominate the one currently in use), the arm's index must be independent of the cost of switching back to it. Thus, the index must depend in a non-trivial way on the characteristics of the other arms, and this is not possible. Property 1. For any a G M, we have A({x(a),0}; 1) = A({x(a),0};0). Proof. Consider a two-armed bandit in which both arms are specified by {x(o^),0}. Then, regardless of the arm currently in use, either arm is an optimal continuation, so the claim follows by the presumed optimality of A. Property 2. For any a G R and e > 0, we have A({x(tt + e),0};0) >
A({x(a),0};0). Proof, Pick any b < a, and consider a three-armed bandit in which arm 1 is {x(^)^0}5 arm 2 is {x(<^).0}) and arm 3 is {x(a + e),0}. Suppose arm 1 is the one currently in use. It is immediate that selecting arm 3 is the uniquely optimal continuation, from which the claim follows. Property 3. For any c > 0, e > 0, and a G R, A({x(a + e),0};0) > A({x(a),c};l). Proof Consider a two-armed bandit in which arm 1 is given by {x(ci),c} and arm 2 is given by {x(<^ + e), 0}. Suppose arm 1 is the one currently in use. The uniquely optimal continuation strategy is obviously to play arm 2 forever, and the claim obtains. Property 4- For any c > 0, it is the case that for almost every a G R, we have A({x(a),0};l)>A({x(a),c};l). Proof Pick any c > 0. By Property 2, A({x(a),0}; 0) is a strictly increasing function of a, and so is continuous almost everywhere. Pick any continuity point d. From Property 3, we have A({x(d+e), 0}; 0) > A({x(d), c}; 1) for every e > 0, so taking limits as e ^ 0, we obtain A({x(d),0}; 0) > A({x(d),c}; 1). By Property 1, A({x(d),0}; 0) =: A({x(d),0}; 1), and the result follows. The remarks leading up to Property 4 relied on a series of comparisons in which coming back to the incumbent arm (if a switch away from it occured)
156
Rangarajan K. Sundaram
was a zero-probability event. The next two claims now use another series of comparisons, in which there is a non-zero chance of coming back to the incumbent arm (depending on the observations from the other arms). These will lead to a contradiction to Property 4 by showing that A({x(<^), c); 1} must be strictly increasing in c for c > 0. Property 5. Suppose a > b. Then, creasing in X.
X{{XX{CL)
+ (1 — x)x(&),c};0) must be in-
Proof. Let Xi, X2 G [0,1] with xi > X2^ Pick a € M to satisfy [axi + 6(1 - xi) - (1 - 5)c] > [ax2 + 6(1 - X2) - (1 - 5)c]. For c* G M, consider a two-armed bandit in which arm 1 is given by {X(Q^), c*} and the second arm is given by {xixici) + (1 — Xi)x{b)^(^}' Suppose also that arm 1 is the one currently in use. When c* is sufficiently large, it is uniquely optimal to switch to the second arm and stay there forever if i = 1; and to stay with arm 1 forever if i = 2. Since A is optimal by hypothesis, we must have: A({xix(a) + ( l - x i ) x ( 6 ) , c } ; 0 ) >A({x(a),c*};l) > A({x2x(a) + (1 - X2)x(^),c}; 0), so we indeed have A({xix(a) + (1 - xi)x(6), c}; 0) > X{{x2x{a) + (1 - X2)x{b). c}; 0), as required. Property 6. For any a G M, A({x(<^),c}; 1) must be increasing in c. Proof. Let ci > C2. Pick a, ^ G M to satisfy a> a (3<{a-{l-5)ci). Since ci > C2, we also have /? < (a — (1 — 5)ci) < (a — (1 — 6)02)- Define Xi G [0,1] by: {l-6){a-p-^5ci) a^l-5)p-5a + 5{l-5)ci Note that 1 > xi > ^2 > 0. Consider the two-armed bandit in which arm 1 is given by {x{o),Ci} and arm 2 by {xixic^) -f (1 — Xi)x(/3),0}. Suppose also that arm 1 is the one currently in use. A simple calculation shows that the choice of Xi implies either arm is an optimal continuation in this problem. It follows that we must have A({x(a),c,};l) - A({xa(^) + (1 - ^i)x(/^), 0}; 0). This establishes the result by Property 5, since Xi > X2'
Generalized Bandit Problems
157
Properties 4 and 6 are in evident contradiction, completing the proof of the theorem.
7 Appendix The proof of Theorem 3 takes the form of several lemmata leading to the key Lemma 4 which asserts that if arm i attains the supremum of the indices at the initial state and j does not, then beginning with i and continuing with the index strategy is strictly superior to beginning with j and continuing with the index strategy. We employ the notation of Theorem 4 throughout. Fix an initial state x at which the conditions of the Theorem are satisfied, and denote by fti the initial values of the Gittins indices iJ,i{xi). Recall the definition of the strategy a^ [m] in the stopping problem defining the Gittins index on arm i, and of the expressions Hi{t,m), pi{t,m), and Ri{t^ m). Let Hi{oo^ m) be the set of histories under which <Ji[m] never accepts the terminal reward (i.e., on which the index stays above m forever), and let Pj(oo,m) be the probability of Hi{oo^m). Finally, let Ri{oo,m) be the total expected discounted reward from arm i conditional on Hi{oo,m). Lemma 1. For any i e I and m eR,
it is the case that
OD
/
^^Pi{r,m)Ri{r,m)
OO
\
+p^(oo,m)i^^(oo,m) > m f 1 - y^(5^p^(r,m) I
r=l
\
if m < fti; that equality holds in (4-^ if'^ inequality holds ifm>j2i.
r=l
(18)
/
— Pi,' ci'^d that the reverse strict
Proof The total discounted reward associated with the strategy crj[m], denoted ^ 4 ^ 1 ' is evidently given by the following: OO
Ai[m] = ^'^Pi{T,m)
{Ri{r,m) + 6'^m) + Pi{oo,m)Ri{oo,m).
(19)
r=l
Since it is uniquely optimal in the stopping problem to select arm i initially when m < fli, we have Ai[m] > m when m < jli] rearranging this inequality yields precisely the first inequality in the statement of the lemma. The other two are obtained similarly: since either arm is an optimal initial choice when m = p,i, we have Ai[m] = m in this case; and finally, since accepting the terminal reward right away is the unique optimal action when m > fti^ we have Ai[m] < m when m > pi. This completes the proof of the lemma. Some further notation is unfortunately required for the next two lemmata. For any j G / , denote by cr{j) the strategy in the bandit problem that begins
158
Rangarajan K. Sundaram
by selecting arm j initially, and after the first period proceeds by choosing the arm with the highest Gittins Index in each period. Additionally, let (j{ij) be the strategy that begins with arm i, stays with arm i until the first point where the index on arm i drops below its original value of /2j, then switches to arm j and stays with arm j until the index on arm j falls below the original index pi of arm z, and then proceeds by choosing the arm with the highest Gittins index in each period. Recall that W{a){x) denotes the worth of the strategy from the initial state x. Since x is fixed, we suppress dependence on it. L e m m a 2. W{a{ij)) = W{a{ji))
if Jli = jlj.
Proof. A little algebra shows that W{a{ij)) (X)
= Pi{oo, ili)Ri{oo, pi) + ^Pi{t, 0
0
/
t=l 00
fli)Ri{t, pi)
\
00
\
T=l
J
00
+ X^ X]Pi(^. f^i)Pj{r^ J^i)Elj{t,r, pi) t=l r = l 00
= MH
00
00
t=l
T=\
+ ^(^V2(^,Mi)^j[Ai] + t=l
^^Vi{'t,Pi)Vj{r,pi)Elj{t,T,pi)
where E*j{t,r,pi) is the continuation value of the strategy cr(ij) conditional on the arm i having fallen below pi for the first time in t periods, and arm j having fallen below pi for the first time in r periods. Similarly, we also have Wiaiji)) OC
= Ajlpi] + Y^5''pj{T,p^)Ai[pi] r=:l
00
CX)
+
J2Y^pj{r,pi)pi{t,pi)E^i{r,t,pi) T= l
t=l
where Ej^ is defined in the same way as E^j with the obvious changes. It is easy to see that, by the independence of the arms, and the assumption that pi = pj^ we must have E*^ = Eji. In turn, this gives us the following: W{a{ij)) - W{<j{ji)) = Ailfii] i 1 - ^ 5 > , ( r , r t ) j - AjlPi] i 1 -J2S*Pi{t,Pi)
J•
Substituting for Ai[m\ and j4j[rn] and appealing to Lemma 1 completes the proof.
Generalized Bandit Problems
159
Lemma 2 establishes in particular that if two arms have the same index, the order in which they are used does not matter in evaluating the index strategy. We will now show that if arm i attains the supremum of the indices while arm j does not, the strategy (j{ij) is strictly preferable to the strategy
Lemma 3. W{a{ij))
> W{a{j))
whenever pi = supf.^j flk > fij-
Proof. Under the stated hypothesis that pi = supj^^j pk, W{a{j)) can be thought of as the strategy that begins with arm j , switches to i at the first time t = 1,2,..., at which the index on j drops below /i^, then stays with i until the index on i also drops below /i^, and proceeds by picking at each subsequent stage the arm with the highest Git tins index at that point. Thus, after some algebra, we obtain, oo
W{aij)) = Ajifci] + J2 S^Pjir,fiMAfli]+ E;^ where E*^ is defined exactly as previously. So,
W{aiij)) - W{a{j))
= Ailfli] I 1 -J2S^Pj{T,/li) j - Ajlfli] j 1 - ^ JV^(^,/i^) J By Lemma 1, Ai[fii] = jli{l - J2t^i ^^Pii^^ fii))^ while since j2i > ftj by hypothesis, another appeal to Lemma 1 gives Aj[fii] < jli{l — Y^^=i S'^PJ{T, fti)). Substituting these in the expression above, we obtain W{a{ij)) — W{a{j)) > 0, as required. In words. Lemma 3 establishes that if arm i is the maximum of the indices at the initial state while arm j is not, then there is a strategy beginning with i that does strictly better than the strategy that begins with j and proceeds by picking in each subsequent period, one of the arms with the highest Gittins index at that point. It is now relatively easy to show that: Lemma 4. If fti — supj^^j p,k, and Jxi > ftj, then W{a{i)) >
W{a{j)).
Proof Under the stated hypotheses, the previous lemma shows that cr(zj) strictly improves on o-{j). Of course, cr{ij) initially selects arm i, which has the highest of the Gittins indices at the initial state. We first claim that if, at any point, (j{ij) recommends continuing with an arm that does not have the highest index at that point, then there is an alternative continuation strategy which recommends continuing with one of the arms with the highest index at that point, and which does strictly better in the continuation than (j{ij).
160
Rangarajan K. Sundaram
To see this, note that under a{ij) the decision-maker always picks the arm with the largest Gittins index at each point in time, at least until the first time a state is reached where the strategy recommends that arm j be selected. For notational ease, call this new state x = {xi)i^i. (Of course, x differs from the initial state x only in the value of the i-th coordinate Xi.) At this point carrying on with a{ij) amounts to using a{j) from the new initial state barx. If, however, arm j is not the arm with the maximal index at x (say there is fc G / such that /j,k = supi^j fii > fij^ where all the indices are evaluated at x), then Lemma 3 shows that it is strictly better to use the strategy a{kj) in the continuation, where, of course, cr{kj) is defined with respect to the new state X. This proves the claim. Iterating on the claim shows that the Gittins index strategy of picking the arm with the highest Gittins index at each point is strictly superior to picking j initially and continuing with the Gittins index strategy, if j is not the arm with the highest Gittins index. This completes the proof of the lemma. To complete the proof of Theorem 3, let a be any strategy in the bandit problem. Consider the strategy a^ which imitates a up to period T, and then proceeds with the Gittins index strategy. It is easy to see that a^ is well defined: after any T-history, at most T of the arms have had their initial states altered. Since the Gittins index strategy was well defined at the initial state Xy it continues to be well defined at this resulting new state which differs from X in at most T coordinates. And since this last statement is true for any T-history, a'^ itself is well defined. For any e > 0, the total discounted expected rewards under a and a^ can be made to differ by less than e by choosing T sufficiently large. (This follows since rewards are uniformly bounded.) Now consider any (T — 1)-history in which in period (T — 1), a (and, therefore a^) picks an arm j that does not have the highest index at that point. Since a^ continues from period T on with the Gittins index strategy. Lemma 2 shows that the total reward conditional on this history can be strictly improved over o*"^, by instead picking an arm with the highest index at this point, i.e., by continuing from period (T — 1) with the Gittins index strategy. It easily follows that cr^^~^^ strictly improves over cr-^, unless a^ almost surely picks only arms with the highest index at all points from period (T — 1) onwards, i.e., unless effectively a^ and cr^-^~^^ are the same. In turn, this implies W{a) — W(cr^-^~^^) < e, also. Iterating back to zero, we obtain W{a) — W{<7^) < e, where, of course, a^ is the Gittins index strategy. Since e is arbitrary, we finally see that the Gittins index strategy does no worse than the strategy a. Finally, since a is also arbitrary, this shows that the Gittins index strategy is, indeed, an optimal strategy. To see the other part of Theorem 3 — that every optimal strategy is also a Gittins index strategy — consider any optimal strategy a, and suppose it is not a Gittins index strategy. Let t G {0,1,2,...} denote any date such that
Generalized Bandit Problems
161
with positive probability a fails to use a Gittins index maximal arm for the first time at time t. (Since a is not a Gittins index strategy, such a date t must exist.) Consider the strategy a that imitates a up to, and including, period t, and then proceeds according to the Gitttins index strategy. Since the Gittins index strategy is also an optimal strategy, and a is optimal by hypothesis, it follows that a must also be an optimal strategy. But by Lemma 4, the Gittins index strategy stictly improves upon a in the continuation reward from period t onwards, since a uses arms that are not Gittins index maximal with positive probability in period t. This implies a, and therefore cr, cannot be optimal strategies, a contradiction, completing the proof.
References [1] Agrawal, R., M. V. Hegde, and D. Teneketzis (1988) Asymptotically efficient adaptive allocation rules for the multiarmed bandit problem with switching costs. IEEE Transactions on Optimal Control, 33(10): 899-906. [2] Berry, D. and B. Pristedt (1985) Bandit Problems: Sequential Allocation of Experiments London: Chapman and Hall. [3] Banks, J. S. and R. K. Sundaram (1992a) Denumerable-armed bandits. Econometrica, 60(5): 1071-1096. [4] Banks, J. S. and R. K. Sundaram (1992b) A class of bandit problems yielding myopic optimal strategies. Journal of Applied Probability, 625632. [5] Banks, J. S. and R. K. Sundaram (1994) Switching costs and the Gittins index. Econometrica, 62(3): 687-694. [6] Basu, A, A. Bose, and J. K. Ghosh (1990) An expository review of sequential design and allocation rules. Mimeo. Purdue University. [7] Blackwell, D. (1965) Discounted dynamic programming. Annals of Mathematical Statistics, 36: 225-235. [8] Feldman, D. (1962) Contributions to the "two-armed bandit" problem. Annals of Mathematical Statistics, 33: 847-856. [9] Feldman, M. and M. Spagat (1993), Optimal learning with costly adjustment. Mimeo. Brown University. [10] Gittins, J. (1979) Bandit processes and dynamic allocation indices. Journal of the Royal Statistical Society, Series B 41: 148-164. [11] Gittins, J. (1989) Allocation Indices for Multi-Armed Bandits London: Wiley. [12] Gittins, J. and D. Jones (1974) A dynamic allocation index for the sequential allocation of experiments. In J. Gani et al. (eds) Progress in Statistics Amsterdam: North Holland. [13] Kolonko, M. and H. Benzing (1983) The sequential design of Bernoulh experiments including switching costs. Mimeo.
162
Rangarajan K. Sundaram
[14 Lai, T. L. and H. Robbins (1985) Asymptotically efficient adaptive allocation rules. Advances in Applied Mathematics, 6: 4-22. [15 Mortensen, D. (1985) Job search and labor market analysis. In O. Ashenfelter and J. Layard (eds) Handbook of Labor Economics Vol, II New York: North Holland. [16: Pressman, E. L. and I. M. Sonin (1990) Sequential Control with Partial Information New York: Academic Press. Rieder, U. (1975) Bayesian dynamic programming. Advances in Applied [ir Probability, 7: 330-348. [is: Rothschild, M. (1974) A two-armed bandit theory of market pricing. Journal of Economic Theory, 9: 185-202. [19 Schal, M. (1979) Dynamic programming and statistical decision theory. Annals of Statistics, 7(2): 432-445. [2o: Vicusi, W. (1979) Job hazards and worker quit rates: an analysis of adaptive worker behavior. International Economic Review, 20: 29-58. [21 Weizman, M. L. (1979) Optimal search for the best alternative. Econometrica, 47: 641-654. [22: Whittle, P. (1981) Arm-acquiring bandits. Annals of Probability, 9(2): 284-292. [23; Whittle, P. (1982) Optimization Over Time: Dynamic Programming and Stochastic Control Vol. I New York: Wiley.
The Banks Set and the Uncovered Set in Budget Allocation Problems'*' Bhaskar Dutta^, M a t t h e w O. Jackson^, and Michel Le Breton^ ^ University of Warwick B.DuttaQwarwick.ac.uk ^ California Institute of Technology jacksonmOhss. c a l t e c h . edu ^ University of Toulouse 1 lebretonOcict .fr S u m m a r y . We examine how a society chooses to divide a given budget among various regions, projects or individuals. In particular, we characterize the Banks set and the Uncovered Set in such problems. We show that the two sets can be proper subsets of the set of all alternatives, and at times are very pointed in their predictions. This contrasts with well-known "chaos theorems," which suggest that majority voting does not lead to any meaningful predictions when the policy space is multidimensional.
1 Introduction McKelvey's [25, 26] celebrated theorems on the genericity of global cycles in majority voting are fundamental to our understanding of the potential outcomes of a social decision process. These results assert t h a t if the set of social alternatives is a multidimensional Euclidean space, then under mild conditions on the profile of voters' preferences, there is a finite chain of alternatives starting at any given x and ending in any other y such t h a t each alternative in the chain is preferred by a majority of voters to its predecessor. These results are often interpreted to show t h a t in multidimensional policy spaces majority voting is chaotic or unstable since no alternative appears to dominate the others. T h e general conclusions drawn from this interpretation of McKelvey's Chaos Theorems have led many people to believe t h a t it is impossible to predict the nature of social decision-making without a detailed specification * This paper was written in fond memory of our dear friend and colleague Jeffrey Scot Banks. Financial support under NSF grant SES-0316493 is gratefully acknowledged. We thank Salvador Barbera for helpful conversations that helped spark some of the ideas behind the model we develop here, and David AustenSmith for detailed comments on an earlier draft.
164
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
of social institutions and rules.^ For instance, Stiglitz [39] writes, "If, however, there are a variety of dimensions-some individuals are liberal on some issues and conservative on others-then the median voter is not well defined, and there may be no equilibrium to the political process." According to Perrson and Tabellini [35], "By the mid-to late 1970s, theorists had clearly demonstrated t h a t searching for a universally applicable theory of political equilibrium is a futile exercise. Further, majority voting would generically lead to cycles, unless the voting agenda was restricted T h e outlook of many researchers at the time was thus quite pessimistic: any positive theory of political choice — whether it was based on majority voting or not — seemingly had to rely on unattractive or arbitrary assumptions." These beliefs have resulted in a growing body of literature t h a t derives social equilibria conditional on explicit specifications of the institutional structure. W i t h o u t denying the importance of t h a t direction of investigation, one is left with t h e impression t h a t except in exceptional cases, very little can be said about social choice outcomes t h a t is institution-free. This is problematic if the equilibrium outcomes are highly sensitive to the fine details of the institutional process. For instance, if the "institution" resembles a bargaining game, then the equilibrium outcome will typically depend on the specific bargaining protocol. Therefore, it appears as if the unpredictability associated with the chaos theorems has been replaced by a predictability t h a t may suffer from a lack of robustness. T h e approach adopted in this paper, following another of McKelvey's [27] influential papers, is to explore whether there are some predictions about majoritarian social choice outcomes t h a t are "relatively" institution-free. More precisely, is it possible to bound the set of "stable social outcomes" in the sense t h a t all social equilibria must lie within these bounds under a wide variety of different institutional arrangements? In this paper, we focus on two such sets: the Uncovered Set of Miller [29] and the Banks Set of Banks [2]. T h e two sets are arguably the most appropriate ones to bound the possible social outcomes. McKelvey [27] demonstrated t h a t in any multidimensional setting where voters have quasi-concave preferences (the so-called spatial setting) ^^ the Uncovered Set contains the outcomes t h a t would arise from equilibrium behavior under three different institutional frameworks: a two-candidate competition in a large electorate,^ cooperative behavior in small committees, and sophisticated behavior in a legislative environment. Hence, in a strong sense the Uncovered Set is a useful generalization of the notion of a ConA version of the Chaos Theorem for the finite unstructured setting has been proved by Bell [6]. We refer the reader to chapter 6 in Austen-Smith and Banks [1] for an illuminating presentation of chaos results in the spatial model. Miller [29] had already shown that under a variety of institutional settings, game theoretic behavior by participants leads to outcomes in the Uncovered Set when the set of alternatives is finite. On this aspect, see also Banks at el. [4].
The Banks Set
165
dorcet winner. The Banks Set is less general than the Uncovered Set in that it only applies to a specific institution. Nevertheless, that institution, voting by amendment agendas (also known as voting by successive elimination), is a very important one which is paradigmatic for most committee voting rules and is the procedure central to Roberts Parliamentary Rules of Order. It is often asserted that a chairman (or sub-committee) can manipulate the agenda so as to ensure the choice of an alternative which is in his (or its) interest. However, Miller and McKelvey showed that the set of sophisticated equilibrium outcomes corresponding to voting by successive elimination must lie in the Uncovered Set; and Banks [2] provided a full characterization of this set of outcomes, which is the Banks Set. Thus, the Banks Set puts some bounds on the monopoly power of an agenda setter in the context of amendment agendas.^ The Uncovered and Banks Sets have been investigated extensively in the case where the set of alternatives is finite and voters have no a priori structure to their preferences, and so some properties of these sets are now well-known.^ However, the explicit computation of these sets is not easy, particularly when the feasible set of alternatives is some subset of multi-dimensional Euclidean space and one might expect some natural structure to voters' preferences. The only real analysis that has made any progress on that issue is in the case of purely distributive politics, where Epstein [14] and Laslier and Picard [22] have shown that the Uncovered Set is the entire set of alternatives;^ and recent papers by Penn [33] and Dutta et al. [12] who show that specific equilibrium notions of agenda formation can lead to some pointed predictions in some circumstances. This paper furthers our understanding of the structure of the behavior of majority rule in spatial settings in two ways. The first is that we identify a broad class of social choice problems that includes many situations that have been analyzed separately in the literature. The common feature of these problems is that an alternative is a feasible allocation of a given budget across finitely many uses or projects. These projects could correspond to individuals if we model private transfers in a distributive politics environment, or to districts or regions or diff'erent types of public expenditures in the context of pork barrel politics and financing of local pubhc goods (Ferejohn et al. [17], Lockwood [24]). Other budget allocation problems covered by our framework include the mixed setting where some private projects compete with a global public project as in Lizzeri and Persico [23]. Our key assumption is that voters' preferences are linear; that is, indiff'erence contour sets are parallel hyperplanes. While a special case of the spatial model, this is a rich setting ^ Miller et al. [30, 31] argue that the interest for studying the Banks Set goes beyond this. ^ See Laslier [21] for a detailed description of this area of research. ^ See also McKelvey [27], Banks et al. [5], Cox [9], De Bonder [10], Feld et al. [16], Fey [18], Hartley and Kilgour [19], Koehler [20].
166
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
that to the best of our knowledge has never been investigated in any generality before. Our second contribution is to provide some (partial) characterizations of the Uncovered, Banks, and Top Cycle sets in this setting, in the context of some important special cases. In particular, we examine in some detail the case where each voter views each project as either being good or had. In the next section, we introduce and illustrate the linear setting and the main concepts which are used in this paper. In Section 3, we introduce the main majoritarian sets which are examined in this paper. Then, in Sections 4-7, we focus on the special case where there are three projects and three voters. In Section 4, we present the simple geometry of the majority relation in the linear setting. We show that the necessary and sufficient condition for the existence of a Condorcet winner in this setting is less stringent than the well-known condition of Plott [36]. We also offer a direct simple proof of McKelvey's chaos theorem on the Top Cycle. Then, we calculate explicitly the Uncovered Set for many important linear settings. We also calculate or describe with some accuracy the Banks Set to show how much it differs from the Uncovered Set. We conclude in Section 8 with a discussion of the general case.
2 A Model of Budget Allocation W h e r e Voters Have Linear Preferences Alternatives The set of alternatives X consists of the set of feasible allocations of a given budget, denoted by M, among K distinct possible uses. The uses may be thought of, depending upon the context, as being districts, regions, individuals, public projects or other criteria. Generic elements of X are denoted X, y, and z, and are X-dimensional vectors. If money is assumed to be perfectly divisible, then the set X is infinite. In this case, X is the simplex XGRH-^ :Y^X^
=
M\
When M = 1, the set of social alternatives could be alternatively interpreted as the set of lotteries over an unstructured finite set of choices. If, instead, money can only be divided into discrete units, with say 1 being the smallest unit of money, then the set X is finite. It is then defined as:-^^ ^° We presume that the entire budget is allocated. This is in line with voters viewing at least one of the projects as not being objectionable.
The Banks Set
167
|xG{0,l,...,M}^:^a;^ = M L We shall alternate between the use of these two settings. While the infinitely divisible setting provides some technical advantages, we stick with the finite world in situations where we analyze the Banks Set, as an uncontroversial definition for the Banks Set has not been given for the case where X is infinite. We shall discuss the limit as the units become small (M becomes large), and this provides some predictions for the infinite case. This also gives us some feel for the importance of divisibilities, as we shall see that at least in some cases the Banks Set changes as the units become relatively small, and the limit may have difi'erent features from situations with substantial indivisibilities. We assume that M > 4 in the indivisible setting, as the case where M < 3 is an easily analyzed special case where the geometry of the problem degenerates. Voters and Preferences The committee or society of voters is described by the finite set N = { 1 , . . . ,N], Voter i has preferences over the set of alternatives represented as follows. There exists a vector Ui G R^ such that the utility to i of an alternative x is simply u^-x. Thus, i prefers an alternative x to an alternative y if and only if Ui' x>
Ui-y.
Thus, u^ denotes i's marginal valuation for project k. So, preferences are R^^. completely described by the matrix U = {U\,..,.,UN)^ The linearity of indifi'erence is obviously special; but, as we now illustrate, it is general enough to cover a large family of interesting problems. Example 1. Private Projects: Divide the Dollar This corresponds to the case where K — N and the matrix u is equivalent to 1 0 •• 0 01 u= '' 0
1
The conventional interpretation of this problem is that an alternative is a division of the amount M among the A^ voters, who are assumed to derive utility exclusively from the amount they receive, the larger the better. A second interpretation views the K dimensions as K public projects in competition
168
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
and assuming that each voter cares exclusively about the amount allocated to a specific project, justifying the terminology "private projects" even if the projects have the features of public projects. Example 2. Goods and Bads Consider a world where each dimension is viewed by a voter as either a "good" or a "bad" project. Goods are equivalent in the voter's view, as are bads. To normalize things, goods have a marginal value of 1 and bads have a marginal value of 0. So each Ui is a vector of O's and I's. A special case of this is the divide the dollar setting described in Example 1, where each player has a different dimension that is a good, and only one dimension, and where the number of dimensions is the number of voters K = N. Another special case is the private and public good example (Example 3, below), in the case where b = 1. More generally, the goods and bads model is one where K might differ from A^, several voters might view any particular dimension as good, and players might consider several dimensions to be "goods." Example 3. Private Projects Versus a Public Project This example, inspired by Lizzeri and Persico [23], is a setting mixing the divide the dollar setting (Example 1) with an extra public project that is a pure public good. Here voters are bargaining between allocating resources to a common public good, and payments directly to the voters themselves. In particular, K = N -\-1 and 10 01 u—''
Ob 06
00 • • 1 6 where 6 is a positive parameter describing the common willingness to pay of each voter for the public project. Example 4- Choice Between Public Projects Consider a society allocating resources to any of a list of projects, which may have private and/or public components. In this case, there are no specific restrictions on the matrix u. The K dimensions are interpreted as K different potentially projects that are in competition for funding. The allocation x^ defines the scale of operation of project k (variations in costs can be incorporated into the marginal utilities). Certainly, voter i would like to see all the budget allocated to his or her "favorite projects" (projects k such that
The Banks Set
169
k \ n.k u > u^ for all k'). However, unlike the goods and the bads model, an agent is not systematically indifferent between projects that are identical in their allocation to the agent's most preferred projects. When K = N, and the K dimensions are interpreted as districts or states in a federation or regions in a country, this model describes pork barrel politics with some form of externalities across projects. Suppose that region i derives a benefit equal to Xi from project i operated at the scale Xj, but also derives some benefits from projects implemented in other regions. The benefits resulting from these other projects are less important, the more "distant" is the region (where distance might or might not be a physical measure). Precisely, u^ = 1 — adik' Knowledge of the intensity a of the externality and the pattern describing the geographical network, is essential for understanding the voting behavior.-^-^ I
—
Example 5. Criteria When bargaining over the split of a budget, it is often the case that the discussion takes place on various criteria that might be used to allocate the budget instead of directly in terms of the allocation itself.^^ For instance, consider a university deciding on how to allocate a budget among a set of departments. The decision might be based on a whole set of criteria including quality of research and teaching measured by various indicators, numbers of students, numbers of researchers, etc. Let K be the number of such criteria. With respect to these criteria, voter i (say department i) is described by the vector A^ = (A|,...., Af-) as to how "much" of each criterion voter i possesses. So, X] might be a measure of department i's research output, A^ might be a measure of the number of students enrolled in the department z's courses, and so on. Here, an alternative x is a decision on the relative weight of each criterion in allocating the budget. Given an x the allocation of the budget is such that voter (department) i receives ^
MX: x' k=i
A ^ Y^l<j
For instance, if x^ = 1/3, then in the university example 1/3 of the budget will be allocated based on the number of students that a department has. There, ^F=^ '-—ri would measure the fraction of all students that a given department 2^1<j
^j
has. Presuming that voter i prefers to be allocated as much of the budget as possible, we end up with preferences for voter i described by ^^ See Lockwood [24] for an analysis of a model with externalities. ^^ We thank Salvador Barbera for having suggested this problem.
170
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton u^^
^
for each k.
3 Majority Voting and Tournaments Let us now discuss how we model the choice of alternatives made by voters. The strict majority preference induced by a profile u of preferences is denoted by T{u) and defined over X as follows: xT{u)y if and only if #{i eM \Ui' x> Ui-y] > 4i^{i eN
'.Ui-y > Ui' x] .
If N is odd and individual preferences are strict, then T = T{u) is complete.^"^ Otherwise, ties may occur and this results in some freedom in how one defines the sets and procedures that we examine next. For the following definitions, T may be an arbitrary asymmetric (and possibly incomplete) binary relation. Condorcet Winners An alternative x is a Condorcet winner if: ioT ally
eX\{x},xTy.
An alternative x is a weak Condorcet winner if: for all y G X, not yTx. Let WC{T) denote the set of weak Condorcet winners associated with T. In the case where T is complete, the two definitions coincide. In fact, it is easy to see that whenever there is a Condorcet winner then that alternative must also be the unique weak Condorcet winner. However, in cases where T is incomplete it is possible for there to exist many weak Condorcet winners, in which case there is no Condorcet winner. The Top Cycle As the majority preference is not necessarily transitive, it can have cycles. A prominent cycle that we refer to in the sequel is the Top Cycle associated with r . -^^ A binary relation which is asymmetric and complete is called a tournament. See Laslier [21] for an illuminating account of the principal results in the vast literature on tournaments and majority voting.
The Banks Set
171
Let a weak T-chain between alternatives x and 7/ be a sequence of alternatives xi,...,Xk such that xi = x, Xk = y^ and not Xj^iTxj for each The Top Cycle of T, denoted by TC{T) is the set:^^ TC{T) — {x I Vi/ G X, 3 a weak T-chain between x and y} Thus, the Top Cycle is the set of alternatives that can reach any alternative in X via some weak T-chain. The Uncovered Set The Uncovered Set of T, denoted UC{T), is the set of maximal elements of the covering relation C{T) defined over X. Defining C{T) by xC{T)y if and only if xTy and for all 2: G X : yTz implies xTz. Let UC{T) = {x \yy G X, not
yC{T)x}.
Again, it should be pointed out that when T is not complete, there are several possible definitions of the Uncovered Set. The definition above, which is the most relevant for our subsequent analysis of the Banks Set, corresponds to UCd in Bordes et al. [3], to Fd in Bordes et al. [7] and to Miller's [8] subset. It does not correspond to the definition of the Uncovered Set which is found in Banks et al. [4, 5], Dutta and Laslier [13] and McKelvey [27]. There, the Uncovered Set is defined as the set of maximal elements of the partial order C{T) defined over X by xC{T)y if and only if xTy and for all z £ X : [yTz implies xTz] and [zTx implies zTy] Let
_ TJC{T)
= {x \\fy G X, not
yC{T)x}.
Since C{T) is a subrelation of C{T), the Uncovered Set that we focus on in this paper, UC{T), is a subset of this other Uncovered Set UC{T). Amendment Agendas and Voting by Successive Elimination A prominent procedure that selects a single allocation out of the feasible set X is that based on amendment agendas, as central to Roberts Parliamentary Rules of Order. This procedure, is also often referred to as voting by successive elimination in the literature, and is defined as follows. ^^ When the majority preference is not complete, there are various possible definitions of the Top Cycle (see Schwartz [37] and Duggan and Le Breton [11]). All of these definitions coincide with the definition of TC considered in this paper when the majority preference is complete.
172
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
Consider an ordering a of the finite set of alternatives X and let a = ( a : i , . . . , X L ) where L denotes the number of alternatives in X. A vote is first taken to eliminate either XM or XM-I- T h e 'winning' alternative from the first round is compared to XM-2^ and a vote is taken to eliminate either surviving alternative from the first vote or XM-2, and so on. After ( M — 1) comparisons, the last surviving alternative is declared to be the voting outcome. At each stage, t h e elimination of one alternative is according to majority voting, or more generally according to the binary relation T. This is wellspecified when T is complete. In cases where there are ties under the majority preference relation, or T is incomplete, the voting procedure needs to be more completely specified. We do so as follows. At each stage allow individuals to vote for one of the two alternatives or to abstain (in the case where they are indifferent). In case of a tie in the voting between alternatives xi and x//, xi is elected if and only if x/ comes before x// in the ordering a of voting; t h a t is, I < I'. This favors alternatives proposed earlier in the agenda under ties, which is a natural way to break ties given t h a t they have not already been broken under T. In order to determine the eventual voting outcome, it is also necessary to describe how voters act. We consider the case where they vote strategically at each stage, and so focus on the sophisticated voting outcome of this binary voting procedure.-^^ This is the outcome under the iterative elimination of weakly dominated strategies t h a t has been well-studied. As demonstrated by Shepsle and Weingast [38],^^ the sophisticated outcome induced by the ordering cj, denoted 5(cr, T ) , is equal to w^ which is t h e last element of the finite sequence described by the following algorithm: a J r u 1 ^ ^ a \ Xi \i xiTwZ for all I' < /, and w^ = Xl, and tor a l l / > 1 : it;/ == < ^ ^, ^ . ^ * [Wi_-^, otherwise.
T h e Banks Set T h e Banks Set, denoted B{T), is t h e subset of alternatives which are sophisticated outcomes for at least one ordering of X. Formally, B{T)
= {xeW
:3a e U such t h a t x = 5(c7, u)} ,
where U denotes the set of permutations of X. For more on sophisticated voting, see Farquharson [15] and McKelvey and Niemi [28]. The Shepsle-Weingast algorithm was defined for the case where T is complete. Our procedure of breaking possible ties in the majority preference relation coming earlier in the ordering a ensures that the sophisticated outcome can be derived from a straightforward variation on the algorithm derived by Shepsle and Weingast, as shown, for instance, in Banks and Bordes [3].
The Banks Set
173
Let a T-chain between alternatives x and y be a sequence of alternatives X i , . . . , x/e such that xi = x, Xk = y, and XjTxj^i for each j = 1 , . . . , /c — 1. Given an alternative x € X, an x-chain of T is a T-chain H with x e H such that xTy for all y G H. The set of all x-chains is denoted H{x, T). Thus, an x-chain is a chain where x beats all the other alternatives in the chain according to T. The characterization provided by Banks [2], stated to accommodate the possible incompleteness, can be stated as follows. Proposition 1. B{T) = {x\3H
e H{x,T)
s.t. Wy ^ H 3z e H s.t. not yTz},
Thus, Banks showed that the outcomes found by varying the ordering (for a fixed tournament) of the amendment agenda when voting by successive elimination correspond to the endpoints of chains, where the chains are such that any alternative not included in the chain is beaten by something in the chain. The intuition behind the characterization is that the alternatives in the chain are those who temporarily "win" at some stage in the voting (the Wk's in the Shepsle-Weingast algorithm), and the remaining alternatives are those who are eliminated at their stages. The following variation on well-known inclusions is helpful in what follows. Lemma 1. If T is an asymmetric binary relation, then WC{T) UC{T) CTC{T).
C B{T) C
The first inclusion is easily seen by noting that any weak Condorcet winner forms a maximal T-chain. This means that if the ordering is such that this weak Condorcet winner appears first in the order, then it will be the outcome of the amendment agenda, as no other alternative beats it. The second inclusion appears as Theorem 4.1 in Banks and Bordes [3]. The third inclusion follows easily from the definitions. In what follows we use the notation WC{u),B{u), UC(u),TC(u) the sets WC{T{u)),B(T{u)),UC{T{u)),TC{T{u)).
to denote
In the following sections, we examine the simplest framework for which the class of allocation problems described in the preceding section is not degenerate. If X = 2 or A^ = 2, there is always at least one weak Condorcet winner and all sets coincide with the set of weak Condorcet winners. When if > 3 and A^ > 3, the set of (weak) Condorcet winners is sometimes empty or some set of points that is not a singleton and not the whole set, and the determination of TC^ UC^ and B becomes more challenging and interesting as we have a true multidimensional problem. Thus, in what follows we restrict attention to the case oi K = N — '^,
174
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
4 T h e Simple Geometry of t h e Majority Relation and t h e Top Cycle In this section, we consider the continuous version of X and assume, without loss of generality, that M = 1. Under the assumption that K = N = 3^ we are in position to use the simple geometry of the triangle to support our formal arguments. Given these dimensionality assumptions, X = {x = (xi,X2) G R + :a;i+X2 < 1 } ; that is, X is the triangle represented on Fig. 1. The three vertices of the triangle are denoted e^, e^ and e^.
1
Xi
Fig. 1. Representation of the alternatives Given u G u^ and x G X, let U{u^x) be the set of alternatives that are considered strictly superior to x by a majority, the so-called win set of x and by L{u, x) the set of alternatives that are considered strictly inferior to x by a majority of voters. When there is no Condor cet winner, these two sets are the union of three simple sets as pictured in Fig. 2. The following simple consequences of the linearity assumption on preferences will be very useful in the sequel. (a) If xT{u)y, z£X
and A > 0, then Ax + (1 - \)z T{u) Ay + (1 - A)^.
(b) If xT{u)y and A,/i G [0,1], then X > fj, implies Xx -{- {I — X)y T{u) fix +
(i-M)y(c) An immediate consequence of (a) is that if U{x,u) 7^ 0, then U{u,x) intersects the boundary of X] a similar observation applies to L{u,x).
The Banks Set
175
Fig. 2. The win set of x (d) We deduce from (b) that if x majority dominates y, then any point belonging to the line segment joining x and y majority dominates any other point of the segment which is farther away from x. In the rest of this section, we rule out preference profiles which either offer little interest or will be examined in some subsequent sections. In particular, we assume that each voter has a unique ideal point. It is straightforward to see that the linearity assumption implies that this ideal point is necessarily a vertex of the triangle; i.e., the ideal point must be one of e^,e^ or e^. We also assume that the three ideal points are all different, as otherwise at least two voters have the same ideal point, which is then the unique Condorcet winner. Without loss of generality, let e^ be the ideal point of voter i for all i = 1,2,3. Finally, we limit our attention to the generic case where a given
176
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
voter is never indifferent between the ideal points of the two other voters. A profile of preferences u displaying these features is described by matrix with three degrees of freedom. A profile of preferences is completely described by a vector u = (t^i, 'i^2, '^3) ^ (0^1) where Vi denotes the intensity of the preference of voter i for his second best choice among the vertices. Within this class of linear preference profiles, two situations may appear: (1) None of the vertices dominates the two other vertices. Up to a permutation of voters' labels, a profile of preferences u in this category is described by a matrix 1 0 V3 u = vi 1 0 0 V2 1
where 0 < fi,i;2,f3 < 1. (2) One of the vertices majority dominates the two other vertices. When this happens, we call such a vertex a vertex Condorcet winner as it would be the obvious winner if competition was limited to the finite set of vertices. Up to a permutation of voters' labels, a profile of preferences u in this category is described by a matrix 1 V2 V3
u = vi 1 0 0 0 1 where 0 < t'i,t'2,'y3 < 1We first examine the conditions under which a Condorcet winner exists. Proposition 2. Let u = (i'i,i'27'^3) ^ (O51) ^^ <^ profile of preferences as described in (1) and (2) above. Then x is a Condorcet winner for u, if and only if it is a vertex and u falls in category (2) and satisfies ^2 + ^3 > 1 (where up to a permutation of labels, vertex e^ is the winner). Proposition 2 departs in a fundamental way from Plott's well-known necessary and sufficient conditions for the existence of a Condorcet winner in the spatial model. His result asserts that for some alternative x to be a Condorcet winner, it has to be that x is the ideal point of some voter i and for any other voter j , there exists a voter k such that the normalized gradients of the utility functions of j and k evaluated in x are exactly opposite. Since such symmetry conditions are not robust to perturbations of preferences, a corollary of Plott's result is that Condorcet winners do not exist generically. It is often forgotten that this applies only if x is in the interior or relative interior of the feasible set, which is vacuously true if X is the entire Euclidean space. If, instead, like here, X is a compact convex subset of the Euclidean space,
The Banks Set
177
then Plott's conditions do not apply to alternatives on the boundary. ^^ This applies systematically in our linear setting, since we just demonstrated that Condorcet winners, when they exist, are on the boundary. The necessary and sufficient condition stated in Proposition 2 is robust to perturbations. Let us make a final comment on the existence of Condorcet winners. The linear setting is a natural generalization of the finite setting and will be at least as complicated as the finite setting in that the majority tournament limited to the set of vertices can take any form. But the linear setting is richer in that a vertex doing well when matched exclusively against the other vertices may be defeated by a majority when compromises are introduced. Suppose u displays the pattern 1 V2 V3
u = vi 1 0 . 0 0 1 Then e^ is a vertex Condorcet winner: voters 2 and 3 cannot agree on another vertex. Can they agree on something else? They can if the intensity of their preference for e^ is not too large, as stated by the inequality t'2 + f3 < 1 in Proposition 2. The condition is fairly intuitive since if V2 and vs are small enough then the gap between their second best and worst choices vanishes, and it becomes possible to find a compromise Ae^ + (1 — A)e^ preferred by both of them to e^. Figs. 3 and 4 illustrate the two conceivable situations. The following proposition provides a complete description of TC{u) when u = (t'l, t^25 '^3) ^ (0,1) is a profile of preferences as described above. Proposition 3. Let u he a profile of preferences as described in (1) or (2) above. Then, either there is a Condorcet winner, or TC{u) = X, or TC{u) = X\ {e*} for some i. Proposition 3 is a version of McKelvey's chaos theorem in our linear setting. The proof offered in the appendix shows how a cycle connecting any two alternatives is constructed, and the problems raised by the existence of a boundary are addressed. In contrast to the conditions leading to the existence of a Condorcet winner, the boundary does not have much impact here, as the only departure from total chaos is the exclusion of a Condorcet loser, when there is one. ^^ Plott [36] applies a budget constraint, but does not impose any nonnegativity constraints and so does not consider boundary issues in the manner considered here.
178
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton X2L
Fig. 3. A Condorcet winner
Fig. 4. A "vertex" Condorcet winner which is not a Condorcet winner
The Banks Set
179
5 T h e Goods and Bads Model In this section, we return to the discrete version of the problem, still keeping with N = K = 3. We focus on t h e goods and bads model of example 2 and characterize all of t h e sets, including t h e Banks and Uncovered Sets. Let us first start with t h e analysis of a prominent case t h a t falls in t h e goods and bads model: t h a t of t h e divide t h e dollar model. P r o p o s i t i o n 4. Consider the divide the dollar model of Example 1. The set of weak Condorcet winners is empty, the Top Cycle is the whole set of alternatives, and the Uncovered Set is the set of alternatives excluding the vertices. The Banks Set includes every x ^ X such that xi < [{xj + Xk)'^ -f 5{xj -\- Xk) — 4]/2, for some i and distinct j , k. Thus, the size of the Banks Set converges to the size of the set of alternatives as the grid becomes finer (limM-^oo 4j=x ~ •'•)• However, the Banks Set is a strict subset of the Uncovered Set for any M > 5; as ( M — 1,1,0) and permutations of these points are not in the Banks Set}^ Proposition 4 provides a different view of the Banks Set t h a n what is previously known. While in some finite settings with arbitrary preferences, one can find examples where the Banks Set is a strict subset of the Uncovered Set (see Banks [2]), it was not known whether this was true in more naturally structured environments. Indeed, Penn [34] shows t h a t in an infinitely divisible version of a divide t h e dollar game with three players, t h e Banks Set and Uncovered Set coincide.^^ Here, in contrast, t h e Banks Set makes a selection from t h e Uncovered Set. As t h e indivisibilities disappear, t h e sets converge to each other, with t h e Banks Set always remaining a strict subset of t h e Uncovered Set. Let us now return to t h e more general analysis of t h e goods and bads model, where players may agree on which dimensions are goods, and may like several dimensions. Let 5^ = ^ . u^ and s == Y^j^s^ = Y.i Ek ^i- Note t h a t 5 G { 0 , 1 , . . . , 9}, and s^ G { 0 , 1 , 2 , 3 } . Thus, s^ is the strength of the support for dimension k. T h e analysis of t h e various sets now depends on the relative strengths of the dimensions. For larger M, one can also check that (M —2, 2,0) and (M —2,1,1) (and permutations) are not in the Banks Set, and so forth; but the proofs become increasingly tedious as the number of chains to be ruled out grow as we move away from the vertices. Penn's definition of the Banks Set in infinite settings is directly in terms of maximal chains rather than in terms of an agenda, and her tie-breaking rule is different from ours. It is not clear that there is an unambiguously appealing definition of the Banks Set in the infinite setting, as without some modifications of tie-breaking there do not exist any maximal chains.
180
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
Proposition 5. Consider the goods and bads model from Example 2, and assume that at least one voter is not completely indifferent. Without loss of generality, label the dimensions so that s^ > s'^ > s^. (1) If one dimension dominates the others (s^ > s'^), then the vertex corresponding to that dimension is a Condorcet winner and all the sets coincide (TC{u) = UC{u) = B{u) = WC{u) = {(M,0,0)};. (2) If there are two dimensions that have the same strength and dominate the third (s^ = s'^ > s^), then there is no Condorcet winner and the sets include all points that allocate only to the first two dimensions (TC{u) = UC{u) = B{u) = WC{u) =^{x\x^ = 0}). (3) In the case where the strength of support for the three dimensions is identical (s^ = s^ = s^): (3a) If some voter is completely indifferent, then no alternative beats any other, and so X = WC{u) = B{u) = UC{u) = TC{u). (3b) If each voter views a different two dimensions as goods, then WC{u) = {(M,0,0),(0,M,0),(0,0,M)}, while TC{u) = X, and B{u) = UC{u) = X \ {(M - 2,1,1), (1, M - 2,1), (1,1, M - 2)}. (3c) If each voter views one dimension as a good then we are back in the divide the dollar game setting as characterized in Proposition 4Proposition 5 states that the analysis of the goods and bads model breaks into five cases, basically depending on how much agreement there is among the voters as to which dimensions are goods. When there is enough agreement (as in (1) or (2)), then the predictions are narrow, while when there is significant disagreement (as in (3a), (3b), and (3c)) then many voting cycles appear and the sets are nearer to the entire space. Interestingly, the only situation where something falls in between is in the divide the dollar game with smaller M (substantial indivisibilities) where the Banks Set is narrower than the Uncovered Set and Top Cycle. More specifically, in the first case, there is some dimension that receives more support than any other, and then giving the full budget to this dimension is a Condorcet winner. In the second case, there are two dimensions that are viewed as goods by an equal number of voters and the third dimension is viewed as a good by a lesser number. Here, the set of weak Condorcet winners is the set of alternatives that give only to the two dimensions with broader support. In the third, fourth, and fifth cases, all of the dimensions have equal support. However, they behave quite diff'erently. In the third case, no alternative beats any other, as the two voters who are not indiff'erent completely disagree on the goods and bads, and so all sets are the whole space. In the fourth case, the three vertices form the set of Weak Condorcet winners. The Top Cycle is the whole set X, while the Banks and Uncovered Sets are almost the entire set X. The fifth case refers to the divide the dollar game, as already discussed.
The Banks Set
181
6 Beyond t h e Goods and Bads Model We have offered a complete description oi WC{u), TC{u) and UC{u), and some bounds on the description of B{u)^ for the goods and bads model. In this section, we return to the more general linear model. In Section 4, we analyzed that model in terms of understanding the Top Cycle. We now return to that analysis to see what we can say about the Uncovered Set. Precisely, we focus on the generic case where there is not a Condorcet winner and the profile of preferences is described by the pattern 1 V2 t'3
u = vi 1 0 , 0 0 1 where 0 < t'i,t'2,'^3 < 1- Thus, e^ is a vertex Condorcet winner, as it beats the other vertices in a majority contest. What does the Uncovered Set look like in such a setting? If e^ is a Condorcet winner, then obviously UC{u) = {^^}- So let us assume that e^ is not a Condorcet winner. Prom Proposition 2, this holds true if and only if '^2 + t^3 < 1. In such a case, TC{u) rules out the Condorcet loser e^, but none of the points arbitrarily close to e^. The following proposition demonstrates that there is a neighborhood of e^ which is outside UC{u). The proof technique is based on the following simple but useful lemma which follows immediately from the definition of covering ((7(T)). Lemma 2. Alternative x € UC{u) if and only if for ally ^Y or there exists z E X such that xTz and not yTz.
either not yTx
This lemma states a version of the two-step principle (a terminology due to Miller and McKelvey). Indeed, the lemma states that to be in the Uncovered Set an alternative x must weakly majority dominate any other alternative in either one step or two steps; and if there are two steps then the first component of the weak T-chain must be strict. Let L^(x) be the set {X\U{u,x)) U {\Jy^L(^u,x){X\U{u,y)))?^ The lemma asserts that x G UC{u) if and only if L^(x) = X. Proposition 6. Let u he as discussed above and x = (a:i,X2) G X. x G UC{u) if and only if Vi
Then,
V2
It is interesting to note that the condition in Proposition 6 does not involve Vs. If f 1 = V2 = V, then the condition in the Proposition is simply that: The notation L^ is justified by the fact that when T is a tournament, L^{x) = L{u,x) U (Uy^L(u,x)L(u,y)).
182
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
Obviously, from Proposition 6 it follows that the Uncovered Set rules out many points around e^. This is a first step in an exploration of UC{u). This provides the interesting conclusion that the Uncovered Set is a subset of the space of alternatives that depends in interesting ways on the utility profile. Likely, a similar analysis can be conducted in the case where there is no vertex Condorcet winner.
7 T h e Mixed Private Versus Public Goods Model In this final section, we investigate the mixed private versus public goods model defined as Example 3. In this model, a profile of preferences is identified by the single positive parameter h describing the common willingness to pay of each voter for the public project. To emphasize this specificity, we use the notation WC{b), TC{h) and UC{b) instead of WC{u), TC(u) and UC{u). The following proposition describes the dependence of the three sets upon the parameter b?^ Proposition 7. Consider the mixed private versus public goods model from Example 3. (1) If the benefit from the public good is large (b > \), then allocating the entire budget to the public good is a Condorcet Winner (and thus, WC{b) = UC{b)=:TC{b) = {0,0,0,M) ) . (2) If the benefit from the public good is intermediate (\ < b < \), then there are no weak Condorcet winners, the Top Cycle is the whole set of alternatives, and the Uncovered Set is the set of alternatives such that at most two voters get a positive amount of the private good and no voter gets the entire supply of the private good (UC{b) = {x ^ X : Xi = 0 for at least one i G {1, 2, 3} and x^ ^ M for any k G {1,2, 3 } } / (3) If the benefit from the public good is small (b < \^), then the sets are as in the divide the dollar game (WC{b) = %, UC{b) = {xeX :X4=0 and x^ 7^ M for any k} and TC{b) - X). Proposition 7 demonstrates that the presence of the public project has an impact on the distributive politics component of the budget allocation. If the benefit from the public good is sufficient, then it swamps the private allocation, as in (1). If it is too small, then the problem becomes similar to the divide the dollar game, as in (3). In the middle case, we see some interesting impact of the public good. One voter among the three should derive his payoff exclusively from public consumption. This is due to the fact that when ^ > ^, ^^ We have not calculated the Banks Set in this model.
The Banks Set
183
the Bowen-Lindahl-Samuelson first order optimality condition rules out any interior allocation. Since the Uncovered Set is a subset of the Pareto set, this provides an upper bound. We prove that, in fact, the two sets coincide.
8 Concluding R e m a r k s and Higher Dimensions We have shown that it is possible to make predictions about the nature of voting equilibria under majoritarian rule that are not too sensitive to specific institutional details, even in multi-dimensional policy space. We did this by analyzing budget allocation problems where voters' preferences are linear. Section 4 describes the geometric structure of the Top Cycle set. Proposition 2 showed that if there is a Condorcet winner, then it must be a vertex. We also found that the conditions under which a Condorcet winner exists extend Plott's analysis because they are applicable even when a Condorcet winner lies on the boundary of the feasible set — which is absent from his analysis. Having a boundary on the problem provides a difi'erent perspective than one gets from Plott's analysis, and the possibility of a Condorcet winner is no longer so extreme. Proposition 3 is the counterpart of Mckelvey's chaos theorems, and shows that if a Condorcet winner does not exist, then the Top Cycle set is virtually the entire set — at most it excludes the three vertices. So, while we still come to the conclusion that the Top Cycle is either a single point or the whole space, the conditions under which it is a single point are no longer so extreme. We went on to consider the goods and bads model, where voters view each dimension as either a good or a bad. Proposition 5 demonstrates that there are cases where the Banks Set and Uncovered Sets are strict subsets of the feasible set, even in situations where no Condorcet winner exists. The Banks Set is generally a strict subset of the Uncovered Set, but the difference between the two sets disappears as the divisibility of the budget becomes finer. In Section 6, we returned to the more general linear preference framework of Section 4, but restrict attention to the analysis of preference profiles which give rise to a vertex Condorcet winner. We characterize the Uncovered Set and show that it excludes a neighborhood of points close to the vertex Condorcet loser. This provides an interesting setting in which the Uncovered Set makes pointed predictions about the outcome of any majority rule based collective decision. Finally, Section 7 looks at the "mixed" public and private goods model (Example 3). Not surprisingly, voters' common willingness to pay for the public good turns out to be the crucial parameter in this model. If this willingness to pay is very high, then the entire budget will be spent in production of the public good under majoritarian rule. Conversely, if the willingness to pay is low, then the Uncovered Set excludes production of the public good. The in-
184
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
teresting case is when the common willingness to pay takes on an intermediate value, and then the Uncovered Set predicts that at least one voter must be excluded from consumption of the private good. The bulk of our analysis was in the special case where there are three projects and three voters, as that case is still tractable and yet introduces the full force of multi-dimensionality. Certainly, it is worthwhile to explore beyond this. While the extension to more than three projects and/or three voters does not raise conceptual difficulties, it is obviously much trickier. One reason is that the linear model is at least as difficult as the finite model and therefore moving to larger K complicates the combinatorics of the problem, as we know from the theory of majority tournaments. We have listed below several directions of investigation that seem promising to explore as a continuation of the analysis performed here. •
What happens to the goods and bads model in higher dimensions? The following conjecture might be considered. Conjecture: Consider the goods and bads model, and a case where there is some dimension k that a strict majority of agents view as a good. Let J be the set of agents who think fc is a good and let A = {k' \ such that u^ = 1 for some i G J } . If x G UC{u) and ^ ^ A, then x^ < K,
•
It seems that Proposition 2 generalizes to higher dimensions. A Condorcet winner will have to be a vertex. What conditions ensure that this vertex Condorcet is a Condorcet winner? Using Farkas' Lemma, it seems that a complete characterization of preference profiles for which this holds is possible! Once again, this will depart from Plott's symmetry conditions.
•
It seems that we can also generalize Proposition 3 to higher dimensions, as follows. Define the Vertex Top Cycle, denoted VTC{u), to be the subset of vertices that are in the Top Cycle of the majority weak tournament restricted to the vertices and the Vertex Bottom. Cycle, denoted VBC{u), to be the subset of vertices that are in the Bottom Cycle of the majority weak tournament restricted to the vertices. We conjecture that TC{u) = {xeX:Xk=^ioi
all e^ G VBC{u)] .
This implies that if a vertex is in the Vertex Top Cycle, then it is in the Top Cycle, but the converse does not hold, as we know already from the case where K = N = 3. •
The computation of the Uncovered Set does not seem out of reach either. One preliminary question we may ask could be the following. Define the Vertex Uncovered Set to be the subset of vertices which are in the Uncovered Set of the majority weak tournament restricted to the vertices. Is it true that a vertex is in the Vertex Uncovered Set must also be in the Uncovered Set? We know that the converse does not hold from Proposition 4.
The Banks Set •
185
Finally, a detailed exploration of Example 3 would be valuable. It is straightforward to check that if there is a Condorcet winner, it must give the whole allocation to the public project. Furthermore, this project is a Condorcet winner if and only if 1
b>
(f)" + l The following conjecture, extending Proposition 7, could be considered. Conjecture: Consider the private versus public goods model. If b > j ^ for some positive integer M, then x G UC{b) =^ # {i : x^ > 0} < M. Furthermore, if 6 < j ^ , then UC{b) = {x e X : Xi = 0 ioi dit least N - M + 1 voters}. Note that the first assertion is true from an analysis of the Pareto set. Only the second assertion remains to be proved.
9 Appendix Proof of Proposition 2. Let a: be a Condorcet winner for u. We first show that X must be on the boundary of the triangle. Assume to the contrary that x is in the interior of X. Then, for all z, j G A/", the indifi'erence lines of voters % and j passing through x must be identical, as otherwise, there would exist y in the neighborhood of x such that Ui-y > Ui-x and Uj -y > Uj • x, contradicting our assumption that x is a Condorcet winner. This implies that the slopes of the indiff'erence lines of voters i and j through x are the same, so that ViVj = 1. This cannot be, as there is no solution (t'i,t'2, "^s) G (0,1) to the system of equations viV2 = l,'i^ii^3 = 1 a n d i;2'i^3 — 1.
So we have shown that a Condorcet winner must be on the boundary of X. Next, we show that a Condorcet winner must be a vertex. We know from above that a Condorcet winner x can be written as x = Ae*4-(1 —A)e-^ for some 0 < A < 1. Then, either u^-e'^ > tt/e -e-^, in which case e* majority dominates e^ via the coalition {i, /c}; or, Uk-e'^ < Uk-e^, in which case e^ majority dominates e* via the coalition {j, k}. Therefore, either A = 0 or A = 1, and the Condorcet winner must be a vertex. We complete the proof by showing that (2) must apply and that Vj-\-Vk > 1 for some j and k. Without loss of generality, let x = e^. Then, u must fall in (2) and it must be that either 1 t'2 '^3
u — vi I 0 or 0 0 1
1 V2 V3
u=0 10. vi 0 1
186
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
Indeed, since e^ majority dominates e^ and e^, then either e^ is the worst choice for 1 and 2 or e^ is the worst choice for 1 and 3. Without loss of generality, consider the first case. For e^ to be Condorcet winner it is necessary and suflacient that there not exist (7/1,2/2) ^ -^ such that V2yi +y2>V2
and {V3 - l)yi - y2 > vs - l-
It is straightforward to check that this system of inequalities is consistent with (2/1,2/2) G X if and only if V2-{-V3 < 1. Proof of Proposition 3. Assume that there is no Condorcet winner. We distinguish two cases. Case 1: There is no vertex Condorcet winner. In this case, up to a permutation, e^T{u)e'^T{u)e^T{u)e^ and from (d) above, the cycle extends to the whole boundary of X: for any two points z and w on the boundary there is a weak T-chain between z and w. Now take x and y in X. Since there is no Condorcet winner, we deduce from (c) above that there exist z and w on the boundary of X such that xT{u)z and wT{u)y. The existence of a weak T-chain between x and y follows from the juxtaposition of the three weak T-chains. This proves that TC{u) = X. Case 2\ There is a vertex Condorcet winner. Without loss of generality, assume that eMs the vertex Condorcet winner. Since e^ is not a Condorcet winner, there exists z on the segment [e^, e^] such that zT{u)e^. From (6) above, we deduce that e^T{u)e^T{u)zT(u)e^ and from {d) above the cycle extends to the entire boundary of the triangle with vertices e \ e ^ and z, as illustrated in Fig. 5. We first show that for all x,y ^ [ e \ e^] U [e^,e^] U [e^,e^] U [ e \ ^ ] such that x ^ e^, there exists a weak T-chain from x to y. For any x G [ e \ e^] U [e^,z] U [:^, e^], the claim follows from the existence of a cycle as in claim. Consider now the case where x G [ e \ e^] U [^, e^] with x ^ e^. The idea is to construct a weak T-chain starting from x and ending in h belonging to the smaller triangle with vertices e^,e^ and z] once there, we just demonstrated that you can anywhere else on the boundary of X. The construction goes as follows. First consider / € [z^,e^'\ and let g be the intersection of [ e \ e^] with the indiff"erence line of voter 2 passing through / . Given the slopes of the indifference line of voters 2 and 3, it is easy to see that this point is well defined and that us - g > u^ • f. Then, define h as being the intersection of the indifference line of voter 3 with either [z_,e^] or [1,6^^]. Given the slopes of the indifference lines of voters 1 and 3, it is easy to see that this point is well defined and that ui - h > ui - g. We have obtained the short weak T-chain fT{u)gT{u)h. This is illustrated in Fig. 6.
The Banks Set
- Xi
Fig. 5. The "cycles" on the boundary of the triangle
Fig. 6. Construction of h
187
188
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
li h e [z,e^], we have completed the desired argument. If instead, h G [z_, e^], it is easy to show that | h — e^ \>\ f — e^ |. Starting from h, we repeat the argument above to obtain g' and /i' G [z^e^] U [^,e^]. If h' G [^, e-^], we are done. Otherwise, we continue this process. After a finite number of steps, we will obtain a point in [^, e^]. This is illustrated in Fig. 7.
Fig. 7. Connection to [e^, Z] The case where / G [e^, e^] follows from (a) above, since / = Ae^ + (1—A)e^ for some A G ]0,1[ and e^T{u)w for all t(; G ]z, e^ [, we deduce that fT{u)Xw + (1 — A)e^. The connection involving points in the interior of X is done as in case 1. This completes the proof of the claim that TC{u) = X\ {e^}Proof of Proposition 4- First, note that yTx implies that y exceeds x on exactly two dimensions. From this it is clear that there are no weak Condorcet winners, as for any alternative there exists some other alternative that gives more to two of the dimensions (given that M > 4). Next, let us check that the Uncovered Set is the set of all points less the vertices. Consider x = (u^v^w) that is defeated by some y — (a^b^c).
The Banks Set
189
Without loss of generality, let a > u^ b > v^ and c < w — 2. Consider z = (M — c — 1,0, c + 1). Here, provided v > 0, x beats z and yet z beats y. Thus, y cannot cover x. This implies that the only covered points could be the vertices. Indeed, the vertices never beat any point, and are beaten by any interior point, and so are covered. To verify that the Top Cycle is X, we only need to check that the vertices are in the Top Cycle, as the other alternatives are all uncovered. We need to check that from any vertex, say x = (M, 0,0), and any other alternative y there is a weak T-chain. If y has a 0 in either of the last two dimensions, then X and y are non-comparable, and so there is a weak T-chain directly. Thus consider any alternative y = (u^v^w), where v > 0 and w > 0. Let z = {u + l^v -{- w — 1^0). Then x is non-comparable to z and z defeats y, so there is a weak T-chain from x to y. This completes the proof of the Top Cycle. The claims about the Banks Set are established as follows. Let us identify a maximal T-chain with x = {u,v,w)^ where u>v>w^Sit the end. Consider the case where u > v -\- w. {u^v,w)^ (u — l,v -\- 2^w — 1), {u — A,v -\- 6,w — 2), . . . {u — Ci^v -\- i + Ci,w — i), . . . where i is the index of the step until w — i hits 0, then {u — Ci, v — i -{- w^ i + Ci) for the remaining steps until V + w ~ i hits 0. Let us define Q , and let z* be the smallest i for which
u-{i^-\-3i~2)/2
< ^ + K;-Z. Then fori < r set Q = {i^+ 3i-2)/2Foi
i > i*
set Q = u — {v-\-w)-\-i. Let us prove that this chain is maximal. Suppose that y = (a, 6, d) beats everything in the chain. It cannot be that 6 < z;, as then there is some point in the chain with middle entry b. Similarly d < w is not possible. Sob > V and d > w. Thus it must be that a < u — 2.lt cannot be that a < u — Q*, as then there is some step with first entry a. So, it must be that a > u — Ci*. Without loss of generality then, take a = 7i —c^ + l, for some i < i'^. Then it must be that b and d beat all the second and third entries above this. This means that 6+d > v-\-w-\-Ci-i-h{i — l)-{-l-^l [either beating the highest second entry and w -\-l^ or the highest third entry and v-{-l if we are already in the second part of the algorithm]. We also know from the value of a that v-\-w-\-Ci — l > b-\-d. This implies that Q > Ci_i+i + 2. This does not hold by the definition of Cj, which solving inductively amounts to Ci > {i^ + 5i — 4)/2, which cannot hold given that u < {{v -\- wY -\- b{v -]- w) — A)/2. So, we have reached a contradiction. So, to complete the proof consider the case where u < v -^ w and let us identify a maximal T-chain with x = {u^v^w) at the end. (u^v^w)^ {u — 1,^ + 2,10 — 1), (u — 2,v -]- A,w — 2), ... [u — i,v -\- 2i,w — i)^ . . . where i is the index of the step until w — i hits 0, then {u — i^v ^ w — i^2i) for the remaining steps until u — i hits 0. Note first that this chain of length u -\- 1 is well defined; indeed, when i = u, v + w — i > 0. Let us prove that this chain is maximal. Assume on the contrary that y = (a, 6, d) beats everything in the chain. It cannot be that a < t^, as then there is some point in the
190
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
chain with first entry a. Similarly d < w is not possible. The same reasoning show that b is such that either b>v or b u-\-v -\-w which is not possible. Consider the second case. Since (a, 6, d) beats all alternatives in the chain, we deduce from b < V + w — u that (a, 6, d) is preferred by voters 1 and 3 to any alternative in the chain. This implies a > u and d > 2u and therefore, since u > v > w^ a^b-\-d>a-^d>3u>u-\-v-\-w, which is not possible. Finally let us show that for any M > 5, (M — 1,1,0) and its permutations are not in the Banks Set, and so B{u) ^ UC{u). The only alternatives that this beats are (fc,0,M — /c) for k < M — 1. The only chains that could conceivably be maximal are then of the form (/c, 0, M - k), (M - 1,1,0). If A: < M - 3, then the alternative (M - 3,2,1) beats both. If fc > M - 3, then (0,2, M - 2) beats both (provided M - 2 > M - (M - 3) = 3, so when M > 5). Proof of Proposition 5. Case (1) is easily checked directly. Let us check (2). If 5^ — 5^ = 3 > s'^, then it must be that every voter weakly prefers any allocation x with x^ = 0 to any allocation y with y^ ^ 0, and some voter has a strict preference between any two such allocations. Moreover, all voters are indifferent between any two allocations that have x^ = 0, and so the set of weak Condorcet winners is the set {x \ x^ = 0}. Since any allocation outside of this set is defeated by one inside this set, this is the Top Cycle. Also, since the set of weak Condorcet winners is a subset of the Banks Set, the claim follows from Lemma 1 noting that {x\x^ = 0} = WC{u) C B{u) C UC{u) C TC{u) = {x \ x^ == 0}. If 5I — 5^ _ 2 > ^3^ then it can be checked that any allocation x with x^ = 0 defeats any allocation y with y^ 7^ 0. [Such a y gets at most one vote versus such an x^ and such an x always gets at least one vote versus such a y. For any configuration of preferences that fits in this case where such a y gets one vote, it must be that such an x gets two votes.] Also, there must be one voter who is indifferent between all allocations in {x \ x^ = 0}^ while the other two agents split on it. Thus, again the set of weak Condorcet winners is the set {a: | x*^ = 0}, and any allocation outside of this set is defeated by one inside this set. So the rest of the proof is as in the case above. If 5I = s"^ — 1 > 5^ — 0, then either there is one voter who thinks both dimensions 1 and 2 are goods, and other voters are completely indifferent, or there are two voters who each like one of the two dimensions, and the other voter is indifferent between all allocations. In either situation it is clear that the set of weak Condorcet winners is the set {x \ x^ = 0}, and any allocation outside of this set is defeated by one inside this set, as in the earlier cases. Next, let us consider (3a). In this case, without loss of generality, suppose that voter 1 views dimension 1 as a good, voter 2 views dimensions 2 and 3 as goods, and voter 3 is completely indifferent. If we consider two alternatives that have the same allocation to dimension 1, then all voters are indifferent
The Banks Set
191
between these alternatives. If we consider two alternatives that have different allocations between dimension 1, then voters 1 and 2 will have opposing preferences over the alternatives. Thus, any two alternatives are non-comparable under T{u). Next, let us consider (3b). Note that no two voters agree on which dimensions are goods. There is one voter who likes dimensions 1 and 2, one who likes 2 and 3, and one who likes 1 and 3. One key observation is that if yT{u)x in this case, it must be that y exceeds x on exactly one dimension and is less than X on the two remaining dimensions. (If it is the same on any dimension then they are non-comparable. If y exceeds x on two dimensions, then the sum of the remaining dimension together with either other dimension is greater under x, and x will win.) This results in the following observations about T{u), (a) Any two alternatives which agree on some dimension are non- comparable to each other. (b) Any vertex will beat any alternative that is positive on the other two dimensions. From (a) and (b) it follows that the vertices are not beaten by any alternative, and from (b), it follows that any other alternative is beaten by some vertex. Thus the set of weak Condorcet winners is exactly the set of vertices. Next, let us show that UC{u) =X\{(M-2,1,1), 2)}-X*.
(1, M - 2 , 1 ) , ( 1 , 1 , M -
First, we show that x = (M — 2,1,1) is not in the Uncovered Set. Let y = (M, 0,0). Then, yTx. Suppose not yTz. Then, from (b), either 2^2 = 0 or zs — 0. Without loss of generality, suppose Z2 = 0. In order for x to beat z, X has to be bigger than z in just one component, and smaller than z in the other two components. Since Z2 = 0, this means that X\ < zi and X3 < 2^3. But this is not possible. So, xTz implies yTz. Hence, x is covered. Analogous arguments establish that (1, M — 2,1) and (1,1, M — 2) are covered. Next, we show that no other element in X* is covered. Each vertex forms a maximal chain as a singleton and so is in the Banks Set and thus the Uncovered Set. Next, consider an alternative x G X* that has two dimensions positive and the other 0. Without loss of generality, say x — {a.,M — a., 0) where a > M — a. This alternative beats (0, M — 1,1). Note also that any alternative y that has y^ > I does not beat (0, M — 1,1) (the voter who likes the last two dimensions is at best indifferent, and the voter who likes the first two dimensions prefers (0, M - 1,1)). Thus only alternatives with y^ = 0 beat (0, M - 1,1). Then, it follows from (a) that forming a chain of x and (0, M — 1,1) is a maximal chain that results in x, and so x is in the Banks Set, and thus the Uncovered
192
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
Set. Next, consider an interior alternative x = (x^,x^,x^) € X*. Without loss of generality, assume that x^ > x'^ > x^. Note that since x G X*, x^ > 2. Suppose yTx and y is an interior alternative in X. Without loss of generality, let y* < x'^,y^ < x^ and y^ > x^. Of course, such 2, j,/c must exist. So, yi <^ x'^ — l,y^ < x^ — 1, y^ > x^-\-2. This must mean that x^ > 2, x^ > 2 since y is interior by assumption. Now, consider z such that z* = 0, z^ = x^-{-x^ — 1, and z^ =0:^ + 1. Then, xTz, but zTy. So, x cannot be covered by y. Next, suppose that yTx and y is not an interior point. If y^ = 0, then choose z = {0, x'^ -\-1, x^ + x^ — 1). Since x^ > 2, we have xTz. But, by (a), y and z are non-comparable. So, y does not cover x. If y2 ^ Q^ ^]^gj^ choose z = {x'^ ^ 1,0, x^ + x^ - 1). Note that since x G X*, x^ > 2. Again, xTz, but 7/ and z are non-comparable. The last possibility is that y^ — 0. Since yTx, there is i such that y* > X* + 2. Choose z such that z* = x* -h 1 < y'^, z^ = 0 where /c ^ 0, and 2;"^ == M — X* — 1. Since x^ > 2, check that z^ > x^. It follows that xTz and yTz. So, X is not covered. Thus, we have shown that UC{u) — X*, and so Lemma 1 implies that X* C TC{u). Now, consider x = (M - 2,1,1). We show that there is a weak T-chain connecting x to each of the vertices. Take (M, 0,0). Then, the weak r-chain is (x, (M - 2,0, 2), (M, 0,0)). Weak T-chains to other vertices are obvious extensions of this weak T-chain. Similarly, there is a weak T-chain from X to any other point in X. Hence, TC{u) — X. Next, let us identify the Banks Set. Our arguments above already show that the Banks Set includes all alternatives that are not in the interior. Consider x = (x\x^,x'^) in the interior. Without loss of generality, let x^ > x^ > x^ > 1, and since B{u) C UC{u) = X*, we know that x^ > 2. Let k'^ be the smallest integer greater than or equal to k, and k~ the greatest integer smaller than or equal to k. Let us build a T-chain that ends in x and argue that it is maximal. This shows that x is in the Banks Set. The first element in the chain is x. The next part of the sequence are the alternatives (x^ + l,x^ - 2,x'^ + 1), (x^ + 2,x^ - 4,x^ + 2), ...,
(xi + (^)-,0,x3 + ( 4 ) + ) . Denote x\j = x'^ -{- (%)~, and xj^ = x^ -\- ( ^ ) ' ^ . The last part of the sequence is ( x ] ^ — 2 , x ^ + l , x ^ + l ) , ..., fo,x^ + ( ^ ) ~ ^ ^M + It is easy to check that this is a chain. Let us show that it is maximal.
The Banks Set
193
Suppose y beats everything in the chain. Consider the case where y^ > x^. The chain contains without any gap everything from x^ to x]^. So, y^ > x\j. But, then y cannot beat (a;j^,0,x|^). The same argument rules out cases where y"^ > x"^. So we are left with the case y^ > x^. In the third dimension, the chain contains all consecutive elements from x^ to x\^ -f- ( ^ possibly x\j — 1 and x\^ + ( ^ )
j
except
- 1-^^
Suppose y^ = x%^ — 1. Since y^ > x\^ — 2, we need y'^ < 3. But, then y^ ^ x\j — 1. So, y does not beat {x\j — l,3,x|^ — 2), which is the element just before {x\j,{),x\j). An analogous argument works if y'^ — x\^ + i ^
j
— 1.
Finally, (3c) follows from Proposition 4. Proof of Proposition 6. Let x = (xi,X2) G X. We examine the indifferences lines of voters 1 and 2 through x. The four possible cases are: •
The indifference line of voter 1 through x intersects [e^,e^] and the indifference line of voter 2 through x intersects [6^,6^^] (this is the case considered in Proposition 6)
•
The indifference line of voter 1 through x intersects [e^, e^] and the indifference line of voter 2 through x intersects [e^,e^]
•
The indifference line of voter 1 through x intersects [e^, e^] and the indifference line of voter 2 through x intersects [6^,6"^]
•
The indifference line of voter 1 through x intersects [e^,e^] and the indifference line of voter 2 through x intersects [ e \ e^]
This leads to a partition of the triangle X into four areas as indicated in Fig. 8. We can check that whenever x belongs to areas 2, 3 and 4, L^(x) = X. From Lemma 2, this implies that the union of these areas is included in the Uncovered Set. Assume now that x belongs to the first area. Then, L{x) is the union of the quadrilateral xAe^B and the two triangles xCD and xEF. This pattern is depicted in Fig. 9 where the hatched area corresponds to L{x)?^ From Lemma 2, to test if x G UC{u), it is enough to calculate L'^{x). From the geometry of the problem, it is straightforward to verify that L^(x) is the union of the two triangles e^FG and e^DH where G is the intersection of [e"^, e^] with the indifference line of voter 1 through F and H is the intersection of [ e \ e^] with the indifference line of voter 2 through D. Let / = {wi,W2) be ^^ There are no gaps if x^ and x\i are even. ^^ Up to the exclusion of the boundaries.
194
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton X21
Fig. 8. The four areas
Fig. 9. L{X) and
L^(X)
The Banks Set
195
the intersection of the lines GF and DH. Therefore, Lp'{x) = X iSwi-\-W2 < 1. The rest of the proof amounts to simple calculus. The first coordinate of F , say / i , is solution of the equation V2fl
= ^^2^1 + X2
/ l ==Xi +
—. V2
Therefore, the line FG is described by the equation X2
yi -\-v1y2 = xi + — . V2
Similarly, the second coordinate of D, say c?2, is solution of the linear equation vid2
= x i ^ 'i;iX2
or d2 =
hX2.
Therefore, the line DH is described by the equation Xi V2yi + 2 / 2 =
^X2. Vi
We deduce that
X2
.
OCi
wi = — a n d W2 = — , V2 Vi
which implies the conclusion. Proof of Proposition 7. (1) Let b > ^ and x e X with X4 < 1. Then, for at least two of the voters, say i and j , Xi < ^ - ^ and Xj < ^ - ^ . Therefore, since ^ > ^, 1 —
Uk ' X = Xk -\- bx4 < —
XA
h bx4 < b = Uk ' {0,0, 0,1) for k € {z, j} .
We deduce that (0,0,0, l)T{b)x and the conclusion follows. (2) Let I < 6 < | . It is clear that WC{b) = 0 as (^, ^,0,0)T(6) (0,0,0,1). Let us prove that UC{b) = {x G X : x^ = 0 for at least one i G {1, 2,3}, x/e 7^ M for any /c{l,2,3}}. (i) UC{b) d {x £ X \ Xi = 0 ioi dX least one i G {1,2, 3}, x/c 7^ M for any fc{l,2,3}}. Let X G X with Xi > 0 for all i G {1,2,3}. Let y = {xi — 5,X2 — S.x^ — 6,X4 + 3(5) where 0 < S < Min (a;i,X2,X3). Since 6 > | , y Pareto dominates
196
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
and therefore covers x. Next, suppose that Xk = M for some k G {1,2,3}. Then, x is covered by (0,0,0,M), as x does not defeat any alternative. (ii) UC{b) D {x G X : Xi = 0 ior dit least one i € {1,2,3} and x^ y^ M for any k G {1,2,3}}. Without loss of generality, consider the case where X3 = 0. Suppose that, contrary to the assertion, x is covered by y. Since we can take y to be uncovered, from (i) we deduce that either y = (^1,^2^ 0^ 2/4) or y = {yi^0,^3,^4) OY y = (0,^2,ys^y^)- Let us consider the case where X4 ^ 0. Case 1: y = (2/1,2/2,0,7/4). Subcase 1: y^ ^ ^4. Since yTx, we deduce that 2/4 7^ ^4 and therefore 2/1 > 0:1 and 2/2 > ^2- It follows that (xi,X2,0,a;4)T(2/i +2/2,0,0,2/4) but not [(2/1,2/2, 0,2/4)T(2/i +2/2,0,0,2/4)]. This shows that y does not cover x. Subcase 2: y4 > X4. The extra public good 2/4 — X4 is financed by voters 1 and 2. Without loss of generality, assume that voter 2 pays at least half of the cost i.e. 2/2 — ^2 £ — ^^^-^. Consider the vector z = (1 — 2/2 — 62/4 — £, 2/2 + 62/4 + e, 0,0) where 5 is a small positive number. Note that for e small enough, 2: is a feasible allocation as 2/2 + 62/4 < 2/2 + ^ < 1 since b < ^, Then, for € small enough, xTz since X4 7^ 0 while however not [yTz]. This shows again that 2/ does not cover x. Case 2: y = (2/1,0,2/3,2/4). Subcase 1: y^ < x^. Then, for sufficiently small but positive, e, we have: {xi, X2,0, X4)T{yi +2/3, e, 0,2/4-e) but not [(2/1,0,2/3,2/4)^(2/1 + 2/3, e, 0,2/4 - e)]. This shows that y does not cover a:. Subcase 2: 2/4 > ^4. Clearly, 3 prefers y to x. Suppose first that 1 also prefers y to x (and therefore that 2 prefers x to 2/), and consider the vector z = (1 — 62/4 — ^,^2/4 + ^,0,0), where 5 is a small positive number. Then, for small enough e, 2; is a feasible allocation and xTz while, not [yTz], If instead 1 prefers a: to 2/, and therefore 2 prefers y to x, consider the vector z = {y2 -\- by4 + 5,0,1 — 2/2 — %4 — ^, 0), where 5 is a small positive number. Then, for small enough 5, z is a feasible allocation and xTz since X4 / 0, while not [yTz]. This again shows that y does not cover x, Case ^S*-. 2/ = (0,2/2,2/352/4)- This case is similar to case 2. Consider the situation where X4 = 0. The analysis has to be changed slightly. Case 1: y = (2/1,2/2,0,2/4). Since yTx and 6 < ^, it must be that 2/4 > 0. Furthermore, either 2/1 + by4 < xi or 2/2 + by4 < X2. Without loss of generality assume that the second inequality holds and let 2: ^ (0,2/2 + %4 + ^, 1 — 2/2 — 62/4,0), where 5 is a small positive number. Then, for small enough e, 2: is a feasible allocation and xTz while not [yTz]. Case 2:y = (2/1,0,2/3,2/4). Subcase 1: y"^ — 0. Since Xi 7^ 0 and x^ i^ 0, we deduce from Proposition 4 that 2/ cannot cover x.
The Banks Set
197
Subcase 2: 2/4 > 0. Clearly, 3 prefers y to x. Suppose that 1 also prefers y to X (and therefore that 2 prefers x to 7/), and consider the vector z = (0, by4 + 5,1 — by4 — e, 0), where 6: is a small positive number. Then, for small enough s, z is a, feasible allocation and xTz while not [yTz]. If on the other hand, 1 prefers x to y, and therefore 2 prefers y to x, consider the vector z = (yi -\- by4 + s, 0,1 — 7/1 — 6^/4 — 5, 0) where £ is a small positive number. Then, for small enough £, 2; is a feasible allocation and xTz since ^2 7^ 0 while not [yTz]. This shows again that y does not cover x. Case 3: y = (0,2/2 5 2/3)^4)- Similar to case 2. (3) Let 6 < | . Then UC{b) - {x G X : x'^ = 0 and x^ 7^ M for any k G {1,2,3}} The inclusion UC[b) C {x G X : X4 = 0 } follows from the fact that if 6 < | , then any x such that X4 > 0 is Pareto dominated and therefore covered. Since any alternative in UC{b \ PO{b)),^^ where PO{b) denotes the set of Pareto undominated allocations is in UC{b),'^^ we deduce from Proposition 4 that UC{b) contains the set {x G X : x^ - 0 } . We now only need to prove that the vertices are not in UC{b). This follows from the fact that for any vertex x, there is no y such that xTy. To complete the proof, it remains to be shown that if 6 < | , then TC{b) = X. From (3c) in Proposition 2 we know that any alternative in the set X = {x = {xi,X2,xs,X4) G X : X4 — 0 } is connected to any other alternative in that set (the weak T-chain is in X). To conclude it remains to prove that X and X\X are connected. Let x £ X with X4 > 0. Since 6 < | , (xi + ^,0:2 -(^,xs,0)Tx. Finally, observe that xT{xi + X4,X2,X3,0).
References [1] Austen-Smith, D. and J. S. Banks (1999) Positive Political Theory I: Collective Preference Ann Arbor: University of Michigan Press. [2] Banks, J. S. (1985) Sophisticated voting outcomes and agenda control. Social Choice and Welfare^ 1: 295-306. ^^ For any A C X, UC{b \ A) denotes the Uncovered Set when the set of alternatives is restricted to the subset A. ^^ We leave the proof of this simple claim to the reader. Note however that the reverse inclusion UC C UC{PO) does not always hold i.e., while the deletion of Pareto undominated alternatives can never hurts an alternative already in the Uncovered Set, the consideration of such alternatives may help some other alternatives which would not be in otherwise!
198
Bhaskar Dutta, Matthew O. Jackson, Michel Le Breton
[3] Banks, J. S. and G. Bordes (1988) Voting games, indifference and consistent sequential choice rules. Social Choice and Welfare^ 5: 31-44. [4] Banks, J. S., J. Duggan, and M. Le Breton (2002) Bounds for mixed strategy equilibria and the spatial model of elections. Journal of Economic Theory, 103: 88-105. [5] Banks, J. S., J. Duggan, and M. Le Breton (2003) Social choice and electoral competition in the general model of elections. Journal of Economic Theory, forthcoming. [6] Bell, C. E. (1981) A random voting graph almost surely has a Hamiltonian cycle when the number of alternatives is large. Econometrica, 49: 1597-1603. [7] Bordes, G. (1983) On the possibility of some reasonable consistent majoritarian choice: some positive results. Journal of Economic Theory, 31: 122-132. [8] Bordes, G., M. Le Breton, and M. Salles (1992) Gillies and Miller's subrelations of a relation over an infinite set of alternatives: general results and applications to voting games. Mathematics of Operations Research, 17: 509-518. [9] Cox, G. W. (1987) The uncovered set and the core. American Journal of Political Science, 31: 408-422. [10] De Donder, P. (2000) Majority voting solution concepts and redistributive taxation. Social Choice and Welfare, 17: 601-627. [11] Duggan, J. and M. Le Breton (2001) Mixed refinements of Shapley's saddles and weak tournaments. Social Choice and Welfare, 18: 65-78. [12] Dutta, B,, M. O. Jackson, and M. Le Breton (2002) Endogenous agenda formation. Social Choice and Welfare, forthcoming. [13] Dutta, B. and J. F. Laslier (1999) Comparison functions and choice correspondences. Social Choice and Welfare, 16: 513-532. [14] Epstein, D. (1997) Uncovering some subtleties of the uncovered set: social choice theory and distributive politics. Social Choice and Welfare, 15: 8 1 93. [15] Farquharson, R. (1969) Theory of Voting New Haven: Yale University Press. [16] Feld, S., B. Grofman, R. Hartley, M. Kilgour, and N. R. Miller (1987) The uncovered set in spatial voting games. Theory and Decision, 23: 129-155. [17] Ferejohn, J., M. Fiorina, and R. D. McKelvey (1987) Sophisticated voting and agenda independence in the distributive politics setting. American Journal of Political Science, 31: 169-194. [18] Fey, M. (2004) Choosing from a large tournament. Social Choice and Welfare, 23: 275-294. [19] Hartley, R. and M. Kilgour (1987) The geometry of the uncovered set Mathematical Social Sciences, 14: 175-183. [20] Koehler, D. H. (1990) The size of the yolk: computations for odd and even-numbered committees. Social Choice and Welfare, 7: 231-245.
The Banks Set
199
[21] Laslier, J. F. (1997) Tournament Solutions and Majority Voting Berlin/Heidelberg: Springer-Verlag. [22] Laslier, J. F. and N. Picard (2002) Distributive politics and electoral competition. Journal of Economic Theory, 103: 106-130. [23] Lizzeri, A. and N. Persico (2001) The provision of public goods under alternative electoral incentives. American Economic Review, 91: 225-239. [24] Lockwood, B. (2002) Distributive politics and the costs of centralization. Review of Economic Studies, 69: 313-338. [25] McKelvey, R. D. (1976) Intransitivities in multidimensional voting models and some implications for agenda control. Journal of Economic Theory, 12: 472-482. [26] McKelvey, R. D. (1979) General conditions for global intransitivities in formal voting models. Econometrica, 47: 1085-1112. [27] McKelvey, R. D. (1986) Covering, dominance and institution-free properties of social choice. American Journal of Political Science, 30: 283-314. [28] McKelvey, R. D. and Niemi, R. G. (1978) A multistage game representation of sophisticated voting for binary procedures. Journal of Economic Theory, 18: 1-22. [29] Miller, N. R. (1980) A new solution set for tournament and majority voting: further graph-theoretical approaches to the theory of voting. American Journal of Political Science, 24: 68-96. [30] Miller, N. R., B. Grofman, and S. L. Feld (1990a) The structure of the Banks set. Public Choice, 66: 243-251. [31] Miller, N. R., B. Grofman, and S. L. Feld (1990b) Cycle avoiding trajectories, strategic agendas, and the duality of memory and foresight: an informal exposition. Public Choice, 64: 265-277. [32] Ordeshook, P. (1993) The development of contemporary political theory. In W. A. Barnett, M. J. Hinich, and N. J. Schofield (eds) Political Economy: Institutions, Competition, and Representation Cambridge: Cambridge University Press. [33] Penn, E. M. (2001) A distributive N-amendment game with endogenous agenda formation. Mimeo. California Institute of Technology. [34] Penn, E. M. (2003) The Banks set in infinite spaces. Mimeo. Carnegie Mellon University. [35] Persson, T. and G. Tabellini (2000) Political Economics Cambridge: MIT Press. [36] Plott, C. (1967) A notion of equilibrium and its possibility under majority rule. American Economic Review, 57: 787-806. [37] Schwartz, T. (1972) Rationality and the myth of the maximum. Nous, 6: 97-117. [38] Shepsle, K. and B. Weingast (1984) Uncovered sets and sophisticated voting outcomes with implications for agenda institutions. American Journal of Political Science, 28: 49-74. [39] Stiglitz, J. E. (1988) Economics of the Public Sector (2nd Ed) New York: Norton.
Experiments in Majoritarian Bargaining"*" Daniel Diermeier^ and Rebecca Morton^ ^ Northwestern University [email protected] ^ New York University [email protected]
Summary. We investigate the predictive success of the Baron-Ferejohn model of legislative bargaining in laboratory environments. In particular, we use a finite period version of the bargaining game under weighted majority rule where a fixed payoff" is divided between three players. We find that our subjects' behavior is not predicted well by the Baron-Ferejohn model. The model predicts hardly better than a coin flip which coalition partner is selected by the chosen proposer, and proposers allocate more money to other players than predicted. A sizable number of proposals are rejected in the first proposal periods, and subjects who vote to reject a proposal on average receive a higher payoff from the new proposal. We find that a simple equal sharing rule yields point predictions that can account for | to | of all accepted proposals.
1 Introduction T h e Baron-Ferejohn (hereafter BF) model of legislative bargaining (Baron and Ferejohn [8]) is one of t h e most widely used formal frameworks in the study of legislative politics. Variants of the model have been used in the study of legislative voting rules (Baron and Ferejohn [8]), committee power (Baron and Ferejohn [9]), pork-barrel programs (Baron [6]), government formation (Baron Support from Stanford University's Graduate School of Business and the Dispute Research and Resolution Center at the Kellogg School of Management, Northwestern University, is gratefully acknowledged. Earlier versions of this paper were presented at the Annual Meeting of the Allied Social Sciences Association, the Annual Meetings of the American Political Science Association, the Economic Science Association, and the Political Science Department at Columbia University. We appreciate the diligent assistance of Kristin Kanthak in administering the laboratory experiments and helpful comments from Jonathan Bendor, Richard Boylan, Randy Calvert, Charles Cameron, Yan Chen, John Duggan, Catherine Hafer, Nolan McCarty, the late Richard McKelvey, Thomas Palfrey, Thomas Rietz, Roberto Weber, and Rick Wilson. All errors are our own.
202
Daniel Diermeier and Rebecca Morton
[5, 7], Baron and Ferejohn [8]), multi-party elections (Austen-Smith and Banks [2], Baron [7]), and inter-chamber bargaining (Diermeier and Myerson [16]). A general analysis of the Baron-Ferejohn framework was recently presented in Banks and Duggan [3, 4]. The BF model is based on bilateral alternating offer bargaining models of Stahl [41], Krelle [24] and especially Rubinstein [38]. The BF model adapts the Rubinstein model to the type of bargaining process that can occur in legislatures by extending the number of actors and incorporating a voting rule to determine when a proposal is accepted. In all variants of the BF model, a proposer is selected according to a known rule. He/She then proposes a policy or an allocation of benefits to a group of voters. According to a given voting rule, the proposal is either accepted or rejected. If the proposal is accepted, the game ends and all actors receive payoffs as specified by the accepted proposal. Otherwise, another proposer is selected, etc.*^ This process continues until a proposal is accepted or the game ends. In many applications the game is potentially of infinite duration. That is, it can only end if a proposal is accepted.^ Consider a simple version of the model where there are three political parties with no party having a majority of the votes in the legislature. The BF model predicts that the party with proposal power will propose a minimal winning coalition consisting of him/herself and one other member, leaving the third party with a zero payoff. As in the Rubinstein model, the proposing party will give a proposed coalition partner just the amount necessary to secure an acceptance. This amount (or continuation value) equals the coalition partner's expected payoff if the proposal is rejected and the bargaining continues. Proposals are thus always accepted in the first round. Note that the proposing party will always choose as a coalition partner the party with the lowest continuation value. The division of spoils will in general be highly unequal, especially if the parties are very impatient. The BF model is attractive for the study of legislative politics, especially when the political issues have important distributive consequences for the representatives' constituencies. It is well known that simple majority voting games over redistribution lack a core. The BF model provides a useful alternative. The model explicitly formalizes the bargaining process and provides both point and comparative statics predictions about the effects of different institutions on bargaining behavior and outcomes. This allows researchers to analyze how different legislative institutions and rules may affect political behavior and outcomes. ^ A variant of this set-up allows (nested) amendments to a proposal before it is voted on. This is the case of an open amendment rule (Baron and Ferejohn [8]). For a thorough analysis of a finitely repeated variant of the Baron-Ferejohn model see Norman [30].
Experiments in Majoritarian Bargaining
203
However, the usefulness of the Baron-Ferejohn model for the study of legislatures depends on how well it explains actual bargaining behavior in political environments. Recent experimental work on simpler two-person bargaining games has shown that such bargaining models' predictions may fail to be supported in the laboratory. For example, a number of experimental studies have examined the "ultimatum game" (Giith et al. [20]) in which one player makes a proposal on the division of a fixed amount of money and the other player must either accept or reject, with rejection implying a zero payoff for both. In experiments on ultimatum games, proposers should take (almost) all of the money, yet the divisions are far more equal than predicted. Moreover, if proposers offer less than a certain amount,^ the other player frequently rejects the offer (even if the offered amount constitutes a significant amount of money) and receives a payoff of zero. Experiments on bargaining games with a series of alternating offers result in similar outcomes. Proposers offer more money than suggested by their subgame-perfect strategy, and bargaining partners consistently reject offers and forgo higher payoffs (e.g., Davis and Holt [15], Forsythe et al. [18], Giith et al. [20], Ochs and Roth [31], Roth [35]).^ Many explanations have been suggested for this anomaly. One of the first hypotheses suggested concerns flaws in the experimental design. Early bargaining experiments were criticized for their lack of anonymity between the players (e.g., Hoffman and Spitzer [22]). Hoffmann et al. [23] suggested that a similar effect might be due to the fact that the subjects' identities are known to the experimenter. Using an elaborate experimental design, Bolton and Zwick [10] show that there is no evidence that the presence of an experimenter inhibits subjects in ultimatum games. Another hypothesis suggests that players are behaving altruistically or trying to follow a fairness norm. This would imply that players are motivated by other factors than their monetary payoffs, even under experimental conditions that guarantee anonymity between players. Forsythe, Horowitz, Savin, and Sefton [18] investigated this hypothesis by comparing ultimatum and dictator games. The dictator game differs from the ultimatum game in that the proposing player proposes a division between the two players and the other player cannot reject the proposal. In ultimatum games almost 60% of the offers observed propose an equal division of payoffs. While there is still a significant percentage of equal divisions in dictator games (less than 20%), the modal division is the subgame-perfect allocation where the proposer keeps the entire payoff. This result suggests that while some of the subjects are primarily motivated by notions of fairness, the high percentage of equal divisions in ultimatum games cannot be attributed to a simple desire to be fair. ^ This amount varies from culture to culture. In experiments conducted by Roth et al. [36] the modal offer varied between 40% of the pay-ofF in Jerusalem, Israel, and 50% in Pittsburgh. ^ See Roth [35] pages 253-292 and Davis and Holt [15] pages 241-274 for reviews of this literature.
204
Daniel Diermeier and Rebecca Morton
Another suggested explanation concerns the equilibrium concept used (Forsythe et. al. [18]). In ultimatum games players did not choose subgame perfect strategies, but play was consistent with Nash equilibria. This suggests a connection to other experiments where subgame perfect equilibria are not observed in the laboratory. The most noteworthy example is the centipede game (McKelvey and Palfrey [29]). However, subgame perfect equilibria are supported in other non-trivial extensive form games. Prasnikar and Roth [34] consider a sequential model called the "best shot" public good provision game. In this game, first a player proposes a quantity of a public good that he will provide, and then a second player (after being told the quantity proposed by the first player) proposes a quantity that she will provide, with the quantity of public good provided being the maximum of the two. Both players are charged for the quantity that he or she proposes to provide, but they are only paid a redemption value based on the quantity actually provided. The perfect equilibrium prediction is for the first player to propose to provide a zero quantity, the second player proposes a positive quantity, and the bulk of the profits are then earned by the first player. Although this prediction appears to have similar unfair consequences and the potential for error and learning as in the ultimatum and other bargaining games, Prasnikar and Roth find that this prediction is supported in laboratory experiments. They suggest that the diflPerences between the two settings are due to off-the-equilibrium-path differences in the best shot game as compared with the ultimatum and other bargaining games.^ The BF bargaining game is similar to these bargaining games in the sense that proposers are expected to offer their coalition partner his/her continuation value. In the last period of a finite game, this continuation value is zero, as in the ultimatum game. Hence the non-proposing coalition partner in the last period of the BF model is like the second player in an ultimatum game. The relationship between the proposer and non-coalition member in the last period, however, is also similar to the dictator game, since the votes of the non-coalition members are not necessary to pass a proposal. The similarity between the two-person sequential bargaining games and the BF games allows us to build on the experimental evidence and methodological expertise accumulated in the years of laboratory research with twoperson bargaining. The only other experimental attempt to directly test the BF approach is McKelvey [28]. McKelvey uses a three-voter, three-alternative stochastic version of the BF model that mimics Baron and Ferejohn [8] for the closed rule case. This game has many subgame-perfect equilibria. In this case it is customary to focus on the (usually) unique stationary equilibria. Stationarity rules out any dependence of the agents' strategies on the history of play and thus avoids the multi-equilibrium problem generated by a Folk-Theorem. McKelvey concludes that the stationary solution to his game at best modestly ^ Roth and Erev [37] have recently proposed a simple learning rule to address this
Experiments in Majoritarian Bargaining
205
explains the data. Proposers usually offer too much and these proposals are accepted with too high a probability. In contrast to McKelvey, we use a finite game under weighted majority rule where a fixed payoff" is divided among three actors.^ This has methodological advantages. First, we avoid the problem of implementing (potentially) infinite games in finite time. Second, our model has a unique subgame-perfect equilibrium.^ This provides a unique model prediction without assuming the much stronger stationarity requirement. This is important, given t h a t in his discussion McKelvey suggests t h a t subjects may try to coordinate on a nonstationary equilibrium t h a t might explain the data. We can thus test the B F model in isolation, not the model with an additional equilibrium refinement. Third, the subgame-perfect equilibrium involves no randomization. This avoids one of the main problems in testing the B F approach.^^ If randomization occurs in equilibrium, one needs a large number of observations to detect a significant difference between predicted and observed frequencies (McKelvey and Palfrey [29]). Fourth, the weighted majority game allows us to test a rich set of comparative statics, not just its point predictions. Comparative statics analyses are of particular interest in testing game-theoretic models, since few game theorists would maintain t h a t the assumptions of a game-theoretic model literally apply to a particular decision problem. Indeed, the failure of experimental or field d a t a to precisely correspond to the point predictions of a model such as the exact equilibrium allocation of monetary payoffs may not be too damaging to a model as long as the qualitative predictions of the model are confirmed.-^"^ T h e next section formally introduces our version of the B F model. Section 3 discusses experimental design and predictions. Section 4 discusses our experimental results and compares our analysis to other experimental evidence on similar bargaining games. In Sections 5 and 6 we discuss alternative explanations of our empirical findings. T h e last section discusses the impUcations of our results for legislative bargaining models. ^ Baron and Ferejohn [8] discuss this case in a footnote. ^ This is not always true in the finitely repeated version of Baron and Ferejohn. Norman [30] shows that the existence of a continuum of subgame-perfect equilibria. The key is that in our model actors have different recognition probabilities. We also assume (as common in simultaneous move voting games) that voters do not use stage-game weakly dominated strategies. This requirement rules out equilibria where every agent votes the same irrespective of his/her preferences. For the rest of the paper we simply will speak of "subgame perfection." ^° This is true in all periods except for the last, which is off-the-equilibrium path. ^^ We address this issue more expansively in our discussion of the experimental analysis.
206
Daniel Diermeier and Rebecca Morton
2 T h e Baron-Ferejohn Model We consider the following version of the BF model. Assume a set N oi n players, indexed by i = l,...,n with a distribution v = (i;(l),... ,t'(n)) of votes per player. We also assume that
E^w
ieN
is odd. Assume further that there is a fixed transferable payoff x{N) to be distributed among the players. Let x{i) be the payoff assigned to z. We assume that x{i) > 0 and n
E x{i)
=^x{N).
We assume a finite multi-period sequential bargaining game with the following structure. At the beginning of each period a proposer is selected among the players according to a time-invariant selection rule. We denote the probability that player i is selected by p{i). Thus, we assume that p{i) > 0 and n
Let j be the chosen proposer. Then j ' s proposal is a vector Xj =
{xj{l),..,,Xj{n))
again with Xj{i) > 0 and
E Xj^)=x{N). A proposal is accepted if and only if it receives a majority of the votes. If a proposal is accepted, the game ends and each player's payoff is determined by the accepted proposal. If not, the game moves to the next period where again a proposer is selected according to p = ( p ( l ) , . . . ,p(n)). This process continues until either a proposal is accepted or the last period T is reached. If no proposal is accepted in the last period, every player receives a payoff of x{i) = O.ln the parlance of BF, we thus consider a finite sequential bargaining game with a weighted majority voting rule, random recognition, and a closed amendment rule. The equilibrium concept is subgame perfection.^^ For a given recognition rule we can then solve the game as follows: in the last period T the unique subgame-perfect equilibrium is for the proposer to keep the entire dollar. Given perfection, such a proposal will be accepted for ^^ Since we use a finite game there is no technical reason to discount future payoffs. Moreover, discounting would lead to less sharp predictions.
Experiments in Majoritarian Bargaining
207
sure.^"^ Note that in the last period a rejection will give each player a payoff of zero. Call this each player i's continuation value for period t: wj. In general, a continuation value equals a player's expected utility if the proposal fails and subsequent play is consistent with subgame-perfect equihbrium. So, in period T we have for all i: wJ = 0. The key to analyzing the BF model is to note that this insight generalizes to each period: in equihbrium a proposer will always make an offer to the nonproposer k with the lowest continuation value (the "cheapest" non-proposer). In particular, the proposer will offer k exactly his/her continuation value.^^ Such a proposal is expected to pass for sure. Consider a three-player version of the game with players z, j and k with x{N) = 1. For each t let wj be the largest continuation value, w^^ the second biggest, and wl the smallest in period t. We can then recursively calculate the continuation value for period t — 1 according to the following formulae.
wl'=p{k){l-wl)
+ {l-p{k))wl).
The intuition underlying these calculations is straightforward. Player i has the highest continuation value. Hence, he will only be included in a coalition if he himself is the proposer. In this case he will offer fc, the player with the smallest continuation value, exactly w^ and keep the rest of x{N) for himself. This event occurs with probability p{i). A similar argument works for j in that player j will, if the proposer, also offer k exactly w^ and keep the rest of x{N) for herself. But in contrast to i, j will also be included in the coalition if k is the chosen proposer, since for A:, a coalition with j is the cheaper of the two potential coalition members. In this case j will receive exactly her reservation value. Because k has the lowest continuation value, she will always be included in the coalition, either as a proposer, in which she receives 1 — i^^, or by being offered her continuation value by either i or j . Note that z's continuation value decreases in the next period, while /c's always increases (the direction of w^j is in general undetermined). This implies that the order of continuation values may change from period to period. Each player may be the cheapest player in some period. This leads to abrupt changes in the predicted probability of being included in a coalition from period to period. The BF model thus leads to sharp predictions both about ^^ To see, why note that the proposer j has a strictly dominant strategy to vote for his proposal. The other two voters are indifferent. So, suppose they randomize, then j could randomly choose one of the non-proposers, say z, and propose an e-amount to i. Then i will accept this proposal for sure. But this cannot be equilibrium since j would prefer to reduce e to zero. Hence, in any perfect equilibrium at least one non-proposer has to accept the proposal for sure. ^^ If there are more than one "cheapest" player the proposer will randomize.
208
Daniel Diermeier and Rebecca Morton
which coalitions will form and what the equilibrium offers will be. Both the expected coalitional composition and the equilibrium offers in general differ for different recognition probabilities and number of remaining periods.
3 Experimental Design We conducted three treatments at the University of Iowa using a large (400 plus) subject pool of undergraduate and MBA students recruited from the Colleges of Business and Liberal Arts. The treatments were conducted via a computer network on terminals. Subjects were separated by dividers and given experiment-specific ID numbers. Subjects were given copies of the instructions and the instructions were read aloud. All questions were answered in public. Subjects were also given a short quiz on the instructions to ensure that they understood the experiment.-^^ Subjects were also provided with personal history forms to record the events that occurred in each bargaining period in which they participated. The treatments used twelve subjects each, giving a total of 36. In each treatment, the subjects were randomly assigned to three-member groups. Within each three-person group, the subjects were then randomly assigned a vote total and a color: Blue, Green, or Orange. The color assigned to a subject was the label for that subject for that bargaining period and bargaining group. Thus, subjects' identities were anonymous. In each treatment we assume that the total number of votes is 99 and that 50 votes are needed to pass a proposal. In treatment 1, the three vote totals assigned were 34, 33, and 32. That is, one player was assigned 34 votes, a second 33, and a third 32. These assignments were made independently of the color assignments. In treatment 2 the three vote totals assigned were 49, 33, and 17, and in treatment 3, the three vote totals assigned were 46, 44, and 9. Subjects were told both their own color and vote total and also the colors and vote totals of the other members of their group, but not the identity (ID numbers or names) of the other members. In each proposal period, one of the group members was selected to make a proposal using known probabilities. Initially, the probability of being selected as a proposer was based on the percentage of votes assigned to the subject. That is,
In treatment 1 this roughly corresponds to recognition probabilities of ^^ A sample copy of the instructions is contained in an extended version of this paper available at http://www.kellogg.northwestern.edu/faculty/diermeier/ personal/papers/ (last visited January 20, 2004).
Experiments in Majoritarian Bargaining
209
i3 ' 3i' 3r In treatment 2 V'
1 1 1 2'3'6
and in treatment 3 we have
4^ 4^ 2.
lo''To'10 Subjects selected to make a proposal were told to allocate $45 (in integers) among the three members of their group (including themselves). They were told that they could allocate any integer amounts as long as the total sum allocated did not exceed $45. The proposal was then revealed to the three members of the proposer's group and the proposer was identified by her color assignment in the group. All group members then voted all of their votes either yes or no on the proposal. A proposal was considered accepted if the proposal received 50 or more yes votes. Thus, in all treatments, any coalition of two subjects voting yes was sufficient for a proposal to be passed. If the proposal was accepted, the bargaining game was considered over for the group. Group members were told the outcome of the vote and how each group member voted (by color assignment). In the treatments, subjects were given five proposal periods. That is, if the fifth proposal was rejected by a group, then all members of the group were given $0 payoff. The treatments differed in vote shares and recognition probabilities. After each group had either accepted a proposal or gone through the maximum number of proposal periods and rejected the final proposal, the subjects were randomly re-assigned to three-member groups again. Subjects were also randomly re-assigned vote totals and colors to minimize repeated game effects. To control for income effects, one bargaining period was chosen at random for payment.^^ There were 18 bargaining periods in each treatment; thus, there were 18 x 4 x 3 = 216 bargaining groups. As discussed above, the BF theory provides precise predictions of the allocations that will occur in the three experimental treatments. These predictions fall into different categories. The first two pertain to initial proposing behavior. They indicate the types of coalitions proposed and the precise division of payoffs in a proposal. First, we expect that a coalition will form between the chosen proposer and the subject of the remaining group members with the lower continuation value. For the three treatments the predictions can be summarized in the following table:-^^ ^^ All subjects were paid an additional $7.00 minimum show-up fee. "^^ Note that since we required the subjects to make proposed allocations in integer amounts, the continuation values are calculated for each period as the next highest
210
Daniel Diermeier and Rebecca Morton Table 1. Coalition Composition Predicted by Baron/Ferejohn Model
Treatment 1 (34,33,32) Vote Allocation Recognition Probability* (0.34,0.33,0.32) Predicted Coalitions (T==5) a —>• {a,b} or {a,c} b^ {a,b} c —> {a,c}
Treatment 2 (49,33,17) (0.49,0.33,0. 17) a ^ {a,c} b^{b,c} c -^ {a,c}
Treatment 3 (46,44,9) (0.46,0.44,0. 09) a -^ {a,c} b^{b,c} c —» {a,c}
* These values are rounded
Second, the B F model provides precise predictions concerning the allocations offered to the coalition members. A coalition partner is expected to be offered a payoff equal to her continuation value. Hence, in the first proposal period of treatment 1, if player b is chosen to make a proposal, she is expected to propose t h a t player a receive $11, keep $34 for herself, and allocate $0 for player c. T h e predictions are summarized in Table 2. Notice t h a t the coalitions, as well as t h e allocations, vary with the treatments. For example, when player b is chosen to make a proposal in the first proposal period in treatments 2 and 3, she is expected to form a coalition with player c instead of a. And while player c is expected to always choose player a as a coalition partner in treatment 1, she is expected to offer a lower allocation t h a n in treatments 2 or 3. Third, we expect t h a t the subjects will vote for a proposal if and only if it provides them with an allocation greater t h a n or equal to their continuation values. Thus, for example, in the first proposal period of treatment 1, we would expect subjects of type a will only vote for proposals t h a t offer them $11 or more. As in all empirical analyses of experiments, we generally would not expect to observe t h e exact allocations in the B F model, even in the controlled laboratory environment. It is well known t h a t some learning may take place in early periods, as subjects become more familiar with the bargaining game and the computer experimental environment. Subjects may make random errors in calculation throughout the experiment. It is thus important not only to consider point predictions, but also to consider general qualitative predictions and comparative statics analyses. integer over the expected value to a player for continuing the game to the next proposal period. As a consequence, even though in treatments 2 and 3 there are significant differences in vote shares, the proposed allocations are very similar. Moreover, in treatment 1 players b and c have the same continuation value because of the effect of the rounding.
Experiments in Majoritarian Bargaining Table 2. Coalition Allocation Predicted by Baron-Ferejohn Model, rij (i's allocation)/(j's allocation)
a b c a b c a b c a b c
Treatments 2 1 Recognition Probabilities 0.34 0.49 0.33 0.33 0.32 0.17 Proposer = a $34 $27 $0 $18* $11 $18* Proposer = b $11 $0 $34 $34 $11 $0 Proposer = c $11 $16 $0 $0 $34 $29
211 =
3 0.46 0.44 0.09 $34 $0 $11 $0 $34 $11 $17 $0 $28
*In treatment 1 player a will randomize between coalition partners b and c (whose continuation values are both $18).
4 Experimental Results Coalition Types. The first important finding concerns minimum winning coalitions. In all three treatments we find a large number of proposals that allocate money to all players ("grand coalitions"). In treatment 1 these are 40% of all allocations; in treatment 2, 41.6% of all allocations; and, in treatment 3, 30.5% of all allocations. This finding may suggest some concern for fairness or altruistic preferences among the subjects. But such a conclusion would be premature. Most of the grand coalition proposals, for instance, do not allocate money equally. ^^ Only 15% in treatment 1 and 11% in treatment 2 and 3 allocate $15 to each subject. This is far less than the percentages of equal allocations reported for the ultimatum game and even less than for the dictator game (Forsythe et al. 1994). To put it diff'erently, what could be considered fair in the BF model is far less obvious than in two-person bargaining games. It may apply to coalitional composition, to the allocation of payoffs within a coalition, or both. However, even if we restrict attention to minimal wining coalition proposals only, the BF model explains little of the observed variations. In treatment We refer to these coalitions as "grand fair coalitions." See also Fig. 1.
212
Daniel Diermeier and Rebecca Morton
1, 51% of the minimal winning coalition proposals are predicted by the BF model, in treatment 2, 55%, and in treatment 3, 46%. If we assume that a chosen proposer will always include himself in a proposed coalition, the BF-model does not predict better than a coin flip. As an obvious alternative to the BF model, one may consider the following "naive" proposal rule: "always propose to the remaining player with the lowest number of votes." We call these coalitions "simple minimal winning." Such a rule successfully predicts 42% of minimal winning coalition proposals in treatment 1, 74% in treatment 2, and 44% in treatment 3. Sometimes, however, BF and the naive proposal rule predict the same coalition. In treatment 2, for instance, both predict the same coalition, except if c is chosen as the proposer. If only comparing the cases where the two rules differ, the BF model predicts 43% of all cases correctly, the naive rule 57%. One may suspect that the number of grand coalitions declines over time as subjects become more experienced. Fig. 1, however, suggests that this is not the case. We find that the percentage of coalitions correctly predicted by the BF model increases in the early bargaining periods, but this increase is also accompanied by an increase in the simple minimal winning coalitions. Two-player proposals also increase with bargaining period. These increases seem to occur early, by the fifth bargaining period. The percent of grand fair coalitions varies but does not increase or decrease significantly with bargaining period. This finding is similar to the persistence of equal distributions in the two-person bargaining models (Ochs and Roth [31]). Proposal Power. Analyzing the observed proposed allocations yields a similar result. Consider for example treatment 1.^^ In the following figures we can see that proposers almost always allocate less money to themselves than predicted by the BF model. Note also that in all three cases, some proposers allocate a positive payoff to all players (these allocations are in the interior of the simplices). We can confirm this finding using a series of difference in means tests. The observed mean allocations are never within one standard deviation of the predicted allocation and rarely within two standard deviations. Our subjects do not exploit their proposal power as predicted by the BF model. As we discussed above, the BF model predicts that proposals will be accepted in the first proposal period if and only if the proposal is at least as high as a player's continuation value. Since there are so few allocations at or below a respondent's continuation value, we can only investigate voting behavior ^^ The distributions for treatments 2 and 3 are very similar and are omitted here. An extended version of the paper, available at http://www.kellogg.northwestern. edu/faculty/diermeier/personal/papers/ (last visited January 20, 2004), contains all figures.
Experiments in Majoritarian Bargaining Figure 1: Coalition Types by Bargaining Period
100.00%
10.00%
0.00% 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18
Bargaining Period •^" — • -
Percent Percent Percent Percent
Two Player Proposals Grand Fair Coalitions BF Predicted Coalitions Simple Minimal Winning Coalitions
Figure 2: Treatment 1 - Proposer a
Allocation to a [•Observed Allocation «> Predicted Allocation |
213
214
Daniel Diermeier and Rebecca Morton Figure 3: Treatment 1 - Proposer b
Allocation to b [•Observed Allocation 3^ Predicted Allocation |
Figure 4: Treatment 1 - Proposer c
\ 40
^"--..^^ 35
30
^^^^-^
^^\
ra
B 25
1 <
0 20
15
10
•
•
'^'•-^.^^^ "^-^^ ^\^
•
5
0
•
1 ^
,
^^^ ^^--..
• Observed Allocation Si Observed Allocation
where proposals exceed the continuation value. The BF models predict that all such proposals will be accepted. Yet, a significant percentage of first period proposals above the continuation value are rejected (from 19% in treatment 2 to 27% in treatment 1), and in some cases, groups go through a number of rejections before reaching agreement. Our findings are consistent with some of the findings from the literature on two-person sequential bargaining games.^° Proposers choose to allocate less to themselves than predicted by the theory, and rejections occur at significant rates even if proposals exceed a respondent's continuation value. However, in contrast to two-person games, where previous research has indicated a tendency for players to receive lower payoffs after a rejection, we find that subjects who voted against a proposal that was subsequently rejected on average reSee Roth [35, pages 304-323].
Experiments in Majoritarian Bargaining
215
Table 3. Mean Allocation to Proposer in First Period (standard deviations in parentheses)
a b c Predicted Observed
Predicted Observed
Predicted Observed
Treatments 1 2 Recognition Probabilities 0.34 0.49 0.33 0.33 0.32 0.17 Proposer = a 34.00 allocation to a 27.00 20.90 allocation to a 21.68 (4.8) (3.8) Proposer = h 34.00 allocation to b 34.00 22.70 allocation to b 22.20 (7.9) (4.4) Proposer = c allocation to c 34.00 29.00 allocation to c 21.80 19.40 (5.7) (3-6)
3 0.46 0.44 0.09 34.00 24.20 (5.13) 34.00 23.10 (4.8) 28.00 20.50 (2.5)
ceived a higher payoff from the subsequent proposal, and that subjects who voted for a proposal which was rejected on average received a lower payoff. Table 4 presents the mean increase in allocation from the subsequent allocation for subjects across treatments when proposals are rejected. Table 4. Mean Gain from Subsequent Proposals when Proposal is Rejected (standard deviations in parentheses)
Voted "no"
Voted "yes"
Increase in a's Allocation $6.34 (13.05) 59 obs. -$4.95 (10.62) 46 obs.
Increase in b's Allocation
Increase in c's Allocation
$5.36 (11.83) 58 obs. -$8.09 (12.84) 47 obs.
$3.26 (13.80) 74 obs. -$10.29 (9.63) 31 obs.
216
Daniel Diermeier and Rebecca Morton
5 Gamson's Law in t h e Lab - An Alternative Explanation of Majoritarian Bargaining Given the poor fit of experimental d a t a and the B F model, we need to consider alternative explanations for the observed payoff distributions. A very simple alternative is a Proportionality Rule for allocations within coalitions, T h e Proportionality Rule (PR) states t h a t the distribution of benefits within a government is proportional to relative vote shares in a coalition. This alternative explanation is particularly attractive because it is consistent with one of the most striking empirical regularities in comparative politics: the proportional division of portfolios among the parties of a governing coalition. T h e original formulation of this proposition is due to Gamson [19]. Subsequent research (e.g., Browne and Franklin [12], Browne and Fendreis [13], Schofield and Laver [40], Laver and Schofield [25]) has confirmed this conjecture to an extent t h a t one can justifiably call it Gamson^s Law?^ Gamson's Law, however, is not the only instance of proportional distributions of legislative offices. Other examples include the proportional assignments to committees in the United States Congress,^^ the distribution of committee chairs in parliamentary democracies, or the allocation of time in debates (e.g., Loewenb e r g e t al. [27]). T h e Proportionality Rule states t h a t the benefits are divided within coalitions by party vote shares. Let C C N with C 7^ 0 be a coalition of players. For every C let
be the weight assigned to coalition C. Then the Proportionality Rule states that x{i) v{i) x{N) v{C) At least two amendments suggested in the subsequent literature, however, ought to be mentioned. Browne and Fendreis [13] found a tendency of small parties to be overrepresented (by about one portfolio). Laver and Schofield [25] report that while cooperative bargaining models like the "bargaining set" (see Osborne and Rubinstein [32]) are outperformed by the simple Proportionality Rule, in some countries predictions derived from the bargaining set are slightly more accurate. Nevertheless, the basic empirical support for Gamson's Law is impressive. In a simple regression model Browne and Fendreis [13], for instance, compared the actual number of portfolios Y to a party's seat share X. Perfect proportionality would imply a regression equation of y = bo -\- biX where 60 = 0, and hi — 1. The estimated coefficients for Browne and Fendreis were Y — 0.97 + 0,83X with B? - 0.93. As in the case of portfolio distributions there are small deviations from proportionality. In some committees such as the House Committee on Rules the members of the majority party are over-represented.
Experiments in Majoritarian Bargaining
217
or
Note that PR does not have any implications on the size of C or the timing of coalition formation. That is, it only predicts the division of payoffs conditional on there being an agreement. It is important to distinguish between PR as a predictive device in a majoritarian bargaining situation and its application to a specific empirical domain such as the distribution of cabinet portfolios. That is, Gamson's Law follows as an application of PR to the case of government formation. To draw this inference one would need additional applications such that each player i represents a political party, v{i) corresponds to party z's seat share in the chamber that elects the government, and C refers to a governing coalition, etc. Finally, it is assumed that x(N) is the number of cabinet portfolios and x{i) the number of portfolios assigned to party i. In our context we are not testing any specific application such as Gamson's Law, but the usefulness of PR in generic bargaining situations. It is evident that from a methodological point of view, the BF model is superior to PR. It has a well-defined behavioral motivation, uses a general solution concept and provides a coherent framework to analyze a variety of issues in legislative politics. Moreover, it has a richer set of empirical implications since it not only predicts allocations, but also which coalitions will form. It is important to note that the Proportionality Rule can be sustained as a Nash equilibrium, albeit not a subgame-perfect one. To see why, note that the non-proposing voters can adopt the following voting strategy: reject every proposal unless it conforms to the Proportionahty Rule. Given this voting strategy, the proposer's best response is to allocate payoff's proportionally. However, many other (non-proportional) payoff allocations can be sustained as Nash equilibria in this way. PR thus could be interpreted as a focal point in majoritarian bargaining games.^'^ We can summarize the PR's predicted allocations in the following table. It will be useful to use the measure lij i— (allocation to i)/(allocation to j). Recall that PR does not rule out coalitions that are not minimal winning coalitions. Yet, if grand coalitions occur, the expected allocations vary across treatments, with only treatment 1 expected to result in grand fair coalitions. In contrast to BF, which predicts initial proposals, PR applies to accepted coahtions only. In the case of treatment 1, PR performs reasonably well in predicting accepted allocations. ^^ As the reader will note, these Nash equilibria are in general sustained by noncredible threats, since they require voters to reject allocations greater than their expected post-rejection payoffs, a strategy that is ruled out by subgameperfect ion.
218
Daniel Diermeier and Rebecca Morton
Table 5. Allocation Predictions from the Proportionality Rule (allocation for a, allocation for b, allocation for c) nj = (i's allocation)/(j's allocation) in two player coalitions Coalitions (a,b) (a,c) (b,c) (a,b,c)
Treatments 1 2 ($27,$18,$0) ($23,$22,$0) rah = 1.50 rah = 1.05 ($33,$0,$12) ($23,$0,$22) Tac = 2.75 Vac = 1.05 ($0,$23,$22) ($0,$30,$15) Uc = 1.05 rhc = 2.00 ($23,$15,$7) ($15,$15,$15) rah — 1.53 rah - 1.00
3 ($23,$22,$0) rah = 1.05 ($38,$0,$7) rac = 5.43 ($0,$37,$8) rhc = 5.43 ($20,$20,$5) rah = 1.00
Figure 5: Treatment 1 - Proportionality Rule (Accepted Allocation)
Allocation to a I • Observed Allocation
S Predicte
However, this is not the case in treatments 2 and 3. Here, the accepted allocations are consistently more egalitarian t h a n expected. We can confirm this result by conducting simple difference in means tests, which are summarized in the following table,^"^
6 A Simple Equal Sharing Rule Neither t h e Baron-Ferejohn model, nor the Proportionality Rule was very successful in explaining our subjects' behavior. A closer inspection of the dis^^ The large standard errors in treatment 1 and 3 for the case of 3-player coalitions are due to "pittance coalitions" where two players each receive $22 and one player receives $1. See also below.
Experiments in Majoritarian Bargaining
219
Figure 6: Treatment 2 - Proportionality Rule (Accepted Allocations)
Allocation to a I •Observed Allocation S? Predicted Allocation |
Figure 7: Treatment 3 - Proportionality Rule (Accepted Proposals)
•
^-.\ 40 -
•1
~-^\
35
^\^^^ ^^-\
30
B 25
1
O 20
•
15 • 10 ^
• *
^^\^
•
^^""^ -.^^
1 I• 1
I •Observed Allocations ^Predicted Allocations |
tribution of two-player accepted proposals in treatment 1-3, however, reveals a surprisingly clear pattern. In all three treatments there is little or no proposer effect. Rather, the mean allocations are close to an equal split of the $45 between the two coalition members, no matter which players compose the coalition. The good fit of PR in treatment 1 thus can be attributed to the fact that for an equal vote distribution, PR also predicts equal allocations. But, if proposers expect to share the benefits equally, they are indifferent on which coalition to pick, which may explain the apparent randomness in the choice of coalition partner. We may inspect this regularity in more detail by assessing the ability of the equal sharing rule to yield accurate point predictions. Consider the following histograms for the three treatments.
220
Daniel Diermeier and Rebecca Morton
Table 6. Table 6: Mean Allocation Ratios for Accepted Two Player Coalitions in First Proposal Periods (standard deviations in parentheses) Treatment 1 1.05 1.06 (0.29) 1.05 1.04 (0.18) 1.05 1.06 (0.17) 1.00 3.60 (6.96)
Predicted rah for {a,b}-coalition Observed Tab for {a,b}-coalition Predicted Vac for {a,c}-coalition Observed rac for {a,c}-coalition Predicted rbc for {b,c}-coalition Observed rtc for {b,c}-coalition Predicted rat for {a,b,c}-coalition Observed Tab for {a,b,c}-coalition
Treatment 2 1.50 1.10 (0.14) 2.75 1.06 (0.11) 2.00 1.10 (0.26) 1.53 1.00 (0.63)
Treatment 3 1.05 1.23 (0.41) 5.43 1.30 (0.38) 5.43 1.31 (0.28) 1.00 3.13 (5.90)
Table 7. Mean Proposer Allocations for Accepted Two Player Coalitions (standard deviations in parentheses) Coalition
{a,b}
{a,c}
{b,c}
Treatment 1 Proposer Proposer 2 1 23.3 23.8 (2.9) (1.6) n= 9 n=7 22.6 23.5 (1.6) (2.3) n= 6 n= 9 23.3 22.6 (1.5) (2.5) n = 10 n — 5
Treatment 2 Proposer Proposer 1 2 24.0 23.0 (1.3) (0.0) n-7 n= 2 23.7 2.8 (0.4) (1.0) n=- 12 n= 6 22.2 24.0 (2.8) (1.5) n=7 n = 10
Treatment 3 Proposer Proposer 2 1 22.7 25.1 (3.4) (0.6) n= 3 n=7 22.0 25.3 (3.1) (0.0) n= 2 n = 13 21.0 25.5 (1.2) (2.4) n=- 19 n == 4
Figure 8: Accepted 2-Player Coalitions - Proposer Allocations Treatment 1 (n=46)
X
0.4-f
n 0.35- ^
X
Percentage 0.25-
A'
y\
1 22-23
24-25 Proposer's Allocation
J O
U
Experiments in Majoritarian Bargaining
221
Figure 9: Accepted 2-Player Coalitions - Proposer Allocations Treatment 2 (n=43)
/
y\
^
^ 22-23
^ ^ 24-25
Proposer's Allocation
Figure 10: Accepted 2-Player Coalitions • Proposer Allocations Treatment 3 (n=48)
^
y
A
X
41
G>
/ 22-23
24-25 Proposer's Allocation
We can see that between roughly \ and | of all accepted proposals allocate either $22 or $23 to the proposer.^^ Recall that proposers are constrained to make integer proposals. Thus, they cannot split the $45 exactly into equal halves, but have to either propose either $22 or $23 to the coalition partner. Equal sharing may also explain the occurrence of some peculiar threeplayer coalitions, where two players receive $22 each and the third player receives $1. In treatment 1 about 42% of all three-player coalitions are such "pittance coalitions," in treatment 3, 25%, while there are no pittance coalitions in treatment 2, which exhibits by far the highest percentage of equal sharing allocations. Since the total payoff ($45) cannot be allocated equally in a two-player coalition, players appear to "discard" $1 (i.e., allocating it , ^^ These percentages are of the same magnitude as the number of equal allocations in ultimatum games.
222
Daniel Diermeier and Rebecca Morton
to the third player) rather than proposing an unequal allocation, even if the allocations differ only by a single dollar! Applying the equal sharing rule to three-player coalitions predicts fair coalitions. In treatment 1, 42% of all accepted three-player coalitions are fair, in treatment 2, 28%, in treatment 3, 33%. If we eliminate pittance coalitions from the set of accepted three-player coalitions, the percentages are 73% in treatment 1, 28% in treatment 2, and 44% in treatment 3.
7 Discussion The subjects' behavior in our experiment is not predicted well by the BaronFerejohn model. Rather, a simple equal sharing rule where subjects share the available amount equally among the coalition members is highly consistent with the data. While this may suggest some intrinsic fairness motivation of our subjects, such a conclusion would be premature. As we discussed above, equal sharing may also serve as a simple focal point. Finally, subjects may be overwhelmed by the complexity of the experiment and rely on simple rules of thumb. We discuss each of these possibilities in turn. Fairness. The equal sharing rule and large number of three player coalitions seems to suggest that players are motivated by factors other than monetary rewards, even under experimental conditions that guarantee anonymity (both with respect to other players and the experimenter). While there is ample evidence that in some experiments players do exhibit behavior that suggests they are following certain fairness norms, our results indicate that the precise nature of these norms is far from obvious. In our experiment, for instance, we saw that almost half of all first-round proposals allocated money to all three players, but only 11% to 15% allocate $15 to all subjects. Moreover, while the suggested equal sharing may be consistent with some notion of fairness, such "fairness" would be restricted to the chosen coalition. The excluded player receives a payoff of zero.^^ Interestingly, players did not consider different vote shares to be important reference points for deciding the allocation. That is, a proportionality rule based on vote share did not explain the data well. Any explanation of Gamson's Law that is based on some "proportionality norm" is not supported in our experiment. ^^ This is consistent with the findings of GUth and van Damme [21] who add a dummy player to the standard ultimatum game who receives a zero pay-off, if the game is played sufficiently often. Fairness norms may also depend on the context. Hoffmann et al. [23], for instance, show that in ultimatum games the percentage of equal distributions decreases if the right to propose is not allocated randomly, but earned, e.g., through a prior test or contest. For recent theoretical accounts of equity, reciprocity, and spitefulness see Levine [26] as well as Bolton and Ockenfels [11] and Fehr and Schmidt [17].
Experiments in Majoritarian Bargaining
223
Alternatively, one may suspect that the occurrence of fair coalitions may be due to a small number of players that persistently propose fair allocations. "Selfish" proposers, for example, then should strategically adapt their strategies to the presence of voters with fairness concerns. However, while it is true in our experiments that some individuals are more likely to propose egalitarian allocations, there is not a single player in our sample who always exhibits this kind of behavior. Rather, even players who predominantly propose fair allocations occasionally propose two-player coalitions. Conventions and Focal Points. An alternative explanation may be based on the rejection of perfection as a useful equilibrium concept.^'' In this case there are many Nash equilibria, and the question is why players coordinate so frequently on the one that splits the payoff" equally. One way to address this question is to interpret the equal distribution as a focal point or a convention (Schelling [39]). But what would rationalize an equal sharing rule? One approach would be to use solutions concepts from cooperative game theory. The equal sharing rule is, for instance, consistent with the Aumann-Myerson value (Aumann and Myerson [1]) in games with endogenous coalition structures. In such games players first non-cooperatively form links between each other (in our case, choose a coalition) and then allocate payoffs within linked sets using the Shapley value. Based on our experimental data we cannot decide between these alternative accounts. However, one could easily design experiments with more than three players that would separate, e.g., the Aumann-Myerson value from the Stable Aspiration Set, and both from a fairness norm. Cognitive Limitations and Learning. A final interpretation of our results is that subjects did not suflficiently understand the incentives in the game. Hence, they relied on simple "rules of thumb" that serve them well outside the laboratory. One experimental strategy that would alleviate this problem is to have a computer calculate each subject's reservation value and then make this information available to subjects. However, experiments that have used this technique find that subjects rarely use this information even if it is easily available (e.g., Camerer [14]). Another possibility is to investigate the presence of learning by subjects as they gain experience with the experiment. However, similar to previous multi-stage bargaining experiments (e.g., Ochs and Roth [31]), we find no evidence of learning effects in our data. It is, of course, possible, that the game is of sufficient cognitive complexity that even multiple rounds of play will not provide sufficiently diverse feedback for subjects to learn. Note, however, that even if the cognitive complexity ^^ Another possible explanation of our findings could rely on the fact that our subjects make non-trivial errors in their proposing and voting choices. If the possibility of errors is common knowledge among the players, then Nash Equilibrium is not the appropriate solution concept. Rather we would have to solve for Bayesian or Quantal Response Equilbria (McKelvey and Palfrey [29]). The latter approach has the advantage of usually identifying a unique equilibrium.
224
Daniel Diermeier and Rebecca Morton
hypothesis is correct, our finding of equal sharing rules may be important, because most real bargaining or voting situations are highly complex, and thus actors may indeed rely on simple sharing rules as cognitive short-cuts. Thus, we should investigate how robust equal sharing rules are in environments that are easy to understand.
8 Conclusion Our laboratory experiments of the Baron-Ferejohn model of sequential majoritarian bargaining indicate that the model has little success in predicting our subjects' behavior. Grand coalitions occur at a significant rate and proposers are not more likely to include coalition partners with lower continuation values. In addition, proposers fail to sufficiently exploit their proposal power and allocate too much money to their coalition partners. Voting behavior also differs from the behavior of the Baron-Ferejohn model. About a quarter of proposals are rejected, although a majority of players is offered more than their continuation value. A simple proportionality rule based on Gamson's Law of cabinet formations predicts reasonably well, as long as the expected allocations are not too unequal. For unequal allocations, the fit is poor. The best account of the subjects' behavior is provided by a simple sharing rule where the proposer chooses any winning coalition and then distributes the pay-off equally among the coalition members. This finding is consistent with various explanations: (i) players incorporate additional fairness concerns in their voting and proposing behavior, (ii) the equal sharing rule may constitute a focal point in majoritarian bargaining games, or (iii) it may be the consequence of using a simple rule of thumb in a complex decision environment.
References [1] Aumann, R. J. and R. B. Myerson (1988) Endogenous formation of links between players and of coalitions: an application of the Shapley value. In A. E. Roth (ed) The Shapley Value Cambridge: CUP. [2] Austen-Smith, D. and J. S. Banks (1988) Elections, coalitions and legislative outcomes. American Political Science Review, 82: 405-422. [3] Banks, J. S. and J. Duggan (2000) A bargaining model of collective Choice. American Political Science Review, 94: 73-88. [4] Banks, J. S. and J. Duggan (2003) A bargaining model of legislative policy-making. Mimeo. University of Rochester. [5] Baron, D. P. (1989) A noncooperative theory of legislative coalitions. American Journal of Political Science, 33: 1048-1084.
Experiments in Majoritarian Bargaining
225
[6] Baron, D. P. (1991a) Majoritarian incentives, pork barrel programs and procedural control. American Journal of Political Science^ 35: 57-90. [7] Baron, D. P. (1991b) A spatial theory of government formation in parliamentary systems. American Political Science Review, 85: 137-165. [8] Baron, D. P. and J. A. Ferejohn (1989a) Bargaining in legislatures. American Political Science Review, 89: 1181-1206 [9] Baron, D. P. and J. A. Ferejohn (1989b) The power to propose. In P. C. Ordeshook (ed) Models of Strategic Choice in Politics Ann Arbor: University of Michigan Press. [10] Bolton, G. E. and R. Zwick (1995) Anonymity versus punishment in ultimatum bargaining. Games and Economic Behavior, 10: 95-121. [11] Bolton, G. E. and A. Ockenfels (2000) ERG: a theory of equity, reciprocity, and competition. American Economic Review, 90: 166-193. [12] Browne, E. G. and M. N. Franklin (1973) Aspects of coalition payoffs in european parliamentary democracies. American Political Science Review, 67: 453-469. [13] Browne, E. G. and J. Fendreis (1980) Allocating coalition payoffs by conventional norm: an assessment of the evidence for cabinet coalition situations. American Journal of Political Science, 24: 753-768. [14] Gamerer, G. F. (2003) Behavioral game theory: experiments in strategic interaction Princeton: Princeton University Press. [15] Davis, D. D. and G. A. Holt (1993) Experimental Economics Princeton: Princeton University Press. [16] Diermeier, D. and R. B. Myerson (1994) Bargaining, veto power, and legislative committees. Mimeo. Northwestern University. [17] Fehr, E. and K. M. Schmidt (1999) A theory of fairness, competition, and cooperation. Quarterly Journal of Economics, 114: 817-868. [18] Forsythe, R., J. L. Horowitz, N. E. Savin and M. Sefton (1994) Fairness in simple bargaining experiments. Games and Economic Behavior, 6: 347-369. [19] Gamson, W. (1961) A theory of coalition formation. American Sociological Review, 26: 373-382. [20] Giith, W., R. Schmittberger and B. Schwarze (1982) An experimental analysis of ultimatum bargaining. Journal of Economic Behavior and Organization, 3: 367-388. [21] Giith, W., E. Van Damme (1998) Information, strategic behavior and fairness in ultimatum bargaining: an experimental study. Journal of Mathematical Psychology, 42: 227-247. [22] Hoffman, E. and M. Spitzer (1982) The Goase theorem: some experimental tests. Journal of Law and Economics, 25: 73-98. [23] Hoffman, E., K. McGabe, K. Shachat and V. L. Smith (1994) Preferences, property rights and anonymity in bargaining games. Games and Economic Behavior, 7: 346-380. [24] Krelle, W. (1976) Preistheorie (Part 2) Tubingen: J. G. B. Mohr (Paul Siebeck).
226
Daniel Diermeier and Rebecca Morton
[25] Laver, M. and N. Schofield (1990) Multiparty Government: the Politics of Coalition in Europe Oxford: Oxford University Press. [26] Levine, D. (1996) Modehng altruism and spitefulness in experiments. Mimeo. UCLA, [27] Loewenberg, G., S. C. Patterson and M. Jewell (1985) Handbook of Legislative Research Cambridge MA: Harvard University Press. [28] McKelvey, R. D. (1991) An experimental test of a stochastic game model of committee bargaining, in T. R. Palfrey (ed). Laboratory Research in Political Economy. Ann Arbor: University of Michigan Press. [29] McKelvey, R. D. and T. R. Palfrey (1995) Quantal response equilibria for normal form games. Games and Economic Behavior^ 7: 6-38. [30] Norman, P. (2002) Legislative bargaining and coalition formation. Journal of Economic Theory, 102(2): 322-353. [31] Ochs, J. and A. E. Roth (1989) An experimental study of sequential bargaining. American Economic Review, 79: 355-384. [32] Osborne, M. J. and A. Rubinstein (1994) A Course in Game Theory Cambridge MA: MIT Press. [33] Prasnikar, V. and A. E. Roth (1989) Perceptions of fairness and considerations of strategy in bargaining: some experimental data. Mimeo. University of Pittsburgh. [34] Prasnikar, V. and A. E. Roth (1992) Considerations of fairness and strategy: experimental data from sequential games. Quarterly Journal of Economics, 107: 865-888. [35] Roth, A. E. (1995) Bargaining experiments. In J. Kagel and A. E. Roth (eds) The Handbook of Experimental Economics Princeton: Princeton University Press. [36] Roth, A. E., V. Prasnikar, M. Okuno-Fujiwara and S. Zamir (1991) Bargaining and market behavior in Jerusalem, Ljubljana, Pittsburgh and Tokyo: an experimental study. American Economic Review, 81: 10681095. [37] Roth, A. E. and I. Erev (1995) Learning in extensive-form games: experimental data and simple dynamic models in the intermediate term. Games and Economic Behavior, 8: 164-212. [38] Rubinstein, A. (1982) Perfect equilibrium in a bargaining model. Econometrica, 50: 97-109. [39] Schelling, T. (1960) The strategy of conflict Cambridge MA: Harvard University Press. [40] Schofield, N. and M. Laver (1985) Bargaining theory and portfolio payoffs in european coalition governments, 1945-1983. British Journal of Political Science, 15: 143-164. [41] Stahl, I. (1972) Bargaining Theory Economics Research Institute at the Stockholm School of Economics, Stockholm.
Legislative Coalitions in a Bargaining Model with Externalities Randall L. Calvert^ and Nathan Dietz^ ^ Washington University in St. Louis calvertOwustl. edu ^ Corporation for National and Community Service ndietzOcns. gov
1 Introduction The recent literature on political parties in Congress features a renewed interest in the nature of parties and new ideas for understanding parties in terms of the political motivations of legislators. Several analysts (including Aldrich [1], Cox and McCubbins [10], Kiewiet and McCubbins [12], and Rohde [14]) have suggested that parties are designed by legislators to overcome complex problems of delegation, collective action, or coalition building in the legislature, and discussed the design of parties, the behavior of members in the context of legislative parties, and the effects of such parties on legislative structure and legislative outcomes. By and large, these studies have constructed or sketched models of a fairly complex legislative process within which the delegation, collective action, and coalition building problems occur. A few authors, however, have tackled the problem of party organization among rational legislators in different ways. Baron and Ferejohn [6] take a more stripped-down approach to the modeling of the legislative process, portraying it simply as a series of distributional offers proposed by individual members and accepted or rejected by a vote of the legislative body. In this context, Baron [4, 5] proceeds to develop the idea of a party as an agreement allowing a given coalition to control legislative outcomes at the expense of nonmembers of the coalition. In an entirely different vein, Krehbiel [13] questions whether we should interpret legislative party as a phenomenon separate from simple rational behavior of individual legislators with similar preferences. He argues that we cannot draw conclusions about the effect, and effectiveness, of parties simply from the observation that legislators having consistently similar goals exhibit consistently similar voting choices; those legislators would be expected to vote similarly anyway, and the proponent of party significance in legislative outcomes must show that parties play a role beyond simply naming a group of
228
Randall L. Calvert and Nathan Dietz
members with similar preferences. Krehbiel believes that parties are significant only to the extent that "individual legislators vote with fellow party members in spite of their disagreement about the policy in question" Krehbiel [13, page 238]. It is possible, however, that the independent eff'ect of parties lies instead in the nature of the bills that are presented for voting, rather than in the voting decisions of legislators, as suggested by Snyder's account [15] of the basis for apparent unidimensionality in roll-call voting. If we want to resolve the question of whether parties have a significant effect on legislative outcomes, independently of legislator preferences, it is necessary to explain how individual members not organized by parties would generate bills for voting as well as how those members would vote. The purpose of this paper is to contribute to such an explanation. Baron and Ferejohn's [4] initial portrayal (see also Baron and Ferejohn [6]) of legislation as an extended bargaining process is not suitable for addressing the issues raised by Krehbiel, since legislative bills in those models are matters of pure distribution: each proposal would simply divide a fixed "pie" among the members, and each legislator cares only about the size of his or her own slice. In that setting, there can be no interesting patterns of preference similarity. Even so. Baron identifies a motivation for legislators to form parties in that setting: by being a member of a coalition whose members always make proposals favoring the coalition at the expense of nonmembers, the individual gains a higher expected payoff. Aldrich [1] discusses this same motivation in a different theoretical context; such considerations go back ultimately to the reasoning of Buchanan and TuUock [7, chapters 9-11] that majority rule allows a majority coalition to exploit a minority coalition. Baron [5], however, extends the "bargaining" model of legislatures to the choice of positions in a multidimensional policy space, over which the participants have quadratic preferences. The stated purpose of that model is to examine the process of coalition formation among parties in a parliamentary government; the same model would in principle be ideal for examining proposals and voting patterns in any legislature. With such a model, one could depict patterns of similarity in the policy goals of legislators, and determine the extent to which such legislators might exhibit party-like similarities in voting without the benefit of any real party mechanisms to control their voting or bill-proposing behavior.^ Unfortunately, calculations in the Baron model become intractable in all but the simplest settings. Baron is able to derive closed-form solutions for stationary equilibria of proposal and voting strategies only for certain very restrictive special cases having only three or four legislators and a specific layout of ideal points. In this paper, we propose a model that retains a bit ^ Other authors have considered further extensions: Jackson and Moselle [11] analyze party formation when, in addition to distribution, a one-dimensional policy is chosen; Banks and Duggan [2, 3] generalize the Baron-Ferejohn framework to general multidimensional policy spaces.
Legislative Coalitions in a Bargaining Model with Externalities
229
more of the simplicity of the pure-distribution setting but still allows for the depiction of preference similarities among legislators. This model allows us to begin to distinguish conceptually between true party-driven legislative behavior and the purely preference-driven, faux-pdiity behavioral similarities recognized by Krehbiel. Our main accomplishment in this paper is to clarify Krehbiel's idea of what ought to be the baseline from which party significance is assessed, using a model that incorporates proposals of bills, as well as voting, among otherwise unorganized rational legislators.
2 T h e Baron-Ferejohn Model of Legislative Politics In their original bargaining model of legislative politics. Baron and Ferejohn [6] set up a simple legislative game having two distinguishing features. First, the only task of the legislature is to decide on the division of a fixed stock of resources among its members. All members value their own allocation of this stock in the same way; none receives any utility from resources allocated to other members. Second, legislative procedure is explicitly modeled as a process of recognition of a member to propose a division of the stock, followed by voting on that proposal. If a proposal passes, the legislators realize their payoffs and the game ends. If not, the game continues with another round of recognition and voting. Baron and Ferejohn consider several variants on this simple procedure in order to study the effects of closed rules, amendment processes, and the like. For present purposes, it suffices to consider the simplest version of their model.^ The Baron-Ferejohn legislature consists of a set of members L = { 1 , 2 , . . . , n] where n — 2m — 1 for some integer m, so m is the size of a minimal majority and n is an odd number. An outcome of the legislative process is any distribution x = ( x i , . . . , x^) of shares of a fixed stock of resources, with each X >0 and ^ ^ Xj — 1. Each legislator i receives utility from a distribution X simply equal to Xi. The game form consists of the following parliamentary procedure "stage game," repeated if necessary: 1. A legislator is recognized at random to propose a distribution. In this recognition process, legislator i is recognized with probability p^, with each p > 0 and J2LPJ ~ ^' 2. The legislature votes on the proposal, with all members voting simultaneously. 3. If a majority votes for the proposal, the game ends and players receive their payoffs; otherwise, the stage game is repeated starting at (1). Note ^ Aside from the two-period example that they introduce for illustrative purposes. The model described here is the potentially infinite-period, "closed-rule" model with no discounting.
230
Randall L. Calvert and Nathan Dietz that the legislator who made the failed proposal is eligible to be recognized again.
Baron and Ferejohn define a stationary strategy Si for legislator i in this game as consisting of: 1. a proposal rule, describing what proposal x^ = (x^,... ,x^) player i will make whenever recognized; and 2. a voting rule, such as "Vote in favor of any proposal having Xi > y, and against any other," which specfies how the player will vote on any proposal. Such strategies are "stationary" in that they are not contingent on the history of play in the game, that is, on previous unsuccessful proposals, who made them, or how various members voted on them. (Of course, nonstationary strategies are possible too; we will have more to say about these subsequently.) A stationary equilibrium in this game is a profile of stationary strategies s — ( s i , . . . , s^) for which, for each z, 1. i could not get a better expected outcome by proposing some other x** at any point; 2. i could not get a better expected outcome by voting otherwise on any proposal at any point; 3. i's strategy is not weakly dominated (due to its voting rule specifying a vote that would harm i if i's vote were pivotal, in an outcome in which i is in fact not pivotal); and 4. conditions (l)-(3) hold in every possible subgame (subgame perfection). Baron and Ferejohn prove that the following properties hold for any stationary equilibrium: 1. the proposing player always offers x'j = 1/n to each player in some randomly chosen group of m — 1 players, keeps 1 — (m — l ) / n for herself, and offers the others zero; 2. each player i always votes for any proposal offering i at least 1/n and against any other proposal; 3. in equilibrium, the first proposal made is always accepted; 4. ex ante, each player can expect to receive a payoff of 1/n; the player selected as the first proposer ends up receiving 1 — (m — l ) / n 5. any minimal winning coalition is as likely to form as any other. In particular, property (5) indicates that there are no legislative parties in this game in any sense.
Legislative Coalitions in a Bargaining Model with Externalities
231
3 Parties in Bargaining Models Baron and Ferejohn [6] also examine briefly what can happen in their model when strategies are not stationary. They begin by assuming t h a t players discount by some small factor any payoffs to be received from the next proposal, should the present one fail. W i t h this proviso Baron and Ferejohn are able to prove a "folk theorem:" any distribution of the stock of resources can be supported as the outcome of some equihbrium strategy profile, provided t h a t the discounting is sufficiently light and t h a t n > 5. A distribution x is implemented through a strategy profile t h a t punishes any player who proposes any other distribution y 7^ x, as follows: some designated majority M{y) votes against the deviant proposal y; the next proposer is to make a new proposal z{y) t h a t gives all members of M{y) what they would have gotten in y, plus a share of what the deviant would have gotten, and gives the deviant zero; and if t h a t second proposer fails to propose z{y)^ the same sort of punishment is then directed at him, and punishment of the initial deviant is abandoned. Although Baron and Ferejohn do not examine parties explicitly in their paper, this result suggests the possibility t h a t a sort of legislative party might be formed, since it is possible to implement a distribution in which a particular coalition would with certainty receive all of the stock of resources. We return subsequently to the related notion t h a t a legislative party consists of a particular repeated-game equilibrium. In a subsequent paper Baron [4] examines a related model t h a t generates party-like behavior in a bargaining legislature, again restricting attention to stationary strategies. In this paper,^ Baron assumes t h a t members of a coalition can commit perfectly in advance to making a certain proposal if recognized. By using this ability to commit to a proposal t h a t , although it still may favor the proposer, distributes the remainder of the benefits only to members of the coalition, a majority "party" can ensure most benefits for itself. In this equilibrium, however, if a coalition non-member is recognized, t h a t proposer will be able to pick off some coalition votes and get her proposal passed. Even so, as Baron shows, there is an advantage to being a member of such a coalition.^ Baron also demonstrates t h a t if the players discount future ^ As in the discussion of Baron and Ferejohn [6], we confine attention to the closedrule case, although Baron also examines a version in which legislators can propose amendments. Within the closed-rule setting, Baron examines cases in which coalition members do and do not have the ability to commit themselves to "party discipline" in voting. Under party discipline, when a non-member makes a proposal, coalition members will vote for it only if it gives every coalition member the share he or she would get from a proposal by another coalition member. Here we examine only the non-discipline case, since it is most relevant to the question at hand; in this case, a non-member need not gain the votes of all coalition members simultaneously in order to pass a proposal. ^ Baron's main result, and his purpose in this model, is to show that the optimal coalition in this setting will, generally, have more than m members. This is be-
232
Randall L. Calvert and Nathan Dietz
payoffs to differing extents, some coalitions will offer a higher payoff advantage to their members t h a n other coahtions will to theirs; in this sense, some groupings of legislators would make more sense as a "majority party" t h a n others. However, this sort of legislative party lacks a feature of primary importance to parties in the real world: since all they are doing is dividing a pie, there is no particular policy-based reason for a legislator to prefer some coalition partners to others. In contrast, one normally thinks of parties as consisting of members who want something in common, and in contrast to party non-members. In "A Spatial Bargaining Theory of Government Formation in Parliamentary Systems," Baron [5] examines one further approach to understanding party formation through a bargaining model. Here, rather t h a n dividing a fixed stock of resources, the players are engaged in choosing a point in a policy space in which all have quadratic preferences; thus they are restricted in the ways they are able to divide the available policy benefits. Although the analysis is couched in terms of forming a coalition government in a parliamentary system, it speaks to legislative proposing and voting in exactly the same way as t h e model in Baron [4], with one important advantage: due to t h e relative locations of their ideal points, the legislators in a spatial bargaining model have more reason to coalesce with some of their colleagues t h a n with others. T h u s there could, in principle, be parties whose members have prior, common interests inherently opposed to those of non-members, beyond simple distributional jockeying. Unfortunately, calculation of equilibrium outcomes in the spatial bargaining model turns out to be extremely complicated. Baron is able to derive a closed-form solution only for the following special cases: (1) three players whose ideal points are co-linear; (2) three players whose ideal points are all equidistant from one another, each having a 1/3 chance of recognition; (3) three players whose ideal points form the vertices of an isosceles triangle, with the player whose ideal point is equidistant from the other two never being recognized to propose; (4) four players, three at the vertices of an equilateral triangle and one at the center of the triangle, all having equal probability of recognition.^ T h e results indicate t h a t legislators whose ideal points are centrally located receive higher payoffs t h a n others, and t h a t legislators usually propose so as to a t t r a c t the vote of the nearest colleague, but in equilibrium must sometimes propose so as to a t t r a c t the vote of a more distant colleague instead. cause optimal coalition size must balance how thinly coalition proposals spread their benefits against the probability that a proposer outside the coalition will be recognized, resulting in a lower ex ante expected payoff to coalition members. Baron also examines a couple of three-player cases in which the members make proposals in a sequence of recognition that is fixed in advance, a setting more relevant to the formation of parliamentary coalition governments than to the formation of voting coalitions on a bill.
Legislative Coalitions in a Bargaining Model with Externalities
233
4 A Bargaining Model with Preference Similarities In order to gain additional analytical leverage on the problem while retaining the possibility of varying preference similarities among legislators, consider a variation on the Baron-Ferejohn, pure-distribution bargaining model of the legislature. Begin with the Baron-Ferejohn setting of a legislature L in which each member i is recognized with probability pi to propose a distribution of a fixed stock of resources of size 1. Now, however, let us broaden our assumption about the utility to a member of a particular distribution x of the stock, as follows:
where an = 1 and, for i :^ j^ aij < 1. Thus the share received by a player may affect the payoffs of other players to varying extents. The direct interpretation of this payoff function is that each legislator cares not only about his or her own allocation of the stock, but also to some extent about the shares of some other members. This would be the case, for example, if the good being distributed were not purely private but rather had spillover effects. A utility function of this form could also result from other constructions. For example: (1) Legislators wish only to gain the approval of interest groups in their districts. Benefits are "distributed" not to legislators but to interest groups, with different legislators having diff'erent concentrations of the groups in their prospective election-winning coalitions at home. This utility function represents simply an adding up of the distributive payoffs to the various interest groups, weighted by the prevalence of that group in member i's constituency. (Obviously the model could be broadened to allow a the number of interest groups to differ from the number of legislators.) (2) This utility function approximates a spatial model. The a^j are inverse distances between legislators' positions in some issue space, and the legislators' indifference curves are straight lines. (Again, obviously the dimensionality of the issue space could be made different from the number of legislators.) Since this utility function is simpler in form than the quadratic preferences examined by Baron [5], there is reason to hope that it will yield more extensive results, allowing us to say more about the nature of legislative parties. Thus far we have been able to obtain some improvement on Baron's results by deriving a closed-form solution to a general three-player case. In what follows, we present an illustrative special case, and then proceed to the analysis of the general three-legislator situation with symmetrical spillovers (a^j — aji always). For simplicity, we assume throughout the remainder of this paper that the legislators' probabilities of recognition are equal: Pi = 1/n.
234
Randall L. Calvert and Nathan Dietz
4.1 An Illustrative Case Let L = {1,2,3} and suppose that the only spillover effect is a symmetrical one between legislators 1 and 2: that is, ai2 = a2i = a for some a G [0,1], while ai3 = asi — a^z — 0,32 = 0. Thus for an allocation x = (xi,X2,X3), the players would receive the following payoffs: ui{x) = xi -{- ax2 U2{x) = a X i + X 2 ^3(0;) = 0:3 == 1 - x i - 0:2.
Several of the main considerations of the Baron-Ferejohn model are important for solving this new game. First, given any profile of strategies for proposing and voting, we can calculate a continuation value for each player (i.e., the expected payoff from the remainder of the game as seen before anyone is recognized to make the next proposal). Second, in order to have her proposal accepted, a player i must gain the vote of at least one other player j by offering j a large enough allocation to offset the possible gain to j from defeating i's proposal and perhaps being able to make j ' s own, more favorable proposal on the next round. Third, it never makes sense to offer a partner an allocation any amount beyond that just large enough to attract his vote. In equilibrium, exactly one player will make a proposal, and that first proposal will be accepted by a majority; off the equilibrium path, a subgame-perfect equilibrium will always feature majority acceptance of the next proposal. Now, however, because of the spillover effect, the third player may also gain utility from the proposed allocation. As we will see, this incidental gain will in general not be sufl&cient to gain the third player's vote as well. As in the previous analyses, then, the proposals in a stationary equilibrium can be calculated by setting each proposal exactly equal to the continuation value of the targeted player, and then solving for the proposed allocations and for each player's mixed-strategy probability of offering to each of the other players. We first solve for the type of equilibrium that corresponds most closely to those derived by Baron and by Baron and Ferejohn: a stationary equilibrium in which all players use mixed strategies in a fashion that respects the symmetries assumed in the payoff functions. Accordingly, assume that legislator 3, if recognized, attempts to gain the vote of either legislator 1 or 2 with equal probability, and that in either case 3 offers the same allocation, w^ to the intended coalition partner (and zero to the other player). Players 1 and 2, meanwhile, will attempt to attract one another's support with probability p, and with probability 1—p will attempt to gain the support of player 3. When player 1 tries to attract player 2's support, she offers him an allocationof y (and nothing to player 3), and vice versa. When player 1 or 2 tries to attract the vote of player 3, they offer z to player 3 and nothing to one another. These assumptions are suflficient to set up equations yielding a particular equilibrium to the game. First of all, the players' continuation values are as
Legislative Coalitions in a Bargaining Model with Externalities
235
follows: for player 1, Vi = -[pui{l
- y,y,0) + {1 - p)ui{l - z,0, z)] +
- [pui (y, 1 - y, 0) + (1 - p)ui (0,1 - z, z)] + -txi(tt;,0,1 — w) + -t^i(0, to, 1 — w) where the three main terms correspond to the possibilities that player 1, 2, or 3, respectively, will be making the next offer, and the two terms within each set of brackets corresponding to the two different pure-strategy actions that could be taken by that proposer. Substituting for ui in this equation and simplifying yields V^=:{l-z
+ pz^ w/2){l -f a ) / 3 .
Due to the symmetries in this case, V2 has the same formula. Using a similar method of calculation, F3 - [ 2 ( 1 - p ) ^ + 1 - ^ 1 / 3 . Now each offer must yield the intended voting partner his continuation value. This means that the following equations must hold: for player I's offer to player 2, ^2(1 -y,2/,o) = V2, that is, y-^a{l-y)
= {l-z
+ pz + w/2){l + a ) / 3 .
(1)
The same constraint applies to 2's offer to 1. For I's or 2's offer to 3,
that is, z= [2{l-p)z
+ l-w]/3.
(2)
Finally, 3's offer to 2 or 1 must satisfy ui{w,0,1 — w) = Vi, that is, w=^{l-z
+ pz + w/2){l -h a ) / 3 .
(3)
Furthermore, the use of mixed strategies implies that each player must be precisely indifferent between offering to either of the other two. This yields the following constraints: for player 1, and equivalently for player 2,
236
Randall L. Calvert and Nathan Dietz ui{l-y,y,0)
=ui{l
-
z,0,z),
that is, 1 — z — 1 — y-\- ay, or equivalently, z = {1 — a)y. For player 3, the mixed-strategy constraint is always satisfied given our assumption that 3 offers the same allocation to 1 or 2 and that the two have the same spillover effect on 3 (namely, none). Equation (4) immediately gives a solution for z in terms of y. Since (1) and (3) have the same right-hand side, we can set their left-hand sides equal to get w = y -\- a ( l — y) = a -\- {\ — a)y. Substituting for w and z in terms of y in (2) and (3) and solving simultaneously for y and p, then, gives the following values of the strategy parameters in terms of a: p={l^a)/2 2/ = l/(c^ + 3) ^ = ( l - a ) / ( a + 3) w = a -{- z and of course Vi = V2 = w and V2, = z. Notice that the value of being the proposer \s 1 — w for player 3 and is 1 — z for players 1 and 2. A little calculation (using the assumption a G [0,1]) shows that for each player the payoff from being the proposer is strictly greater than the continuation value, so, as initially assumed, each player will prefer to make these successful offers when recognized rather than take the chance of passing along the proposalmaking opportunity to another player. Since this condition is indeed satisfied, the solution is an equilibrium when combined with the voting strategy, "vote for any proposal that yields at least the continuation value." The outstanding feature of this solution is that, when the value of a is reasonably large, legislators 1 and 2 will usually propose allocations that favor each other, so the apparent winning coalition will usually consist of those two. Consider some numerical examples. First, a = 0 is the baseUne case equivalent to the Baron-Ferejohn model. In that case, p = 1/2, and w = y = z = 1/3: the proposer chooses each of the other players with equal probability as a coalition partner, and offers that player 1/3 of the "pie," keeping 2/3 for herself Baron and Ferejohn [6, pages 1191-92]. In this case, each of the three possible two-player coalitions forms the majority for passage with probability 1/3. If we introduce a modest amount of spillover between legislators 1 and 2, those two players begin to favor one another in their proposing behavior. Suppose a = 1/4. Then the computed solution gives p = 5/8, w = 25/52, y — 4/13, and z = 3/13. Legislators 1 and 2 offer to form the voting coalition with one another slightly more often than with legislator 3; they offer each other
Legislative Coalitions in a Bargaining Model with Externalities
237
slightly more than they offer legislator 3, and, when legislator 3 is recognized to propose the allocation, she has to offer even more in order to attract the vote of 1 or 2. In this case, {1,2} provides the majority for passage with probability 5/12, and {1,3} and {2,3} form the majority with probability 7/24 each. A higher level of congruence between the preferences of players 1 and 2 makes them truly predominate as the winning coalition. Suppose oc — 1/2. Then with probability p = 3/4, legislator 1, when recognized, proposes to allocate 2/7 to legislator 2 and keep the rest for herself, and legislator 2 does the same. The proposer receives a payoff of 5/7 + (1/2) (2/7) or 6/7 from this arrangement, and the coalition partner receives 9/14. Overall, {1,2} form the majority for passage with probability 1/2. With probability 1/4, legislator 1 or 2 will make the offer to legislator 3 instead, offering only 1/7 to get 3's vote, and realizing a payoff of 6/7 while 3 gets 1/7 (the left-out legislator, 2 or 1 respectively, realizes a payoff of 3/7 - less than his continuation value, but, if we allow an interpersonal comparison of utility, more than 3 gets!). As a increases toward 1, the probability that {1,2} is the winning coalition increases toward 2/3. However, in the extreme, if the payoffs of players 1 and 2 are equal (a == 1), then they always propose to one another, and when legislator 3 is recognized, she can gain the vote of 1 or 2 only by offering one of them the whole pie - in which case both of them vote in favor of the proposal. This resembles the behavior of the "majority party" in Baron [4], in that members of the {1,2} coalition propose exclusively to one another, but here their behavior involves no kind of commitment, enforced or otherwise, to propose only to party colleagues or to vote only for party proposals. The partylike behavior is due simply to individual rational behavior in the presence of strong similarities of interest. Fig. 1 graphs the probability of {1, 2} being the winning coalition (simply 2p/3 or (1 -h <^)/3) against the value of a.
p r o b of 1
{1,2}
0.8 0.6 0.4 0.2 0.2
0.4
0.6
0.
a
value
Fig. 1. Probability that {1,2} is the winning coalition as a function of a.
238
Randall L. Calvert and Nathan Dietz
4.2 T h e G e n e r a l T h r e e - L e g i s l a t o r M o d e l Now consider the general three-legislator case with equal probabilities of recognition and symmetric spillovers. T h a t is, L = { 1 , 2 , 3 } , Pi = 1/3, ai2 = a2i = a, a23 = as2 = /?, and ais = asi = 7. To identify the set of stationary, fully mixed-strategy equilibria we can use essentially the same method as in the special case above, and as used by Baron and Ferejohn and by Baron. Let p be legislator I's probability of offering to player 2, let q be player 2's probability of offering to player 1, and let r be player 3's probability of offering to player 1. Finally, let Xij be the share t h a t player i proposes to allocate to player j . T h e continuation values can be expressed as follows: 3Vi = p[l - (1 - a)xi2] + (1 - p)[l - (1 - 7)^13] + q[a + (1 - a)x2i] + (1 - q)[a + (7 - a):r23] + r[-f + (1 - 7)x3i] + (1 - r)[7 -f ( a - 7)x32] 3^2 = p[a + (1 - a)xi2] + (1 - p)[a + {p - a j x i s ] + q[l - (1 - a)x2i] + (1 - g)[l - (1 - /3)X23] + r[p + (a-/?)x3i] + (l-r)[/3+(l-/3)x32]
3^3 - p[7 + iP-
7)^12] + (1 - p)b + (1 - 7)^13] -f
q[P + (7 - /?)a;2i] + (1 - q)[P + (1 - /?)x23] + r [ l - (1 - 7)0:31] + (1 - r ) [ l - (1 - P)X32]T h e requirement t h a t an offer intended to solicit a player's vote yield t h a t player his continuation value defines six continuation-value constraints t h a t the equilibrium must satisfy: a + (1 - a)a;2i = Vi
(4)
7 + (1 - 7)x3i - Vi
(5)
a + (1 - a)xi2
= V2
(6)
P+{l-P)xs2
= V2
(7)
P+{1-P)X23^V3
(8)
7 + (l-7)^i3-V^3.
(9)
T h e indifference constraints for mixed strategies are: l-(l-a)a:i2 = l-(l-7)^13 I - {1 - a)x21 = 1 - (l - I3)X23 1-(1-7)0:31 =l-(l-/?)a:32.
(10) (11) (12)
Legislative Coalitions in a Bargaining Model with Externalities
239
Using MAPLE, a computer program for symbolic manipulation of equations, we obtained the following solution to the system of equations 4-9 and equations 10-12:
-a + /3 + 7 —1 f 7 —1
{a — l)r
\—a —/3-h7H-l
- a + /? + 7 - l
r/
IX
,
o^
(l-/3)(a-/3 + 7 - l ) '
""'^^ X\3
^31)5
(l-/?)(a-/? + 7 - l ) ' (7 - l)D
2:21
(l-7)(-Q-/g + 7+l)' (a - l)D
X2Z
(l-7)(-a-/? + 7+l)^ (/? - 1 ) ^ ( l - a ) ( - Q + /? + 7 - l ) 2
X3\
a;32
(7-1)^ (l-a)(-a + ^ + 7-l)2 (/3-l)D
where i:> = a^ + /?^ + 7^ - 2a/? - 2a7 - 2/?7 + 2a + 2/3 + 27 - 3 - ( - a + /? + 7 - l ) ' - 4 ( l - / 3 ) ( l - 7 ) . Notice that this equilibrium has one free parameter among the probability variables; here, it is taken to be the probability r. As is always the case for equilibria in the Baron-Ferejohn model, in this model particular values of a, /?, and 7 often yield a whole family of fully mixed equilibria (parameterized here by r), whose members differ in the probabilities with which each member makes an offer designed to attract the vote of each other member. Across the equilibria in this family, the offer values Xij and the continuation payoffs Vi are unchanging. Changing one of the mixed-strategy probabilities while keeping the Vi constant necessitates offsetting changes in the other mixed-strategy probabilities. This effect is most easily illustrated using the equilibrium in the case of P = ^ = {) computed above; there we assumed in effect that r = 0.5, but
240
Randall L. Calvert and Nathan Dietz
different values of r would also have generated equilibria, albeit with different values for the other mixed-strategy probabihties (which would no longer be equal). In particular, letting r take on different values gives the following generalized formulas for p and q: p = (a + l)r and q = {a + 1){1 — r). Thus a higher probability of 3 offering to 1 is offset by a higher probability of 1 offering to 2 and a lower probability of 2 offering to 1. Note that, unlike what happens in the Baron-Ferejohn model, there are limits to how far we can vary r: if r lies outside the interval [I/{ex + 1), a/{a + 1)] then either p or q will no longer be a valid probability. Similar limits apply to our general fully mixed equilibrium. 4.3 Existence of Fully Mixed Equilibria Contrary to what happened in either the Baron-Ferejohn model or in the equilibrium we computed for the special case of /? = 7 = 0, fully-mixed equilibria in our general game may fail to exist for all values of r; and for certain values of a, /?, and 7, there may be no fully mixed equilibrium at all.^ This happens when the offer values or mixed-strategy probabilities given by the formulas above are outside the admissible range [0,1]. To get an idea of the combination of parameter values for which fully mixed equilibrium exists, consider what happens when a = p. Inspection of the formula for p above reveals that when a = /?, p is a decreasing linear function of 7, equal to one when 7 = (1 — a)r 4- 2a — 1 and equal to zero when 7 = (1 — a)r + a. Thus 7 must lie between these two values in order to yield a valid p for the mixed strategy. When a = /3, g is a slightly more complicated function of 7, having a singularity at 7 = 2a — 1. Below that value, however, q is always greater than 1, so we can focus on 7-values above the singularity point. In that region, q is increasing; it reaches the value 0 at 7 — (1 — a ) r - f 2a — 1 and asymptotes to 1 (unless a = /? = 1, in which case q = I identically). Thus we get a valid probability q in all cases in which p is valid. We can do similar calculations for the values of the offers made by the players, which also must lie between 0 and 1, inclusive. When a = P^ X12 and 0:32, which are identical, lie in [0,1] for all values of 7 that yield valid g-values. The values of xis and X31, which are identical, are positive only when 7 is above the singularity point of 4a — 3; there, the offer values are monotonically declining in 7, reaching 1 when 7 = 3a — 2, and asymptoting to 0. Thus any 7 above 3a — 2 generates valid offer values. Finally, X21 and X23 are identical and equal to a rather complicated fraction of polynomials even when a = /3. However, it is still easy to calculate that these two offers are valid exactly when 7 meets two conditions: first, 7 > 4a — 3 (which is less than the lower bound for X13 and X31); and second, 72 + (3a -f 1)7 + 3a — 2 < 0. The latter An example of the latter phenomenon occurs when a — 0.1, /3 = 0.5, 7 == 0.9, where the computed value for q is negative for all values of r G [0,1].
Legislative Coalitions in a Bargaining Model with Externalities condition, required when OL and 7 are and vice-versa; and greater t h a n (\/3 -
241
for the offer values to be no greater t h a n 1, is satisfied not too large; for example, if 7 = 1 then ot must be zero, if a, /?, and 7 are all equal, then their value must be no l ) / 2 , or 0.37.
T h e upshot of all this is t h a t , when a = /? < 1/3, fully mixed equilibria exist only when 7 is not too large; and if a = /3 > 1/3, 7 must be neither too large nor too small. T h e more general lesson is t h a t if any of the three parameters is large, t h a t is, if any of the legislators' consumption externalities is great, then all externalities must be similar in magnitude in order for the fully mixed equilibrium to exist. T h e leeway available to these three parameter values also depends upon the value of the free mixed-strategy probability r t h a t is used: higher values of a and /? require lower values of r in order to allow a maximum-sized range for 7.^ This general behavior can be illustrated through numerical analysis of the formulas for fully mixed equilibria. If all three parameters take on moderate or small values, there is more leeway for them to differ. For example, if a = 0.05, /? = 0.25, and we consider equilibria with r = 0.75 (which is among the most favorable r-values) then all values of 7 from zero up to 0.8 generate valid strategies.^° If a = 0.25, /3 = 0.5, and r — 0.5, then 7 values from about 0.1 to 0.75 generate valid strategies, only a slightly smaller range.-^^ If a == 0.7 and /? == 0.8, valid strategies result only when 7 is between about 0.58 and 0.9 even for the most favorable r-values, around 0.25.^^ Numerical analysis indicates that the limiting factor is the generation of p- and qvalues between 0 and 1; in all the examples we have constructed with a ^ [5 ^ ^, whenever p and q are valid probabilities, the values of all the also valid. We have been unable to prove this as a general fact, however, and it seems likely that in extreme cases one or more of the offer values could be independently constraining. ^° For smaller values of r, the range of 7 generating valid strategies is more restricted and often more complicated. For r-values of 0.5 or less, valid strategies result from sufficiently small or sufficiently large 7 values (although always only for 7 < 0.8), but not for values in the middle. When r = 0.25, for instance, valid strategies result from 7 of 0.4 and below, near 0.8, but not in the neighborhood of 0.6. For all r-values, 0.8 is the exact upper limit of 7-values allowing fully-mixed equilibrium. ^^ Here too, for some r the range of allowable 7 may be in two pieces. For r = 0.1, 7-values around 0.67 fail to generate valid strategies, although 7 from zero up to that neighborhood, and also between that neighborhood and 0.75, do generate valid strategies. For all r-values, 0.75 is the exact upper limit of 7-values allowing fully mixed strategies. ^^ For such large values of a and (5, only small values of r seem to cause the abovementioned bifurcation of the range of on which fully-mixed equilibrium exists. At r = 0.1, fully mixed equilibrium exists for all 7 from 0.53 to 0.9, except for a tiny interval around 7 = 0.85. For higher r-values, the range of compatible 7 becomes a single interval that shrinks as r increases; for r — 0.9, fully mixed equilibria exist only for 7 between about 0.78 and 0.9. Whatever the value of r, the exact upper bound on compatible 7 is 0.9.
242
Randall L. Calvert and Nathan Dietz
4.4 Other Stationary Equilibria in the General Three-legislator Game We have examined the possibility that stationary, partially mixed or pure strategy equilibria might exist in the general three-legislator model. No such equilibria exist in the Baron-Ferejohn model except as extreme members of the family of fully mixed equilibria, that is, equilibria that satisfy all the continuation-value constraints and mixed-strategy indifference constraints, but that happen to have one or more of the probabilities equal to 0 or 1 (in the Baron-Ferejohn model, necessarily either all or none of the probabilities would be strictly between 0 and 1.) The existence of externalities in our model complicates the matter, and we have been unable thus far to prove analytically that such additional equilibria do not exist apart from the fully mixed class, that is, equilibria that are not fully mixed and do not satisfy all the constraints 4-9 and constraints 10-12. On the other hand, we have also tried but failed to identify any such partially mixed stationary equilibria. Such equilibria would have to obey any relevant constraints from the full set above. In addition, an equilibrium pure strategy (i.e., a strategy that specifies making offers exclusively to one of the other players) for any player would have to satisfy the condition that there is no way for that player to offer to the other potential partner instead, giving that new partner at least his continuation value and retaining enough "pie" to be better off than if she had offered to the partner originally specified by the pure strategy. Although we can identify many pure and partially-mixed strategies that satisfy the relevant constraints, all have thus far failed this equilibrium test. The possibility remains that there may be non-stationary equilibria even for the values of a, /?, and 7 incompatible with stationary, fully mixed equilibrium. If the "folk theorem" of Baron and Ferejohn is any indication, there will in fact be many non-stationary equilibria. 4.5 "Parties" Under Fully Mixed, Stationary Equilibrium We can now examine the appearance of consistent proposing and voting blocs in the general three-legislator model, generalizing our earlier description of party-like behavior in the illustrative case where two of the externality parameters were set to zero. In the family of fully mixed equilibria for any given a, /3, and 7, the continuation payoffs for all three players are fixed. However, by and large a player sharing stronger externalities with others enjoys a higher expected payoff. Moreover, if one pair of players shares stronger externalities than the others, that pair will tend to form the winning coalition more often than the others. As a result, in repeated legislative interactions, that pair of players may behave as though they were an imperfect party, even though they
Legislative Coalitions in a Bargaining Model with Externalities
243
have no internal organization beyond their mutual expectation about stationary equilibrium behavior. There is really no effect of party as such; rather, the predominance of one voting coalition, and its members' tendency to propose policies they mutually desire at the expense of the non-member, is purely the result of individual myopic, rational action. To illustrate these effects clearly, we consider a few specific cases in which the externality within two of the legislator pairs is identical and relatively small, and that in the other pair is relatively large. First, suppose that a — 0.5 while /? == 7 = 0.2. These parameter values allow a family of equilibria parameterized by r, where r ranges from about 0.273 to 0.727. As Table 1 indicates, the probability with which 1,2 forms the winning coalition is 0.458 for any value of r, while the other two winning coalitions appear with probabilities between 0.242 and 0.299; larger r are offset by larger p and smaller q. The situation is really symmetric from the standpoint of legislator 3 - she has the same level of shared interest with each of them. If we set r = 0.5 accordingly, then we see that legislators 1 and 2 make their offers to one another with probability 0.688 each. Once again, this resembles (imperfectly) the behavior of "parties" in Baron [4], but involves no kind of commitment to propose to party colleagues or to vote only for party proposals. r 0.273 0.3 0.4 0.5 0.6 0.7 0.727
P 0.375 0.413 0.550 0.688 0.825 0.963 1.000
Q
1.000 0.963 0.825 0.688 0.550 0.413 0.375
Pr(l,2) 0.458 0.458 0.458 0.458 0.458 0.458 0.458
Pr(l,3) 0.299 0.296 0.283 0.271 0.258 0.246 0.242
Pr(2,3) 0.242 0.246 0.258 0.271 0.283 0.296 0.299
Table 1. Probability of each winning coalition, as a function of the equilibrium mixed-strategy probabilities, when a = 0.5 and /3 = 7 — 0.2.
Table 2 illustrates how this party-like behavior varies when we vary the strength of the externality between 1 and 2 and the difference between it and the strength of the externality between them and legislator 3. Either increasing the externality between 1 and 2 or decreasing the common externality between 1 and 3 and between 2 and 3 increases the expected frequency with which 1,2 appears as the winning coalition. When a = 0.9 while /? = 7 = 0.1, this probability reaches as high as 0.63. For those externality levels, legislators 1 and 2 are offering to each other with probability 0.944 each; when they do offer to legislator 3, they allocate him only 0.029, and when 3 proposes, he has to allocate 0.917 of the "pie" to legislator 1 or 2 in order to gain their vote.
244
Randall L. Calvert and Nathan Dietz a P= 7 0.2 "05" 0.2 0.6 0.2 0.7 0.2 0.8 0.1 0.8 0.2 0.8 0.3 0.8 0.4 0.8 0.5 0.8 0.6 0.8 0.7 0.8 0.1 0.9
Pr(l,2) 0.458 0.500 0.542 0.583 0.593 0.583 0.571 0.556 0.533 0.500 0.444 0.630
Pr(l,3)-Pr(2,3) 0.271 0.250 0.229 0.208 0.204 0.208 0.214 0.222 0.233 0.250 0.278 0.185
Table 2. Probability of each winning coalition, as a function of the externality parameters, when 0 = j and r = 0.5.
5 Discussion The above examples indicate that party-like behavior may in principle be a consequence of nothing more than preference similarity. When two players are more similar in preferences to one another and less similar to the third, as indicated by the externality parameters, the similar players propose "bills" that favor each other disproportionately often, and vote so as to form the eventual winning coalition with a higher probability than do other possible coalitions. When the relative similarity between the two players is large, this party-like effect is fairly dramatic. If an analyst were to observe roll-call votes from repetitions of such legislative situations, and the two similar players had been previously labeled as a "party," the frequency with which they vote together might fool the analyst into thinking that some significant form of party action underlay their voting. If the analyst also had a means of observing the proposals made and how favorable those proposals were to the various legislators, he might be even more inclined to invoke party as an explanation of legislative outcomes. Keith Krehbiel would rightly object. The observed voting patterns would be completely the result of legislators having similar preferences about bills. And the party analyst could not respond that this is only because parties controlled the proposal process, generating proposals that exploited certain divisions among legislators while papering over others. Rather, even the partyfavoring proposals in the modeled world are generated by legislators acting independently, solely in their own interests, without any organizing, gatekeeping, or sanctioning by a party hierarchy.
Legislative Coalitions in a Bargaining Model with Externalities
245
We have no intention of arguing, however, t h a t parties are not important, even in t h e U.S. Congress, even perhaps at the modern nadir of party voting in the 1970s. T h e models we have begun to analyze here serve to clarify the baseline for attributing of significance to parties, combining the analysis of endogenous proposals and voting as pioneered by Baron and Ferejohn [6] with the possibility of varying preference similarities, which, as Krehbiel [13] makes clear, are basic to evaluating the independent effect of party. It is entirely possible t h a t a version of our model with a realistic number of legislators would require much larger differences between within- and across-group externalities in order to generate high probabilities of the majority "party" providing the winning coalition; perhaps only extreme preference similarities and differences would generate the kind of false party effects t h a t Krehbiel discusses, and the appropriate baseline level of party voting would be much lower. There is no reason to conclude t h a t preferences alone will be sufficient to account for legislative outcomes as we observe them in modern American and Americantype legislatures. T h e main promise of the model we present, however, lies in clarifying this party baseline in a context in which real parties themselves can also be modeled in a fully endogenous manner. Institutions such as legislative parties can be productively studied through models t h a t portray the institution as a feature of equilibrium behavior in an appropriately chosen underlying game Calvert [8]. This might be accomplished in the one-bill legislative bargaining game studied here, or even better, in a model of repeated legislative sessions. Baron [4] merely assumes t h a t members of his parties can commit themselves to proposing only bills t h a t concentrate all benefits on the party. As we have seen, extreme preference similarities could lead to behavior substantially like this even in stationary equilibrium.-^^ But real partiesas-equilibrium-institutions would go further t h a n this, employing sanctions across stages of the game (nonstationary equilibria) to enforce a higher level of party-favoring proposals t h a n could be assured by preference similarities alone. In such an equilibrium, a party member who fails to propose as prescribed could be cut out of the coalition in future iterations and replaced by a party non-member waiting in the wings, similar to the device used in ^^ Since a selection among multiple equilibria is necessary to reach this state of affairs in our model, one might be tempted to say that even our stationary equilibrium represents a basic sort of institution. But we would have to maintain the Krehbiel no-real-party line in this case: as our examples show, in the ideal case of one large externality and two smaller ones all of the equilibria exhibit the same degree of party-like behavior; moreover, failure to coordinate on one of these equilibria would present no real problem, since there is no need to identify and punish deviation from the equilibrium path. The result, in observational terms, would simply be added noise in the frequency of the off- "party" coalitions and in the frequency of intra- and inter-"party" offers made by individual legislators.
246
Randall L. Calvert and Nathan Dietz
the Baron-Ferejohn "folk theorem" to enforce an arbitrary pattern of benefit distribution.^^ The examination of repeated-game models of real party behavior is a matter for future research. For now, based on what we have learned from the present model, we are inclined to elaborate on Krehbiel's definition of "significant party behavior" in the following terms Krehbiel [13, page 236]: Legislatively significant parties, theoretically speaking, are nonstationary equilibrium institutions in the legislative game that use retaliation and coordination mechanisms to make members propose and vote in a manner contrary to what they would do in a stationary equilibrium.
References [1] Aldrich, J. H. (1995) Why Parties? Chiceigo: University of Chicago Press. [2] Banks, J. S. and J. Duggan (2000) A bargaining model of collective choice. American Political Science Review^ 94: 73-88. [3] Banks, J, S. and J. Duggan (2003) A bargaining model of legislative policy-making. Mimeo. University of Rochester. [4] Baron, D. P. (1989) A noncooperative theory of legislative Institutions. American Journal of Political Science^ 33: 1048-1084. [5] Baron, D. P. (1991) A spatial bargaining theory of government formation in parliamentary systems. American Political Science Review^ 85: 137164. [6] Baron, D. P. and J. A. Ferejohn (1989) Bargaining in legislatures. American Political Science Review, 83:1181-1206. [7] Buchanan, J. M. and G. Tullock (1962) The Calculus of Consent Ann Arbor: University of Michigan Press. [8] Calvert, R. L. (1995) Rational actors, equilibrium and social institutions. In J. Knight and I. Sened (eds) Explaining Social Institutions Ann Arbor: University of Michigan Press. [9] Calvert, R. and J. Fox (2000) Effective parties in a model of repeated legislative bargaining. Mimeo. Washington University. [10] Cox, G. W. and M. D. McCubbins (1993) Legislative Leviathan Berkeley: University of California Press. [11] Jackson, M. and B. Moselle (2002) Coalition and party formation in a legislative voting game. Journal of Economic Theory^ 103:49-87. [12] Kiewiet, D. R. and M. D. McCubbins (1991) The Logic of Delegation Chicago: University of Chicago Press. [13] Krehbiel, K. (1993) Where's the party? British Journal of Political Science, 23: 235-266. ^^ Some progress along these lines, incorporating a more appealing account of the retaliation against unfaithful party members, is reported in Calvert and Fox [9]
Legislative Coalitions in a Bargaining Model with Externalities
247
[14] Rohde, D. W. (1991) Parties and Party Leaders in the Postreform House Chicago: University of Chicago Press. [15] Snyder, J. M. (1992) Committee power, structure-induced equilibria and roll call votes. American Journal of Political Science, 36: 1-30.
Testing Theories of Lawmaking Keith Krehbiel^, Adam Meirowitz^, and J o n a t h a n Woon^ ^ Stanford University krehbielQstcinf ord.edu ^ Princeton University ameirowiOprinceton.edu ^ Stanford University [email protected]
S u m m a r y . Tests of formal models of legislative politics have become increasingly common, and have tended to draw confident and positive inferences about focal theories. This is not a particularly satisfactory development, however, inasmuch as the supposedly supported theories are quite different from one another, and the tests that generate the support tend overwhelmingly to focus on one theory rather than competing theories. We develop and employ a method of comparative theory-testing using estimates of cutpoints on final passage results. The findings are inconclusive in part because the theories, while substantively different, are often operationally nearly observationally equivalent.
1 Introduction Explaining the policies t h a t result from collective choice in legislatures has been a prominent objective of positive political theory. Two p a t h s of complementary research have been traversed. Social choice theory a t t e m p t s to identify stable policies by focusing on characteristics of legislative preferences and the choice space. Noncooperative game theory more explicitly relies upon characterizations of institutions to identify regularities in behavior. Jeffrey Banks made fundamental contributions to both important strands of theory. Prom the social choice perspective, a few seminal papers have clarified the mathematical structure of policies t h a t are supported by stable coalitions. Plott [28] demonstrated t h a t the existence of stable policies is critically dependent on symmetry conditions. McKelvey and Schofield [27] established t h a t these conditions generically do not hold as long as the policy space is of sufficiently high dimensionality. Banks [6] provided a more precise treatment of the problem, correcting the results from earlier papers, and Saari [32] closed the question by providing tight dimensional bounds. Institutionalism and noncooperative game theoretic approaches began to flourish in the late 1970s and early 1980s, and Banks, even in the infancy of his
250
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon
career, was an immediate, accelerating force. By recognizing and exploiting the fact that there is a modicum of structure even in environments where social choice theory implies the absence of stable policies. Banks [5] characterized the set of policies that can emerge as undominated equilibrium outcomes from amendment agendas. In characterizing what has become known as the Banks Set, the article "Sophisticated voting outcomes and agenda control" illuminated the interplay between sophisticated behavior and a fixed agenda structure, thereby paving the way for the subsequent focus on characterizing the types of agendas developed endogenously. Thereafter, Austen-Smith [4] considered sophisticated voting and optimal agenda construction and showed that for an empirically plausible type of agenda formation, sincere voting and sophisticated voting are observationally equivalent. Similarly, Banks and Gasmi [9] shed light on both strategic proposing and voting. Most recently, Banks and Duggan [7] pushed on to the frontier while also circling back to the past. They consider a large class of endogenous agenda problems and find important relationships between solution concepts in social choice and equilibria in bargaining games. Concomitant with these developments, a related but more applied literature progressed in the area of legislative studies. Scholars intent on understanding the role of parties in Congress developed and occasionally tested models of endogenous agenda formation. Aldrich [1, 2], Aldrich and Rohde [3], Shepsle and Weingast [34], Weingast [38], and Krehbiel and Meirowitz [25], for example, all analyzed models similar to that of Banks and Gasmi but with an explicit status quo. Within this set of paradigmatically similar works, the range of implications and testable hypotheses regarding rights and power within legislatures is diverse. For example, one particularly lively debate concerns whether political parties are first- or second-order forces in the shaping of law. One set of researchers argues that the majority party in Congress exerts disproportionate control over legislative choices (see, for example. Cox and McCubbins [14], Cox and McCubbins [15], Rohde [31], Sinclair [35]). An opposing stance is taken by an assortment of researchers whose theoretical and empirical work suggests that the evidence for party strength in Congress is overstated and that nonpartisan theories provide better first-order accounts than partisan theories of systematic patterns in the data (e.g., Brady and Volden [11], Krehbiel [22], Schickler [33]). Only with rare and noteworthy exceptions, such as Banks and Duggan [8], do researchers appreciate the intricate and crosscutting relationships between social choice, noncooperative game theory, institutionalism, nonpartisan and partisan models, and jointly theoretical and empirical research. The connections can be portrayed roughly as follows. In the debate within legislative studies on the role of parties, the two sides are, in eff*ect, competing endogenous agenda models. The models, furthermore, are unidimensional special cases within the class of models that Banks and Duggan consider. From a theoretical perspective, the question of whether legislative policy making is
Testing Theories of Lawmaking
251
better characterized by strong or weak parties is a question about the type of aggregation rule(s) and the distribution and sequence of proposal rights in the bargaining problem. In the party cartel model, Cox and McCubbins contend that the majority party leadership can veto any and all changes from the status quo that the party dislikes. In the pivotal politics model, Krehbiel contends that either of two different agents — a veto pivot or a filibuster pivot — has a key, constraining influence on outcomes. Both models implicitly assume that amendment rules are sufficiently open that core convergence is possible. However, because of super majority requirements and more extreme veto rights, core convergence is not assured. Adjudicating between the two competing models is methodologically difficult for several related reasons. Chief among these is the challenge of distinguishing preference effects from party effects (Krehbiel [20], McCarty et al. [26], Snyder and Groseclose [37]). Other problems involve difficulties in measuring preferences (Clinton and Meirowitz [12, 13], Groseclose et al. [17], Heckman and Snyder [18], Pool and Rosenthal [29]), diflSculties in interpreting results based on conventional measures, such as "party voting" and "party cohesion" (Krehbiel [23]), and, not least, the near observational equivalence of the theories in question (Cox and McCubbins [15], Krehbiel [24]). Eventually, tests of these simple models may also need to confront an even more serious challenge: how meaningful are comparisons between specific models — say partisan versus nonpartisan — if neither such model explains a great deal of legislative behavior? One reason for this may be that a single dimension is insufficient for characterizing the lawmaking environment, whereas both pivot and cartel theories explicitly rely on this assumption. Of course, views on the dimensionality of the congressional policy space and how it should be portrayed in models are unsettled, too. Social choice theoretic results clearly demonstrate the analytical significance of dimensionality. Practical significance may be another matter, however. For example, recent applied research in legislative studies has tended to rely on unidimensional models, in part because Poole and Rosenthal find that a large quantity of roll call voting in Congress can be reduced statistically to one primary dimension of conflict. This finding is not without qualification or controversy either (see Clinton and Meirowitz [12], Heckman and Snyder [18], Snyder [36] among others), yet most empirically testable models of legislative choice remain unidimensional. Furthermore, such models are tested under the implicit assumption that legislative choice is also unidimensional. The danger of a heavy reliance on unidimensional models is that, even if preferences across dimensions are highly correlated, proposals and voting behavior in a higher dimensional world may be dramatically different than proposals and voting behavior in a one-dimensional world. This suggests that most direct horse-race tests between two theories that have not been shown to depart from a credible null model run the risk of pro-
252
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon
ducing a Pyrrhic victor. Exerting great effort to select the better of theories A and B when neither is demonstrably superior to null hypothesis C is comparable to an optometrist being compulsive about the astigmatism numbers in a prescription for Stevie Wonder's sunglasses. Mindful of the relationship between the competing specific models of legislative policy making and the much larger subsuming class of endogenous agenda models, we approach the associated obstacles and caveats as follows. We not only extract falsifiable predictions from the cartel and pivot models, as others have done, but also test them against an even simpler unidimensional model of endogenous agenda formation—a chamber operating under an open rule agenda with majority rule. This latter model predicts that the median legislator's ideal point will be enacted. This approach leads to some modest but unique results. First, while the cartel, pivot, and median theories have proven observationally similar, tracing out predictions about the types of cutpoints that should be observed leads to a straightforward, potentially-discriminating test of the theories. The theoretical overview in the next section identifies the situations in which the predictions diverge. Thereafter, use of DW-Nominate cutpoints and ideal points allow for a seemingly straightforward test between three competing theories: median voter (the baseline model), party cartel, and pivotal politics. In light of the discussion above and some additional subtleties introduced below, however, the test is not adequate for making confident inferences. Specifically, we note that this test may not place the theories on an equal footing and also that their predictive abilities seem neither very impressive nor very diff'erent from one another. As a result, we devise another test in which we compare their predictions to those of a naive null model. Again, the findings are inconclusive. None of the unidimensional models perform particularly well compared to the naive random null model based on prediction rates. Nonetheless, visual inspection of the distribution of cutpoints reveals more clearly that gridlock-interval theories have an empirical basis. Deviations from normally distributed cutpoints occur, as hypothesized, in both theories' gridlock intervals. A concluding discussion attempts to add coherence to the admittedly mixed findings and to forecast future work in this area with a sense of optimism befitting of the honoree of this volume.
2 Models Broadly construed, endogenous agenda models may be considered bargaining games in which agents select a policy from some set of alternatives. An exogenous rule assigns individuals the right to make proposals in particular periods. In each period, before a decision is reached, either a proposal is made or a binary vote is taken. The game form specifies the voting rule that is used for
Testing Theories of Lawmaking
253
each decision. Banks and Duggan [8] present a class of models in which the reversion policy may be strictly preferred to some of the feasible alternatives and demonstrate that the pivot model is a member of this class. Clearly, the cartel model is as well. The cartel and pivot models have much more in common analytically than their substantive conclusions suggest. As noted above, both models are unidimensional, and each model has a specified pair of actors whose ideal points define a gridlock interval inside which the status quo policy is predicted to remain unchanged. Therefore, each theory partitions the policy space into different intervals that nevertheless have common qualitative properties. For status quo points sufl3ciently extreme on either side of the policy spectrum, in equilibrium, the new policy gravitates to the ideal point of the median voter and is, therefore, centrist. However, for status quo points that are centrally located, gridlock occurs. In contrast, the diff'erences between the theories are fewer in number, more subtle, yet of great substantive significance. In a nutshell, the party-centered cartel theory predicts that, at any given time, public policy can be significantly out-of-step with what the median legislator wants, while in a world of pivotal politics, relatively moderate (hence, representative) policies will prevail. Fig. 1 illustrates these expectations. The horizontal axis represents policy space and underscores the fundamental dependence of outcomes on status quo. The top part of the horizontal axis is the characterization of gridlock interval under cartel theory, while the bottom portion represents pivot theory's gridlock interval.
the the the the
Consider cartel theory first. There are two stages of the game, as explicated by Cox and McCubbins [15]. First, an agent of the majority party decides whether to report a proposal to the entire legislature, or whether instead to exercise "negative agenda power" (a.k.a. "gatekeeping"). Second, if the party leader makes a proposal, then the entire legislature chooses a final policy under an open rule. Accordingly, let m represent the median voter and j represent the party median agent. (For purposes of exposition, suppose m < j.) Then 2j — m is the reflection point of m with respect to j . With this notation, the interval (m, 2j — m) exhaustively defines the status quo policies that a majority of the majority party prefers to the median's ideal point. Therefore, any such status quo elicits majority-party gatekeeping, hence gridlock. On the other hand, for any status quo point outside this cartel gridlock interval, the majority party chooses to report a proposal knowing that it will be amended under an open rule and result in a policy at the chamber median's ideal point. The vertical axis in Fig. 1 is interpreted as the outcome of the legislative process. We can then represent the cartel theory as a mapping from status quo points, ^, into outcomes, x. The graph of the outcomes for the cartel theory begins on the far left side at the median voter's ideal point, m, and is flat while q increases, all the way to the beginning of the cartel gridlock interval.
254
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon o
Cartel theory
. • - - Pivot theory
/ / /
/
/
/
//
Cartel gridlock interval m Q. CNI
T
2pL-m
PL
j
PR
2j-m
2pp>-m
Status Quo
Pivot gridlock interval
Fig. 1. Equilibrium outcomes in cartel and pivot theories The graph of the outcomes then turns upward 45 degrees and continues to climb throughout the gridlock interval. At the right boundary of the interval (2j — m), the majority party median is indifferent between the status quo and the chamber median. At all subsequent points, however, the party-median prefers the chamber median outcome and, as in the case of the status quo points to the left of the gridlock interval, sends a proposal to the floor, which is then amended until m is passed. The pivot theory is only slightly more complex (Krehbiel [21, 22]). Like the cartel theory, it has a gridlock interval, although it is more centrist than the cartel gridlock interval. Let pi be a generic left-of-center pivot (e.g., the ideal point of the 41st senator from the left, whose vote is necessary and sufficient for invoking cloture) and let PR be a generic right of center pivot (e.g., the 67th senator from the left, whose vote is necessary to override a presidential veto). The gridlock interval is then (PLIPR)- For status quo points within this interval, the equilibrium policy is the status quo policy. Outside of the pivot theory's gridlock interval, however, two separate types of behavior may occur, depending upon the extremity of the status quo point relative to the median voter's ideal point. For the most extreme status quo points, convergence to the median voter's ideal point is complete, just as it was in the cartel theory. For two intervals adjacent to the gridlock interval, however, this convergence is limited by the equilibrium requirement that the
Testing Theories of Lawmaking
255
optimal proposal leaves the pivotal voter indifferent between the status quo point and the proposal. The mapping of outcomes for the pivot theory is also depicted in Fig. 1. It has a trough in one area in which the cartel theory is flat, and a peak in the adjacent area within the cartel theory's gridlock interval. In addition to the pivot and cartel theories, we consider two baseline or null models. The median model is characterized by an open amendment rule and majority rule voting. It is well known that the equilibrium outcome from this process is the median legislator's ideal point (Black [10]). Theorems 5 and 6 of Banks and Duggan [7] establish that all no-delay stationary equilibria in unidimensional bargaining models involve selection of the median voter's ideal point. The other baseline model is random and assumes that agendas are generated by an exogenous stochastic process. Specifically, it assumes that each yea and nay location pair is drawn from a bivariate normal distribution. The median-voter and random-alternatives models are sensible baselines for somewhat diff'erent reasons. The defense for the median-voter-theoretic baseline is that most research in legislative studies argues (or assumes) that institutional features are rich and textured and that such features have significant eS'ects on choice behavior. By implication, institutionally sparse models are inadequate in accounting for variation. It stands to reason that models with institutional features should predict better than relatively institutionfree counterparts. The defense for the random null is similar but somewhat more extreme. The median baseline, while sparse, is nevertheless grounded in collective choice theory and has a modicum of credibility as a predictive theory. In contrast, the random null model has no grounding in explicit institutions and behavior. It, therefore, provides a more fundamental opportunity to determine whether the signals extracted from knowingly noisy data are consistent with any model.
3 Cutpoints While Fig. 1 clarifies the analytic difi'erences between cartel and pivot theories, it is not data-ready, because status quo points are not observable and are diflScult to estimate. This obstacle is somewhat less debilitating in the presence of cutpoint estimates — essentially roll-call-specific midpoints between the estimated bill and status quo locations. Consider the case of a vote on final passage of a bill b implicitly paired against the status quo q. If legislators vote sincerely (as theory suggests they will at the final stage) and have symmetric single-peaked preferences, then the cutpoint for such a vote is c = (6 + ^)/2. Everyone on the bill side of this cutpoint votes yea and everyone on the status quo side votes nay. This simple observation provides considerable leverage on the problem, because it is now straightforward to supplement the graphs of outcomes in Fig. 1 with a graph of implied cutpoints. Fig. 2 does this.
256
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon CO
-^ "o
V Both
IV Not Cartel III Neither
• • • . Pivot theory —
— Cartel theory
O
CL
E II Not Pivot
'&MM&m::M0iM
'/7//y/y///:^^
A
'%.'%. '\.'%.'%.' %.'%.'%:%.'%. ••v%'%.'%.'%.' '•%.'%. -\.'%.'%' w^
mmmmimm. Q.
1 Both
m
1
2pL-m
^ ~ PL
^ ^ ^ ^ Wi 2j-m miiPi
PR
1 2pR-m
status Quo
Fig. 2. Cutpoint consistency in the cartel and pivot theories
Two observations reveal the logic of cutpoint analysis. First, for any interval of the status quo, g, define 6*(g) to be the equilibrium proposal. Then, the cutpoint implied by the theory on the vote on final passage is simply ^{b''{Q)^Q) — (^*(9) + ^ ) / 2 . Thus, the theory implies a mapping from status quos to cutpoints t h a t circumvents the need to identify status quos directly. Second, any theory with a gridlock interval is falsifiable via some cutpoints within its gridlock interval. For example, any cutpoint within {PL^PR in the pivot theory implies t h a t either the bill location is out of equilibrium or the s t a t u s quo is in t h e gridlock interval in which case we ought not to observe a proposal. Similarly, any cutpoint in (m, j ) within cartel theory implies, contrary to the theory, t h a t either the bill does not lie at the chamber median or t h a t majority party leaders ought not to have let the bill be considered. Within this cutpoint framework, the implications of the two theories jointly partition the policy space as shown on the vertical axis of Fig. 2. In the two extreme intervals, I and V, the existence of cutpoints is consistent with b o t h theories, and, so, too, are the large coalition sizes t h a t form when the status quo is extreme relative to the proposed, moderating bill, 6*(Q'). Similarly, in the center and moderate interval. III, no discrimination between t h e theories is possible. Here, however, the reason is t h a t both theories predict no cutpoints. Finally, in intervals II and IV, a cutpoint-based test can discriminate cleanly between the two theories. Interval II defines cutpoints t h a t are inconsistent with pivot theory but consistent with cartel theory. Substantively, these cutpoints are on the minority party side of the median voter but less
Testing Theories of Lawmaking
257
extreme t h a n the veto or filibuster pivots.^ Conversely, interval IV defines cutpoints t h a t are inconsistent with cartel theory but consistent with pivot theory. These cutpoints are on the majority party side of the median and more extreme t h a n t h a t side's veto or filibuster pivot but less extreme t h a n the filibuster or veto pivot. This interval is typically very small.^
4 Data Within this theoretical framework, a direct test of the theories with gridlock intervals via cutpoints is possible. T h e ideal point estimates allow us to construct Senate specific intervals t h a t correspond to intervals T V on the vertical axis of Fig. 2. In the simplest and most direct test, then, we can simply compare theories based on their respective frequencies of inconsistent (theorycontradicting) cutpoint estimates. T h e master dataset includes all roll-call votes in the U.S. Senate for the 80th through the 107th Congresses. For 86 percent of these roll calls, DW-Nominate ideal point and cutpoint estimates were calculated.^ It is not abundantly clear whether all or a subset of votes should be analyzed. An emerging convention in roll call analysis is to focus on final passage votes. Theoretically, one reason is t h a t only once voting reaches its last stage does it simplify to a binary choice, when we can then be certain t h a t sophisticated voting does not occur. If the voting on an individual policy were not sincere, then the estimated cutpoint may not accurately measure what we think it does. On the other hand, the argument for excluding votes upon The relevant pivot depends on the location of the president. In unified government the filibuster is the binding constraint; in divided government, the veto pivot is. The median and random models can generate any possible cutpoint. In the median model every final-passage yea location is, by construction, at the median legislator's ideal point, but since the status quo or nay locations are unconstrained any cutpoint is possible. In the random model, cutpoints are normally distributed because they are simply midpoints between two normally distributed random variables. We use the estimates provided by Keith Poole and available at the web site: http://k7moa.uh.edu/. Even though use of the DW-Nominate yea and nay location estimates would yield a more direct test of the mappings in Fig. 2, there is a compelling reason to use the cutpoint estimates instead of the yea and nay location estimates. The classification procedure that recovers the ideal points also recovers the bill cutpoints (and yea and nay locations). It is well known, however, that yea and nay locations are not well-identified. Procedures like DW-Nominate recover these parameters by leveraging off of the postulated functional forms of the loss function and error distributions. In contrast the cutpoints are the dual of the ideal points and are as reliably estimated as the ideal points. Accordingly, the cutpoint estimates are much more robust than the yea and nay location estimates. See the appendix of Poole et al. [30] for more information about DW-Nominate.
258
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon
which voting may be strategic exposes a limitation of using estimates from Nominate or any other scaling procedure that uses votes other than final passage types. If the sophisticated-voting-based defense for selecting only final passage votes is compelling, then it must be that sophisticated voting occurs with sufiicient regularity that one should exclude pre-passage votes. However, if sophisticated voting passes this threshold, then tests using Nominate (or other large-N) ratings must be treated with suspicion, too, because our usual interpretations of the recovered ideal points and cutpoints rest on an assumption that the inputs for the rating scheme are sincere votes. Even if all voting is sincere, however, another reason to focus on final passage votes is that the theories describe the policy alternatives that ultimately emerge from the legislative process and are silent about the amendments that would be observed on the equilibrium path. Moreover, pre-passage votes would typically not be between the equilibrium proposal 6*(^) and the status quo q. In that case, we would not necessarily be able to cleanly partition the policy space as in Fig. 2. For example, in a pre-passage vote, two alternatives (perhaps an amendment and a substitute amendment) may produce a cutpoint within the gridlock interval but the alternative that survives this vote would later lose to 6*(^). A related complication is the absence of a clear consensus about how "votes on final passage" should be defined and operationalized. We will consider three definitions: (A) expansive, (B) moderate, and (C) restrictive. The expansive definition (A) considers any of the following a final-passage vote: relatively unambiguous motions (such as "to pass S.1234"), a vote on a veto override, a vote on a motion to suspend the rules in order to pass a measure (uncommon in the Senate), final votes on resolutions (even though most of these are measures of strictly intra-Senate relevance), final votes on procedural initiatives (e.g., to initiate investigations, to discharge a committee), final votes on nonbinding measures such as resolutions of disapproval), votes on engrossment and third reading of bills, votes on conference reports, and on amendments between the chambers. The moderate definition (B) tightens up these criteria (and reduces the number of false positives) by not counting as final passage votes: resolutions, nonbinding measures, procedural votes, and amendments between the chambers. The most restrictive definition (C) does not count any of the above-listed as final passage votes but rather demands that legislation have the possibility of becoming Public Law. The trade-offs are more or less obvious: the more expansive is the definition, the more likely it is that not-really-finalpassage votes slip in; the more restrictive is the definition, the more likely it is that truly substantive votes that are de facto the last action taken by the chamber on significant legislation are omitted.^ ^ Selection of roll calls was implemented via Microsoft Access queries of ICPSR and CQ Washington Alert roll call records.
Testing Theories of Lawmaking
259
Table 1. Cutpoint-inconsistency of two theories II III IV I V Inconsistent with : Neither Pivot Both Cartel Neither N All roll calls Final passage A Final passage B Final passage C
5,940 37% 1,611 53% 1,395 56% 1,076 56%
4,067 26% 511 17% 381 15% 266 14%
2,439 15% 252 8.3% 190 7.6% 151 7.9%
2924 573 3.6% 18% 47 619 1.5% 20% 510 37 1.5% 2 0 % 29 390 1.5% 20%
15,943 3,040 2,513 1,912
We address these complications by reporting various combinations of votes, ranging from all roll calls on the inclusive end of the spectrum {N — 15,943) to narrowly defined final passage votes on the exclusive end of the spectrum ( i V - 1,912).
5 A Direct Test All cutpoints are, of course, consistent with the median and random models, because neither of these has a gridlock interval, which is a necessary condition for refutation using the direct cutpoint approach. The analysis discussed in this section, accordingly, focuses on the refutable implications of only the cartel and pivot theories. Table 1 shows the distribution of roll calls across the cutpoint intervals as defined on the vertical axis of Fig. 2. While potentially significant diff'erences between all roll calls and final-passage roll calls are evident, it seems not to matter whether expansive or restrictive definitions of final passage votes are assessed. Unless differences are striking, we henceforth report results only for the moderate definition (B). Columns I and V of Table 1 show clearly that most cutpoints are extreme relative to pivotal Senators' ideal points. Even though fewer than 30 percent of cutpoints lie in the interior segments, however, these observations have discriminatory potential. In this phase of the analysis, the findings seem clearly to favor the cartel over the pivot theory. Cutpoints that are inconsistent with both theories (interval III observations) occur roughly ten percent of the time, but the remaining two intervals (II and IV), in which only one theory is rejected, are consistently in favor of the partisan model. As noted above, intervals II and IV are necessarily on the opposite sides of the median voter in the legislature, with the rejection interval IV for the cartel theory residing considerably farther from the median than the medianadjacent interval II. Furthermore, interval II is on the minority party side of the median, while interval IV is on the majority party side. Accordingly, we
260
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon
can view the comparative frequencies in these two intervals as a record of which party tends to win the figurative pohcy tug-of-war. This interpretation is similar to the increasingly prominent research methodology focusing on "roll rates:" the proportion of times that one party defeats another as indicated by intra-party homogeneity or cohesiveness and inter-party opposition. Upon first glance, the two methods — roll rate analysis and cutpoint analysis — seem to validate one another. Here, as elsewhere, the ratios range roughly from 4:1 to 8:1 (see also Cox and McCubbins [15] and Groseclose et al. [16]).
6 Strong Consistency Although the direct horse-race setup is intuitive and convenient, things are not as simple as it seems upon first glance. Three caveats need to be addressed. First, the data in Table 1 may include a large number of false positives: namely, roll calls in which the cutpoint may be consistent with the theory but in which the bill and status quo locations are not consistent with the theory. Second, the inclusive data set may be distorted by the presence of motions that do not pass, because the theories are silent about policy-inconsequential proposals and voting. Third, and most challenging, the two theories differ substantially in the degree to which they are exposed to refutation. Specifically, cutpoints throughout the entire gridlock interval in the pivot theory are inconsistent with the theory, while cutpoints throughout only half of the gridlock interval in the cartel theory are inconsistent with it. As an empirical matter, the average ratio of pivot-to-cartel widths in terms of rejection intervals is 1.66 to 1. A further and even more subtle bias exists in the likely case that cutpoints are likely to be approximately normal, in which case the centrality of the pivot rejection interval relative to the cartel rejection interval exacerbates the asymmetry in exposure of the theories. Fortunately, additional analysis can address these obstacles, at least in part. Table 2 addresses the false positives obstacle by adding two filters to the location-based definition of consistency employed in Table 1. Specifically, three percentages are reported for each of three theories: median, cartel, and pivot. The first such number is the consistency rate based solely on whether the estimated cutpoint lies outside of the respective theory's rejection interval. So, for example, because the median voter theory has no rejection interval, its consistency rate is necessarily 100 percent. For the cartel theory, the locationbased consistency rate is the sum of percentages from Table 1, row 1, columns I, II, and V. For the pivot theory, the rate is the sum from columns I, IV, and V. For both sets of votes — all roll calls or type B final passage votes — the ordering is necessarily the same as in Table 1 because nothing has changed about the data. Specifically, the irrefutable median voter theory necessarily performs best, and the cartel theory outperforms the pivot theory.
Testing Theories of Lawmaking
261
The filters are applied in the second and third columns (for each of two sets of roll calls). For all three theories, when votes occur, the consequence should be convergence toward the median voter's ideal point on the primary dimension. Although we share Poole and Rosenthal's skepticism about the point precision of their estimates of yea and nay locations, it seems clear we may assume that the estimates are quite accurate in assigning the direction of the yea alternative. Therefore, in the columns titled Direction, we tabulate the percentage of roll calls both whose cutpoints are theory-consistent and whose yea alternative lies closer to the median than the nay location. That is, the conditions are each necessary and cumulatively suSicient. The result is a shocking drop-off in the success rates of all three theories, particularly in the larger data set. The corresponding drop-offs from the direction criterion in final passage votes is much lower — only about 10 percent. This suggests that focusing on final passage votes may, indeed, yield somewhat more confident inferences than the much more inclusive sample. Finally, the third and sixth columns add the requirement that the motion obtain the requisite number of votes to carry. The drop-off here is lower than when the direction criterion was applied, but the final result is discouraging in the case of the larger data set. The median voter theory does no better than a flip of a fair coin in predicting whether motions were successful and moved the status quo closer to the median voter. The cartel and pivot theories perform even worse. More precisely, at least half and as many as two thirds of roll calls in the Senate failed to meet the most stringent cumulative criterion based on cutpoint locations, direction of change, and success. The situation is much better, however, for final passage votes. Here, the drop-off is minor by comparison, and the most-stringent-criterion findings are more sanguine. The range of prediction rates across theories is 68 percent for pivot theory to 81 percent for median voter theory. In summary, it seems that this class of models does not perform well over the entire set of roll calls, but it does perform well on final passage votes. Table 2. Cutpoint-consistency under increasingly stringent definitions All roll calls Final passage roll calls Location Direction Success Location Direction Success Median voter theory 100% Cartel theory 81% Pivotal politics theory 59%
59% 49% 37%
50% 44% 33%
100% 91% 77%
86% 81% 70%
81% 77% 68%
262
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon
7 Exposure-adjusted Consistency Although the more stringent criteria for selecting observations tend to bring together the three models in terms of predictive performance on final passage votes, the models are still not quite on even footing with the previous forms of analysis. Specifically, we have still not addressed the fact that, on average, the pivot rejection region is 1.6 times the size of the cartel rejection region. An implication of this fact is that if the cutpoints were uniformly generated, we would expect the pivot model to perform only 62.5 percent as well as the cartel model. In fact, the number of cutpoints that are inconsistent with pivot is not far from 1.6 times the number of cutpoints that are inconsistent with cartel theory. Twenty percent of the cutpoints are inconsistent with the cartel theory, and 30 percent are inconsistent with the pivot theory, so the empirical ratio is 1.5. The problem of diff'erential exposure is the primary reason for introducing the random model. The approach is to compare three cutpoint-generating processes: cartel-theoretic, pivot-theoretic, and an atheoretic random draw from a normal distribution. Because the null model assigns more cutpoints to large and central rejection intervals than to small and one-sided intervals, a comparison of the actual percentage of cutpoints in a model's rejection region to the expected percentage of cutpoints that the null model would put in the model's rejection region makes a neutral test possible. Though not a formal statistical test, this procedure is conceptually similar. It captures the idea that, if a theory does not explain the data appreciably better than a simple naive model, then we can conclude that the model's fit is underwhelming. In the spirit of comparing logit regression models to a null of tossing an unfair coin, we compare pivot and cartel to a similarly exposure-calibrated null model. We utilize the normal distribution because of an a priori aflinity which is reinforced by the actual distribution of cutpoints. We rely on the data to choose the mean and variance parameters for each Congress's distribution. For each Congress, therefore, the null model is that the cutpoints are drawn from a normal distribution where the first and second moments correspond to the estimated sample moments for that Congress's cutpoints. For this analysis we focus on the votes satisfying the moderate (B) final passage criterion. Table 3 presents the first of two sets of results that address the differential exposure problem. The first row gives the percentage of roll calls whose cutpoints would be consistent with the respective theory, if cutpoints were distributed normally.^ As asserted earlier, the cartel theory's expected percentage {86) is greater than the pivot theory's (73), because the former's ^ Table 3 uses weighted averages across Congresses. The results are only very slightly different than the unweighted averages. We again require that consistent votes move in the right direction throughout this section. While this may seem unfair to the cartel and pivot models because the null model faces no such test, in fact this concern is unwarranted. Recall, that the real null model is that
Testing Theories of Lawmaking
263
Table 3. Exposure-adjusted consistency Pivot Cartel Null 73% 86% Observed 82% 94% Improvement 12% 10%
exposure is not as great as the latter's. T h e second row gives the actual observed percentages for the two theories. Finally, the third row calculates the percentage improvements in prediction over the null or baseline percentage. T h e difference between theories, while small (2 percent), is in the opposite direction of the direct-test finding. In other words, relative to expectations, the pivot theory slightly outperforms the cartel theory. A second, closely related method for assessing the exposure-adjusted strengths or limitations of the theories is based on Congress-level observations. Specifically, for each Congress we calculate a proportional improvement measure of each theory over its baseline expectation, given the width and location of its gridlock interval. T h e scatterplot in Fig. 3 displays this information. T h e horizontal axis represents pivot-theoretic improvement over the null, while t h e vertical axis measures the corresponding values for the cartel model. Therefore, observations t h a t are to the northeast of the point (0,0) represent Congresses in which both models improve on the random null. Observations below (above) the 45 degree line represent congresses in which the pivot's relative performance is better (worse) t h a n t h a t of the cartel model's performance. Consistent with Table 3, Fig. 3 indicates t h a t the pivot theory outperforms the cartel theory in most Congresses — 19 of 27 to be precise. However, in four Congresses the performance of the pivot theory is abysmal, and, when standard deviations are calculated for the two variables, it is clear t h a t one cannot reject the hypothesis t h a t the theories' average proportionalimprovement rates are equal. T h e results as a whole, then, are mixed and inconclusive regarding cartel and pivot theories, b u t either of these seems to perform better t h a n the median voter theory. If there is a surprise performance in these exercises, it would seem to be the random model. Can we confidently say t h a t the two gridlock theories are responsible for observed deviations from normality in the distributions the yea and nay locations are themselves normally distributed. As a consequence the cutpoints are normal (as the weighted average of two normal variables). In addition, by altering the means and variances of the yea and nay location distributions, it is possible to alter the expected percentage of yea and nay locations which would satisfy the directional tests. Accordingly, for any desired normal distribution of cutpoints, we can construct distributions for yea and nay locations that give us the appropriate distribution of cutpoints and would do exactly as well as the actual data on the directional tests.
264
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon
of cutpoints? Although a rigorous statistical test is beyond the scope of the analysis, we can get some insight into this question simply by observing kernel density plots of the cutpoint distributions. Presenting all such distributions would unnecessarily clutter the presentation, so we select one congress as an example (noting that nearly all of the congresses are qualitatively quite similar). The kernel density plot of the final passage cutpoints for the 94th Congress is shown in Fig. 4.^ The corresponding normal distribution for this set of cutpoints is superimposed on the figure, and the vertical lines correspond to the intervals labeled I-V in Fig. 2. Neither models' rejection region consistently matches up with the troughs in the data. Nevertheless, there is striking evidence that troughs are present and qualitatively they are in the correct part of the policy space. Overall, this statement applies to 23 out of 27 congresses which seems to be genuine support for some type of gridlock theory and, likewise, more-than-coincidental evidence against the baseline random cutpoint model.
8 Discussion Our conclusions are partly methodological and partly substantive. As a methodological exercise, the study has been humbling, both in comparison with other similar research and throughout the various phases of Again, these cutpoints satisfy type B restrictiveness and the direction criterion.
0
0.2 o
o
o
o
y ^ y^ o
y^
O
O
o
o CO
c
g o o 0
y ^
o
y^ /
O o
o
Q.
O
y^
O
^
1
1
1
0 .1 .2 Pivot Proportional Improvement Over Null
T"
Fig. 3. Comparative improvements in predictions over a naive null model
Testing Theories of Lawmaking LO
^
• -
1
265
Pivot rejection interval
Cartel rejection interval
/ // IT) _
/
f
//
" / // / '/
^^ \ "^"^ \\ \ "^ \\ \ \ \ \ \\
/ \ N
O -
• f r - ^ ^ — " " "
-.5
0 Outpoints
Fig. 4. Kernel and normal distributions of outpoints, 94th Congress analysis in this project. Prior research, of course, has studied partisan and nonpartisan theories of legislative choice in considerable detail. Researchers have tended to come down in favor of one theory over the other with considerable confidence. Often, however, the reason for their confidence is that "the other" theory is either ignored, or not put on equal footing with the focal theory. The intended neutral stance taken here is based on the premise that, if discrimination is to occur convincingly, the high degree of observational equivalence of pivot and cartel theories requires joint, neutral tests. In this project, we have pursued joint tests of two sorts: direct and indirect. The direct test — a comparison of consistency (or inconsistency) of cutpoints with the respective theories — suggests quite strongly that the cartel theory is better than the pivot theory. However, the direct test can be questioned in terms of the equal-footing criterion. Furthermore, it is unclear whether either gridlock-based theory is significantly better than the simple median-voter baseline. A series of indirect tests was then devised, and the results were painfully inconclusive. The pivotal politics theory fares better than the cartel theory relative to their respective baseline expectations. However, the variance in performance levels is sufliciently large that we cannot be confident that the difference is real. While this analysis suggests somewhat more convincingly that gridlock-based theories provide marginal value over median voter theory, their marginal value over the random-normal null model is questionable. In short, the visuals are suggestive, but the data remains ambiguous. In light of all of the above, the study at least underscores the value of multiple methodological approaches prior to making definitive conclusions. Kramer [19] reminds us
266
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon
that well formulated theoretical propositions are fragile. We have focused on well-formulated theoretical propositions and illustrated their fragility. As a substantive exercise, the study fails in the obvious respect that discrimination among the four models proves to be much more difficult than anticipated. The interplay between theory, method, and data, however, is such that we can offer some remarks about why our findings are inconclusive. The two primary possibilities are problematic data or problematic theory. To dwell on the first of these is admittedly cowardly insofar as we ought not to have conducted the tests had we not thought the data were up to the task. Still, the possibility that we were mistaken deserves some retrospection. The nature of the exercise — here and elsewhere — is to assume not only that there is a primary dimension of conflict on all issues but also that legislators are arrayed in the same way on a primary dimension on all issues. Clearly, this stretches credulity. One important avenue of future methodological inquiry is to assess rigorously the degree and nature of bias on various tests of errors or imperfections in the array of preferences projected from a multidimensional onto a unidimensional space. Although this may seem like a theory problem rather than a data problem, there is a subtle reason that it is more of the latter than the former. Unidimensional theories assume only that preferences exist and are well behaved on a single dimension in any given choice situation. They do not assume, as does the implementation of tests of the sort we conducted, that well-defined preferences of Senators maintain the same orderings and locations on the same primary dimension across all roll calls. In other words, unidimensional theories can, in principle, predict perfectly well even in a truly multidimensional world. The second possibility — inadequate theory — brings us back to the concerns and contributions of our late colleague Jeffrey Banks. Banks and Duggan [8, page 26] conclude: While we have considered mainly foundational issues here, we have proposed a general framework in which more substantive questions, about the nature of public goods provided or the coalitions that form to pass proposals, for example, can be taken up in special cases with more structure. The pivot and cartel models are exactly that: special cases with structures that are justified by observations about the nature of legislative institutions. Our examination of these examples leads to the recommendation that additional structures and/or special cases ought to be investigated. We share Banks's hope and expectation that future models will improve upon those considered here and shed a brighter light on legislative policy-making.
Testing Theories of Lawmaking
267
References [1] Aldrich, J. (1994) A model of a legislature with two parties and a committee system. Legislative Studies Quarterly^ 19: 313-340. [2] Aldrich, J. (1995) Why parties? The origin and transformation of party politics in America Chicago: University of Chicago Press. [3] Aldrich, J. and D. Rohde (1998) Measuring conditional party government. Mimeo. Duke University. [4] Austen-Smith, D. (1987) Sophisticated sincerity: voting over endogenous agendas. American Political Science Review^ 81: 1323-1330. [5] Banks, J. S. (1985) Sophisticated voting outcomes and agenda control. Social Choice and Welfare^ 1: 295-306. [6] Banks, J. S. (1995) Singularity theory and core existence in the spatial model. Journal of Mathematical Economics, 24: 523-536. [7] Banks, J. S. and J. Duggan (2000) A bargaining model of collective choice. American Political Science Review, 94: 73-88. [8] Banks, J. S. and J. Duggan (2003) A bargaining model of legislative policy-making. Mimeo. University of Rochester. [9] Banks, J. S. and F. Gasmi (1987) Endogenous agenda formation in three person committees. Social Choice and Welfare, 4: 133-152. [10] Black, D. (1958) The theory of committees and elections London: Cambridge University Press. [11] Brady, D. W. and C. Volden (1998) Revolving gridlock: politics and policy from Carter to Clinton New York: Westview Press. [12] Clinton, J. D. and A. Meirowitz (2001) Agenda constrained legislator ideal points and the spatial voting model. Political Analysis, 9: 242-259. [13] Clinton, J. D. and A. Meirowitz (2003) Integrating voting theory and roll call analysis: a framework. Political Analysis, 11: 381-396. [14] Cox, G. W. and M. D. McCubbins (1993) Legislative leviathan: party government in the House Berkeley: University of California Press. [15] Cox, G. W. and M. D. McCubbins (2002) Agenda power in the US House of Representatives, 1877 to 1986. In D. W. Brady and M. D. McCubbins (eds) Party, process and political change in Congress: new perspectives on the history of Congress Stanford: Stanford University Press. [16] Groseclose, T. and J. M. Snyder (2003) Interpreting the coefficient of party influence: comment on Krehbiel. Political Analysis, 11: 104-107. [17] Groseclose, T., S. Levitt and J. M. Snyder (1999) Comparing interest group scores across time and chambers: adjusted ADA scores for the U.S. Congress. American Political Science Review, 93: 33-50. [18] Heckman, J. and J. M. Snyder (1997) Linear probability models of the demand for attributes with an empirical application to estimating the preferences of legislators. RAND Journal of Economics, 28: 142-189. [19] Kramer, G. H. (1986) Political science as science. In H. Weisberg (ed) Political science: the science of politics New York: Agathon Press.
268
Keith Krehbiel, Adam Meirowitz, and Jonathan Woon
[2o; Krehbiel, K. (1993) Where's the party? British Journal of Political Sci-
ence, 23: 235-266. [21 Krehbiel, K. (1996) Institutional and partisan sources of gridlock: a theory of divided and unified government. Journal of Theoretical Politics, 8: 7-40. [22 Krehbiel, K. (1998) Pivotal Politics: A Theory of U.S. Lawmaking Chicago: University of Chicago Press. [23 Krehbiel, K. (2000) Party discipline and measures of partisanship. American Journal of Political Science, 44: 206-221. [24 Krehbiel, K. (2003) Cartels and pivots reconsidered. In E. S. Adler and J. Lapinski (eds) Macropolitics of Congress. [25 Krehbiel, K. Meirowitz A (2002) Minority rights and majority power: theoretical consequences of the motion to recommit. Legislative Studies Quarterly, 27: 191-217. [26 McCarty, N., K. T. Poole and H. Rosenthal (2001) The hunt for party influence in Congress. American Political Science Review, 95: 673-693. [27; McKelvey, R. and N. Schofield (1987) Generalized symmetry conditions at a core point. Econometrica, 55: 923-934. [28; Plott, C. (1967) A notion of equilibrium and its possibility under majority rule. American Economic Review, 57: 787-806. [29; Poole, K. T. and H. Rosenthal (1985) A spatial model for legislative roll call analysis. American Journal of Political Science, 29: 357-384. [30; Poole, K. T. and H. Rosenthal (1997) Congress: A Political-Economic History of Roll Call Voting Oxford: Oxford University Press. [31 Rohde, D. W. (1991) Parties and Leaders in the Postreform House Chicago: University of Chicago Press. [32; Saari, D. (1997) The generic existence of a core for q-rules. Economic Theory, 9: 219-260. [33 Schickler, E. (2000) Institutional change in the House of Representatives, 1867-1998: a test of partisan and ideological power balance models. American Political Science Review, 94: 269-289. [34; Shepsle, K. and B. Weingast (1987) The institutional foundations of committee power. American Political Science Review, 81: 85-104. [35; Sinclair, B. (1997) Unorthodox lawmaking: new legislative processes in the U.S. Congress Washington D.C.: Congressional Quarterly Press. [36; Snyder, J. M. (1992) Committee power, structure induced equilibria, and roll call votes. American Journal of Political Science, 36: 1-30. [37; Snyder, J. M. and T. Groseclose (2000) Party influence and congressional roll-call voting. American Journal of Political Science, 44: 193-211. Weingast, B.R. (1989) Floor behavior in the U.S. Congress: committee power under the open rule. American Political Science Review, 83: 795816.
Deliberation and Voting Rules'' David Austen-Smith^ and Timothy Feddersen^ ^ Northwestern University dasmOkellogg.northwestern.edu ^ Northwestern University tfedQkellogg.northwestern.edu
S u m m a r y . We analyze a formal model of decision-making by a deliberative committee. There is a given binary agenda. Individuals evaluate the two alternatives on both private and common interest grounds. Each individual has two sorts of private information going into committee: (a) perfect information about their personal bias and (b) noisy information about which alternative is best with respect to a (commonly held) normative criterion. Prior to a committee vote to choose an alternative, committee members engage in deliberation, modeled as a simultaneous cheap-talk game. We explore and compare equilibrium properties under majority and unanimity voting rules, paying particular attention to the character of debate (who influences who and how) and quality of the decision in each instance. On balance, majority rule induces more information sharing and fewer decision-making errors than unanimity. Furthermore, the influence and character of deliberation per se can vary more under majority rule than under unanimity.
1 Introduction This paper concerns the relationship between deliberation and voting in committees. Prom at least one perspective, the issue is moot. In contrast to Condorcet, who saw the role of deliberation and debate largely in positive terms,^ the recent political theory literature on "deliberative democracy" is more expressly normative, being concerned with questions of legitimacy and achieving a consensus sufficient to make voting irrelevant (see, for example, the essays in Bohman and Rehg [7] and in Elster [16]). Thus, when summarizing the most optimistic view of deliberation, Elster [17] writes t h a t "there would not be any need for any aggregating mechanism, since a rational discussion would tend to * Jeff Banks was a coauthor of one of us, a teacher of the other, and a friend to both. This paper is for him. ^ Deliberation for Condorcet is necessary both to clarify individual interests and to formulate coherent agendas over which to vote: see Marquis de Condorcet, 1793; translated by Ian McLean and Fiona Hewitt [32, page 193].
270
David Austen-Smith and Timothy Feddersen
produce unanimous preferences" .^ And there is some support for this view in a very recent contribution to the formal mechanism design literature. Exploiting the revelation principle for Bayesian communication games, Gerardi and Yariv [23] prove an equivalence result for all binary voting rules of the form, "choose X over y if at least r of n individuals, 1 < r < n, vote for x against y". Specifically, suppose a non-unanimous binary voting rule r of the sort described is used to choose one of two fixed alternatives and suppose t h a t , prior to voting, all individuals have the opportunity to make cheap talk statements about any decision-relevant private information they may have regarding the relative value of these alternatives;^ an outcome is then a probability of winning for each alternative as a function of the underlying distribution of private information about these alternatives. T h e theorem Gerardi and Yariv prove is the following: under very general conditions on preferences and information, the set of sequential equilibrium outcomes achievable by augmenting a voting game with a prior communication stage is constant in the voting rule, r (Gerardi and Yariv [23, Proposition 2]). In other words, although there is no assurance t h a t all of the private information held by voters is necessarily realized in debate (the cheap talk communication stage), it is the case t h a t the voting rule governing the final decision is immaterial, so apparently justifying the deliberative democracy thesis t h a t voting becomes insignificant once talk is permitted. T h e central observation used to prove the Gerardi-Yariv result is t h a t all individuals voting unanimously can be a component of a sequential equilibrium. Because the voting rule r is presumed non-unanimous, it immediately follows t h a t if n — 1 individuals are voting for the same alternative, the n*^ individual can do no better t h a n to vote for this alternative as well. And because this is true for any non-unanimous voting rule r, unanimous voting at any given profile of information effectively renders all such rules equivalent: if, at some rule r, an individual is willing to reveal information under the expectation t h a t voting will be unanimous following such revelation, then t h a t individual must also be willing to reveal exactly the same information at a rule r ' conditional on the same expectation over voting. Specifying unanimous Not all normative theorists writing on deliberative democracy are enthusiastic about the value of the process. Particularly coherent critiques are offered by Christiano [9], Knight and Johnson [27] and Sanders [37]. Moreover, "tending to produce" unanimity is not the same as producing unanimity: "Even under ideal conditions there is no promise that consensual reasons will be forthcoming. If they are not, then deliberation concludes with voting, subject to some form of majority rule." (Cohen [10, page 23]). In a recent contribution, Dryzek and List [14] argue that deliberation can (but need not) induce conditions under which democratic aggregation procedures are well-behaved and free of opportunities for strategic manipulation. Formally, a statement is "cheap talk" if the statement itself does not affect the speaker's payoffs from outcomes (although, of course, cheap talk statements can influence, and are intended to influence, which outcome is realized).
Deliberation and Voting Rules
271
voting under any non-unanimous voting rule r is equivalent to saying the voting stage is irrelevant. It follows that, conditional on deliberation, any voting rule is in principle as a good as any other. On the other hand, if individuals behave as if voting mattered then the Gerardi-Yariv theorem does not hold in general. To see this, consider an example with seven voters in which the only private information is the individuals' prior preference for x over y; in particular, suppose any additional information regarding the relative value of the alternatives is decision-irrelevant in that the ordinal preference profile over alternatives is independent of such information, pooled or otherwise. Suppose further that each individual conditions on the event that their vote is pivotal in the collective decision. Then whatever speeches individuals hear in debate, every individual must vote 'sincerely' for their most preferred alternative. Consequently, if four of the seven voters have a prior preference for x over y and the remaining three individuals have the opposite preference then, under majority rule r == 4, the unique outcome is X with probability one but, under a supramajority rule r = b, the unique outcome is y with probability one: the voting rule matters. Because we are concerned in what follows with understanding how details of the voting rule affect the character of deliberation, insisting only on sequential equilibria is inappropriate and, therefore, we assume throughout that individuals condition voting decisions on events at which their vote matters, that is, on being pivotal. In such a world, little is yet understood about optimal committee voting rules conditional on deliberation. Although we are not in a position to offer any sort of definitive analysis of the problem here, we do address two subsidiary questions with respect to majority rule and unanimity rule: first, "How does deliberation affect collective decisions under different voting rules?"; and second, "Can deliberation result in worse collective choices than those made in its absence?". It is useful to distinguish committee decision-making with deliberation from that with debate^ more broadly considered. As understood in this paper, in deliberation at least two privately informed individuals engage in unmediated cheap talk over a collective decision to be made by these same individuals under a given voting rule; in debate at least two privately informed individuals engage in unmediated cheap talk prior to a decision being made. So deliberation here is understood as a subset of debate: in debate there is no requirement that those engaged in deliberation have any direct responsibility for the final decision, which might be made unilaterally by some quite distinct individual; in deliberation, however, the decision is necessarily by voting and those eligible to vote are precisely those eligible to deliberate. Although the existing formal strategic literature on decision-making with debate is now quite con-
272
David Austen-Smith and Timothy Feddersen
siderable,^ the corresponding literature on decision-making with deliberation is as yet very limited. In addition to Gerardi and Yariv [23], Austen-Smith [2, 3] analyzes a model of deliberative committee decision making with endogenous agenda-setting; Calvert and Johnson [8] look at the coordinating role of debate in a complete information model of committee decision-making; Meirowitz [34] also worries about debate and coordination but under incomplete information; Doraszelski et al. [13] study a two-person unanimity problem in which both agents are known to share identical information conditional on full information; CoughIan [11] explicitly includes a cheap talk communication stage to the standard model of voting in juries; and Hafer and Landa [25] develop a non-Bayesian model of argument. For Condorcet at least, perhaps the most important role of deliberation is in agenda-setting; nevertheless, we assume there is an exogenously fixed agenda. Although it is fairly natural to begin by asking what happens with fixed alternatives and then back up to ask how the alternatives for consideration might be chosen, the Condorcet Jury Theorem along with recent results on information aggregation through voting over fixed binary agendas (e.g., Ladha [29]; McLennan [33]; Duggan and Martinelh [15]; Feddersen and Pesendorfer [21, 22]) raise a more concrete question about whether deliberation over given agendas is a salient issue. At least asymptotically as the electorate gets large, any majority or supermajority voting rule short of unanimity almost surely selects the alternative that would be chosen under the given rule were all voters fully informed and surely voted. However, committees in which deliberation is feasible are typically too small for asymptotic results to be useful.^ Thus there remains room for decision-relevant information sharing and argument in committees. We consider a three person committee that must choose between two alternatives. Committee members' preferences have a private interest and a common interest component. An individual's private interest is her bias toward one or other of the alternatives and is private information to the individual; net of private interests, individuals' have a common interest in choosing the correct alternative for the realized state of nature, presumed unknown at the time of the decision. Each committee member observes a noisy private signal about the true state. We compare majority rule with unanimity rule both with and without debate. Examples include Battaglini [6], Glazer and Rubinstein [24], Ottaviani and Sorensen [35], Krishna and Morgan [28], Diermeier and Feddersen [12], Lipman and Seppi [30], Matthews and Postlewaite [31], and Austen-Smith [4, 5]. Condorcet surely felt this to be significant when claiming that the second form of debate he identifies (in which general questions are refined into "a number of clear and simple questions") "could not take place outside an assembly without becoming very time-consuming" and "is of use only to men who are required to prepare or pronounce a joint decision" (trans. [32, page 193]).
Deliberation and Voting Rules
273
There are two kinds of mistakes a committee can make given the information of its members. First, when the information available to the committee is suflicient to convince all members regardless of bias to support an alternative, the committee may choose the other alternative. We call this an error in common interest. Second, when the information available to the committee is insuflScient to convince members to vote against their bias, the committee can choose an alternative that is not preferred by a majority. We call this second error an error in bias. Finally, when the committee never makes either kind of error in equilibrium we say that the equilibrium satisfies full information equivalence, Overall, the results point to majority rule being superior both with respect to the expected quality of committee decisions and to the quality of debate it induces. Among other things, we show that with respect to pure strategy equilibria: (1) Without debate there are no equilibria under either rule that satisfy full information equivalence. However, there are equilibria under majority rule with debate that satisfy full information equivalence but there are no such equilibria under unanimity rule with debate. (2) Under majority rule with debate there are no equilibria that result in errors in common interest that would not also occur without debate. In contrast, there are equilibria under unanimity rule with debate that result in errors in common interest that would not occur without debate. (3) The only circumstances under which an equilibrium in the game with debate under majority rule produces an error in bias are those in which an equilibrium satisfying full information equivalence is possible but not played. And although there exist circumstances under which debate weakly improves on no-debate under unanimity rule, we do not yet know whether this is a general property of debate equilibria under unanimity. In the next three sections we describe our model and results for majority and unanimity rule. We conclude with a brief discussion of paths for future research.
2 A Deliberative C o m m i t t e e Consider a three person committee, N = {1,2,3}, that has to choose an alternative z € {x^y}] let x be the status quo policy. Individual preferences over the feasible alternatives can be decomposed into two parts, one reflecting purely private interests and one reflecting a notion of common good or fairness. Specifically, for any i G N, i^s private interests are given by a utility u{x;bi) = l-u{y\hi)
G {0,1},
274
David Austen-Smith and Timothy Feddersen
where bi G {x^y} is i's bias^ defined by u{z; bi) = 1 if and only if bi = z. The common good value of an alternative z G {x,y} is f{z\Lo) G {0,1}, describing which alternative is fair in state cu G {X, F } . Then for any z G {x^y}, bi G {x,y} and A G [0,1], assume i's preferences can be represented by:^ Uiz; bi) = Xuiiz; bi) + (1 -
X)f{z\uj).
To save on notation, hereafter we write u{z;bi) = Ui{z), leaving the dependency of an individual's payoff on bi implicit in the individual subscript. In general, different individuals can be expected to have different moral systems or senses of what constitutes the common good. For example, suppose individuals are either Benthamite Utilitarians or Rawlsian Maximinimizers. Then reasons for choosing one alternative over another that are germane to the former can be utterly irrelevant to the latter and conversely. In this setting, productive debate might proceed either by a discussion of principles along, say, axiomatic grounds, or by seeking out reasons and arguments that are decisionrelevant to both conceptions of how to evaluate the common good. Although such issues are, we think, quite important and worth thinking about more deeply, for now it is convenient simply to ignore such differences. So assume the evaluation function / is the same for everyone and satisfies f{z\uj) = 1 if and only if c<; = Z. Similarly, without suggesting the assumption describes reality, it is convenient to suppose individuals value the common good in the same way, so A is common across committee members. There are two substantive sources of incomplete information. First, individual i's bias bi G {x^y} is known only to i: for all i G A', assume the probability that bi — x is 1/2. The second informational incompleteness concerns which of the two alternatives is most in the common interest, modeled as uncertainty over the realized state a; G {X, F } . The common prior belief over {X, y } is assumed uniform. With probability (1 — ^) G (0,1) an individual i G A" is either uninformed^ observing no further information denoted 5j = 0, or, with probability q is informed and observes a noisy signal Si G {—1,1} from a common state-dependent distribution. Whether or not any i G A' has observed any signal and, if so, which signal he or she received, is private information to i. Conditional on observing a signal, let p = Fi[si = 1|X, Si^O]=
Fi[si = - l | y , Si ^ 0]
and assume p G (^, 1). A pair {p,q) G (^, 1) x (0,1) is called an information structure. In sum, therefore, an individual's type going into the committee decisionmaking process is a pair {b,s) where b G {x^y} is the alternative most in the individual's private interests and s G {—1,0,1} is the individual's signal See Kami and Safra [26] for an axiomatic justification of separable preferences for individuals with both private interests and a preference for fairness.
Deliberation and Voting Rules
275
regarding which alternative is fair. Hence, for each alternative z G {^^y}^ i e N has induced preferences going into committee given by E[U{z; bi)\s] = Xuiiz) + (1 - A) FT[Z\S] G [0,1]. Clearly, if A > 1/2 then no type ever cares sufficiently about the common good for it to be decision relevant; therefore assume hereon that A G (0,1/2). Let 7T = FT[(JJ = X] and define 7r^(A) - min {TT : A + (1 - A)7r > (1 - A)(l - TT)} ; 7r^(A) = min {TT : (1 - A)7r > A + (1 - A)(l - TT)} . Then 7ra;(A) [respectively, %(A)] is the decreasing [respectively, increasing] curve in Fig. 1 below, illustrating induced preferences in (TT, A)-space. If TT < 7ra;(A), an x-biased individual (i.e., i such that hi = x) nevertheless strictly prefers y to x on grounds of expected fairness and, similarly, if TT > 7ry{X) then a ^/-biased individual (i.e., i such that hi = y) strictly prefers x to y. The more individuals focus on their private interests (the higher is A), the more evidence on the relative fairness of the two alternatives they require for such interests to be dominated.
Fig. 1. Induced preferences
It is analytically useful to define critical values for A, /i(p), hiv) ^ (0^ 1/2), by h{v) = -^
and hip) = -li{p) zp p where there is no ambiguity, write Ij = lj{p). To interpret /i(p), let S = ^ieN ^i ^^ ^^^ ^ ^ ^ ^f ^^^ individuals' signals. If this sum were common knowledge and if A < h then, irrespective of the distribution of signals across committee members, all individuals strictly prefer x [respectively, y] when S > 1
276
David Austen-Smith and Timothy Feddersen
[respectively, S < —1]. And for A G {h^h)^ ctll individuals strictly prefer x [respectively, y] only when 5 > 2 [respectively, S < —2]. When A is below the threshold /i, individuals' induced preferences and behaviour in committee are in principle most sensitive to the opportunities offered by deliberation. Prom this perspective increases in p at a given A are analogous to reductions in A at a given p. Hence it suffices for the most part to focus on X < li. Once types are fixed, the committee decision-making process has two stages: the final stage is a vote (with no abstention) to choose between the two alternatives; this may be preceded by a "debate" in which committee members simultaneously send a cheap talk message about the committee choice. 2.1 Strategies and Equilibrium The solution concept is a refinement of Perfect Bayesian Equilibrium in undominated (anonymous) strategies; although details of the refinement are discussed later, any subsequent reference to "equilibrium" or "equilibrium behaviour" refers to this solution concept. Anonymous strategies are imposed by definition and assumed throughout; anonymous strategies do not depend on the names of the agents. A message strategy is a map fi: {x,y}x
{-1,0,1}-^M
where A4 is an arbitrary uncountable list of messages or speeches. Thus, any individual i G N with bias bi and signal Si makes a speech (sends message) /^{bi,Si) G M. Let M^ = [U{^,j^}x{-i,o,i}/^(^5)] C M denote the range of fi. A debate is a list of messages m == {mi,mj,mk) for any i G N and debate m, let M-i = ^j^i TTIJ.
G M^;
A (pure) voting strategy is a map V : {x, y} X {-1,0, l]xM^
^ {x, y).
So for any individual with bias bi and signal 5j, v{bi, si, m) G {x,y} describes the individual's vote conditional on 6^, si and the debate m = {rrii, rrij^mk) G M^] by convention, the message listed first is invariably that of the individual i. There are two sorts of constraint that any equilibrium strategy pair (/x, v) must satisfy. In any equilibrium, rational individuals vote for that alternative they most prefer conditional on their type, on the equilibrium messages heard in debate and on the event that they are pivotal in the vote under the committee decision rule in effect. That is, for any i e N and signal 5^, given the
Deliberation and Voting Rules
277
voting behaviour v-i of committee members other than z, given the voting rule and given a message strategy // yielding debate m e M^, v must satisfy: E[U{z',bi)\si,m,(i,z,v-i,vpiv]
>
E[U{z';bi)\si,m,fi,z\v-i,vpW]
implies i surely votes Vi = z rather than v^^ — z' ^ where vpiv denotes the event that i's vote is pivotal with respect to x and y at the voting stage. Consequently,the first set of constraints, the pivotal voting constraints^ insure that all individuals' voting behaviour is optimal conditional on their vote being pivotal at every information set. In some cases, such voting recommends voting with an individual's private interests and it is useful to have a term for this: say an individual votes her bias iibi = z implies i surely votes z. On the other hand, say that i votes her signal if, irrespective of bias, Si = 1 (respectively, Si = —1) implies i votes for x (respectively, y). The second set of constraints, the pivotal message constraints, insure that every individual's message is optimal conditional on that message being pivotal for the final committee outcome, given individuals' voting strategies. Specifically, for any i e N and signal s^, given the voting strategy v, the voting rule, and message strategies /i-i for individuals other than i, // must satisfy: E[U{z; bi)\si,mi,fi-i,v, mpiv(mi, m-)] > E[U{z'; 6i)|5i, m-, /i_^,^;, mpiv(mi, m-)]. Thus, i surely makes the speech rrii rather than the speech m^, where inpiv(m^, mQ denotes the event that the messages rrii and m^ produce difi'erent outcomes at the voting stage. Satisfying both sets of constraints gives rise to a variety of equilibria. To simplify the analysis and develop some intuition for how deliberation and voting interact, we focus on equilibria involving three important forms of debate. A (pure) message strategy fj, is: Separating in common interest if, for all 5,5' G { — 1,0,1}, s ^ s' implies [U6e{x,2/}/^(^,5)] n [u6e{x,y}M(^,5')] = 0 ; Semi-pooling in common interest if, for all b G {x,y], fi{x, 0) = /i(6,1) ^ /i(6, - 1 ) - /i(y, 0); and Pooling in common interest if, for all 5,5^ G { — 1,0,1}, /i(x, s) = fi{x, s') and //(y, s) = fi{y, s'). Further, say that fi is separating in private interests if, for all 5 G { — 1,0,1}, IJ^{x,s) y^ /j.{y,s).
278
David Austen-Smith and Timothy Feddersen
An equilibrium (//, v) is said to be a separating debate equilibrium if /i is separating in common interest; analogously, define semi-pooling and poohng debate equilibria. Because debate is cheap talk there is always a pooling equilibrium in which no information is revealed in debate (Farrell [20]). Further, private interest information (bias) matters in debate only insofar as it influences the audience's interpretation of any information offered regarding the common good. With respect to separating equilibria we need only consider equilibria that are separating in common interests and pooling in private interests. This is without loss of generality because if there exists an equilibrium with /x separating in common interests, then speakers are able to include a credible statement of their private interests too: information on bias does not matter under separation in common interest since all of the decision-relevant information is shared. The restriction to semi-poohng equilibria in which agents reveal how they would vote without deliberation is entirely ad hoc, adopted to reduce complexity and provide a method for comparing voting rules.^ We wish to understand how deliberation influences subsequent voting behaviour and thereby the quality of committee decisions. It is not enough simply to look for equilibria exhibiting more or less informative message strategies: any given informative message strategy can in principle be consistent in equilibrium with many voting strategies. Although some of the variation in voting behaviour at any given parameterization is eliminated through reflnement, some remains and is substantively interesting. Perhaps not surprisingly, given any attitude toward fairness (A), the influence of deliberative argument on voting depends on the distribution of private bias in the committee and the relative likelihoods of any individual being informed (q) and the quality of the information conditional on being informed (p). Exactly how these features of the environment interact and the character of deliberative influence that they support, however, is not immediately apparent and turns out to be quite subtle. More detailed discussion of the particular sorts of influential equilibrium behaviour that can arise is deferred until the analysis, and we conclude this section with a brief description of the refinement (the formal definition is given in the Appendix). The possibility, at any given parameterization of the model, of out-ofequilibrium messages and of undominated equilibrium voting profiles under which no individual is pivotal, motivates using a refinement to sharpen predictions. The refinement is essentially technical and has two components. First, individual vote decisions are subjected to individual-invariant trembles and we report behaviour in the limit as the trembles become vanishingly small. This insures that all individuals' equilibrium strategies are the limit of a sequence of best response strategies chosen conditional on being pivotal with strictly posIn particular, we make no claim that these are the only sorts of equilibria; exploring alternatives is left for future work.
Deliberation and Voting Rules
279
itive probability. The second component is a restriction on out-of-equilibrium beliefs at the debate stage. The issue here arises only for semi-pooling debate equilibria in which uninformed individuals are supposed to speak in support of their bias. There is little guidance on how best to proceed here and we simply assume that listeners hearing an out-of-equilibrium speech "I am uninformed" treat it as equivalent to hearing the speech "I believe y is likely the best choice" (where choosing y is without loss of generality)."^^ Although the comparative equilibrium properties of deliberative committee decision making with majority and unanimity voting are a main concern, it is necessary (and of independent interest) to analyze behaviour under the two rules separately. Begin with majority rule.
3 Majority Rule Under majority rule, the alternative receiving at least two votes at the voting stage is the committee decision. This rule is inherently symmetric and, therefore, we consider symmetric strategies here. For any bias b G {x,y} let -b = {x, y}\{b}] recall s e {-1,0,1} and m G {-1,0,1}^. Definition 1. (1) A message strategy JJL is symmetric if and only if, for all (6, s), /j.{b, s) = -/i(-6,-5). (2) A vote strategy v is symmetric if and only if, for all (6,5, m), v{b, 5, m) — a: 4=^ f (—6, —5, —m) = y. Imposing symmetry on fi clearly adds little: by definition, if jd is separating, semi-pooling or pooling in common interest then // is symmetric. Requiring symmetric voting strategies, however, although a mild restriction for the present model, has more bite. Suppose there is no debate or that the message strategy is pooling in common interest (which, given debate is cheap talk here, always constitutes an equilibrium strategy). Then it is easy to check the symmetry assumptions imply that, for all A < 1/2, the unique equilibrium voting profile is for all individuals to vote for their most preferred alternative conditional on their private signal and on being pivotal for the decision. ^° We also considered an alternative specification: that out-of-equilibrium speeches claiming no information were believed surely. Although there turn out to be some differences, they are inconsequential; the derivations required, however, are considerably more tedious.
280
David Austen-Smith and Timothy Feddersen
Proposition 1. Suppose there is no debate or that // is pooling in common interests. Then, up to behaviour on the boundary X = h, there is a unique symmetric voting equilibrium under majority rule; further, for any z G {x, y}: (1) if X > h then, for all s G {—1,0,1}, v{z, 5,0) = z; (2) if X < h then v{z, 1,0) = x, v{z, - 1 , 0 ) = y and v{z, 0, 0) = z. Proofs for Proposition 1 and all subsequent results (save Proposition 5, where the proof is by example) are collected in the Appendix. Say that a committee decision for z G {a;, y} is right if z is chosen under the voting rule conditional on s = (si, 52, 53) being common knowledge at the time of the vote.^-^ Then, for X /i, however, the likelihood of error jumps to 1/2. Now suppose individuals have an opportunity for debate prior to voting. Because information on common interests is intrinsically imperfect, we abuse language somewhat and say there is "full information" at the voting stage if the realized list of signals s G {—1,0,1}"^ is common knowledge. An equilibrium (/i, v) exhibits full information equivalence if and only if, for all s, the equilibrium committee decision is always right. It is worth noting that full information equivalence does not imply all information is revealed in debate but only that, along the equilibrium path, committee decisions are those that would be made under common knowledge that s = m. 3.1 Separating Debate Equilibria It is evidently possible for there to exist separating debate equilibria in which deliberation has no impact at all on individual voting behaviour. For example, suppose A is sufficiently high relative to the quality of private information p; then no feasible private signal or deliberative argument can outweigh any individual bias and, therefore, fully revealing private information in debate can be an equilibrium strategy precisely because it is inconsequential. More interesting are those separating debate equilibria in which voting behaviour is responsive to deliberation. ^^ An alternative definition of the "right" decision is the alternative most likely in the common interest, conditional on the realized list of signals. When A < /i and voting is majority rule, the two definitions recommend the same alternative but, in general, they are distinct because the definition in terms of common interest alone is insensitive to private bias.
Deliberation and Voting Rules
281
In separating debate equilibria, speeches regarding the relative merits of the two alternatives are completely untainted by private interest and deliberation can generate consensus in individuals' induced preferences over {x,y} whenever warranted by the realized list of signals. Moreover, if a separating debate equilibrium (//, v) exhibits full information equivalence, then the right committee decision is guaranteed because v is uniquely defined by all individuals voting sincerely conditional on s being common knowledge. Perhaps unfortunately, therefore, existence of such equilibria cannot always be assured. Proposition 2. Fix an information structure (p^q). There is a unique value X{p,q) < h{p) such that there exists a full information equivalent symmetric separating debate equilibrium if and only if \ < )^{p^q)' Moreover, (1) for all p G (1/2,1)^ X{p^q) is strictly single-peaked in q on (0,1) with peak A*(p) < li{p) such that limp_i A*(p) = limp_^i li{p) — 1/2; (2) for all q G (0,1); A(p, g) is strictly increasing in p on (^,1) with maximum A*(g) < 1/2 such that limg_>i X*{q) — 0. At first glance, statements (1) and (2) of the proposition, taken together, may appear contradictory. However, they simply indicate that the order of limits is consequential: Fig. 2 illustrates the function A(p, q) for three values of q. An implication of the proposition is that, for any signal quality p < 1,
Fig. 2. A(p, q) for various q
there exists a full information equivalent separating debate equilibrium only if the probability of individuals being uninformed, g, is neither too high nor too low. To see the intuition here, let (/i, v) be a full information equivalent separating equilibrium and fix p < 1; then A < /i(p). Because A < li{p) and fi is separating in common interest, there is no difliiculty satisfying the pivotal
282
David Austen-Smith and Timothy Feddersen
voting constraints; it is the pivotal message constraints that bind. Specifically, from the proof to Proposition 2, it is the pivotal message constraint on the uninformed individuals that defines those (A,p, g) for which the full information equivalent separating debate equilibria exist. So consider an uninformed, y-biased individual i and suppose i is considering rrii G { 0 , - 1 } conditional on all others following the prescribed separating equilibrium strategies. Then z's speech in debate is message pivotal in three events: If either event (a) or (b)
event Sj = rrij Sk = ruk (a) (b) (c)
0 1 0
0 -1 1
bj
bk Pr[(.)|si = 0]
a: X y
X X
y
1(1-9f ia'p(i - p) i(i-9)<j
occurs, z's preferred outcome is y and, given equilibrium voting v, this is the committee decision if and only if i sends the message m^ = — 1 rather than the truthful message rrii == 0. On the other hand, i's most preferred outcome at event (c) is x and this is the committee decision if and only if i sends message rrii = 0. The critical pivotal event as q goes to one is clearly (b), the case in which both of the other committee members are almost surely informed but with opposing signals. As the probability of being uninformed becomes negligible, the likelihood of event (b) being true conditional on i being pivotal increases in relative importance to the point that i chooses to deviate from reporting her lack of information (inducing all individuals to vote their bias for x) in favour of influencing the committee to support y}'^ Similarly, when q goes to zero the most likely message pivot event is (a) with both j and k being uninformed; in this case i believes that the committee decision depends almost surely on the distribution of bias in the case i reports rrii = Si = 0 but is (conditional on the event (a)) surely y if she sends message m[ = —1. An alternative perspective on the separating equilibria identified in Proposition 2 is useful. Fixing A = 1/10, Fig. 3 identifies the set of parameter values (p, q) for which there is a full information equivalent separating debate equilibrium. As A ^ 1/2, this region shrinks toward a neighbourhood of the point (1,1) and, as A -^ 0, the region expands to fill the space of all information structures. Loosely speaking, given any probability of being informed, g, high p is equivalent to low A. For A < ^i and message strategy separating in common interest, the pivotal vote constraints are surely satisfied by individuals' voting on the basis of their full information induced preferences. Therefore the boundary of the region in Fig. 3 for which the relevant equilibria exist is the set of informational structures at which uninformed individuals are just ^^ Of course, at p and q sufficiently large, the overall probability of an uninformed committee member being signal pivotal is negligible.
Deliberation and Voting Rules
283
separating debate equilibria exhibiting full information equivalence
/ Fig. 3. Separating debate equilibria indifferent between revealing their lack of information and making a speech in support of their private interests. Of course, because making the latter speech is designed to encourage others to vote for the speaker's bias by influencing beliefs, the speech itself is not in terms of the speaker's bias per se but rather in terms of the common interest. Proposition 2 claims that when individuals value the common good sufficiently highly and only a minimal amount of evidence in favour of an alternative being more in the common good is required to induce an individual to support that alternative, all private information on the common good can be credibly revealed in equilibrium and the subsequent voting behaviour results in full information equivalence. None of these properties, however, necessarily hold for semi-poohng debate equilibria. 3.2 Semi-pooling Debate Equilibria By definition, in semi-pooling (SP) debate equilibria relatively informed individuals - that is, those for whom 5^ 7^ 0 - continue to make speeches advocating the alternative supported by their information, irrespective of their private interests, but uninformed individuals - those for whom Si = 0 - now make speeches advocating the alternative they favour on private interests alone. In view of Proposition 1, therefore, speeches in SP debate equilibria involve everyone effectively announcing how they would have voted without debate. Beliefs regarding a speaker's private bias are not the only thing that distinguish interpretation of speech under SP from that under separating message strategies. In a separating debate equilibrium, the particular values of the
284
David Austen-Smith and Timothy Feddersen
parameters q and p play an important part in defining when full information equivalent voting constitutes equilibrium behaviour, but have nothing to do with the interpretation of debate speeches per se}^ In SP debate equilibria, however, this is no longer true: the likelihood of being informed and the quality of any information in fact received bear both on the interpretation of speech and on subsequent voting decisions. Not surprisingly therefore, there can be a variety of SP debate equilibria that, at least observationally, differ exclusively in voting behaviour. Depending on the information structure, identical debates (that is, any m G M^^ or permutation thereof) can influence diff'erent individuals in different ways and lead to various profiles of voting decisions. The different sorts of symmetric SP debate equilibria identified reflect different degrees to which individuals can be influenced by debate and signals. Some more language is useful. Fix a semi-pooling debate equilibrium {ji^v). An uninformed individual i is influenced in debate if i's vote under v is not constant in the speeches of others. An informed individual i is marginally influenced in debate if either (1), (2) or (3) are true: (1) Vs signal is against i's bias, both of the others argue in favour of i's bias, and i votes for her bias (2) i's signal is for i's bias, both of the others argue against i's bias, and i votes against her bias (3) i's signal and both of the others' speeches are against i's bias, and i votes against her bias. An informed individual i is influenced in debate if all of (1), (2) and (3) obtain. The set of pure voting strategies, u, for the (symmetric) SP debate equilibria that exist is described in the Appendix, where their respective location in (p, g)-space is also Ulustrated for A = 1/10. And it is worth remarking here that, whereas the binding constraint for the separating debate equilibrium is at the debate stage through the pivotal message constraints, the binding constraints on the SP equilibria are at the voting stage, through the pivotal voting constraints. Insofar as debate in SP equilibria is constant and only the voting responses to debate changes with the information structure, this shift in which constraints bite has some intuition. Broadly speaking the distribution of semi-pooling debate equilibria on the space of information structures exhibits the following the pattern. Conditional on the quality of information, p, not being too low, in which case semi-pooling debate cannot be influential at all, if the probability of being informed, Q', is: •
low, then at most the uninformed are influenced in debate;
^^ Of course, A < li{p) is necessary for full information equivalent separating debate equilibria to exist at all. Nevertheless, any interpretation of (equilibrium) speech conditional on this constraint not binding is independent of p.
Deliberation and Voting Rules
285
•
moderate, then informed individuals can be marginally influenced in debate at all save the lowest values of p, but uninformed individuals are influenced in debate only if p is moderate to high;
•
high, then the informed and uninformed individuals are influenced in debate at all save the lowest values of p.
When the probability of being informed is high and all individuals are influenced in semi-pooling debate, the equilibrium voting strategy v is consistent with all individuals treating all speeches as true. Thus, all voting behaviour is "sincere" in that it reflects the balance of bias, signal and debate for each committee member; nevertheless, individual debate behaviour is not so sincere since the uninformed misrepresent their knowledge in debate, arguing for their bias by adopting the speech of those informed in favour of that bias. Consequently, not all of the decision-relevant information is offered in semipooling debate and full information equivalence is not assured. For example, let (/i, v) be a semi-pooling debate equilibrium in which both informed and uninformed individuals are influenced; suppose s — (1,0,0) and individuals 2 and 3 are ^/-biased. Then by deflnition of /i, the debate is m = ( 1 , - 1 , - 1 ) and the equilibrium voting strategy v yields a unanimous vote for y. But (as shown in the Appendix, Fig. 6) this sort of equilibrium only exists for A < hip) which implies the right committee decision at s is x. On the other hand, by Proposition 1, the outcome in this case without debate is also x. In other words, despite some speeches in semi-pooling debate occasionally failing to reflect any private information, such strategic speech-making turns out not to lead to any worse an outcome than the one under no debate. The preceding example is not an artifact. Before making this assertion precise, it is useful to check some intuitive properties of voting behaviour in any (anonymous although not necessarily symmetric) SP debate equilibrium under majority rule. Lemma 1. In any SP debate equilibrium under majority rule, i;(y,0,-l,-l,-l)=y. Assuming the committee makes decisions under majority rule, therefore, an uninformed y-biased agent surely votes her bias following any (semi-pooling) debate in which everyone argues for choosing y over x and, evidently, a completely symmetric argument applies for x-biased individuals; that is, v{x, 0,1,1,1)=:?: also. In other words, the symmetry of majority rule coupled with that of the semi-pooling message strategy fi implies a considerable degree of symmetry in SP equilibrium voting. The next result, Lemma 2, says that SP debate equilibrium vote decisions are signal (claim 1) and bias (claim 2) monotonic.
286
David Austen-Smith and Timothy Feddersen
Lemma 2. Let {^,v) be any SP debate equilibrium and m G M.^^. Then, under majority rule, for all signals s > s' and each bias b: (1)
= y], ( [v{b,s, m) =y^v{b,s',Yn) , m) = X ^ v(b^ 5, m) = x\;
(2)
( [v{x, s,xn) =y=^ v{y, 5, m) = y], \ [u(y, 5, m) = X =^ v{x, 5, m) = x].
It seems sensible that in addition to bias and signal monotonicity, debate equilibria should also exhibit some sort of monotonicity in messages: if, given a distribution of bias and information in the committee, the only difference between two debates (mi,m_-i), (mi,m'_J from i's perspective is that the speeches of others m'_^ are both more favourable to the individual's bias than are m_i, then i should vote his bias following (mj,m'_j) if i votes his bias following (mi,m_i). This sort of monotonicity is satisfied by all of the debate equilibria considered so far and, as will become apparent shortly, all of those discussed in the next section on unanimity rule. But messages and votes are strategic decisions and, at least as far as we know at present, this form of monotonicity is not implied by the current assumptions on equilibrium behaviour. Definition 2. A voting strategy v satisfies debate monotonicity if and only if, for all {bi.Si.rrii), v{bi^Si,mi,mj,mk) = bi, bivn'^ > biUij and bivn'^^ > bimk imply Vi{bi,Si,mi,m'^,m'^) = bi. In words, if an agent is voting consistent with his bias after observing a signal and some debate then he must also be voting for his bias if he observes the same signal (and therefore sends the same message) and a debate that is more favourable for his bias. Note that debate monotonicity requires holding constant the agent's bias, signal and message. Moreover, debate monotonicity does not imply, for instance, that an uninformed y-biased individual who sends a message m = —1 and hears a split debate m_i = (1,-1) surely votes his bias. Requiring debate monotonicity, then, is a substantively weak restriction; it is nevertheless very useful analytically. Lemma 3. In any symmetric SP debate equilibrium in which voting is debate monotonic, either there is a positive probability the vote of agent i is pivotal given debate {rrii.mj.mk) = (1, —1, —1) or agents j and k are both voting for y-
Recall the definition of a "right" committee decision as being the decision reached under decision making with full information on s; a "wrong" committee decision is any decision that is not "right". There are, therefore, two sorts of error in committee decision making, depending on the distribution of induced preferences at any realized list of signals s.
Deliberation and Voting Rules
287
Definition 3. The equilibrium {/J^^v) induces an •
error in common interest at s if all individuals' induced preferences over {x, y] are unanimous at s, but the equilibrium committee decision is for the less preferred alternative;
•
error in bias at s if each individual's induced preference over {x,y] at s is completely defined by his or her bias, but the equilibrium committee decision is for the minority's preferred alternative.
Say that committee decision-making subject to a set of debate and voting rules a can make an error in common interest (bias) at {X,p,q) if, under a, there exists an equihbrium {fi,v) at (X^p^q) that induces an error in common interest (bias). Definition 4. With respect to common interest (bias), committee decisionmaking subject to a set of rules a weakly dominates committee decision-making subject to a set of rules (3 at {X,p,q) if (1) ot can make an error in common interest (bias) at {X,p,q) implies P can make an error in common interest (bias) at {X,p^q) and (2) the converse is false. In words, an institutional arrangement a weakly dominates another, /3, with respect to making a particular sort of error if, first, whenever there exists an equilibrium under a at which the committee makes the wrong decision for some s, there also exists an equilibrium under /? at which the committee makes the same mistake; and second, there exist circumstances under which P yields an error but every equilibrium under a results in the right decision. Proposition 3. Assume only separating, semi-pooling and pooling debate equilibria are played and that committee decisions are made by majority rule. If equilibrium voting is debate monotonic then, (1) with respect to common interest, committee decision making with debate weakly dominates committee decision making without debate at almost all {X,p,q); (2) if separating equilibria are played whenever available, then (1) holds also with respect to bias. Two things are worth emphasizing about Proposition 3, the main result of this section. First, the result applies quite generally to all symmetric pure strategy SP debate equilibria exhibiting debate monotonicity (both with and without the technical refinement); and second. Proposition 3(1) does not say that for every feasible (A,p, g) there exists a debate equilibrium that is, with respect to yielding "right" decisions at (A,p, g), weakly better with respect to errors in common interests than the no-debate equilibrium, but rather that every separating, semi-pooling and pooling equilibrium has this property at
288
David Austen-Smith and Timothy Feddersen
any feasible (A,p, g). But as indicated in the statement of Proposition 3(2), debate monotonicity is not enough to extend the result for errors in common interest to errors in bias. Errors in bias can only occur if (up to permutations) SG {(0,0,0),(0,1,-1)}, implying the probability of Y being the true state is 1/2. A priori^ there seems little reason to think that such errors in bias are any less important than errors in common interest. Although majority rule without debate is not immune to errors in bias (an example is given below), it is possible for debate to yield such an error where none would be made in its absence. To see why the restriction to playing separating debate equilibria where available is necessary for Proposition 3(2), assume y is the right decision at the situation s = (0,0,0). Since y is the right decision by assumption, at least two of the committee are y-biased (say, z = 1,2), so i/ is surely the nodebate equilibrium decision (Proposition 1). Now let (fJ^^v) be a symmetric semi-pooling debate equilibrium satisfying debate monotonicity. There are two debates possible in equilibrium, depending on individual 3's bias, li bs = y then m = ( —1,—1,—1) and Lemma 1 implies there is no error; so suppose ^3 = ^1 yielding m — (—1, —1,1). By anonymity, if {ix.v) results in an error at this debate, each uninformed ^/-biased individual i G {1,2} must vote for X conditional on sending a message rrii = —1 and hearing a split debate m_j = (—1,1); that is an error in bias implies t;(2/, 0 , - 1 , - 1 , 1 )
=x.
And it turns out that indeed there exist SP debate equilibria in which the uninformed adopt such voting decisions (see the Appendix, Lemma 4); that is, uninformed individuals speak in favour of their bias in debate but vote against that bias if they hear a split debate among the other committee members. Although perhaps unusual, such voting behaviour by uninformed individuals may not be absurd. Given that an uninformed individual i delivers the speech rrii = —1, when i hears a split debate {mj^rrik) = ( 1 , - 1 ) , she might reason that if her vote is pivotal, it is most likely to be the speaker who presented the minority opinion, rrij = 1, who is voting against the majority position, ^ i = ^/c == — 1? advocated in debate. In turn, this suggests that j is relatively more likely to be informed in which case, conditional on being pivotal, i voting for the minority deliberative opinion is the best thing to do. Similar reasoning applies to the remaining possibility, s = (0,1, — 1): if the bias distribution is b = (y^x.y), then the debate has to be m = (—1,1, —1) and, in the SP debate equilibrium in which uninformed individuals vote against their bias conditional on hearing a split debate, the first two individuals vote for X to produce an error. In this case, however, there is no guarantee that the no-debate equilibrium outcome is right: if b ' = (x,y,y), the right decision is y but the no-debate outcome is x.
Deliberation and Voting Rules
289
It is not hard to see from the preceding discussion t h a t a necessary condition for an error in bias (not involving errors in common interest) to result from debate is t h a t uniniformed individuals vote against their bias conditional on observing a split debate. In the Appendix, we show t h a t all of the symmetric semi-pooling debate equilibria exhibiting such behaviour exist only if the separating debate equilibrium also exists. Proposition 3(2) then follows.
4 Unanimity Rule In this section we ask how variations in the voting rule influence the character and extent of deliberation. Further, much of the more normative literature on deliberation suggests t h a t requiring unanimity at the voting stage promotes more information sharing and argument in debate, since some form of consensus is now essential for pro-active committee decision.^^ In this section we demonstrate t h a t such reasoning is mistaken. Unlike with majority rule, the status quo policy is consequential under unanimity rule: the status quo x can be rejected in favour of y only if all three committee members vote for y against x}^ And since unanimity rule is evidently not symmetric, there is no good reason to insist, or even focus, on symmetric equilibria; in fact, quite the contrary is true. Consequently, we no longer look for symmetric voting strategies, although we maintain the presumption of anonymity. Moreover, there are (as discussed momentarily) multiple voting equilibria without debate under unanimity rule. It follows t h a t any sort of comparative statement across majority and unanimity rules is necessarily tempered by these fundamental diS'erences between the two decision schemes. Suppose there is no debate stage and note t h a t an individual is pivotal in voting only if b o t h of the other committee members are voting for y. T h e n it cannot be the case t h a t all types surely vote for y in any equilibrium, ^^ Particularly direct examples include: "It should be remembered that veto power or unanimity represents a constraint that induces deliberation: when parties can block outcomes, actors have incentives to find reasons that are convincing to all, not just to the majority" (Eriksen [18, page 15/16]); "Hence the unanimity requirement in jury verdicts, which is intended to encourage through deliberation as necessary for a conviction" (Shapiro [38, page 12]); "The necessity of a consensus of all jurors which flows from the requirement of unanimity, promotes deliberation and provides some insurance that the opinions of each of the jurors will be heard and discussed" (South Australian High Court 1993; quoted in Walker and Lane [39, page 2]) ^^ A seemingly plausible alternative assumption here, is to take a fair lottery over {x, y} as the status quo and require a unanimous vote to insure either alternative surely. But then decision making is over three, rather than two, alternatives, a quite different scenario.
290
David Austen-Smith and Timothy Feddersen
irrespective of their bias or signal. It is easy to see why: suppose the claim false and consider an individual i with signal 5^ = 1 and bias for x. Then the event that i is vote pivotal under unanimity contains no additional decision-relevant information for i, in which case, given signal 5^ = 1, voting for x surely is the best decision. Depending on the parameters {X,p,q) there can exist distinct no-debate equilibria with unanimity rule and, for some (A,p, g), the only nodebate equilibria are in mixed strategies. The possible pure voting strategies are described in the table below, where the entries in each cell are the vote pairs [u(^,5^,0),u(x,5i,0)]; all no-debate equilibrium mixed strategies have support in this set of pure strategies.^^ Table 1. Pure strategy no-debate voting equilibria Si
[v{y,Si,(l)),v{x,Si,(D)] 1 2
3
4
5
-1
y,y y,y y,y y,yy,x
0
y,y y,y y,x y,xy,x
1
x,x
y , x x^x y , x y , x
In common with the story for SP debate equilibrium voting under majority rule, broadly speaking, the better the quality of the information the more willing are uninformed individuals to vote for y, effectively delegating the committee decision to the informed committee members, who likewise are more willing to vote their signal irrespective of bias (1). And again, the presence of uninformed individuals is important here. As is apparent from Fig. 7 in the Appendix, as signal quality declines individuals with an x-bias become increasingly unwilling to vote against their bias. It is immediate that no-debate equilibrium under unanimity can yield errors in bias: suppose the equilibrium is 1 and all individuals are both uninformed and x-biased; then there is a unanimous vote for y where in fact x is the right decision. The multiplicity of no-debate equilibria under unanimity rule makes an unequivocal statement about the likelihood of an equilibrium committee decision being "right" contingent on the particular equilibrium played. However, the bounds are clear. The smallest likelihood of error is when voting strategy 1 is equilibrium behaviour. Here, an error occurs only in state Y when all individuals are informed but one sees an incorrect signal; this occurs with probability q^p'^{l—p)/2. At the other extreme for A < /i, when 4 is equilibrium behaviour ^^ A description of the mixed strategy equilibria and of the sets of information structures for which the various no-debate equilibria exist can be found in the Appendix.
DeUberation and Voting Rules
291
there are multiple events at which error can occur; doing the (tedious) calculation gives the likelihood of error as [4pq {1 — 2q + pq) — pq^ {p + 2p^ ~ 4)] / 1 6 . And for A > / i , under 5, the likelihood of error is 1/2. More interesting, is what happens when deliberation precedes any committee vote. P r o p o s i t i o n 4. There exists no separating debate equilibrium ity.
under
unanim-
Comparing this result with Proposition 2 undermines any general claim t h a t requiring unanimity to make policy changes induces more deliberation in committee t h a n requiring only a majority. Depending on individual attitudes toward the common interests and on the information structure, deliberation can be fully informative under majority rule but not under unanimity.^^ Although full separation, and thereby full information equivalence, is impossible under unanimity, deliberation can nevertheless be informative under some circumstances. We prove the claim by example. Under unanimity rule, any individual who so chooses can guarantee a committee decision for x with her vote alone. This suggests a debate strategy in which a ^/-biased individual argues for y in debate irrespective of any private information: if such an individual is persuaded, either by her private information or by deliberation, t h a t X is most in her interests then her own vote insures this outcome whatever she says in debate; b u t if she is left preferring y over x then her deliberative argument can be pivotal. Similarly, an individual for whom x is most in his private interests has nothing to lose by sharing his information on the relative common good properties of the two alternatives. Formally, the suggested p a t t e r n of deliberation is described by the asymmetric message strategy, fl: for all distinct 5,5' G { - 1 , 0 , 1 } , fL{x, s) ^ jl{x, s') and /i(y, s) = Jl{x, —1). 17
A similar impossibility result is proved by Doraszelski, Gerardi and Squintani [13] (DGS), the only other model, to the best of our knowledge, that considers deliberation under unanimity rule. DGS study a two-person committee that is choosing between a status quo and a given alternative policy; rejection of the status quo requires unanimous approval. There are two states of the world from the common interest perspective, say X and F , and both individuals strictly prefer x (respectively, y) in state X (respectively, Y), Where they differ is in the attitudes about making errors and these attitudes (parametrized by some real number from the unit interval) are private information. In addition to learning their particular attitude to error, each individual also observes a noisy binary signal regarding the true state of the world. Inter alia, DGS study what happens when both individuals can give cheap-talk signals about their signals prior to voting. Their main results are that there is no separation in debate and deliberation is influential only in the case when an individual's signal conflicts with her disposition and prior belief: "When there is a conflict between a player's preferences and her private information about the state, she votes in accordance with her private information only if it is confirmed by the message she receives from her opponent" (p.2).
292
David Austen-Smith and Timothy Feddersen
Thus, under p,^ all y-biased individuals pool in common interests and all xbiased individuals separate in common interest. For want of a better term, then, call any debate equilibrium {ft, v) a bias-driven debate equilibrium, In any bias-driven debate equilibrium, those who are most likely to want change (y-biased individuals) argue consistently for this alternative irrespective of their signal, so suppressing any information they might have in support of the status quo x; against this obscurantism, those most likely to resist change (x-biased individuals) are willing to reveal all of their information in debate, whether or not it suggests that in fact y is the better alternative on common interest grounds. But despite this willingness on the part of x-biased committee members to make a case for y when appropriate, the only credible arguments are those who argue (at least weakly) on behalf of x; any effort by an x-bias individual to argue for y is confounded by the incentives for those with a private interest for y also arguing that case. So there is small hope here of achieving any sort of consensus through deliberation alone. But such a lack of deliberative consensus need not imply that deliberation cannot yield unanimous voting in committee. Equilibria involving such asymmetric deliberation do exist; Fig. 4 illustrates an example for A = 1/10. As indicated in the diagram, a necessary but not sufficient condition on the information structure (p, q) for the bias-driven equilibrium to exist is that the no-debate voting equilibrium 1 also exists at {p.q)' 1
bias-driven debate equilibrium
p
1
Fig. 4. Bias-driven debate equilibrium (/i, v) The identified bias-driven equilibrium is the pair (/i, v). The message strategy ft is defined above and the voting strategy v is described in Table 2, where the pairs in the two "mi"-columns are the votes, (v{y,'),v{x,')).The binding constraints on {ft.v) are two pivotal voting constraints: the lower
DeUberation and Voting Rules
293
Table 2. Voting strategy v Si
-1
M-i
rrii e
{-1 .0}
rrii — X
1 2
y.y y,x x,x
y,x x,x
0
< -1 [-1 + 1] [0+0] >1
y,y y^x y^x x,x
y^y x,x y^x x,x
1
> -2
x,x
x,x
y->y
boundary illustrated in Fig. 4 describes the locus of information structures at which an x-biased individual with signal against his bias is indifferent at v between voting for y (as required) or x conditional on hearing a split debate, {rrij.mk) = (—1,1); the upper boundary describes the locus of information structures at which an ^/-biased individual with signal against her bias is indifferent at V between voting for x (as required) or y conditional on hearing a uniform debate, {rrij.mk) — (—1, —1). The most interesting thing to note about the strategy v is that an uninformed 2/-biased individual i votes for y against x if i makes any speech rrii G {—1,0} (weakly) in support of choosing y, but votes for x against y if, for some reason, i advocates choosing x^rui = 1, and the others are divided in debate, (mj.mk) ~ (—1,1). In other words, under v, an individual with a given signal, hearing given speeches by others in debate, nevertheless votes differently depending on the particular cheap talk speech she delivers; in this case, the individual "talks herself into voting against her bias". Such behaviour is not, it turns out, unreasonable: because the subsequent votes of others depend in part on the arguments they hear in debate, the pivotal voting constraint facing an individual following one speech does not necessarily coincide with that following a different speech. In fact, although, in the equilibrium {ft^v), this particular behaviour is off-equilibrium-path, it proves essential to support existence of (/i, v) as equilibrium behaviour at all. If, as seems intuitive, the uninformed y-biased individual's vote is independent of her own message at any debate (in particular, at the debate {mj.mk) = (—1,1)), then not all of the message pivotal constraints can be satisfied along the equilibrium path. Similar considerations apply, although less evidently, elsewhere in the equilibrium voting profile. From Table 2, an x-biased individual with a signal against her bias [si = —1) is required to vote for y conditional on M-i = —1 whatever speech she makes. However, if the probability of others being informed, q, is sufficiently low, then such an individual strictly prefers to vote for X in the event she sends the off-path message m[ = 1 supporting her pri-
294
David Austen-Smith and Timothy Feddersen
vate bias rather than her signal, /i(x, — 1) = — 1 (or a speech m^ — 0) but not otherwise. Moreover, if the individual is presumed to vote for x conditional on sending m\ — \^ then (]x,v) cannot describe equilibrium behaviour at any information structure. Recall that the probability of the committee choosing the wrong alternative in the no-debate equilibrium 1 is g^p^(l — p)/2. Under the bias-driven debate equilibrium {fi,v), the probability of the committee making an error in common interest falls to zero but that of making an error in bias remains strictly positive: if bi = y^ b2 = b^ = x and all individuals are uninformed, the debate under (/i, i)) is m = (—1,0,0) and, given D, all individuals vote for the wrong outcome, y, exactly as in the no-debate equilibrium 1. Nevertheless, it is clear by inspection that the debate equilibrium {fi,v) also weakly dominates the no debate equilibrium 1 with respect to bias (ft.v), that is, when s G {(0,0,0), (0, —1,1)}. Whenever there is an error in bias alone under (/2, v) there is also an error under 1 without debate, but the converse is false: let s = (0,0,0) and bi = x all z; then without debate the wrong decision y is made but with debate the decision is x. It follows that the bias-driven debate equilibrium (/i, v) weakly dominates the no debate equilibrium 1. This is perhaps to be expected: debates supported by ft necessarily make committee members strictly more informed at the voting stage than they are without debate.^^ In general, however, the weak dominance result for {j2,v) does not extend to all bias-driven debate equilibria. Proposition 5. Assume committee decisions are made by unanimity rule. There exist (A,p, Q') at which the committee makes the wrong decision under a bias-driven debate equilibrium (ft^v), but makes the right decision under a no-debate equilibrium, v^. Proof. We show by example that bias-driven debate can support errors in common interest in settings where the committee decision under the relevant (pure strategy) no-debate equilibrium is the right decision. Assume A = 1/10 (this particular value is inessential). The message strategy ft is defined above; the vote strategy v is described in Table 3 where, as usual, the "mj"-columns are the votes, {v{y, ')^v{x, •)). Insisting on the technical equilibrium refinement (individually independent trembles) leads to difficulties off" the (postulated) equilibrium path here. In particular, the strategy pair (/i, -0) is an equilibrium and survives the refinement only on a line cutting through the set (p,g)[l] C (^,1) x (0,1) on which the no-debate voting equilibrium 1 exists. However, the no-debate equilibrium 1 obviously exists without insisting on the refinement and lifting the refinement further results in {ft,v) constituting equilibrium behaviour on a nonempty set of information structures having strictly positive measure: see ^^ This is true even if the debate is m = (—1,—1,—1); in this case all individuals know there is no x-biased committee member with a signal 5 > 0.
Deliberation and Voting Rules
295
Table 3. Voting strategy v Si
M-i
rrii = —1 rrii = 0 rrii = 1
-1
< -1 [-1 + 1] [0 + 0] 1 2
y,y y,y y,y y,x x,x
y.y y,y y.y y,x x,x
y,y y,x y.y y^x x,x
0
-2 -1 [-1 + 1] [0 + 0] > 1
y^y y,x y,x y.x x,x
y,y y^x y,x y^x x^x
y^y y,y y.y x,x x,x
1
> -2
x,x
x,x
x,x
Fig. 5. ^ In the figure, the region below the two intersecting thick lines is
p
1
Fig. 5. Bias-driven debate equilibrium (/i, v)
the set of information structures for which (/i, 0) is an (unrefined) equilibrium; the downward sloping thin line is the lower boundary of (p, g)[l]. If the technical equilibrium refinement is imposed, t h e set of information structures delineating those (/2,0) debate equilibria surviving the refinement is precisely ^^ As indicated in the Appendix, establishing these claims formally is both tedious and computationally demanding, so we omit the details. All of the derivations supporting this example and the figures in the text, however, are available from the authors on request.
296
David Austen-Smith and Timothy Feddersen
the downward sloping thick line, that is, the upper boundary of the unrefined set. Consider any {p\q') G (p, ^)[1] for which (ft^v) is a bias-driven debate equilibrium (the information structure (0.68,0.30864) works for the refined, non-generic, case). Assume the realized profile of signals is s == (—1,0,0) so the right committee decision is y. Assume that the two uninformed individuals, i = 2,3, are both x-biased. Then under the no-debate voting equilibrium 1, the committee unanimously votes for the right alternative y. Under the debate equilibrium (/i,'0), however, the realized debate is m = s = (—1,0,0) but subsequent equilibrium voting has both individuals 2 and 3 voting for x, thus vetoing y and leading to the wrong committee decision. The reason for the error in the example establishing Proposition 5 is not hard to see. In the relevant information structure, the probability of any individual being informed is sufl&ciently low that a single noisy speech for y is insuflScient to ofi'set any private bias for x. When there is no debate, however, the uninformed x-biased individuals condition on being pivotal, that is, on the event that both of the other committee members are surely voting for y, in which event there is positive probability on both individuals observing signals for Y being the true state. On balance, the ex ante possibility of there being two signals in favour of Y conditional on being pivotal without debate, is stronger support for choosing y than knowing as a result of debate that there is at most one signal in favour of Y. Proposition 5 implies that an analogous claim to Proposition 3 (which holds with or without the trembles refinement) is not available. The result does not imply that deliberation is on balance detrimental to the quality of committee decision making under unanimity rule and it seems unlikely that this is the case. What is true, is that, in comparison with majority rule, requiring unanimous voting induces quite distinct sorts of deliberation and incentives to share information in debate. And on balance, majority rule offers more opportunity for credible deliberation and symmetric information sharing.
5 Discussion Despite the fact that the role of deliberation in agenda-setting per se may likely prove the most important, there is still a great deal to be learned about deliberation over fixed agendas. Assuming a fixed agenda, the particular issue we address in this paper concerns the connection between the voting rule adopted by a committee for making a decision and the character of any deliberation preceding the vote. In this regard, the informal literature on deliberation and consensus claims (among other things) that unanimity rule promotes deliberation. The intuition underlying this argument is that because
Deliberation and Voting Rules
297
it is necessary to arrive at a consensus to reach a decision, more information is revealed by committee members in an effort to persuade the last individual. As our model demonstrates, however, such reasoning is fallacious. Although unanimity rule creates incentives for supporters of the status quo to reveal information, it likewise creates incentives for others to conceal information favouring that status quo. This in turn generates an externality rendering information from all members of the committee suspect. In contrast, majority rule balances the incentives of those biased for and against the status quo to the extent that, under at least some circumstances, everyone can truthfully reveal their private information in debate. Moreover, even when majority rule cannot support purely truthful communication we show that majority rule (for a reasonable subclass of semi-pooling strategies) does not result in the committee making decisions that everyone in the committee would oppose if all information were shared; this is not true of deliberation under unanimity rule where such mistakes can occur. The analysis underlying our results depends on what is, at least from a standard game-theoretic perspective, a fairly natural conception of committee deliberation, specifically, deliberation as strategic information transmission. And within this framework, there are some fairly obvious extensions, including sequential speechmaking, consequential variation in the relative weights individuals' place on private interests, and so on. However, the usual apparatus of incomplete information games may in fact to be too restrictive to address some of the important questions considered in the normative political theory literature. And a key issue in this regard concerns whether or not all consequential deliberation is inherently informational. If it turns out that in fact arguments predicated on strategic information transmission models fail to capture the salient features of committee deliberation precisely because these features are not intrinsically informational, then the relevance of our discussion to the normative literature becomes moot. There seem to be two principal ways in which deliberation might not be informational. Loosely speaking, the first involves equilibrium selection in coordination games (Farrell [19]; Rabin [36]; Calvert and Johnson [8]) and the second involves argument through analogy and precedent (Aragones et al. Although it is surely the case that coordination and argument through analogy do not concern information of the sort considered in the model here, they are both intrinsically concerned with some form of informational imperfection. This is most evident for coordination games; here, no new information regarding the state of the world is produced in debate but the extent to which speech is informative is the extent to which any strategic uncertainty is resolved. Thus speech can lead to ex post Pareto efficiency gains by facilitating coordination on a particular equilibrium and, in the typical case where the dis-
298
David Austen-Smith and Timothy Feddersen
tribution of payoffs is not neutral across equilibria, any tension in deliberation involves the equilibrium on which to coordinate. Aragones, Gilboa, Postlewaite and Schmeidler [1] (AGFS) observe that not all persuasive arguments involve changes in beliefs through information sharing. Rather, many arguments are by analogy, whereby the speaker makes explicit to the listener relations between known facts that the listener may not have seen. As an example, they suggest an individual, initially predisposed against US intervention in the Iraqi invasion of Kuwait, may be induced to change her mind after an analogy is drawn between Hussein's actions toward Kuwait and Hitler's actions toward Foland. It is, AGFS claim, perfectly reasonable to assume that while both individuals are fully aware of the cases involved, only one of them has made any connection between the two. There is a strong intuition for analogies being important for debate and it seems apparent that the setting is not one usefully captured by orthodox Bayesian theory. Nevertheless, analogic arguments still seem to be fundamentally concerned with information transmission, albeit of qualitatively different sort to that in the standard framework: the speaker in the example is pointing out a connection of which the listener was previously unaware. From this perspective, information asymmetries remain critical to any notion of consequential debate and what AGFS, along with those looking at the role of debate in coordination games, make explicit is that we are going to have look for new tools if we hope to model all of the relevant forms such information asymmetries might take. On the other hand, if AGFS are correct in claiming both that information is not the issue and that it is the relations between known sets of facts, or "cases", that form the basis of much persuasive rhetoric, then models permitting failures of logical, as well as informational, omniscience are going to prove important. For it seems that logically omniscient individuals under complete and full information are going to know all possible connections between facts.
6 Appendix 6.1 Proofs We first derive some important threshold inequalities exploited in some of the formal arguments below. Given a message strategy fx and debate m € A4^, any equilibrium vote strategy v has to satisfy the pivotal voting constraints: that is, conditional on being pivotal at v, a 6-biased agent i who observes a signal s G { — 1,0,1} weakly prefers to vote for z rather than z' under majority rule if and only if E[U{z;b)\s,in,/j,,z,v^i,vpiv]
> E[U{z^]b)\s,in, fj., z\v-i,
vpiv]
Deliberation and Voting Rules
299
and, by definition of being pivotal, if the individual votes z in this event then z surely wins. With this in mind, X^i h = z •= y, z' — x and substitute for preferences U{';y) into the inequality to yield
= Pr[y|5, m, vpiv] H- Pr[X|5, m, vpiv]A -Pr[X|5,m,vpiv](l-A) = Pr[y|5, m, vpiv] + (1 - Fi[Y\s, m, vpiv])(2A - 1), where the strategy pair (/i,u_j) is understood and, in obvious notation, we write Pr[Z|-] = Pr[u; = Z|-], Z e {X,Y}.'^^ It follows that a y-biased individual votes for y rather than for x at v only if ^^ 1 / _ Pr[y|5,m,vpiv] 2 V 1 -Pr[y|s,m,vpiv] By Bayes rule, Pr[y|5,in, vpiv] Pr[F|5]Pr[vpiv|y,m] ' Pr[y |5] Pr[vpiv|y, m] + Pr[X|5] Pr[vpiv|X, m] f2{s) '' i 7 ( 5 ) + ^ ( m ) where
^, , Pr[y|5] ,^, , Pr[vpiv|X,m] i7(s) = : \ and #(m) = ^^^— ^ Pr[X|5] ' ' Pr[vpiv|y,m]' So a ^/-biased individual votes for y rather than for x at v only if 2 V
^(m)
Similarly, an x-biased (6 = x) individual who observes signal 5 weakly prefers to vote for x at v only if 2 V
^{s)
Further, if voting strategies are symmetric and committee decision making is by majority rule, the following are easily checked:
{Hi)
^(0,m, - m ) = 1.
^° An analogous inequality can be derived for the pivotal signaling constraints in similar fashion (although it is important in this case to fix the vote strategy v across all individuals, including the one to whom a particular constraint applies).
300
David Austen-Smith and Timothy Feddersen
Proof of Proposition 1, Let v^ be any symmetric pure strategy voting equilibrium and, without loss of generality, let i € A'' have y-bias {hi = y) and signal Si G {—1,0,1}. For j y^ i and k ^ i, (sj.Sk) must satisfy exactly one of the following: (a) Sj = s^ = 0; (b) Sj = —Sk 7^ 0; (c) Sj = Sk 7^ 0; (d) Sj -\- Sk — —1; (e) Sj -\- Sk = I. Then for each to G {X, Y}, Pr[a|w] = (1 - q)\
Pr[6|c^] = q^l
- P), Pr[c|a.] = ^q\2p
- 1);
and Pr[d\X] = Pr[e|y] = ^(1 - q)q{l - p), Pr[d\Y] = Pv[e\X] = ^(1 - q)qp. Now suppose i is vote pivotal. Then j ^ i and k ^ i must be voting for different alternatives. Furthermore, v^ symmetric implies that, conditional on i being pivotal, (d) can be true of {sj.Sk) if and only if (e) can be true of (sj.Sk). Hence, although not every possibility in {(a),...,(e)} need have strictly positive probability conditional on i vote pivotal, v^ symmetric implies Pr[vpiv|i;°,y] = Pr[vpiv|2;^X]. By Bayes rule, therefore, Pr[y|5^,'L'^, vpiv] in this case is simply Pr[vpiv|^^y]Pr[y|5i Pr[vpiv|i;°]
Pr[Y\si].
Substituting for U{'\y) into the pivotal voting constraint (with debate ignored) and collecting terms, voting for y is a best response for i if and only if: E[U{y\ y)\s,y,i;^^, vpiv] - E[U{x\ y)\s,x,i;^.,
vpiv]
- Pr[y|5, vpiv] + Pr[X|5, vpiv]A - Pr[X|5, vpiv](l - A) - Pr[y|5, vpiv] + (1 - Pr[y|s, vpiv])(2A - 1) > 0 where the dependency on v^_^ is understood. It follows that a y-biased individual votes for y rather than for x at v^ only if ^^
l-2Pr[y|5,,vpiv] 2(1-Pr[y|5,,vpiv]) ^ (l-2Pr[y|^,]) 2(1-Pr[y|5,])'
li Si = 1, Pr[y|5i] = {1 — p) and the constraint for voting y is A > h{p); if Si < 0, Pr[y|5i] > 1/2 and the constraint for voting y is A > 0. This proves the proposition. Proof of Proposition 2. Let (/i, v) be a full information equivalent separating debate equilibrium at (p, g'). Given // is separating in common interests, it
Deliberation and Voting Rules
301
is immediate from the definition of h{p) that A < li{p) is necessary and sufficient for v to satisfy the pivotal voting constraints and be full information equivalent voting. We therefore have to check the pivotal message constraints, given A < li{p). Without loss of generality, consider a ^/-biased individual i G A^. It is straightforward to check that if Sj = —1 then m^ = —1 is the unique best response to fi-i. Suppose i has signal Si = 0. Given {fi-i,v) and Si = 0, it is clear that i never strictly prefers sending message m'l = 1 rather than sending rrii — 0; and i is willing to send the message rrii = Si = 0 rather than deviate to a speech m^ = — 1 < 5^ if and only if E[U{z', y)|0,0, /i_,, i;, mpiv(0, -1)] > E[U{z'', y)\0, - 1 , /x_,,^, mpiv(0, - 1 ) ] . Given (/i, i;), i is message pivotal at 5j = 0 between rrii = 0 and m^ = — 1 if either (a) both j and k are uninformed, have a bias for x, and send messages rtij = m/c = 0, or (b) both j and k are informed, have a bias for x, and send messages rrij = —ruk = 1, or (c) j is uninformed and sends ruj = Sj = 0, k is informed and sends message rrik = 5^ = 1, and both j ,fchave a bias for y. Suppose i sends the truthful message rrn = Si = 0. Then the committee decision is surely x. On the other hand, if i sends the message m^ = —1, the committee decision is surely y. With these remarks in mind, compute Pr[y|5i,/^_i,'L',mpiv(m,m')] = Pr[mpiv(m,m')|/i_i,'L',y] Pr[y|5j] Pr[inpiv(m, m^)\ii-i,v, Y] Pr[y|5^] -\- Pr[inpiv(m, m')|//_i, i;, X] Fv[X\si] where Pr[mpiv(0, -l)\fi-uv,
Y] = [J(1 - qf + \q^p{l - p) + \qil - p)(l - q)],
Pr(mpiv(0, -l)\^X-^, V, X] = [\{l - q? + \q\l
- p)p + ^9P(1 - q)]-
Since Pr[y|s, = 0] = 1/2, i is willing to send m^ = 0 rather than m^ = —1 only if ^^
l-2Pr[y|0,M-i,t^,mpiv(0,-l)] 2(l-Pr[y|0,M-i,t^,mpiv(0,-l)]) ^ g(l-g)(2p-l) [{l-q)^ + 2q^l-p)p + 2qp{l-q)]
< hip)Now suppose. Si = 1. If ever i prefers to send a message m'^ = 0 rather than the message m^ = 1, then i surely prefers to send a message m[ = —1 rather than the message m^ = 1. So it suffices to identify when sending m^ = 1 is a best
302
David Austen-Smith and Timothy Feddersen
response for i. Given (//, i;), z is message pivotal between rui = 1 and m[ = —1 at events (a') both j and k are uninformed and send messages rrij = ruk = 0, or (b') both j and k are informed and send messages rrij — —rrik = 1, or (c') j is uninformed and sends rrij = Sj = 0, k is informed and sends message ^/c = -s/c == 1, and both j , k have a bias for y, or (d') where j is uninformed and sends rrij = Sj = 0, k is informed and sends message rrik = Sk — —1, and both j , k have a bias for x. Then whichever event obtains, if i sends the truthful message mi = Si — 1, the committee decision is surely x and, if i sends the message m[ = —1, the committee decision is surely y. Thus Pr[mpiv(l,-l)|/i_,,i;,y] = [(1 - qf + ^qMl - P) + ^^(1 - P){1 -q) + \qp{l - Q)]. and Pr[mpiv(l,-l)|/i_i,i;,X] = [(1 - qf + ^q\l - p)p + \qp{l - q) + \q{l - p)(l - q)]. Rehearsing the same argument as before, mutatis mutandis, yields that i is wiUing to send m, = 1 rather than m[ = —1 only if A<
l-2Pr[y|l,M-i.^,mpiv(l,-l)] 2(l-Pr[y|l,M-i,y,mpiv(l,-l)]) (2p-l) = h{p)2p
Therefore the binding message pivot constraint is that on the uninformed individual, in which case there exists a full information equivalent separating debate equilibrium if and only if ^^
q{l-q){2p-l)
Maximizing the RHS of this inequality with respect to q and p in turn, substituting back and taking limits appropriately yields Proposition 2(1) and 2(2), completing the proof. Proof of Lemma 1. Let {fJi^v) be an SP debate equilibrium and suppose the lemma is false at (fJ^.v). Assume individual i is uninformed {si = 0), has bias hi — y and that {rrii,rrij,rrik) = (—1 — 1,—!)• Given /i is semi-pooling in common interests, a message m = — 1 is sent in debate only if the sender has a signal 5 = —1 or is y-biased and uninformed. By supposition u(2/,0,-l,-l,-l)=a;.
Deliberation and Voting Rules By ^ semi-pooling, it must be that for j^k
303
^i^
(5,-,s^)€{(0,0),(0,-l),(-l,-l)}. If ever 5^ = 0, then ^ semi-pooling implies j is ^/-biased and the supposition requires j to vote surely for x. Hence, individual i cannot be vote pivotal if (sj^Sk) = (0,0). And if i is vote pivotal under majority rule and (sj^Sk) = ( 0 , - 1 ) , it must be that k votes for y; and if (sj^Sk) = (—1,-1), j , k must (given majority rule) have opposite bias. In any case, the pivotal voting constraints imply that a y-biased individual is willing to vote for x at m — (—1, —1, —1) if and only if
2V
^(-1,-1,-1)
where, for any signal s and debate m € A^^, 0(s) = ^'^[^1^1 and $(xn) = Pr[vpiv|X,^,u,m] Given i7(0) = 1, there exist A G (0,1/2) satisfying the inequality only if ^( —1,—1,—1) > 1. Because i can be pivotal at m — (—1,—1,—1) given 5^—0 only at the events identified above, we have
^
{l-p){l-qp) p{l-q + qp)
But p> 1/2 implies ^(—1,—1,—1) < 1. This contradiction proves the lemma. Proof of Lemma 2. The pivotal voting constraints imply a y-biased individual is willing to vote for y at m G A^^ given a signal s if and only if
2 V
^(m)^
Similarly, an x-biased individual who observes signal 5 weakly prefers to vote 3 if and only if for X at m G A1^
A>iri-^('") 2 V
^(5)
By assumption, p > 1/2; hence, i? is strictly decreasing in s. The claims now follow directly. Proof of Lemm.a 3. Suppose (m^, m^, m/c) = ( 1 , - 1 , - 1 ) and assume individual i who sends message rrii = 1 cannot be pivotal and that both agents j and
304
David Austen-Smith and Timothy Feddersen
k always vote for x irrespective of their bias and signal. Then it must be the case that ^(y,-l,-l,l,-l)-x. (1) Consider such a y-biased individual who has observed signal s = —1 and sent message m = —1 and observes a split debate (1,-1) and who is supposed to vote for x. There can be no event such that this agent's vote is pivotal for this debate since otherwise he must vote for y. To see this note that the observed split debate and the assumption of the SP signalling strategy implies at most one other agent has observed the signal 1 so it follows that if there is a positive probability the agent is pivotal he should vote for y. To ensure such an agent votes for x it must be the case that his vote cannot be pivotal. But then, since the other agent sending message rrik == — 1 is always voting for x by assumption we get the requirement that '?;(1,1,1,-1,-1) =:x. Symmetry and anonymity implies ^(-l,-l,-l,l,l)-y.
(2)
But equations (1) and (2) imply a violation of debate monotonicity. Proof of Proposition 3(1). Fix any feasible information structure (p, g). By Proposition 1, there is a unique equilibrium in pure strategies without debate: when A < /i(p), all informed individuals surely vote their signal and all uninformed individuals vote their bias; when A > li{p) all individuals vote their bias. Let v^ denote this no-debate voting strategy and let {(i, v) be any pure strategy debate equilibrium (in undominated strategies and subject to the maintained technical refinement). Then the proposition is trivial if ^ is either separating or pooling in common interest. Suppose {IJL.V) is a semipooling debate equilibrium. Under a semi-pooling equilibrium, all individuals offer make speeches that reveal how they would have voted without debate. For a committee decision distinct to the no-debate decision, therefore, at least one person must change their vote as a consequence of the debate. As a consequence of debate, that is, either an informed individual votes against her signal or an uninformed individual votes against her bias. Moreover, if the outcome is going to be worse with debate than without, it must be that an individual who changes her vote switches to the worse outcome. Let y be the right outcome; then the committee can make an error in common interest by choosing x following debate only if y is defined by unanimous induced preferences at s. So there can be an error in common interests only if (up to permutations) SG { ( - 1 , - 1 , - 1 ) , ( 0 , - 1 , - 1 ) , ( 1 , - 1 , - 1 ) , ( - 1 , 0 , 0 ) } .
Deliberation and Voting Rules
305
We consider each case in turn. Throughout, the SP debate equilibrium (/i, v) is fixed and taken as understood. Let v = (i'i,'i'25'^3) ^ {^^y}^ denote a list of votes. (I) (51,52,53) = ( - 1 , - 1 , - 1 ) . Under v^ all individuals vote for y and, given the signal profile and definition of /i, the debate must be m = (—1,-1,-1). Consequent on m, therefore, there are essentially two possible voting outcomes v = (f 1,1^2, t's) that result in a mistake: (a) V = (x,x,x). In this case, all agents are supposed to vote for x. Then, by Lemma 1, i;(y,0, —1, —1, —1) = y and so, by signal monotonicity (Lemma 2.1), v{y,—l^—l,—l,—l) = y. Hence, hi — x for all i, so we must have 'L'(x,—1,-1,—1,-1) = x. Consider any x-biased agent who is supposed to vote for x here. For this debate, there is a positive probability of being pivotal and ^(—1,—1,—1) is defined. Specifically, 5(-l,-l,-l) =
Pr[vpiv|X,M,t;,-1,-1,-1] Pr[vpiv|r, /u, V, - 1 , - 1 , -1] \q{l - q)p + Iq^p^
= CI -
^
'I
1- 9 P
^>p{l-q
+ qpy
Now i7(—1) = p / (1 — p) so the agent is willing to vote for x only if
2 V ^ 2 > 1
p p{'^-q + qp) l{2p-l){l-qp{l-p)) p2(i-q^qp)
But then, by Proposition 1, the no-debate equihbrium v^ requires all individuals to vote their bias which makes x the right outcome and contradicts the supposition of an error here. (b) V = {x,x,y) or V == {y,x,x). For either of these possibilities to constitute equilibrium behaviour here requires 'L'(X,0, —1, —1, —1) = x. But then the same logic as for (a) applies and we obtain a contradiction. (II) (51,52,53) = ( 0 , - 1 , - 1 ) . It must be the case that the uninformed agent is x-biased since otherwise all the messages are — 1 and the argument in case 1(a) applies. As indicated (and without loss of generality), assume 5i = 0 and therefore, by yu semi-pooling, (7711,7712,7773) = (1, —1, —1). By Lemma 3, if 1 is not pivotal then the right decision must be made. So if there is an error, 1 must have positive probability of being pivotal here. And for I's vote to be pivotal it must be the case that individuals 2 and 3 are diff'erent (if they have the same bias, send the same message and observe the same messages
306
David Austen-Smith and Timothy Feddersen
from others then they vote the same way). But since 7712 = m^ = — 1, /x semipooling implies Sj < 0, j = 2,3, and moreover rrij = —1 and Sj = 0 imply bj = y. There can be only one such agent j € {2,3} in the pair if a vote is pivotal, so the other agent, /c, must have observed Sk = —1. In this case the uninformed x-biased agent, i = I, who sends message mi = I must believe that, conditional on being pivotal, exactly one other agent k has observed signal Sk = —1. In which case, by /i > A, individual i = 1 prefers to vote for y-
(III) (51,52,53) = ( 1 , - 1 , - 1 ) . Then (mi,7712,ms) - ( 1 , - 1 , - 1 ) . By the same argument as for (II), if a vote is pivotal it must be the case that agents 2 and 3 are different. Consequently, at least one of these agents must have observed the signal Sj = —1; let j = 2. It follows that agent 1 cannot be ^-biased: for if bi = y, then he would vote for y conditional on being pivotal because he knows the third agent has not seen 5 = 1. Either b2 = x or bs = x; assume b2 = x. Since 2 and 3 must be different, it must be that bs = y and 53 < 0. It follows that k = 3 votes for y, implying both individuals 1 and 2 are voting for x. Now individual 1 is a:-biased and 5i = 1; therefore 1 prefers to vote for x only if 1/ ^(1,-1.-1)' Because A < /i the above inequality can be satisfied only if ^ ( 1 , - 1 , - 1 ) > 1 but, given the voting strategies described above,
^'
'
'
( l - p ) ( 5 V + i9(l-9)p)
since p > 1/2. (IV) (51,52,53) = ( 0 , 0 , - 1 ) . By fj. semi-pooling, li bi = b2 = y then 7771 = ^ 2 = —1 and, therefore, by Lemmas 1 and 2(1), both individuals surely vote y. On the other hand, because informed individuals vote their signal and uninformed individual vote their bias when \ < li and there is no debate, if bi = b2 — X then the decision under no debate is x and evidently a debate equilibrium cannot do worse. To obtain a mistake therefore, it is necessary that 61 7^ 62; without loss of generality, assume bi = x and 62 = V- Then the debate is (7711,7712,^3) = (1, —1, —1). By Lemma 3, if there is an error there must be positive probability of i = 1 being pivotal at this debate. But then individuals 2 and 3 must be voting differently and therefore, by 7772 = 7713 and M-2 = M-s, have different biases. By semi-pooling debate, 771^ = —1 implies either Sj = 0 and bj = y 01 Sj = —1. Hence, individual 1 knows surely that •52 + 53 < — 1 in which case, since X < li and 5i = 0, 1 surely votes y. Because (I) through (IV) exhaust the possibilities for errors in common interest, we are done.
Deliberation and Voting Rules
307
The following lemma is useful for proving Proposition 3(2). Let /i be the semi-pooling message strategy and define the (SP equilibrium path) voting profile R by the following table,
M-i
Si
-1 -2 0 2
y,y y,x
0
1
y^y x,y
y.x x,x
where the entries in each cell describe the vote pair, a n d X ^ = {-1,1}.
[v{y,Si,m),v{x,Si,m)]
Lemma 4. / / (/i, v) and (//, v') are both symmetric semipooling debate equilibria under which uninformed their bias on hearing a split debate. Then, along the and equilibrium voting decisions are described by the
and debate monotonia individuals vote against equilibrium path, v — v' profile R.
Proof. Consider equilibrium path voting behaviour. By hypothesis, along the equilibrium path uninformed individuals vote against their bias on hearing a split debate ( — 1,1); that is, v{y, 0, - 1 , - 1 , 1 ) - X and v{x, 0,1, - 1 , 1 ) = y
(3)
By (3) and debate monotonicity, v{y, 0, - 1 , 1 , 1 ) = X and v{x, 0,1, - 1 , - 1 ) - y
(4)
By (3) and Lemma 2 (signal monotonicity), v{y, 1,1, - 1 , l)^x
and v{x, - 1 , - 1 , - 1 , 1 ) = y
(5)
v{y, 1,1,1,1) = X and v{x, —1, —1, —1, —1) = y
(6)
By (5) and debate monotonicity,
By (5) and Lemma 2 (bias monotonicity), v{x, 1,1, - 1 , l) = x and v{y, - 1 , - 1 , - 1 , 1 ) = y
(7)
Similarly, by (6) and Lemma 2 (bias monotonicity), v{x, 1,1,1,1) = X and v{y, - 1 , - 1 , - 1 , - 1 ) = y And by Lemma 1,
(8)
308
David Austen-Smith and Timothy Feddersen v{y, 0, - 1 , - 1 , - 1 ) - y and v{x, 0,1,1,1) - x,
(9)
There remain two (equihbrium path) decisions to be determined; specifically, for each z e {x^y} v{z, —1, —1,1,1) and v{z, 1,1, —1, —1). Suppose first that individual i e N has L'(X,—1,-1,1,1) = y. Then both of the other two committee members observe a split debate. Hence, (3) through (9) imply there exists a unique event at which z's vote is pivotal: there exists an uninformed {sj = 0) x-biased individual j who has sent message m^ = 1, hears a split debate and votes for y; and there exists an informed {sk — 1) individual k who has sent message rrik — 1, hears a split debate and votes for X. But then z's unique undominated vote decision is to vote for x. Therefore, v{x, —1, —1,1,1) = X in which case, by symmetry u(?/, 1 , 1 , - 1 , - 1 ) =y. Now suppose that individual i G N has ^'(y,—1,-1,1,1) = x. Then i's vote is pivotal in exactly the same case as above; but since i is now presumed 2/-biased, we conclude ^ ( ^ , - 1 , - 1 , 1 , 1 ) =y so by symmetry v{x, 1,1, —1, —1) = X. And because there exist no further unspecified equilibrium path voting decisions, this proves the lemma. Proof of Proposition 3(2). To prove the result, it suffices to show there exists a symmetric and debate monotonic semipoohng debate equilibrium at (A,p, g) in which uninformed individuals vote against their bias on hearing a split debate only if there exists a separating debate equilibrium at (A,p, g). A necessary condition for any such semipoohng debate equilibrium to exist is for the pivotal constraints to hold along equilibrium path. So consider a ybiased individual who has signal 5 = 0, sends message m = — 1 and observes a split debate (—1,1). By hypothesis, v{y, 0, —1, —1,1) = x. By Lemma 4, the unique equilibrium voting path in any such semipoohng debate equilibrium is described by the strategy R, defined above. Therefore, there are three events at which the vote of an uninformed individual z, having sent message rrii = —1 and observed a split debate {mj.mk) — (—1,1), is pivotal: Either both j and k are uninformed: j is y-biased, M-j = 0 and votes x\ k IS 2;-biased, M-k = — 2 and votes y\
Deliberation and Voting Rules
309
Or j is uninformed, y-biased and votes x given M_j = 0; A: is informed with Sk = I, ^/-biased and votes y given M-k = —2; Or j is informed with Sj = —1, y-biased and votes y given M_j == 0; k is informed with s^ = I, x-biased and votes x given M_/c = —2. Substituting into the pivotal voting constraint and rearranging yields v{y, 0, —1, —1,1) = X can be an undominated best response only if X<Xr, = "" 2[{l-qf
q{l-q){2p-l) + qp{l-q) +
2q^p{l-p)]'
Prom the proof to Proposition 2, the binding pivotal message constraint for the separating debate equiUbrium requires
A
^a-^)(2p-i) + 2qp{l-q) + 2qMl-p)]'
Hence, at any information structure {p,q) G (^, 1) x (0,1), XR<Xs^O<{l-qf
+
2q^p{l-p)
which is obviously true. This fact proves the result. Proof of Proposition 4- Suppose by way of contradiction that (//,f) is a separating debate equilibrium. Then no new information is revealed by the fact that a vote is pivotal and, therefore, the sincere voting strategy is weakly dominant; in particular, given m G A1^, m = s and X < li implies v[y, s^, mj - j ^ otherwise
^"""^ ''^'^' ^^' ""^ " \ x otherwise
with the sincere strategy being defined analogously for A > /i. We show that an uninformed {s = 0) individual with ^/-bias strictly prefers to send message m = —1 to message m = 0, thus violating the relevant pivotal message constraint for /j, separating in common interests. Because the event that an individual is message or vote pivotal under unanimity implies that both the other committee members are making similar decisions, to prove the result it suflices to check the case A < /i. Given X < h, fi separating and v sincere, an uninformed y-biased individual i is message pivotal between rrii = 0 and m^ = — 1 under unanimity rule with status quo x if and only if (a) (ruj^mk) = (sj.Sk) — (0,0) and at least one of j . A: has an x-bias, or (b) (mj^rrik) = (sj^Sk) = (—1,1) and at least one of j , k has an x-bias. Therefore, recalling
Pr[mpiv(m, mQI/x-^, f, Y] Fi[Y\si] Pr[mpiv(m, m')|/i_i,i;, F] Pr[y|5i] + Pr[mpiv(m,m')|/i_^,'L',X] Pr[X|5i]
310
David Austen-Smith and Timothy Feddersen
and Pr[mpiv(0, -l)\fi-i,v,
Y] = [ j ( l - qf + ^qM^
Pr[mpiv(0,-l)|M-i,t;,X] = [ ^ ( 1 - 9 ) ^ +
-P)l
^q'{l-p)p].
By V sincere, in either event (a) or (b), individual i votes her bias whatever message she delivers. On the other hand, both j and k vote surely for y in these events if m^ = —1 and at least one of them votes for x otherwise. Therefore, since Pr[y|5i = 0] = 1/2, substituting into the relevant message pivot constraint implies that i is willing to send rrii = 0 rather than m[ = —1 only if l-2Pr[y|0,/x-,,u,mpiv(Q,-l)] ^ - 2(l-Pr[r|0,M-z,^,mpiv(0,-l)]) which contradicts A > 0. 6.2 Refinement and Derivations In this Appendix we define the technical (trembles) equilibrium refinement and describe the approach to identifying particular classes of equilibria discussed in the paper. With some abuse to the notation in the text, it is useful to begin by redefining some variables. Fix an individual i £ N and hereafter suppress any individual-specific subscripts. Assume also that committee decision making is by majority rule; similar constructions apply to the case of unanimity. Let 6 = — 1 if the individual's bias is for y and let 6 = 1 if her bias is for X. Similarly, let CJ = — 1 if the state of the world is Y and let a; = 1 if the state is X. Let m, m' etc denote the relevant individual's messsage in any debate and let 9,p denote the messages of the other two committee members; by convention, when writing any debate m = {m^O^p) G M^ = {—1,0,1}^, the relevant individual's message is always listed first. Refinement For any profile {b,s,m^9^p) G {—1,1} x { — 1,0,1}^ and any z G {x^y}, let v{b,s,m,0,p)e
[0,1]
be the probability that an individual with bias b and signal s, having sent debate message m and heard messages 9, p, votes for alternative y. Since there is no abstention, 1 — u(6, s, m, ^, p) is the probability that the individual votes for X. Similarly, for any (6,5) G { — 1,1} x { — 1,0,1} and any m G { — 1,0,1} let
Deliberation and Voting Rules
311
/i(6,5,m) G [0,1] be the probability that an individual with bias b and signal s sends debate message m; by assumption, X]mG{-i,o,i} f^i^^^i'^) — 1With this notation, an anonymous message strategy is a triple 11 - (/i(6, 5, - 1 ) , fi{b, 5,0),fi{b, 5,1)) and an anonymous voting strategy is simply a pair of vote-probabilities sufficiently described by V = v{b,s,m,9,p). Let (/i, v) be an equilibrium in pure strategies, i.e. the adding up constraints are satisfied and ii{b^s,m) G {0,1} for each message m and v{b,s^m^6^p) G {0,1}. It is irrelevant to apply any refinement to separating debate equilibria as there is no out-of-equilibrium behaviour to worry about. So assume for this discussion that p is semi-pooling in common interests. Then the only messages supposed to be sent in equilibrium are m = — 1 and m' = 1. However, if ever a message m = 0 is observed in a semi-pooling equilibrium, we assume all individuals surely identify the (out of equilibrium) message with the message m = — 1. Now consider the voting strategy v. Given v^ define a perturbed voting strategy component-wise by 1 — e li v{b, 5, m, O^p) — 1 ^(6,.,m,e,p;.)^|^ otherwise where £ > 0 and small. For each £, let v{e)) = ((i;(6,5,m,6',p;£)). Then the pure strategy pair {fi^v) survives the technical refinement (individual-invariant trembles) if \im{p,v{e))
= {p.,v).
Derivations For each rule and any conjectured equilibrium strategy pair (/x, f), we have to identify the message pivot and vote pivot constraints for each possible event. Typically, there are a great many such events to check To see why, consider an individual with bias b and signal 5 who is supposed to send message m in debate; then there are two possible deviations from m and, for each deviation, there are multiple distinct pivot events. And given a realized debate, the vote pivot constraints for the individual have to be checked for each possible message he might have sent, both in and out of equilibrium, and for each possible
312
David Austen-Smith and Timothy Feddersen
debate that might be realized. Finally, this family of constraints has to be checked for consistency. Not surprisingly, the algebra becomes very cumbersome and tedious very rapidly. We therefore wrote a program using the Maple V symbolic manipulation package in Scientific Workplace 4.1 to identify the relevant pivot events and do the algebra. This is available from the authors on request. 6.3 Semi-pooling Equilibria for Majority Rule Table 4 describes the voting strategies, v^ for the (symmetric) SP debate equilibria that exist. As indicated, each column headed by a bold-faced letter is a particular SP debate equilibrium and the voting behaviour is described in terms of an individual's signal, 5j, and the sum of the others' debate messages, M-i; the ^/-biased individual's prescribed vote is listed first. Table 4. SP debate equilibrium voting strategies under majority rule Si M-i
[v{y,Si,m),v{x,Si,ni)] A
B
CI
C2
R
M l M2
U
-1
-2 0 2
y,y y , y y , y y , y y , y y , y y , y y , x y,y y,y y,y y,y y,y y,x y,xy,x y,y y,x y,x x,x y,x y,x y,xy,x
0
-2 0 2
y^x y,x y^x
y,x y.y y^x y,x y,x x,x
y,y y,x x,x
y.y x,y x,x
y,y y,x x,x
y^x y,x y,x
y,x y,x y,x
1
-2 0 2
x,x x,x x,x
y^x y,x x,x x,x x,x x,x
y,y x,x x,x
y,x x,x x,x
y,x y,x x,x
y,x y.x x,x
y,x y.x y,x
SP equilibrium A has all informed individuals surely vote their signals and the uninformed always vote their bias. Under B, uninformed individuals always vote their bias and only those informed individuals with a signal against their bias who are marginally influenced in debate. In SP equilibrium Cl, the informed agents' voting behaviour is the same as under B but now the uninformed individuals can be influenced in debate. •
C2 is the 'most influential' of the SP equilibria available; along the equilibrium path, every individual's voting behaviour in C2 coincides with that in a separating debate equilibrium. Indeed, this last property is true of all uninformed voters in both Cl and C2 SP equilibria.
Deliberation and Voting Rules
313
•
In the equilibrium R, informed individuals are marginally influenced in debate exactly as in B and Cl; the singular feature of R is the voting behaviour of the uninformed. Although they make speeches in support of their bias, they vote against their bias unless (like the informed individuals) they hear two speeches in debate that favour their bias.
•
Debate in both equilibria Ml and M2 has very little impact. Under Ml the uninformed are influenced in debate but the informed are only marginally influenced in debate: they vote their bias unless they have a signal against their bias and both of the other speeches in debate support the alternative favoured by that signal. And although the voting behaviour of informed agents is the same in M2 as in Ml, uninformed individuals under M2 surely vote their bias irrespective of the debate.
•
Finally, debate in U is informative but utterly uninfiuential. Despite the message strategy being semi-pooling in common interest, all individuals always vote their bias.
Fig. 6 describes those information structures for which the various pure strategy SP equilibria exist; the dotted outline in the figure is the boundary for the full information equivalent separating equilibrium.
/2"'(A)
l^\X) Fig. 6. Semi-pooling debate equilibria
6.4 No-debate Equilibria for Unanimity Rule Recall the pure strategy no-debate equilibria under unanimity rule from Table 1 of the text:
314
David Austen-Smith and Timothy Feddersen Si
[v{y^Si, 0),t^( X^
Si^
0)1
1
2
3
4
5
-1
y^y
y.y
0
y,y x,x
y,y
y^y y,x
y.y y^x
y,x y^x
y,x
x,x
y,x
y,x
1
For any pair of pure strategy no-debate equilibria, a, b G { 1 , . . . , 5}, let ab denote a mixed voting strategy profile that involves individuals randomizing between their respective vote decisions under pure strategies a and b (in fact, only one type of person is ever required to use a non-degenerate lottery). Fig. 7 describes the distribution of voting equilibria under unanimity rule with no debate, assuming a typical value, A = 1/10.
1
p
1
Fig. 7. Voting equilibria under unanimity with no debate
References [1] Aragones, E., I. Gilboa, A. Postlewaite and D. Schmeidler (2001) Rhetoric and analogies. Mimeo. University of Pennsylvania. [2] Austen-Smith, D. (1990) Information transmission in debate. American Journal of Political Science^ 34: 124-152. [3] Austen-Smith, D. (1990) Credible debate equilibria. Social Choice and Welfare, 7: 75-93.
Deliberation and Voting Rules
315
[4] Austen-Smith, D. (1993) Interested experts and policy advice: multiple referrals under open rule. Games and Economic Behavior^ 5: 3-43. [5] Austen-Smith, D. (1993) Information acquisition and orthogonal argument. In W. Barnett, M. Hinich and N. Schofield (eds) Political Economy: Institutions, Competition and Representation Cambridge: CUP. [6] Battaglini, M. (2002) Multiple referrals and multidimensional cheap talk. Econometrica, 70: 1379-1402. [7] Bohman, J. and W. Rehg (eds) (1997) Deliberative Democracy: Essays on Reason and Politics Cambridge MA: MIT Press. [8] Calvert, R. and J. Johnson (1998) Rational argument, political argument and democratic deliberation. Mimeo. Washington University, St. Louis. [9] Christiano, T. (1997) The significance of public dehberation. In J. Bohman and W. Rehg (eds) Deliberative Democracy Cambridge MA: MIT Press. [10] Cohen, J. (1989) Deliberation and democratic legitimacy. In A.Hamlin and P.Pettit (eds) The Good Polity Oxford: Basil Blackwell. [11] Coughlan, P. (2001) In defense of unanimous jury verdicts: mistrials, communication and strategic voting. American Political Science Review, 94: 375-93. [12] Diermeier, D. and T. Feddersen (2000) Information and congressional hearings. American Journal of Political Science, 44: 51-65, [13] Doraszelski, U., D. Gerardi and F. Squintani (2001) Communication and voting with double-sided information. Mimeo. Northwestern University. [14] Dryzek, J. and C. List (2003) Social choice theory and deliberative democracy: a reconciliation. British Journal of Political Science, 33: 1-28. [15] Duggan, J. and C. Martinelh (2001) A Bayesian model of voting injuries. Games and Economic Behavior, 37: 259-94. [16] Elster, J. (ed) (2000) Deliberative Democracy Cambridge: CUP. [17] Elster, J. (1986) The market and the forum. In J. Elster and A. Hylland (eds) Foundations of Social Choice Theory Cambridge: CUP [reprinted in J. Bohman and W. Rehg (eds) 1997: 3-33]. [18] Eriksen, E. (2001) Democratic or technocratic governance? Mimeo. NYU School of Law. [19] Farrell, J. (1987) Cheap talk, coordination and entry. RAND Journal of Economics, 18: 34-39. [20] Farrell, J. (1983) Meaning and credibility in cheap talk games. Games and Economic Behavior, 5: 514-31. [21] Feddersen, T. and W. Pesendorfer (1996) The swing voter's curse. American Economic Review, 86: 408-24. [22] Feddersen, T. and W. Pesendorfer (1997) Voting behavior and information aggregation in elections with private information. Econometrica, 65: 1029-1058. [23] Gerardi, D. and L. Yariv (2002) Putting your ballot where your mouth is - an analysis of collective choice with communication. Mimeo. Yale University.
316
David Austen-Smith and Timothy Feddersen
[24 Glazer, J. and A. Rubinstein (2001) Debates and decisions: on a rationale of argumentation rules. Games and Economic Behavior, 36: 158-173. [25 Hafer, C. and D. Landa. (2003) Deliberation as self-discovery. Mimeo. New York University. [26; K a m i , E. and Z. Safra (2002) Individual sense of justice: a utihty representation. Econometrica, 70: 263-284. [27; Knight, J. and J. Johnson (1997) W h a t sort of equality does deliberative democracy require? In J. Bohman and W. Rehg (eds) Deliberative Democracy Cambridge MA: M I T Press. [28; Krishna, V and J. Morgan (2001) A model of expertise. Quarterly Journal of Economics, 116: 747-75. [29; Ladha, K. (1992) T h e Condorcet jury theorem, free speech and correlated votes. American Journal of Political Science, 36: 617-634. [30 Lipman, B. and D. Seppi (1995) Robust inference in communication games with partial provability. Journal of Economic Theory, 66: 3 7 0 405. [31 Matthews, S. and A. Postlewaite (1995) On modeling cheap talk in Bayesian games. In J.Ledyard (ed) The Economics of Informational Decentralization: Complexity, Efficiency and Stability Amsterdam: Kluwer Nijhoff. [32; McLean, I. and F. Hewitt (1994) Condorcet: Foundations of Social Choice
and Political Theory Aldershot: Edward Elgar Publishing. [33; McLennan, A. (1998) Consequences of the Condorcet jury theorem for
[34;
[35; [36; [37' [38' [39^
beneficial information aggregation by rational players. American Political Science Review, 92: 413-418. Meirowitz, A. (2004) In defense of exclusionary deliberation: communication and voting with private beliefs and values. Mimeo. Princeton University. Ottaviani, M. and P. Sorensen (2001) Information aggregation in debate: who should speak first? Journal of Public Economics, 81: 393-422. Rabin, M. (1994) A model of pregame communication. Journal of Economic Theory, 63: 370-391. Sanders, L. (1997) Against deliberation. Political Theory, 25: 347-376. Shapiro, I. (2000) Optimal deliberation. Mimeo. Yale University. Walker, J. and D. Lane (1994) The jury system. Victorian Council for Civil Liberties, Criminal Justice DP 25/02/94.
Index
^-history, 135 action space, 148 activist valence, 77 activists, 59 additive bias model, 22 aggregate strict concavity, 22, 64 amendment agendas, 165, 171, 173 arms of the bandit, 134 asymmetric information, 94-96, 110, 111 bandit problem, 134 Banks set, 115, 163-167, 171, 172, 179, 180, 182, 183, 190-192 Baron-Ferejohn model, 206, 229 bias, 274 bias-driven debate equilibrium, 292 Borda electoral competition, 114 boundary condition, 86 campaign funding, 128 candidate quality, 93, 95, 96, 110-112 closed graph, 39 closed rule, 229, 231 CNE, 63 coalition, 227, 228, 230-232, 234, 236, 237, 242-245 coalition types, 211 cognitive limitations, 223 common interest, 272 comparative statics, 96, 103, 104, 106, 108-111 Condorcet jury theorem, 114
Condorcet winner, 117, 128, 165, 166, 170, 174-177, 180-186 continuation value, 207, 234 conventions, 223 convergence coefficient, 59, 66 core, 42, 43 criteria, 166, 169 critical Nash equilibrium, 63 cutpoint, 98, 99, 255 debate, 271, 276 debate monotonicity, 286 deliberation, 271 deterministic model, 42 divide the dollar, 116, 167, 168, 179, 180, 182 dynamic allocation index, 141, 142 electoral game, 20 electoral variance, 60 equilibrium bias-driven debate, 292 critical Nash, 63 local pure strategy Nash, 62, 83 local strict Nash, 63 locally isolated strict Nash, 86 mixed strategy, 238, 239, 241, 242 mixed strategy electoral, 20, 24, 45 mixed strategy Nash, 61 pure strategy Nash, 59 pure strategy electoral, 20, 25-28, 35 pure strategy Nash, 62 separating debate, 278
318
Index
stationary, 230, 234, 238, 242, 243, 245 error in bias, 273, 287 error in common interest, 273, 287 Euler characteristic, 87 experimental research, 112
local pure strategy Nash equilibrium, 62, 83 local strict Nash equilibrium, 63 locally isolated, 86 locally isolated LSNE, 86 LSNE, 63
fairness, 222 final passage votes, 257 fixed utility model, 21, 26, 29, 30 focal points, 223 folk theorem, 231, 242, 246 full information equivalence, 273, 280
majority preference, 172 mean voter theorem, 34, 59, 64 min-max equilibrium, 117 mixed strategy, 234, 235, 238, 240, 242 electoral equilibrium, 20, 24, 45 equilibrium, 238, 239, 241, 242 Nash equilibrium, 61 model additive bias, 22 Baron-Ferejohn, 206, 229 deterministic, 42 divide the dollar, 116, 167, 168, 179, 180, 182 fixed utility, 21, 26, 29, 30 goods and bads, 168 multiplicative bias, 23 pivotal pohtics, 251, 254 plurality, 114 random utility, 21 spatial voting, 232, 233 utility difference, 22, 27, 28, 32-34, 37, 43, 45, 48 utility ratio, 23, 50 Morse inequalities, 87 multiplicative bias model, 23 myopic strategies, 139
game electoral, 20 plurality, 114-116, 118, 120, 128 repeated, 231, 242, 245, 246 tournament, 114-119, 128 Gamson's law, 216 generalized Euclidean preferences, 34 Gittins index, 141-143 Gittins index strategy, 131 goods and bads, 168, 179-181, 183, 184 gridlock interval, 253 Hessian, 59 index for bandit problem, 136 index strategy, 136 induced mixed strategy, 98-100, 102, 104 inequality, 115, 122, 123, 128 influence, 284 information aggregation, 114 asymmetric, 94-96, 110, 111 private, 94, 98, 107, 111 structure, 274 informed individual, 274 learning, 223 legislative bargaining, 228, 229, 231-233, 245 organization, 227, 228, 243, 244 parties, 227, 229, 231, 232, 242, 244, 245 voting, 227-229, 231, 232, 244, 245 LNE, 62
Nash equilibrium, 59 nationalism, 60 optimal strategy, 117, 135 outcome, 270 parliamentary government, 228, 232 parties legislative, 227, 229, 231, 232, 242, 244, 245 political, 227, 228, 232, 233, 244-246 party cartel model, 251, 253 pivotal message constraints, 277 politics model, 251, 254 voting constraints, 277
Index plurality, 113, 114, 116, 117, 122, 128 game, 114-116, 118, 120, 128 model, 114 polarization, 95, 103, 111 policy coincidence, 20, 37 political institution, 87 political parties, 227, 228, 232, 233, 244-246 pooling in common interest, 277 private information, 93, 94, 98, 107, 111 private interest, 272 proportionality rule, 216 proposal power, 212 public project, 165, 167, 168, 182, 185 pure strategy electoral equilibrium, 20, 25-28, 35 pure strategy Nash equilibrium, 62 purification, 94 random utility model, 21 redistributive politics, 115 repeated game, 231, 242, 245, 246 reward, 148 right, 280 robustness, 40 semi-pooling in common interest, 277 separating debate equilibrium, 278 in common interest, 277 in private interest, 277 social welfare optima, 29, 30, 32-34 sophisticated voting, 115 spatial voting model, 232, 233 states of arm z, 134 stationary equilibrium, 230, 234, 238, 242, 243, 245 Markovian strategy, 136 strategy, 230, 231, 246 status quo, 273, 289, 292, 297 strategy
319
equilibrium, 228 mixed, 234, 235, 238, 240, 242 myopic, 139 optimal, 117, 135 stationary, 230, 231, 246 stationary Markovian, 136 strict local concavity, 59, 64 strictly monotonic, 21 strong, 45 strongly externally stable, 42 top cycle, 166, 170, 171, 174, 179-184, 189, 190 tournament, 113, 114, 122 tournament game, 114-119, 128 transversality, 61, 85 uncovered set, 115, 165, 166, 171, 179, 182, 193, 197 uninformed individual, 274 utility difference model, 22, 27, 28, 32-34, 37, 43, 45, 48 utility ratio model, 23, 50 valence, 59 value, 117 value function, 135 voter covariance matrix, 66 ideal points, 65 rationality, 82 voting by successive elimination, 165, 171 weak T-chain, 171, 181, 186, 189, 192, 197 weak Condorcet winner, 170, 173, 179, 180, 182, 188, 190, 191 weak* topology, 20 weakly externally stable, 42 weakly monotonic, 21 worth of a strategy, 135
