The consistent preferences approach to deductive reasoning in games

To Kristin

viii

CONSISTENT PREFERENCES

5.3

Admissible consistency of preferences

62

6. RELAXING COMPLETENESS 6.1 Epistemic modeling of strategic games (cont.) 6.2 Consistency of preferences (cont.) 6.3 Admissible consistency of preferences (cont.)

69 69 73 75

7. BACKWARD INDUCTION 7.1 Epistemic modeling of extensive games 7.2 Initial belief of opponent rationality 7.3 Belief in each subgame of opponent rationality 7.4 Discussion

79 82 87 89 94

8. SEQUENTIALITY 8.1 Epistemic modeling of extensive games (cont.) 8.2 Sequential consistency 8.3 Weak sequential consistency 8.4 Relation to backward induction

99 101 104 107 113

9. QUASI-PERFECTNESS 9.1 Quasi-perfect consistency 9.2 Relating rationalizability concepts

115 116 118

10. PROPERNESS 10.1 An illustration 10.2 Proper consistency 10.3 Relating rationalizability concepts (cont.) 10.4 Induction in a betting game

121 123 124 127 128

11. CAPTURING FORWARD INDUCTION THROUGH FULL PERMISSIBILITY 11.1 Illustrating the key features 11.2 IECFA and fully permissible sets 11.3 Full admissible consistency 11.4 Investigating examples 11.5 Related literature

133 135 138 142 149 152

List of Figures

2.1

G1 (“battle-of-the-sexes”).

12

2.2

G2 , illustrating deductive reasoning.

13

2.3

G3 , illustrating weak dominance.

2.4

G03

and a corresponding extensive form tipede” game).

2.5

G02 G01

2.6

13 Γ03

(a “cen-

and a corresponding extensive form Γ02 . and a corresponding extensive form Γ01 (“battle-

14 16

of-the-sexes with an outside option”).

17

3.1

Γ4 and its strategic form.

25

4.1

The basic structure of the analysis in Chapter 4.

39

7.1

Γ5 (a four-legged “centipede” game).

93

8.1

Γ6 and its strategic form.

111

8.2

Γ06

and its pure strategy reduced strategic form.

112

10.1

G7 , illustrating common certain belief of proper consistency.

123

10.2

A betting game.

129

10.3

The strategic form of the betting game.

130

11.1

G8 , illustrating that IEWDS may be problematic.

134

11.2

G9 , illustrating the key features of full admissible consistency.

136

G10 , illustrating the relation between IECFA and IEWDS.

142

Γ11 and its pure strategy reduced strategic form.

156

11.3 12.1

xii

CONSISTENT PREFERENCES

12.2 12.3 12.4 12.5

Reduced form of Γ12 (a 3-period “prisoners’ dilemma” game). G13 (the pure strategy reduced strategic form of “burning money”). Γ001 and its pure strategy reduced strategic form. Γ14 and its pure strategy reduced strategic form.

166 169 171 172

List of Tables

0.1 2.1 2.2 3.1 7.1 7.2 10.1 12.1

The main interactions between the chapters. xvi Relationships between different equilibrium concepts. 18 Relationships between different rationalizability concepts. 19 Relationships between different sets of axioms and their representations. 29 0 An epistemic model for G3 with corresponding extensive form Γ03 . 89 An epistemic model for Γ5 . 93 An epistemic model for the betting game. 131 Applying IECFA to “burning money”. 170

Preface

During the last decade I have explored the consequences of what I have chosen to call the ‘consistent preferences’ approach to deductive reasoning in games. To a great extent this work has been done in cooperation with my co-authors Martin Dufwenberg, Andr´es Perea, and Ylva Søvik, and it has lead to a series of journal articles. This book presents the results of this research program. Since the present format permits a more extensive motivation for and presentation of the analysis, it is my hope that the content will be of interest to a wider audience than the corresponding journal articles can reach. In addition to active researcher in the field, it is intended for graduate students and others that wish to study epistemic conditions for equilibrium and rationalizability concepts in game theory.

Structure of the book This book consists of twelve chapters. The main interactions between the chapters are illustrated in Table 0.1. As Table 0.1 indicates, the chapters can be organized into four different parts. Chapters 1 and 2 motivate the subsequent analysis by introducing the ‘consistent preferences’ approach, and by presenting examples and concepts that are revisited throughout the book. Chapters 3 and 4 present the decision-theoretic framework and the belief operators that are used in later chapters. Chapters 5, 6, 10, and 11 analyze games in the strategic form, while the remaining chapters—Chapters 7, 8, 9, and 12—are concerned with games in the extensive form. The material can, however, also be organized along the vertical axis in Table 0.1. Chapters 5, 8, 9, and 10 are concerned with players that are endowed with complete preferences over their own strategies. In con-

xvi

CONSISTENT PREFERENCES

Table 0.1.

The main interactions between the chapters.

Chapter 11 ⇒ Chapter 12 ⇑ Chapter 1 ⇓ Chapter 2

Chapter 4 ⇒ &

⇑

&

⇒ Chapter 3 ⇒

Chapter 6

↑ ⇒

↑ Chapter 5

↓ ⇒

⇓ Chapter 10 ← Motivation

Preliminaries

Strategic games

Chapter 7 Chapter 8 ⇓ Chapter 9 Extensive games

strast, Chapters 4, 6, 7, 11, and 12 present analyses that allow players to have incomplete preferences, corresponding to an inability to assign subjective probabilities to the strategies of their opponents. The generalization to possibly incomplete preferences is motivated in Section 3.1, and is an essential feature of the analysis in Chapter 11. Note also that the concepts of Chapters 7, 8, 9, and 10 imply backward induction but not forward induction, while the concept of Chapters 11 and 12 promotes forward induction but not necessarily backward induction.

Notes on the history of the research program While the arrows in Table 0.1 seek to guide the reader through the material presented here, they are not indicative of the chronological development of this work. I started my work on non-equilibrium concepts in games in 1993 by considering the games that are illustrated in Figures 12.1–12.4. After joining forces with Martin Dufwenberg—who had independently developed the same basic intuition about what deductive reasoning could lead to in these examples—we started in 1994 work on our joint papers “Admissibility and common belief” and “Deductive reasoning in extensive games”, published in Games and Economic Behavior and Economic Journal in 2003, and incorporated as Chapters 11 and 12 in this book.1 1 “Deductive

reasoning in extensive games” was awarded the Royal Economic Society Prize for the best paper published in the Economic Journal in 2003.

Chapter 1 INTRODUCTION

This book presents, applies, and synthesizes what my co-authors and I have called the ‘consistent preferences’ approach to deductive reasoning in games. Briefly described, this means that the object of the analysis is the ranking by each player of his own strategies, rather than his choice. The ranking can be required to be consistent (in different senses) with his beliefs about the opponent’s ranking of her strategies. This can be contrasted to the usual ‘rational choice’ approach where a player’s strategy choice is (in different senses) rational given his beliefs about the opponent’s strategy choice. Our approach has turned out to be fruitful for providing epistemic conditions for backward and forward induction, and for defining or characterizing concepts like proper, quasi-perfect and sequential rationalizability. It also facilitates the integration of game theory and epistemic analysis with the underlying decision-theoretic foundation. The present text considers a setting where the players have preferences over their own strategies in a game, and investigates the following main question: What preferences may be viewed as “reasonable”, provided that each player takes into account the rationality of the opponent, he takes into account that the opponent takes into account the player’s own rationality, and so forth? And in the extension of this: Can we develop formal, intuitive criteria that eventually lead to a selection of preferences for the players that may be viewed as “reasonable”? The ‘consistent preferences’ approach as such is not new. It is firmly rooted in a thirty year old game-theoretic tradition where a strategy of a player is interpreted as an expression of the belief (or the “conjecture”)

2

CONSISTENT PREFERENCES

of his opponent; cf., e.g., Harsanyi (1973), Aumann (1987a), and Blume et al. (1991b). What is new in this book (and the papers on which it builds) is that such a ‘consistent preferences’ approach is used to characterize a wider set of equilibrium concepts and, in particular, to serve as a basis for various types of interactive epistemic analysis where equilibrium assumptions are not made. Throughout this book, games are analyzed from the subjective perspective of each player. Hence, we can only make subjective statements about what a player “will do”, by considering “reasonable” preferences (and the corresponding representation in terms of subjective probabilities) of his opponent. This subjective perspective is echoed by recent contributions like Feinberg (2004a) and Kaneko and Kline (2004), which however differ from the present approach in many respects.1 To illustrate the differences between the two approaches—the ‘rational choice’ approach on the one hand and the ‘consistent preferences’ approach on the other—in a setting that will be familiar to most readers, Section 1.1 will be used to consider how epistemic conditions for Nash equilibrium in a strategic game can be formulated within each of these approaches. The remaining Sections 1.2 and 1.3 will provide motivation for the ‘consistent preferences’ approach through the following two points: 1 It facilitates the analysis of backward and forward induction. 2 It facilitates the integration of game theory and epistemic analysis with the underlying decision-theoretic foundation.

1.1

Conditions for Nash equilibrium

To fix ideas, consider a simple coordination game, where two drivers must choose what side to drive on in order to avoid colliding. In an

1 In

the present text, reasoning about hypothetical events will be captured by each player having an initial (interim – after having become aware of his own “type”) system of conditional preferences; cf. Chapters 3 and 4. This system encodes how the player will update his beliefs as actual play develops. In contrast, the subjective framework of Feinberg (2004a) does not represent the reasoning from such an interim viewpoint, and beliefs are not constrained to be evolving or revised. Instead, beliefs are represented whenever there is a decision to be made based on the presumption that beliefs should only matter when a decision is made. In Feinberg’s framework, only the ex-post beliefs are present and all ex-post subjective views are equally modeled. Even though also Kaneko and Kline (2004) consider a player having a subjective view on the objective situation, their main point is the inductive derivation of this individual subjective view from individual experiences.

3

Introduction

equilibrium in the ‘rational choice’ approach, a driver chooses to drive on the right side of the road if he believes that his opponent chooses to drive on the right side of the road. This can be contrasted with an equilibrium in the ‘consistent preferences’ approach, where a driver prefers to drive on the right side of the road if he believes that his opponent prefers to drive on the right side of the road. As mentioned, this follows a tradition in equilibrium analysis from Harsanyi (1973) to Blume et al. (1991b). This section presents, as a preliminary analysis, how these two interpretations of Nash equilibrium can be formalized. First, introduce the concept of a strategic game. A strategic twoplayer game G = (S1 , S2 , u1 , u2 ) consists of, for each player i, a set of pure strategies, Si , and a payoff function, ui : S1 × S2 → R. Then, turn to the epistemic modeling. An epistemic model for a strategic game within the ‘rational choice’ approach will typically specify, for each player i, a finite set of types, Ti , a function that assigns a strategy choice to each type, si : Ti → Si , and, for each type ti in Ti , a probability distribution on the set of opponent types, µti ∈ ∆(Tj ), where ∆(Tj ) denotes the set of probability distributions on Tj . When combined with i’s payoff function, the function sj and the probability distribution µti determine player i’s preferences at ti over his own strategies; these preferences will be denoted ºti : X X µti (tj )ui (si , sj (tj )) ≥ µti (tj )ui (s0i , sj (tj )) . si ºti s0i iff tj

tj

This in turn determines i’s set of best responses at ti , which will throughout be referred to as i’s choice set at ti : Siti := {si ∈ Si | ∀s0i ∈ Si , si ºti s0i } . Finally, in the context of the ‘rational choice’ approach, we can define the set of type profiles for which player i chooses rationally: [rati ] := {(t1 , t2 ) ∈ T1 × T2 | si (ti ) ∈ Siti } . Write [rat] := [rat1 ] ∩ [rat2 ].

4

CONSISTENT PREFERENCES

It is now straightforward to give sufficient epistemic conditions for a pure strategy Nash equilibrium: (s1 , s2 ) ∈ S1 × S2 is a pure strategy Nash equilibrium if there exists an epistemic model with (t1 , t2 ) ∈ [rat] such that (s1 , s2 ) = (s1 (t1 ), s2 (t2 )) and, for each i, µti (tj ) = 1 . In words, (s1 , s2 ) is a pure strategy Nash equilibrium if there is mutual belief of a profile of types that rationally choose s1 and s2 . In fact, we need not require mutual belief of the type profile: in line with the insights of Aumann and Brandenburger (1995) (cf. their Preliminary Observation) it is sufficient that there is mutual belief of the strategy profile, as we need not be concerned with what one player believes that the other player believes (or any higher order beliefs). Consider next how to formulate epistemic conditions for a mixed strategy Nash equilibrium. Following, e.g., Harsanyi (1973), Armbruster and B¨oge (1979), Aumann (1987a), Brandenburger and Dekel (1989), Blume et al. (1991b), and Aumann and Brandenburger (1995), a mixed strategy Nash equilibrium is often interpreted as an equilibrium in beliefs. According to this rather prominent view, a player need not randomize in a mixed strategy Nash equilibrium, but may choose some pure strategy. However, the other player does not know which one, and the mixed strategy of the one player is an expression of the belief (or the “conjecture”) of the other. The ‘consistent preferences’ approach is well-suited for formulating epistemic conditions for a mixed strategy Nash equilibrium according to this interpretation. An epistemic model for a strategic game within the ‘consistent preferences’ approach will typically specify, for each player i, a finite set of types, Ti , and for each type ti in Ti , a probability distribution on the set of opponent strategy-type pairs, µti ∈ ∆(Sj × Tj ). Hence, instead of specifying a function that assigns strategy choices to types, each type’s probability distribution is extended to the Cartesian product of the opponent’s strategy set and type set. We can still determine type i’s preferences at ti over his own strategies, XX XX si ºti s0i iff µti (sj , tj )ui (si , sj ) ≥ µti (sj , tj )ui (s0i , sj ) , sj

tj

sj

tj

and i’s choice set at ti : Siti := {si ∈ Si | ∀s0i ∈ Si , si ºti s0i } .

5

Introduction

However, we are now concerned with what i at ti believes that opponent types do, rather than with what i at ti does himself. Naturally, such beliefs will only be well-defined for opponent types that ti deems subjectively possible, i.e., for player j types in the set ¯ n o ¯ Tjti := tj ∈ Tj ¯ µti (Sj , tj ) > 0 , P where µti (Sj , tj ) := sj ∈Sj µti (sj , tj ). Say that the mixed strategy pjti |tj is induced for tj by ti if tj ∈ Tjti , and for each sj ∈ Sj , pjti |tj (sj ) =

µti (sj , tj ) . µti (Sj , tj )

Finally, in the context of the ‘consistent preferences’ approach, we can define the set of type profiles for which ti i nduces a r ational mixed strategy for any subjectively possible opponent type: ¯ n ¡ 0 ¢o 0 ¯ [iri ] := (t1 , t2 ) ∈ T1 × T2 ¯ ∀t0j ∈ Tjti , pjti |tj ∈ ∆ Sjtj . If the true type profile is in [iri ], then player i’s preferences over his strategies are consistent with the preferences of his opponent. Rather than player j actually being rational, it entails that player i believes that j is rational. Write [ir] := [ir1 ] ∩ [ir2 ]. Through the event [ir] one can formulate sufficient epistemic conditions for a mixed strategy Nash equilibrium, interpreted as an equilibrium in beliefs: (p1 , p2 ) ∈ ∆(S1 ) × ∆(S2 ) is a mixed strategy Nash equilibrium if there exists an epistemic model with (t1 , t2 ) ∈ [ir] ¡ ¢ such that (p1 , p2 ) = p1t2 |t1 , p2t1 |t2 and, for each i, µti (Sj , tj ) = 1 . In words, (p1 , p2 ) is a mixed strategy Nash equilibrium if there is mutual belief of a profile of types, where each type induces the opponent’s mixed strategy for the other, and where any pure strategy in the induced mixed strategy is rational for the opponent type. Since any pure strategy Nash equilibrium can be viewed as a degenerate mixed strategy Nash equilibrium, these epistemic conditions are sufficient for pure strategy Nash equilibrium as well. Again, we need not require mutual belief of the type profile; it is sufficient that there is mutual belief of each player’s belief about the strategy choice of his opponent. It is by no means infeasible to provide epistemic conditions for mixed strategy Nash equilibrium, interpreted as an equilibrium in beliefs, with-

6

CONSISTENT PREFERENCES

in the ‘rational choice’ approach. Indeed, this is what Aumann and Brandenburger (1995) do through their Theorem A in the case of twoplayer games. One can still argue for the epistemic conditions arising within the ‘consistent preferences’ approach. If a mixed strategy Nash equilibrium is interpreted as an expression of what each player believes his opponent will do, then one can argue—based on Occam’s razor—that the epistemic conditions should specify these beliefs only, and not also what each player actually does. In particular, we need not require, as Aumann and Brandenburger (1995) do, that the players are rational.

1.2

Modeling backward and forward induction

This book is mainly concerned with the analysis of deductive reasoning in games—leading to rationalizability concepts—rather than the study of steady states where coordination problems have been solved— corresponding to equilibrium concepts. Deductive reasoning within the ‘consistent preferences’ approach means that events like [ir] will be made subject to interactive epistemology, without assuming that there is mutual belief of the type profile. Backward induction is a prime example of deductive reasoning in games. To capture the backward induction procedure, one must believe that each player chooses rationally at every information set of an extensive game, also at information sets that the player’s own strategy precludes from being reached. As will be indicated through the analysis of Chapters 7–10—based partly on joint work with Andr´es Perea—this might be easier to capture by analyzing events where each player believes that the opponent chooses rationally, rather than events where each player actually chooses rationally. The backward induction procedure can be captured by conditions on how each player revises his beliefs after “surprising” choices by the opponent. Therefore, it might be fruitful to characterize this procedure through restrictions on the belief revision policies of the players, rather than through restrictions on their behavior at all information sets (also at information sets that can only be reached if the behavioral restrictions at earlier information sets were not adhered to). As will be apparent in Chapters 7–10, the ‘consistent preferences’ approach captures the backward induction procedure through conditions imposed directly on the players’ belief revision policies. In certain games—like the “battle-of-the-sexes with outside option” game (cf. Figure 2.6)—forward induction has considerable bite. To model forward induction, one must essentially assume that each player

Introduction

7

believes that any rational choice by the opponent is infinitely more likely than any choice that is not rational. Again, this might be easier to capture by analyzing events relating to the beliefs of the player, rather than events relating to the behavior of the opponent. Chapters 11 and 12 will report on joint work with Martin Dufwenberg that shows how the ‘consistent preferences’ approach can be used to promote the forward induction outcome. For ease of presentation only two-player games will be considered in this book. This is in part a matter of convenience, as much of the subsequent analysis can essentially be generalized to n-player games (with n > 2). In particular, this applies to the analysis of backward induction in Chapter 7, and to some extent, the analysis of forward induction in Chapters 11 and 12. On the other hand, in the equilibrium analysis of Chapters 5, 8, 9, and 10, a strategy of one player is interpreted as an expression of the belief of his opponent. This interpretation is straightforward in two-player games, but requires that the beliefs of different opponents coincide in games with more than two players—e.g., compare Theorems A and B of Aumann and Brandenburger (1995). Moreover, by only considering two-player games we can avoid the issue of whether (and if so, how) each player’s beliefs about the strategy choices of his opponents are stochastically independent. Throughout, player 1 will be referred to in the male gender (e.g., “he chooses among his strategies”), while player 2 will be referred to in the female gender (e.g., “she believes that player 1 . . . ”). Also, in the examples the strategies of player 1 will be denoted by upper case symbols (e.g., L and R), while the strategies of player 2 will be denoted by lower case symbols (e.g., ` and r).

1.3

Integrating decision theory and game theory

When a player in a two-player strategic game considers what decision to make (i.e., what strategy to choose), only his belief about the strategy choice of his opponent matters for his decision. However, in order to form a well-judged belief regarding the choice of his opponent, he should take her rationality into account. This makes it necessary for the player to consider his belief about her belief about his strategy choice. And so forth. Hence, the uncertainty faced by a player i concerns (a) the strategy choice of his opponent j, (b) j’s belief about i’s strategy choice, and so on; cf. Tan and Werlang (1988). A type of a player i corresponds to (a) a belief about j’s strategy choice, (b) a belief about j’s belief about i’s strategy choice, and so on. Models of such infinite hierarchies

8

CONSISTENT PREFERENCES

of beliefs—see, e.g., B¨oge and Eisele (1979), Mertens and Zamir (1985), Brandenburger and Dekel (1993), and Epstein and Wang (1996)—yield S1 × T1 × S2 × T2 as the ‘belief-complete’ state space, where Ti is the set of all feasible types of player i. Furthermore, for each i, there is a homeomorphism between Ti and the set of beliefs on Si × Sj × Tj . In the decision problem of any player i, i’s decision is to choose one of his own strategies. For the modeling of this problem, i’s belief about his own strategy choice is not relevant and can be ignored. This does not mean that player i is not aware of his own choice. It signifies that such awareness plays no role in the analysis, and is thus redundant.2 Hence, in the setting of a strategic game the belief of each type of player i can be restricted to the set of opponent strategy-type pairs, Sj × Tj . Combined with the payoff function specified by the strategic game, a belief on Sj × Tj yields preferences over player i’s strategies. As discussed in Section 5.1, the above results on ‘belief-complete’ state spaces are not needed (since only finite games are treated without ‘belief-completeness’ being imposed) and not always applicable in the setting of the present text (since some of the analysis—e.g. in Chapters 6, 7, 11, and 12—allows for incomplete preferences). Indeed, infinite hierarchies of beliefs can be modeled by an implicit but ‘belief-incomplete’ model—with a finite type set Ti for each player i—where the belief of a player corresponds to the player’s type, and where the belief of the player concerns the opponent’s strategy-type pair. If we let each player be aware of his own type (as we will assume throughout), this leads to an epistemic model where the state space of player i is Ti × Sj × Tj . For each player, this is a standard decisiontheoretic formulation in the tradition of Savage (1954), Anscombe and Aumann (1963), and Blume et al. (1991a): Player i as a decision maker is uncertain about what strategy-type pair in Sj × Tj will be realized. Player i’s type ti determines his belief on Sj × Tj Player i’s decision is to choose a (possibly mixed) strategy pi ∈ ∆(Si ); each such strategy determines the (randomized) outcome of the game as a function of the opponent strategy sj ∈ Sj .3 2 Tan

and Werlang (1988) in their Sections 2 and 3 characterize rationalizable strategies without specifying beliefs about one’s own choice. 3 Hence, a strategy for a player corresponds to an Anscombe-Aumann act, assigning a (possibly randomized) outcome to any uncertain state; cf. Chapter 3.

Introduction

9

The model leads, however, to a different state space for each player, which may perhaps be considered problematic. In the framework for epistemic modeling of games proposed by Aumann (1987a)—applied by Aumann and Brandenburger (1995) and illustrated in Section 1.1—it is also explicitly modeled that each player is aware of his own decision (i.e., his strategy choice). This entails that, for each player i, there is function si from Ti to Si that assigns si (ti ) to ti . Furthermore, it means that the relevant state space is T1 × T2 , which is identical for both players. In spite of its prevalence, Aumann’s model leads to the following potential problem: If player i is of type ti and in spite of this were to choose some strategy si different from si (ti ), then the player would no longer be of type ti (since only si (ti ) is assigned to ti ). So what, starting with a state where player i is of type ti , would player i believe about his opponent’s strategy choice if he were to choose si 6= si (ti )? In line with the defense by Aumann and Brandenburger (1995) on pp. 1174-1175, one may argue that Aumann’s framework is purely descriptive and contains enough information to determine whether a player is rational and that we need not be concerned about what the player would have believed if the state were different. An alternative is, however, to follow Board (2003) in arguing that ti ’s belief about his opponent’s strategy choice should remain unchanged in the counterfactual event that he were to choose si 6= si (ti ). The above discussion can be interpreted as support for the epistemic structure that will underlie this book, and where the state space of player i is Ti × Sj × Tj . This kind of epistemic model describes the factors that are relevant for each player as a decision maker (namely, what his opponent does and who his opponent is), while being silent about the awareness of player i of his own decision. Also in this formulation, a different choice by player i changes the state, as an element of S1 × T1 × S2 × T2 , but it does not influence the type of player i, as a specific strategy is not assigned to each type. Hence, a different choice by player i does not change his belief about what the opponents do. In this setting, the epistemic analysis concerns the type profile, and not the strategy profile. As we have seen in Section 1.1, and which we will return to in Chapter 5, this is, however, sufficient to state and prove, e.g., a result that corresponds to Aumann and Brandenburger’s (1995) Theorem A, provided that mutual belief of rationality is weakened to the condition that each player believes that his opponent is rational. As we will see in Chapters 5 and 6 it also facilitates the introduction of

10

CONSISTENT PREFERENCES

caution, which then corresponds to players having beliefs that take into account that opponents may make irrational choices, rather than players trembling when they make their choice. Chapters 3 and 4 are concerned with the decision-theoretic framework and epistemic operators derived from this framework. Chapter 3 spells out how the Anscombe-Aumann framework will be used as a decision-theoretic foundation. Following Blume et al. (1991a), continuity will be relaxed. Moreover, two different kinds of generalizations are presented. On the one hand, completeness will be relaxed, as this is not an integral part of the backward induction procedure, and cannot be imposed in the epistemic characterization of forward induction presented in Chapters 11 and 12. On the other hand, flexibility concerning how to specify a system of conditional beliefs will be introduced, leading to a structure that encompasses both the concept of a conditional probability system and conditionals derived from a lexicographic probability system. This flexibility turns out to be essential for the analysis of Chapters 8 and 9. Chapter 4 reports on joint work with Ylva Søvik which derives beliefoperators from the preferences of decision makers and develop their semantics. These belief operators will in later chapters be used in the epistemic characterizations. First, however, motivating examples will be presented and discussed in Chapter 2.

Chapter 2 MOTIVATING EXAMPLES

Through examples this chapter illuminates the features that distinguish the ‘consistent preferences’ approach from the ‘rational choice’ approach (cf. Chapter 1). The examples also illustrate issues of relevance when capturing backward and forward induction in models of interactive epistemology. The same examples will be revisited in later chapters. Section 2.1 presents six different games, and contains a discussion of how suggested outcomes in these games can be promoted by different solution concepts. This discussion leads in Section 2.2 to an overview of the solution concepts that will be covered in subsequent chapters. While Section 2.1 will illustrate how various concepts work in the different examples, Section 2.2 will relate the different concepts to each other, and provide references to relevant literature.

2.1

Six examples

Consider the “battle-of-the-sexes” game, G1 , illustrated in Figure 2.1. This game has two Nash equilibria in pure strategies: (L, `) and (R, r). In the ‘rational choice’ approach, the first of these Nash equilibrium is interpreted as player 1 choosing L and player 2 choosing `, and these choices being mutual belief. It is a Nash equilibrium since there is mutual belief of the strategy choices and each player’s choice is rational, given his belief about the choice of his opponent. In the ‘consistent preferences’ approach, in contrast, this Nash equilibrium is interpreted as player 1 believing that 2 chooses ` and player 2 believing that 1 chooses L, and these conjectures being mutual belief. It is a Nash equilibrium since there is mutual belief of the conjectures about opponent choice and each

12

CONSISTENT PREFERENCES

r ` L 3, 1 0, 0 R 0, 0 1, 3

Figure 2.1.

G1 (“battle-of-the-sexes”).

player believes that the opponent chooses rationally given the opponent’s conjecture. The preferences of player 1—that he ranks L about R—is consistent with the preferences of player 2—that she ranks ` above r, and vice versa. More precisely, that player 1 ranks L above R is consistent with his beliefs about player 2, namely that he believes that she ranks ` above r and she chooses rationally (i.e., chooses a top ranked strategy). The ‘consistent preferences’ interpretation of Nash equilibrium carries over to the mixed strategy equilibrium when interpreted as an equilibrium in beliefs—cf. the Harsanyi (1973) interpretation discussed in Section 1.1. If player 1 believes with probability 1/4 that 2 chooses ` and with probability 3/4 that 2 chooses r and player 2 believes with probability 3/4 that 1 chooses L and with probability 1/4 that 1 chooses R, and these conjectures are common belief, then the players’ beliefs constitute a mixed-strategy Nash equilibrium. It is a Nash equilibrium since there is mutual belief of the conjectures about opponent choice and each player believes that the opponent chooses rationally given the opponent’s conjecture. Rationalizability concepts have no bite in the “battle-of-the-sexes” game, G1 : Interactive epistemology based on rationality alone cannot guide the players to one of the equilibria. Hence, to illustrate the force of deductive reasoning in games—leading to rationalizability concepts— we must consider other examples. In game G2 of Figure 2.2, there is a unique Nash equilibrium, (L, `). Furthermore, deductive reasoning will readily lead player 1 to L and player 2 to `. In the ‘rational choice’ approach this works as follows: If player 1 chooses rationally, then he chooses L. This is independent of his conjecture about 2’s behavior since L strongly dominates R (as 4 > 3 and 1 > 0). Therefore, if player 2 believes that 1 chooses rationally, and 2 chooses rationally herself, then she chooses ` (since 1 > 0). This argument shows that L is the unique rationalizable strategy for player 1 and ` is the unique rationalizable strategy for player 2. In the ‘consistent preferences’ approach, we get: Player 1 ranks L above R, independently of his conjecture about 2’s behavior. If player 2 believes

13

Motivating Examples

r ` L 4, 1 1, 0 R 3, 0 0, 3

Figure 2.2.

G2 , illustrating deductive reasoning.

r ` L 1, 3 4, 2 R 1, 3 3, 5

Figure 2.3.

G3 , illustrating weak dominance.

that 1 chooses rationally, then she believes that 1 chooses L and ranks ` above r. Therefore, if player 1 believes that 2 chooses rationally, and he believes that she believes that 1 chooses rationally, then he believes that 2 chooses `. As we will return to in Chapters 5 and 6, this is an alternative way to establish L and ` as the players’ rationalizable strategies. In any case, the deductive reasoning leading to rationalizability corresponds to iterated elimination of strongly dominated strategies (IESDS). In game G3 of Figure 2.3, there is also a unique Nash equilibrium, (L, `). However, deductive reasoning is more problematic and interesting in the case of this game. For each player, both strategies are rationalizable, meaning that rationalizability has no bite in this game. In particular, if player 1 deems it subjectively impossible that 2 may choose r, then R is a rational choice. Moreover, if player 2 believes that 1 chooses R, then r is a rational choice. Still, we might argue that 1 should not rule out the possibility that 2 might choose r, leading him to rank L above R (since L weakly dominates R) and player 2 to rank ` above r. Such deductive reasoning leads to permissible strategies in the terminology of Brandenburger (1992). Permissibility corresponds to one round of elimination of all weakly dominated strategies followed by iterated elimination of strongly dominated strategies—the so-called Dekel-Fudenberg procedure, after Dekel and Fudenberg (1990). It can be formalized in two different ways. On the one hand, within an analysis based on what players do, one can postulate that players make ‘almost’ rational choices by, in the spirit of Selten (1975) and his “trembling hand”, assuming that ‘mistakes’ are made with (infinitely) small probability. B¨orgers (1994) shows how such

14

CONSISTENT PREFERENCES

r ` Out 2, 0 2, 0 InL 1, 3 4, 2 InR 1, 3 3, 5 Figure 2.4.

1c Out 2 0

In

2s ` 1 3

r

1s L 4 2

R

3 5

G03 and a corresponding extensive form Γ03 (a “centipede” game).

an approach does indeed correspond to the Dekel-Fudenberg procedure and thus characterizes permissibility. On the other hand, within an analysis based on what players believe, one can impose that players are ‘cautious’, in the sense of deeming no opponent strategy as subjectively impossible. This approach to permissibility—which is in the spirit of Blume et al. (1991b) and Brandenburger (1992)—combines such caution with an assumption that each player believes that the opponent is rational. It is shown in Chapters 5 and 6 how this yields an alternative characterization of permissibility, where one need not consider whether players in fact are rational. Let us then turn to an expanded version of G3 , namely the game G03 illustrated in Figure 2.4 with a corresponding extensive form Γ03 . Following Rosenthal (1981) Γ03 is often called a “centipede” game. Here, (Out, `) is normally suggested as a solution for this game. In the strategic form G3 , this suggestion can be obtained by iterated (maximal) elimination of weakly dominated strategies (IEWDS), and in the extensive form Γ03 , it is based on backward induction. While epistemic conditions for the procedure of IEWDS have been given by Brandenburger and Keisler (2002)—see also the related work by Battigalli and Siniscalchi (2002)— IEWDS will fall outside the class of procedures that will be characterized in this book. The procedure of backward induction, on the other hand, will play a central role in Chapters 7–10. Permissibility, which corresponds to the Dekel-Fudenberg procedure, does not promote only (Out, `) in the games of Figure 2.4. While the Dekel-Fudenberg procedure eliminates the weakly dominated strategy InR, this procedure does not allow for further rounds of weak elimination. Hence, since r is not strongly dominated by ` even after the elimination of InR, r will not be eliminated by the Dekel-Fudenberg procedure. Hence, InL as well as Out are permissible for player 1, and r as well as ` are permissible for player 2. In the extensive game, Γ03 , one can give the following intuition for how InL and r are compatible with the deductive reasoning underlying

Motivating Examples

15

permissibility: If player 1 believes that player 2 will choose `, then he prefers Out to his two other strategies. Similarly, if player 2 assigns probability one to player 1 choosing Out, and revises her beliefs by assigning probability one to InL conditional on being asked to play, then she prefers ` to r. However, if player 2 assigns probability one to player 1 choosing Out, but revises her beliefs so that InL and InR are equally likely conditional on being asked to play, then she prefers r to `. So if player 1 assigns sufficient probability to player 2 being of the latter type and believes—conditional on her being of this type—that she will be rational by choosing her top-ranked strategy r, then he will prefer InL to his two other strategies. Following Ben-Porath (1997), Chapter 7 demonstrates within a formal epistemic model how such interactive beliefs are consistent with the assumptions underlying permissibility. As shown by Ben-Porath (1997), when permissibility is applied to an extensive game like Γ03 , each player must believe that her opponent chooses rationally as long as the opponent’s behavior is consistent with the player’s initial beliefs. However, conditional on finding herself at an information set that contradicts her previous belief about his behavior, she is allowed to believe that he will no longer choose rationally. E.g., in Γ03 it is OK for player 2 to assign positive probability to the irrational strategy InR conditional on being asked to play, provided that she had originally assigned probability one to player 1 rationally choosing Out. An alternative is that the player should still believe that her opponent will choose rationally, even conditionally on being informed about “surprising” moves. Chapters 7–9 will consider the event that each player believes that her opponent chooses rationally at all his information sets within models of interactive epistemology. Building on joint work with Andr´es Perea, this provides epistemic conditions for backward induction and definitions for the concepts of sequential and quasi-perfect rationalizability. Note that imposing that a player believes that her opponent chooses rationally at all his information sets is a requirement imposed on her belief revision policy, not on her actual behavior. It therefore fits well within the ‘consistent preferences’ approach. If we move to an expanded version of G2 , namely the game G02 illustrated in Figure 2.5 with a corresponding extensive form Γ02 , not even the event that each player believes that the opponent chooses rationally at all his information sets, will be sufficient for reaching the solution that

16

CONSISTENT PREFERENCES

r ` 2, 2 2, 2 Out InL 4, 1 1, 0 InR 3, 0 0, 3

Figure 2.5.

2 2

1c ¡@

Out InL ¡

¡ s ¡ `¡@ r ¡ @

4 1

1 0

@ InR @ 2 @s ` ¡ @r ¡ @

3 0

0 3

G02 and a corresponding extensive form Γ02 .

one would normally suggest, namely (InL, `). This outcome is supported by the following deductive reasoning: Since InL strongly dominates InR, implying that player 1 prefers the former strategy to the latter, player 2 should deem InL much more likely than InR conditional on being asked to play, and hence prefer ` to r. This in turn would lead player 1 to prefer InL to his two other strategies if he believes that player 2 will be rational by choosing her top-ranked strategy `. However, the concepts of sequential and quasi-perfect rationalizability only preclude that player 2 unconditionally assigns positive probability to player 1 choosing InR. If player 2 assigns probability one to player 1 choosing Out, then she may—when revising her beliefs conditional on being asked to play—assign sufficient probability to InR so that r is preferred to `. If player 1 assigns sufficient probability to player 2 being of such a type, then he will prefer Out to his two other strategies. The outcome (InL, `) can be promoted by considering the event that player 2 respects the preferences of her opponent by deeming one opponent strategy infinitely more likely than another if the opponent prefers the former to the latter. Respect of opponent preferences was first considered by Blume et al. (1991b) in their characterization of proper equilibrium. Being a requirement on the beliefs of players, it fits nicely into the ‘consistent preferences’ approach. Within a model of interactive epistemology Chapter 10 characterizes the concept of proper rationalizability by considering the event that each player respects opponent preferences. Proper rationalizability implies backward induction. However, even though it yields conclusions that coincide with IEWDS in all of the examples above, this conclusion does not hold in general, as will be shown by the next example and further discussed in Chapter 10. Lastly, turn to an expanded version of G1 , namely the game G01 illustrated in Figure 2.6 with a corresponding extensive form Γ01 . The exten-

17

Motivating Examples

r ` 2, 2 2, 2 Out InL 3, 1 0, 0 InR 0, 0 1, 3

1c ¡@ @ InR InL ¡ ¡ @ 2 s ¡ @s `¡@ r ` ¡ @r ¡ @ ¡ @ 2 2

Out

3 1

0 0

0 0

1 3

Figure 2.6. G01 and a corresponding extensive form Γ01 (“battle-of-the-sexes with an outside option”).

sive game Γ01 is referred to as the “battle-of-the-sexes with an outside option” game. This game was introduced by Kreps and Wilson (1982) (who credit Elon Kohlberg) and is often used to illustrate forward induction, namely that player 2 through deductive reasoning should figure out that player 1 has chosen InL and aims for the payoff 3 if 2 is being asked to play. Respect of preferences only requires player 2 to deem InR infinitely less likely than Out since the latter strategy strongly dominates the former; it does not require 2 to deem InR infinitely less likely than InL and thereby prefer ` to r. In contrast, IEWDS eliminates all strategies except InL for player 1 and ` for player 2, thereby promoting the forward induction outcome. Chapter 11 contains a critical assessment of how iterated weak dominance promotes forward induction in this and other examples. Based on joint work with Martin Dufwenberg, it will be suggested how forward induction can be promoted by strengthening the concept of permissibility to our notion of full permissibility. Full permissibility is characterized by conditions levied on the beliefs of players, and therefore fits naturally into the ‘consistent preferences’ approach. In the final Chapter 12 this notion will be further illustrated through a series of extensive games, illustrating how it yields forward induction, while not always supporting backward induction (indeed, Γ03 is an example of an extensive game where full permissibility does not promote the backward induction outcome).

2.2

Overview over concepts

To provide a structure for the concepts that will be defined and characterized in the subsequent chapters, it might be useful as a roadmap to present an overview over these concepts and their relationships.

18

Table 2.1.

CONSISTENT PREFERENCES

Relationships between different equilibrium concepts.

Proper equilibrium Myerson (1978) ↓ Strategic form Quasi-perfect perfect equil. ← equilibrium Selten (1975) van Damme (1984) ↓ ↓ Nash Weak sequential Sequential equi- ← equilibrium ← equilibrium librium Reny (1992) Kreps & Wilson (1982)

First, consider the equilibrium concepts of Table 2.1. Here, weak sequential equilibrium refers to the equilibrium concept—defined by Reny (1992)—that results when each player only optimizes at information sets that the player’s own strategy does not preclude from being reached. Moreover, quasi-perfect equilibrium is the concept defined by van Damme (1984) and which differs from Selten’s (1975) extensive form perfect equilibrium by having each player ignore the possibility of his own future mistakes. The arrows indicate that any proper equilibrium corresponds to a quasi-perfect equilibrium and so forth. Nash equilibrium and (strategic form) perfect equilibrium will be characterized in Chapter 5, while sequential equilibrium, quasi-perfect equilibrium, and proper equilibrium will be characterized in Chapters 8, 9, and 10, respectively. The non-equilibrium analogs to these equilibrium concepts are illustrated in Table 2.2. Again, the arrows indicate that proper rationalizability implies quasi-perfect rationalizability and so forth. Of course, the notion of rationalizability due to Bernheim (1984) and Pearce (1984) is a non-equilibrium analog to Nash equilibrium. Likewise, the notion of permissibility due to B¨orgers (1994) and Brandenburger (1992) corresponds to Selten’s (1975) strategic form perfect equilibrium, and the notion of weak sequential rationalizability due to Ben-Porath (1997)—coined ‘weak extensive form rationalizablity’ by Battigalli and Bonanno (1999)—is a non-equilibrium analog of weak sequential equilibrium. Furthermore, sequential rationalizability due to Dekel et al. (1999, 2002), quasi-perfect rationalizability due to Asheim and Perea (2004), and proper rational-

19

Motivating Examples

Table 2.2.

Relationships between different rationalizability concepts.

Common cert. belief that each player . . . . . . is cautious and respects preferences

. . . is cautious

. . . is not necessarily cautious

. . . believes the oppon. chooses rationally only initially, in the whole game

. . . believes the oppon. chooses rationally at all reachable info. sets

[n.a.]

[n.a.]

[n.a.]

Rationalizability Bernh. (1984) Pearce (1984) [Chapters 5–6] Does not imply backward ind.

←

Permissibility B¨ orgers (1994) Brandenb. (1992) Dek. & Fud. (1990) [Chapters 5–6] ↓ Weak sequential rationalizability Ben-Porath (1997) [Chapter 8] Does not imply backward ind.

. . . believes the oppon. chooses rationally at all info. sets

←

←

Proper rationalizability Schuhm. (1999) [Chapter 10] ↓ Quasi-perfect rationalizability Ash. & Per. (2004) [Chapter 9] ↓ Sequential rationalizability Dekel et al. (1999, 2002) [Chapter 8] Implies backward ind.

izability due to Schuhmacher (1999) are non-equilibrium analogs to sequential equilibrium, quasi-perfect equilibrium, and proper equilibrium, respectively. As indicated by Table 2.2, these concepts will be treated in Chapters 5, 6, 8, 9, and 10, and they are characterized by on the one hand, whether each player is cautious and respects opponent preferences, and on the other hand, whether each player believes that his opponent chooses rationally only initially (in the whole game), or at all reachable information sets, or at all information sets. This taxonomy defines events which are made subject to common certain belief, where ‘certain belief’ is the epistemic operator that will be used for the interactive epistemology. This operator is defined in Chapter 4 and will have the following meaning: An event is said to be ‘certainly believed’ if the complement is deemed subjectively impossible.

20

CONSISTENT PREFERENCES

Throughout this book, we will analyze assumptions about players’ preferences, leading to events that are subsets of type profiles. We can still make subjective statements about what a player “will do”, by considering the preferences (and the corresponding representation in terms of subjective probabilities) of the other player. For the concepts in the left and center columns of Table 2.2, we can do more than this, if we so wish. E.g., when characterizing weak sequential rationalizability, we can consider the event of rational pure choice at all reachable information sets, and assume that this event is commonly believed (where the term ‘belief’ is used in the sense of ‘belief with probability one’). These assumptions yield subsets of strategy profiles, leading to direct behavioral implications within the model. This does not carry over to the concepts in the right column. It is problematic to define the event of rational pure choice at all information sets, since reaching a non-reachable information set may contradict rational choice at earlier information sets. Also, if we consider the event of (any kind of) rational pure choice, then we cannot use common certain belief, since this—combined with rational choice—would prevent well-defined conditional beliefs after irrational opponent choices. However, common belief (with probability one) of the event that each player believes his opponent chooses rationally at all information sets does not yield backward induction in generic perfect information games, as shown in the counterexample illustrated in Figure 7.1. Common certain belief is essential for our analysis of the concepts in the right column of Table 2; this complicates obtaining direct behavioral implications. Before defining the various belief operators that will be used in the later chapters, the decision-theoretic framework will be presented and analyzed in Chapter 3.

Chapter 3 DECISION-THEORETIC FRAMEWORK

In the ‘consistent preferences’ approach to deductive reasoning in games, the object of the analysis is each player’s preferences over his own strategies, rather than his choice. The preferences can be required to be consistent (in different senses) with his beliefs about the opponent’s preferences over her strategies. The player’s preferences depend on his belief about the strategy choice of his opponent. Furthermore, in order for the player to consider the preferences of his opponent, her belief about his strategy profile matters, and so forth. What kind of decision-theoretic framework is suited for such analysis? This chapter spells out how the framework proposed by Anscombe and Aumann (1963) will be used as a decision-theoretic foundation. Following Blume et al. (1991a), the Archimedean property will be relaxed. Moreover, two different kinds of generalizations will be presented: (i) Completeness will be relaxed, as this is not an integral part of the backward induction procedure (cf. the analysis of Chapter 7), and cannot be imposed in the epistemic characterization of forward induction presented in Chapters 11 and 12. (ii) Flexibility concerning how to specify conditional preferences, leading to a structure that encompasses both the concept of a conditional probability system and conditionals derived from a lexicographic probability system. This flexibility turns out to be essential for the analysis of Chapters 8 and 9. Section 3.1 motivates these generalizations, as well as providing reasons for the choice of the Ascombe-Aumann framework. Section 3.2

22

CONSISTENT PREFERENCES

introduces the different sets of axioms that will be considered, while the final Section 3.3 presents the corresponding representation results.

3.1

Motivation

Standard decision theory under uncertainty concerns two different kinds of decisions. 1 In the first kind, the object of choice is lotteries. There is a given set of outcomes, and a lottery is an objective probability distribution over outcomes. If the decision maker satisfies the von NeumannMorgenstern axioms—cf. von Neumann and Morgenstern (1947)— then one can assign utilities to outcomes, so that the decision maker prefers one lottery to another if the former has higher expected utility. 2 In the second kind, the object of choice is acts. There is a given set of outcomes and a given set of uncertain states, and an act is a function from states to outcomes. If the decision maker satisfies the Savage (1954) axioms, then one can assign utilities to outcomes and subjective probabilities to states, so that the decision maker prefers one act to another if the former has higher (subjective) expected utility. An act in the sense of Anscombe and Aumann (1963) is a function from states to objective randomizations over outcomes.1 By considering acts in this sense they are able to extend the von Neumann-Morgenstern theory so that the utilities assigned to outcomes are determined solely from preferences over lotteries, while the subjective probabilities assigned to states are determined when also acts are considered. A strategy in a game is a function that, for each opponent strategy choice, determines an outcome. A pure strategy determines for each opponent strategy a deterministic outcome, while a mixed strategy determines for each opponent strategy an objective randomization over the set of outcomes. Hence, a pure strategy is an example of an act in the sense of Savage (1954), while a mixed strategy is an example of an act in the generalized sense of Anscombe and Aumann (1963). Allowing for objective randomizations and using Anscombe-Aumann acts are convenient for two reasons in the present context: The Anscombe-Aumann framework allows a player’s payoff function to be a von Neumann-Morgenstern (vNM) utility function deter1 Anscombe

and Aumann (1963) use the term ‘roulette lottery’ for what we here call ‘lotteries’, ‘horse lotteries’ for acts from states to deterministic outcomes, i.e., acts in the Savage (1954) sense, and ‘compound horse lotteries’ for what we here refer to as Anscombe-Aumann acts.

Decision-theoretic Framework

23

mined from his preferences over randomized outcomes, independently of the likelihood that he assigns to the different strategies of his opponent. This is consistent with the way games are normally presented, where payoff functions for each player are provided independently of the analysis of the strategic interaction.2 When relaxing completeness, it turns out to be important to allow mixed strategies as objects of choice when determining maximal elements of a player’s incomplete preferences, for similar reasons as domination by mixed strategies is needed for dominated strategies to correspond to strategies that can never be best replies. We will consider three kinds of generalizations of the AnscombeAumann framework. First, as mentioned in the introduction to this chapter, throughout this book we will follow Blume et al. (1991a) by imposing the conditional Archimedean property (also called conditional continuity) instead of Archimedean property (also called continuity). This is important for modeling caution, which requires a player to take into account the possibility that the opponent makes an irrational choice, while assigning probability one to the event that the opponent makes a rational choice. I.e., even though any irrational choice is infinitely less likely than some rational choice, it is not ruled out. Such discontinuous preferences will also be useful when modeling players’ preferences in extensive games. Second, we will relax the axiom of completeness to conditional completeness. While complete preferences will normally be represented by means of subjective probabilities (cf. Propositions 1, 2, 3, and 5 of this chapter), incomplete preferences are insufficient to determine the relative likelihood of the uncertain states. One possibility is, following Aumann (1962) and Bewley (1986), to represent incomplete preferences by means of a set of subjective probability distributions. Subjective probabilities are not part of the most common deductive procedures in game theory—like IESDS, the Dekel-Fudenberg procedure, and the backward induction procedure. One can argue that, since they make no use of subjective probabilities, one should seek to provide 2 This

argument is in line with the analysis of Aumann and Dreze (2004), who however depart from the Anscombe-Aumann framework by considering preferences—not over all functions from states to randomized outcomes—but only on the subset of mixed strategies. The Ascombe-Aumann framework requires that the decision maker has access to objective probabilities; however, Machina (2004) points to how this requirement can be weakened.

24

CONSISTENT PREFERENCES

epistemic conditions for such procedures without reference to subjective probabilities. Indeed, subjective probabilities play no role in epistemic analysis of backward induction by Aumann (1995). In Chapters 6 and 7 we follow Aumann in this respect and provide epistemic conditions for IESDS, the Dekel-Fudenberg procedure, and the backward induction procedure through modeling players endowed with (possibly) incomplete preferences that are not represented by subjective probabilities. Moreover, for the modeling of forward induction in Chapters 11 and 12, it is a necessary part of the analysis that preferences are incomplete. Third, we will allow for flexibility concerning how to specify conditional preferences. Such flexibility can be motivated in the context of the modeling of sequentiality and quasi-perfectness in Chapters 8 and 9. Sequential rationalizability will be defined and sequential equilibrium characterized by considering the event that each player believes that the opponent chooses rationally at all her information sets. Adding preference for cautious behavior to this event yields the concepts of quasiperfect rationalizability and equilibrium. For these definitions and characterizations, we must describe what a player believes both conditional on reaching his own information sets (to evaluate his rationality) and conditional on his opponent reaching her information sets (to determine his beliefs about her choices). In other words, we must specify a system of conditional beliefs for each player. There are various ways to do so. One possibility is a conditional probability system (CPS) where each conditional belief is a subjective probability distribution.3 This is sufficient to model sequentiality. Another possibility, which is sufficient to model quasi-perfectness, is to apply a single sequence of subjective probability distributions—a so-called lexicographic probability system (LPS) as defined by Blume et al. (1991a)— and derive the conditional beliefs as the conditionals of such an LPS. Since each conditional LPS is found by constructing a new sequence, which includes the well-defined conditional probability distributions of the original sequence, each conditional belief is itself an LPS. However, quasi-perfectness cannot always be modeled by a CPS since the modeling of preference for cautious behavior may require lexicographic probabilities. To see this, consider Γ4 of Figure 3.1. In this 3 This

is the terminology introduced by Myerson (1986). In philosophical literature, related concepts are called Popper measures. For an overview over relevant literature and analysis, see Hammond (1994) and Halpern (2003).

25

Decision-theoretic Framework

1c D 1 1

F

2s d 1 1

Figure 3.1.

f

0 0

f d 1, 1 0, 0 F D 1, 1 1, 1

Γ4 and its strategic form.

game, if player 1 believes that player 2 chooses rationally, then player 1 must assign probability one to player 2 choosing d. Hence, if each (conditional) belief is associated with a subjective probability distribution—as is the case with the concept of a CPS—and player 1 believes that his opponent chooses rationally, then player 1 is indifferent between his two strategies. This is inconsistent with quasi-perfectness, which requires players to have preference for cautious behavior, meaning that player 1 in Γ4 prefers D to F . Moreover, sequentiality cannot always be modeled by means of conditionals of a single LPS since preference for cautious behavior is induced. To see this, consider a modified version of Γ1 where an additional subgame is substituted for the (0, 0)–payoff, with all payoffs in that subgame being smaller than 1. If player 1’s conditional beliefs over strategies for player 2 is derived from a single LPS, then a well-defined belief conditional on reaching the added subgame entails that player 1 deems possible the event that player 2 chooses f , and hence, player 1 prefers D to F . This is inconsistent with sequentiality, under which F is a rational choice. Therefore, this chapter will present a new way of describing a system of conditional beliefs, called a system of conditional lexicographic probabilities (SCLP), and which is based on joint work with Andr´es Perea; cf. Asheim and Perea (2004). In contrast to a CPS, an SCLP may induce conditional beliefs that are represented by LPSs rather than subjective probability distributions. In contrast to the system of conditionals derived from a single LPS, an SCLP need not include all levels in the sequence of the original LPS when determining conditional beliefs. Thus, an SCLP ensures well-defined conditional beliefs representing nontrivial conditional preferences, while allowing for flexibility w.r.t. whether to assume preference for cautious behavior.

26

3.2

CONSISTENT PREFERENCES

Axioms

Consider a decision maker under uncertainty, and let F be a finite set of states. The decision maker is uncertain about what state in F will be realized. Let Z be a finite set of outcomes. For each φ ∈ 2F \{∅}, the decision maker is endowed with a binary relation (preferences) over all functions that to each element of φ assign an objective randomization on Z. Any such function is called an act on φ, and is the subject of analysis in the deciscion-theoretic framework introduced by Anscombe and Aumann (1963). Write pφ and qφ for acts on φ ∈ 2F \{∅}. (For acts on F , write simply p and q.) A binary relation on the set of acts on φ is denoted by ºφ , where pφ ºφ qφ means that pφ is preferred or indifferent to qφ . As usual, let Âφ (preferred to) and ∼φ (indifferent to) denote the asymmetric and symmetric parts of ºφ . Consider the following five axioms, where the numbering of axioms follows Blume et al. (1991a).

Axiom 1 (Order) ºφ is complete and transitive. Axiom 2 (Objective Independence) p0φ Âφ (resp. ∼φ ) p00φ iff γp0φ +(1 − γ)qφ Âφ (resp. ∼φ ) γp00φ + (1 − γ)qφ , whenever 0 < γ < 1 and qφ is arbitrary. Axiom 3 (Nontriviality) There exist pφ and qφ such that pφ Âφ qφ . Axiom 4 (Archimedean Property) If p0φ Âφ qφ Âφ p00φ , then ∃0 < γ < δ < 1 such that δp0φ +(1 − δ)p00φ Âφ qφ Âφ γp0φ + (1 − γ)p00φ . Say that e ∈ F is Savage-null if p{e} ∼{e} q{e} for all acts p{e} and q{e} on {e}. Denote by κ the non-empty set of states that are not Savagenull; i.e., the set of states that the decision maker deems subjectively possible. Write Φ := {φ ∈ 2F \{∅}| κ ∩ φ 6= ∅}. Refer to the collection {ºφ | φ ∈ Φ} as a system of conditional preferences on the collection of sets of acts from from subsets of F to outcomes. Whenever ∅ 6= ε ⊆ φ, denote by pε the restriction of pφ to ε.

Axiom 5 (Non-null State Independence) p{e} Â{e} q{e} iff p{f } Â{f } q{f } , whenever e, f ∈ κ, and p{e,f } and q{e,f } satisfy p{e,f } (e) = p{e,f } (f ) and q{e,f } (e) = q{e,f } (f ). Define the conditional binary relation ºφ|ε by p0φ ºφ|ε p00φ if, for some qφ , (p0ε , qφ\ε ) ºφ (p00ε , qφ\ε ). By Axioms 1 and 2, this definition does not depend on qφ . The following axiom states that preferences over acts on ε, ºε , equal the conditional of ºφ on ε, whenever ∅ 6= ε ⊆ φ.

Decision-theoretic Framework

27

Axiom 6 (Conditionality) pε Âε (resp. ∼ε ) qε iff pφ Âφ|ε (resp. ∼φ|ε ) qφ , whenever ∅ 6= ε ⊆ φ. It is an immediate observation that Axioms 5 and 6 imply non-null state independence as stated in Axiom 5 of Blume et al. (1991a).

Lemma 1 Assume that the system of conditional preferences {ºφ |φ ∈ Φ} satisfies Axioms 5 and 6. Then, ∀φ ∈ Φ, pφ Âφ|{e} qφ iff pφ Âφ|{f } qφ whenever e, f ∈ κ ∩ φ, and pφ and qφ satisfy pφ (e) = pφ (f ) and qφ (e) = qφ (f ). Turn now the relaxation of Axioms 1, 4, and 6, as motivated in the previous section. Axiom 10 (Conditional Order) ºφ is reflexive and transitive and, ∀e ∈ φ, ºφ|{e} is complete. Axiom 40 (Conditional Archimedean Property) ∀e ∈ φ, if p0φ Âφ|{e} qφ Âφ|{e} p00φ , then ∃0 < γ < δ < 1 such that δp0φ +(1−δ)p00φ Âφ|{e} qφ Âφ|{e} γp0φ + (1 − γ)p00φ . Axiom 60 (Dynamic Consistency) pε Âε qε whenever pφ Âφ|ε qφ and ∅ 6= ε ⊆ φ. Since completeness implies reflexivity, Axiom 10 constitutes a weakening of Axioms 1. This weakening is substantive since, in the terminology of Anscombe and Aumann (1963), it means that the decision maker has complete preferences over ‘roulette lotteries’ where objective probabilities are exogenously given, but not necessarily complete preferences over ‘horse lotteries’ where subjective probabilities, if determined, are endogenously derived from the preferences of the decision maker. Say that e ∈ κ is deemed infinitely more likely than f ∈ F (and write e À f ) if p{e,f } Â{e,f } q{e,f } whenever p{e} Â{e} q{e} . Consider the following two auxiliary axioms. Axiom 11 (Partitional priority) If e0 À e00 , then ∀f ∈ F , e0 À f or f À e00 . Axiom 16 (Compatibility) There exists a binary relation º∗F satisfying Axioms 1, 2, and 4 0 such that p Â∗F |φ q whenever pφ Âφ qφ and ∅ 6= φ ⊆ F . While it is straightforward that Axiom 1 implies Axiom 10 , Axiom 4 implies Axiom 40 , and Axiom 6 implies Axiom 60 , it is less obvious that

28

CONSISTENT PREFERENCES

Axiom 1 together with Axioms 2, 40 , 5, and 6 imply Axiom 11, and Axiom 6 together with Axioms 1 2, 40 , and 5, imply Axiom 16. This is demonstrated by the following lemma.

Lemma 2 Assume that (a) ºφ satisfies Axioms 1 and 2 if φ ∈ 2F \{∅}, and Axiom 4 0 if and only if φ ∈ Φ, and (b) the system of conditional preferences {ºφ |φ ∈ Φ} satisfies Axioms 5 and 6. Then {ºφ |φ ∈ Φ} satisfies Axioms 11 and 16. Proof. Part 1: Axiom 11 is implied. We must show, under the given premise, that if e0 À e00 , then, ∀f ∈ F , e0 À f or f À e00 . Clearly, e0 À e00 entails e0 ∈ κ, implying that e0 À f or f À e00 if f ∈ / κ or e00 ∈ / κ. The case where f = e0 or f = e00 is trivial. The case where f 6= e0 , f 6= e00 , f ∈ κ and e00 ∈ κ remains. Assume that e0 À f does not hold, which by completeness (Axiom 1) entails the existence of p0{e0 ,f } and q0{e0 ,f } such that p0{e0 ,f } ¹{e0 ,f } q0{e0 ,f } and p0{e0 } Â{e0 } q0{e0 } . It suffices to show that f À e00 is obtained; i.e., p{f } Â{f } q{f } implies p{e00 ,f } Â{e00 ,f } q{e00 ,f } . Throughout we invoke Axiom 6 and Lemma 1, and choose φ ∈ Φ so that {e0 , e00 , f } ∈ φ. Let pφ Âφ|{f } qφ . Assume w.l.o.g. that pφ (d) = qφ (d) for d 6= f, e00 , and p0φ (d) = q0φ (d) for d 6= e0 , f . By transitivity (Axiom 1), p0φ ¹φ|{e0 ,f } q0φ and p0φ Âφ|{e0 } q0φ imply p0φ ≺φ|{f } q0φ . However, since ºφ satisfies Axioms 2 and 40 , ∃γ ∈ (0, 1) such that γpφ + (1 − γ)p0φ Âφ|{f } γqφ + (1 − γ)q0φ . Moreover, pφ (e0 ) = qφ (e0 ) and p0φ Âφ|{e0 } q0φ entail that γpφ + (1 − γ)p0φ Âφ|{e0 } γqφ + (1 − γ)q0φ by Axiom 2, which implies that γpφ + (1 − γ)p0φ Âφ|{e0 ,e00 } γqφ + (1 − γ)q0φ since e0 À e00 . Hence, by transitivity, γpφ +(1−γ)p0φ Âφ|{e0 ,e00 ,f } γqφ +(1−γ)q0φ — or equivalently, γpφ + (1 − γ)p0φ Â γqφ + (1 − γ)q0φ . Now, q0φ ºφ|{e0 ,f } p0φ means that γpφ + (1 − γ)q0φ º γpφ + (1 − γ)p0φ by Axiom 2, implying that γpφ + (1 − γ)q0φ Â γqφ + (1 − γ)q0φ by transitivity (Axiom 1), and pφ Â qφ — or equivalently, pφ Âφ|{e00 ,f } qφ — by Axiom 2. Thus, pφ Âφ|{f } qφ implies pφ Âφ|{e00 ,f } qφ , meaning that f À c. Part 2: Axiom 16 is implied. We must show, under the given premise, that here exists a binary relation º∗F satisfying Axioms 1, 2, and 40 such that p Â∗F |φ q whenever pφ Âφ qφ and ∅ 6= φ ⊆ F . Clearly, since Axiom 6 is satisfied, ºF fulfil these requirements. We end this section by stating for later use an axiom which is implied by Axiom 4 and which implies Axiom 40 . Also for this axiom the numbering follows the one used by Blume et al. (1991a).

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Table 3.1.

Relationships between different sets of axioms and their representations.

Complete and continuous

123456 Prob. distr. ↓

Complete and partitionally continuous

1 2 3 400 5 6 LCPS ↓

Complete and discontinuous

1 2 3 40 5 6 LPS ↓

Incomplete and discontinuous

→ 1 2 3 4 5 60 16 CPS ↓

→ 1 2 3 40 5 60 16 SCLP

10 11 2 3 40 5 6 Conditionality

Dynamic consistency

Axiom 400 (Partitional Archimedean Property) There is a parti0 } of κ ∩ φ such that tion {π10 , . . . , πL|φ ∀` ∈ {1, . . . , L|φ}, if p0φ Âφ|π`0 qφ Âφ|π`0 p00φ , then ∃0 < γ < δ < 1 such that δp0φ + (1 − δ)p00φ Âφ|π`0 qφ Âφ|π`0 γp0φ + (1 − γ)p00φ , and 0 ∀` ∈ {1, . . . , L|φ − 1}, pφ Âφ|π`0 qφ implies pφ Âφ|π`0 ∪π`+1 qφ .

Table 3.1 illustrates the relationships between the sets of axioms that we will consider. The arrows indicate that one set of axioms implies another. The figure indicates what kind of representations the different sets of axioms correspond to, as reported in the next section.

3.3

Representation results

In view of Lemma 1 and using the characterization result of Anscombe and Aumann (1963), we obtain the following result under Axioms 1, 2, 3, 4, 5, and 6; cf. Theorem 2.1 of Blume et al. (1991a). For the statement of this and later results, denote by υ : Z → R aPvNM utility function, and abuse notation slightly by writing υ(p) = z∈Z p(z)υ(z) whenever p ∈ ∆(Z) is an objective randomization. In this and later results, υ is unique up to positive affine transformations.

Proposition 1 (Anscombe and Aumann, 1963) The following two statements are equivalent.

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1 (a) ºφ satisfies Axioms 1, 2, and 4 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {ºφ |φ ∈ Φ} satisfies Axioms 5 and 6. 2 There exist a vNM utility function υ : ∆(Z) → R and a unique subjective probability distribution µ on F with support κ that satisfies, for any φ ∈ Φ, X X µ|φ (e)υ(pφ (e)) ≥ µ|φ (e)υ(qφ (e)) , pφ ºφ qφ iff e∈φ

e∈φ

where µ|φ is the conditional of µ on φ. In view of Lemma 1, and using Theorem 3.1 of Blume et al. (1991a), we obtain the following result under Axioms 1, 2, 3, 40 , 5, and 6. For the statement of this and later results, we need to introduce formally the concept of a lexicographic probability system. A lexicographic probability system (LPS) consists of L levels of subjective probability distributions: If L ≥ 1 and, ∀` ∈ {1, . . . , L}, µ` ∈ ∆(F ), then λ = (µ1 , . . . , µL ) is an LPS on F . Denote by L∆(F ) the set of LPSs on F . Write supp λ := ∪L `=1 supp µ` for the support of λ. If supp λ ∩ φ 6= ∅, 0 denote by λ|φ = (µ1 , . . . µ0L|φ ) the conditional of λ on φ.4 Furthermore, for two utility vectors v and w, denote by v ≥L w that, whenever w` > v` , there exists k < ` such that vk > wk , and let >L and =L denote the asymmetric and symmetric parts, respectively.

Proposition 2 (Blume et. al, 1991a) The following two statements are equivalent. 1 (a) ºφ satisfies Axioms 1, 2, and 4 0 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {ºφ |φ ∈ Φ} satisfies Axioms 5 and 6. 2 There exist a vNM utility function υ : ∆(Z) → R and an LPS λ on F with support κ that satisfies, for any φ ∈ Φ, pφ ºφ qφ iff ³X ´L|φ ³X µ0` (e)υ(pφ (e)) ≥L e∈φ

4 I.e.,

`=1

e∈φ

´L|φ µ0` (e)υ(qφ (e)) , `=1

∀` ∈ {1, . . . , L|φ}, µ0` = µk`|φ , where the indices k` are given by k0 = 0, k` = min{k|µk (φ) > 0 and k > k`−1 } for ` > 0, and {k|µk (φ) > 0 and k > kL|φ } = ∅, and where µk`|φ is given by the usual definition of conditional probabilities; cf. Definition 4.2 of Blume et al. (1991a).

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where λ|φ = (µ01 , . . . µ0L|φ ) is the conditional of λ on φ. In view of Lemma 1 and using Theorem 5.3 of Blume et al. (1991a), we obtain the following rusult under Axioms 1, 2, 3, 400 , 5, and 6. For the statement of this results, we need to introduce the concept that is called a lexicographic conditional probability system in the terminology that Blume et al. (1991a) use in their Definition 5.2. A lexicographic conditional probability system (LCPS) consists of L levels of non-overlapping subjective probability distributions: If λ = (µ1 , . . . , µL ) is an LPS on F and the supports of the µ` ’s are disjoint, then λ is an LCPS on F .

Proposition 3 (Blume et. al, 1991a) The following two statements are equivalent. 1 (a) ºφ satisfies Axioms 1, 2, and 4 00 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {ºφ |φ ∈ Φ} satisfies Axioms 5 and 6. 2 There exist a vNM utility function υ : ∆(Z) → R and a unique LCPS λ on F with support κ that satisfies, for any φ ∈ Φ, pφ ºφ qφ iff ³X ´L|φ ³X µ0` (e)υ(pφ (e)) ≥L e∈φ

`=1

e∈φ

´L|φ µ0` (e)υ(qφ (e)) , `=1

where λ|φ = (µ01 , . . . µ0L|φ ) is the conditional of λ on φ (with the LCPS λ|φ satisfying, ∀` ∈ {1, . . . , L|φ}, suppµ0` = π`0 ). Say that ºφ is conditionally represented by a vNM utility function υ if (a) ºφ is non-trivial and (b) pφ ºφ|{e} qφ iff υ(pφ (e)) ≥ υ(qφ (e)) whenever e is deemed subjectively possible. Under Axioms 10 , 2, 3, 40 , 5, and 6 conditional representation follows directly from the vNM theorem of expected utility representation.

Proposition 4 Assume that (a) ºφ satisfies Axioms 1 0 , 2, and 4 0 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {ºφ |φ ∈ Φ} satisfies Axioms 5 and 6. Then there exists a vNM utility function υ : ∆(Z) → R such that, ∀φ ∈ Φ, pφ ºφ|{e} qφ iff υ(pφ (e)) ≥ υ(qφ (e)) whenever e ∈ κ ∩ φ. Under Axioms 1, 2, 3, 40 , 5, 60 , and 16 we obtain the characterization result of Asheim and Perea (2004). For the statement of this result, we need to introduce the concept of a system of conditional lexicographic probabilities. For this definition,

32

CONSISTENT PREFERENCES

if λ := (µ1 , . . . , µL ) is an LPS and ` ∈ {1, . . . , L}, then write λ` := (µ1 , . . . , µ` ) for the LPS that includes only the ` top levels of the original sequence of probability distributions.

Definition 1 A system of conditional lexicographic probabilities (SCLP) (λ, `) on F with support κ consists of an LPS λ = (µ1 , . . . , µL ) on F with support κ, and a function ` : Φ → {1, . . . , L} satisfying (i) supp λ`(φ) ∩ φ 6= ∅, (ii) `(ε) ≥ `(φ) whenever ∅ = 6 ε ⊆ φ, and (iii) `({e}) ≥ ` whenever e ∈ supp µ` . The interpretation is that the conditional belief on φ is given by the conditional on φ of the LPS λ`(φ) , λ`(φ) |φ = (µ01 , . . . µ0`(φ)|φ ). To determine preference between acts conditional on φ, first calculate expected utilities by means of the top level probability distribution, µ01 , and then, if necessary, use the lower level probability distributions, µ02 , . . . , µ0`(φ)|φ , lexicographically to resolve ties. The function ` thus determines, for every event φ, the number of levels of the original LPS λ that can be used, provided that their supports intersect with φ, to resolve ties between acts conditional on φ. Condition (i) ensures well-defined conditional beliefs that represent nontrivial conditional preferences. Condition (ii) means that the system of conditional preferences is dynamically consistent, in the sense that strict preference between two acts would always be maintained if new information, ruling out states at which the two acts lead to the same outcomes, became available. To motivate condition (iii), note that if e ∈ supp µ` and `({e}) < `, then it follows from condition (ii) that µ` could as well ignore e without changing the conditional beliefs.

Proposition 5 (Asheim and Perea, 2004) The following two statements are equivalent. 1 (a) ºφ satisfies Axioms 1, 2, and 4 0 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {ºφ |φ ∈ Φ} satisfies Axioms 5, 6 0 , and 16 . 2 There exist a vNM utility function υ : ∆(Z) → R and an SCLP (λ, `) on F with support κ that satisfies, for any φ ∈ Φ, pφ ºφ qφ iff ³X ´`(φ)|φ ³X µ0` (e)υ(pφ (e)) ≥L e∈φ

`=1

e∈φ

´`(φ)|φ µ0` (e)υ(qφ (e)) , `=1

33

Decision-theoretic Framework

where λ`(φ) |φ = (µ01 , . . . µ0`(φ)|φ ) is the conditional of λ`(φ) on φ. Proof. 1 implies 2. Since ºφ is trivial if φ ∈ / Φ, we may w.l.o.g. assume that Axiom 16 is satisfied with º∗F |φ being trivial for any φ ∈ / Φ. Consider any e ∈ κ. Since º{e} satisfies Axioms 1, 2, 3, and 40 (implying Axiom 4 since {e} has only one state), it follows from the vNM theorem of expected utility representation that there exists a vNM utility function υ{e} : ∆(Z) → R such that υ{e} represents º{e} . By Axiom 5, we may choose a common vNM utility function υ to represent º{e} for all e ∈ κ. Since Axiom 16 implies, for any e ∈ κ, º∗F |{e} satisfies Axioms 1, 2, 3, and 40 , and furthermore, p Â∗F |{e} q whenever p{e} Â{e} q{e} , we obtain that υ represents º∗F |{e} for all e ∈ κ. It now follows that º∗F satisfies Axiom 5 of Blume et al. (1991a). By Theorem 3.1 of Blume et al. (1991a) º∗F is represented by υ and an LPS λ = (µ1 , . . . , µL ) on F with support κ. Consider any φ ∈ Φ. If pφ Âφ qφ iff p Â∗F |φ q, then pφ ºφ qφ iff ³X

´L|φ

µ0 (e)υ(pφ (e)) e∈φ `

`=1

≥L

³X

µ0 (e)υ(qφ (e)) e∈φ `

´L|φ `=1

,

where λ|φ = (µ01 , . . . µ0L|φ ) is the conditional of λ on φ, implying that we can set `(φ) = L. Otherwise, let `(φ) ∈ {0, . . . , L − 1} be the maximum ` for which it holds that pφ Âφ qφ if ³X e∈φ

´`|φ ³X µ0k (e)υ(pφ (e)) >L

e∈φ

k=1

µ0k (e)υ(qφ (e))

´`|φ k=1

,

where the r.h.s. is never satisfied if ` < min{k |supp λk ∩φ 6= ∅}, entailing that the implication holds for any such `. Define a set of pairs of acts on φ, I, as follows: (pφ , qφ ) ∈ I iff ³X ´`(φ)|φ ³X µ0` (e)υ(pφ (e)) =L e∈φ

`=1

e∈φ

´`(φ)|φ µ0` (e)υ(qφ (e)) , `=1

with (pφ , qφ ) ∈ I for any acts pφ and qφ on φ if `(φ) < min{` |supp λ` ∩φ 6= ∅}. Note that I is a convex set. To show that υ and λ`(φ) |φ represent ºφ , we must establish that pφ ∼φ qφ whenever (pφ , qφ ) ∈ I. Hence, suppose there exists (pφ , qφ ) ∈ I such that pφ Âφ qφ . It follows

34

CONSISTENT PREFERENCES

from the definition of `(φ) and the completeness of ºφ (Axiom 1) that there exists (p0φ , q0φ ) ∈ I such that p0φ ºφ q0φ and X e∈φ

µ`(φ)+1 (e)υ(p0φ (e)) <

X e∈φ

µ`(φ)+1 (e)υ(q0φ (e)) .

Objective independence of ºφ (Axiom 2) now implies that, if 0 < γ < 1, then γpφ + (1 − γ)p0φ Âφ γqφ + (1 − γ)p0φ ºφ γqφ + (1 − γ)q0φ ; hence, by transitivity of ºφ (Axiom 1), γpφ + (1 − γ)p0φ Âφ γqφ + (1 − γ)q0φ .

(3.1)

However, by choosing γ sufficiently small, we have that X µ`(φ)+1 (e)υ(γpφ (e) + (1 − γ)p0φ (e)) e∈φ X < µ`(φ)+1 (e)υ(γqφ (e) + (1 − γ)q0φ (e)) . e∈φ

Since I is convex so that (γpφ + (1 − γ)p0φ , γqφ + (1 − γ)q0φ ) ∈ I, this implies that γp + (1 − γ)p0 ≺∗F |φ γq + (1 − γ)q0 . (3.2) Since (3.1) and (3.2) contradict Axiom 16, this shows that pφ ∼φ qφ whenever (pφ , qφ ) ∈ I. This implies in turn that `(φ) ≥ min{` |supp λ` ∩φ 6= ∅} since ºφ is nontrivial. By Axiom 60 , `(ε) ≥ `(φ) whenever ∅ 6= ε ⊆ φ. Finally, since, υ represents º{e} for all e ∈ κ, it follows that p{e} Â{e} q{e} iff p Â∗F |{e} q. Hence, we can set `({e}) = L, implying `({e}) ≥ ` whenever e ∈ supp µ` . 2 implies 1. This follows from routine arguments. By strengthening Axiom 40 to Axiom 4, we get the following corollary. For the statement of this result, we need to introduce formally the concept of a conditional probability system. A conditional probability system (CPS) consists of a collection of subjective probability distributions: If, for each φ ∈ Φ, µφ is a subjective probability distribution on φ, and {µφ | φ ∈ Φ} satisfies µ² (δ) · µφ (²) = µφ (δ) whenever δ ⊆ ² ⊆ φ and ², φ ∈ Φ, then {µφ | φ ∈ Φ} is a CPS on F with support κ.

35

Decision-theoretic Framework

Corollary 1 The following three statements are equivalent. 1 (a) ºφ satisfies Axioms 1, 2, and 4 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φ, and (b) the system of conditional preferences {ºφ | φ ∈ Φ} satisfies Axioms 5, 6 0 , and 16 . 2 There exist a vNM utility function υ : ∆(Z) → R and a unique LCPS λ = (µ1 , . . . , µL ) on F with support κ that satisfies, for any φ ∈ Φ, X X µφ (e)υ(pφ (e)) ≥ µφ (e)υ(qφ (e)) , pφ ºφ qφ iff e∈φ

e∈φ

where µφ is the conditional of µ`(φ) on φ and `(φ) = min{`| suppλ` ∩ φ 6= ∅}. 3 There exist a vNM utility function υ : ∆(Z) → R and a unique CPS {µφ | φ ∈ Φ} on F with support κ that satisfies, for any φ ∈ Φ, X X pφ ºφ qφ iff µφ (e)υ(pφ (e)) ≥ µφ (e)υ(qφ (e)) . e∈φ

e∈φ

Proof. 1 implies 2. By Proposition 5, the system of conditional preferences is represented by an SCLP (λ, `) on F with support κ. By the strengthening Axiom 40 to Axiom 4, it follows from the representation result of Anscombe and Aumann (1963) that only the top level probability distribution is needed to represent each conditional preferences; i.e., for any φ ∈ Φ, `(φ) = min{`| suppλ` ∩ φ 6= ∅}. This implies that any overlapping supports in λ can be removed without changing, for any φ ∈ Φ, the conditional of λ`(φ) on φ, turning λ into an LCPS. Furthermore, the LCPS thus determined is unique. 2 implies 1. This follows from routine arguments. 2 implies 3. {µφ | φ ∈ Φ} is a CPS on F with support κ since µ² (δ) · µφ (²) = µφ (δ) is satisfied whenever δ ⊆ ² ⊆ φ and ², φ ∈ Φ. If an alternative CPS {˜ µφ | φ ∈ Φ} were to satisfy, for any φ ∈ Φ, X X pφ ºφ qφ iff µ ˜φ (e)υ(pφ (e)) ≥ µ ˜φ (e)υ(qφ (e)) , e∈φ

e∈φ

˜ = (˜ then one could construct an alternative LCPS λ µ1 , . . . , µ ˜L ) such ˜ that, for any φ ∈ Φ, µ ˜φ is the conditional of µ ˜`(φ) on φ, where `(φ) := ˜ min{`| supp˜ µ` ∩ φ 6= ∅}, contradicting the uniqueness of λ. 3 implies 2. Construct the LCPS λ = (µ1 , . . . , µL ) by the following algorithm: (i) µ1 = µF , (ii) ∀` ∈ {2, . . . , L}, µ` = µφ , where φ = a L F \∪`−1 k=1 suppµk 6= F \κ, and (iii) ∪k=1 suppµk = κ. Then, for any φ ∈ Φ ,

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µφ is the conditional of µ`(φ) on φ, where `(φ) := min{`| suppµ` ∩φ 6= ∅}, and λ is the only LCPS having this property. A full support SCLP (i.e., an SCLP where κ = F ) combines the structural implication of a full support LPS—namely that conditional preferences are nontrivial—with flexibility w.r.t. whether to assume the behavioral implication of any conditional of such an LPS—namely that the conditional LPS’s full support induces preference for cautious behavior. A full support SCLP is a generalization of both (1) conditional beliefs described by a single full support LPS λ = (µ1 , . . . , µL ) (cf. Proposition 2): Let, for all φ ∈ Φ, `(φ) = L. Then the conditional belief on φ is described by the conditional of λ on φ, λ|φ . (2) conditional beliefs described by a CPS (cf. Corollary 1): Let, for all φ ∈ Φ, `(φ) = min{`| suppλ` ∩ φ 6= ∅}. Then, it follows from conditions (ii) and (iii) of Definition 1 that the full support LPS λ = (µ1 , . . . , µL ) has non-overlapping supports—i.e., λ is an LCPS— and the conditional belief on φ is described by the top level probability distribution of the conditional of λ on φ. This corresponds to the isomorphism between CPS and LCPS noted by Blume et al. (1991a) on p. 72 and discussed by Hammond (1994) and Halpern (2003). However, a full support SCLP may describe a system of conditional beliefs that is not covered by these special cases. The following is a simple example: Let κ = F = {d, e, f } and λ = (µ1 , µ2 ), where µ1 (d) = 1/2, µ1 (e) = 1/2, and µ2 (f ) = 1. If `(F ) = 1 and `(φ) = 2 for any other non-empty subset φ, then the resulting SCLP falls outside cases (1) and (2).

Chapter 4 BELIEF OPERATORS

Belief operators play an important role in epistemic analyses of games. For any event, a belief operator determines the set of states where this event is (in some precise sense) believed. Belief operators may satisfy different kinds of properties, like if one event implies another, then belief of the former implies belief of the latter (monotonicity), if two events are believed, then the conjunction is also believed, an event that is always true is always believed, an event that is never true is never believed, if an event is believed, then the event that the event is believed is also believed (positive introspection), and if an event is not believed, then the event that the event is not believed is believed (negative introspection). Belief operators satisfying this list are called KD45 operators.1 In epistemic analyses of games, it is common to derive belief operators from preferences, leading to what can be called subjective belief operators. Examples of subjective KD45 operators are ‘belief with probability one’, as used by, e.g., Tan and Werlang (1988), ‘belief with primary probability one’, as used by Brandenburger (1992), and ‘conditional belief with probability one’, as used by Ben-Porath (1997). More recently, 1A

KD45 operator satisfies that belief of an event implies that the complement is not believed, but need not satisfy the truth axiom—i.e. that a believed event is always true.

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CONSISTENT PREFERENCES

Brandenburger and Keisler (2002), Battigalli and Siniscalchi (2002) and Asheim and Dufwenberg (2003a) have proposed non-monotonic subjective belief operators called ‘assumption’, ‘strong belief’ and ‘full belief’, respectively. With the exception of Asheim and Dufwenberg’s (2003a) ‘full belief’, these operators have in common that they are based on subjective probabilities—arising from a probability distribution, a lexicographic probability system, or a conditional probability system—that represent the preferences of the player as a decision maker. An alternative approach to belief operators, applied by e.g. Stalnaker (1996, 1998), is to define belief operators by means of accessibility relations, as used in modal logic. Of particular interest is Stalnaker’s non-monotonic ‘absolutely robust belief’ operator. Reproducing joint work with Ylva Søvik—Asheim and Søvik (2004)— this chapter integrates these two approaches by showing how accessibility relations can be derived from preferences and in turn be used to define and characterize belief operators; see Figure 4.1 for an illustration of the basic structure of the analysis in this chapter. These belief operators will in later chapters be used in the epistemic analysis. Morris (1997) observes that it is unnecessary to go via subjective probabilities to derive subjective belief operators from the preferences of a decision maker. This suggestion has been followed in Asheim (2002) and Asheim and Dufwenberg (2003a), the content of which will be reproduced in Chapters 7 and 11. Epistemic conditions for backward induction are provided in Chapter 7 without the use of subjective probabilities (since one can argue that subjective probabilities play no role in the backward induction argument), while Chapter 11 promotes forward induction within a structure based on incomplete preferences that cannot be represented by subjective probabilities. When deriving belief operators from preferences, it is essential that the preferences determine ‘subjective possibility’ (so that it can be determined whether an event is subjectively impossible) as well as ‘epistemic priority’ (so that one allows for non-trivial belief revision). As we shall see, preferences need not satisfy completeness in order to determine ‘subjective possibility’ and ‘epistemic priority’. This chapter shows how belief operators corresponding to those used in the literature can be derived from preferences that need not be complete. We assume that preferences satisfy Axioms 10 , 11, 2, 3, 40 , 5, and 6, entailing that preferences are (possibly) incomplete, but allow conditional representation (cf. Proposition 4 of Chapter 3). Following the structure illustrated in Figure 4.1, Section 4.1 shows how a binary acces-

39

Belief operators

Preferences over acts (functions from states to randomized outcomes)

. & ‘Infinitely more likely’ ‘Admissibility’ ↓ ↓ Q Accessibility relation (R1 , . . . , RL ) Vector of nested of epistemic priority accessibility relations .characterizes defines& Belief operators s certain belief s conditional belief s robust belief Figure 4.1.

The basic structure of the analysis in Chapter 4.

sibility relation of epistemic priority Q can be derived from preferences satisfying these axioms, by means of the ‘infinitely-more-likely’ relation. The properties of this priority relation are similar to but more general than those found, e.g., in Lamerre and Shoham (1994) and Stalnaker (1996, 1998) in that reflexivity of Q is not required.2 Furthermore, it is shown how preferences through “admissibility” give rise to a vector of nested binary accessibility relations (R1 , . . . , RL ), where, for each `, R` fulfills the usual properties of Kripke representations of beliefs; i.e., they are serial, transitive and Euclidean. Finally, we establish that the two kinds of accessibility relations yield two equivalent representations of ‘subjective possibility’ and ‘epistemic priority’. In Section 4.2 we first use the accessibility relation of epistemic priority Q to define the following belief operators: Certain belief coincides with what Morris (1997) calls ‘Savage-belief’ and means that the complement of the event is subjectively impossible. Conditional belief generalizes ‘conditional belief with probability one’. 2 The

term ‘epistemic priority’ will here be used to refer to what elsewhere is sometimes referred to as ‘plausibility’ or ‘prejudice’; see, e.g., Friedman and Halpern (1995) and Lamerre and Shoham (1994). This is similar to ‘preference’ among states (or worlds) in nonmonotonic logic—cf. Shoham (1988)—leading agents towards some states and away from others. In contrast, we use the term ‘preferences’ in the decision-theoretic sense of a binary relation on the set of functions (‘acts’) from states to outcomes.

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Robust belief coincides with what Stalnaker (1998) calls ‘absolutely robust belief’. We then show how these operators can be characterized by means of the vector of nested binary accessibility relations (R1 , . . . , RL ), thereby showing that the concept of ‘full belief’ as used by Asheim and Dufwenberg (2003a) coincides with robust belief. Section 4.3 establishes properties of these belief operators. In particular, the robust belief operator (while poorly behaved) is bounded by certain and conditional belief, which are KD45 operators. Section 4.4 shows how the characterization of robust belief corresponds to the concept of ‘assumption’ as used by Brandenburger and Keisler (2002), and observes how the definition of robust belief is related to the concept of ‘strong belief’ as used by Battigalli and Siniscalchi (2002). We thereby reconcile and compare these non-standard notions of belief which have recently been used in epistemic analyses of games. The proofs of the results in this chapter are included in Appendix A.

4.1

From preferences to accessibility relations

The purpose of this section is to show how two different kinds of accessibility relations—see, e.g., Lamerre and Shoham (1994) and Stalnaker (1996, 1998)—can be derived from preferences. Consider the decision-theoretic framework of Chapter 3. However, as motivated below, assume that the decision maker’s preferences may vary between states. Hence, denote by ºdφ the preferences over acts on φ at state d, and use superscript d throughout in a similar manner. Assume that, for each d ∈ F , (a) ºdφ satisfies Axioms 10 , 2, and 40 if φ ∈ 2F \{∅}, and Axiom 3 if and only if φ ∈ Φd (recalling from Chapter 3 that Φd denotes {φ ∈ 2F \{∅}| κd ∩ φ 6= ∅}), and (b) the system of conditional preferences {ºdφ |φ ∈ Φd } satisfies Axioms 5, 6, and 11. In view of Axiom 6 we simplify notation and write p ºdφ q instead of p ºdF |φ q ⇔ pφ ºdφ qφ , and simplify further by substituting ºd for ºdF . By Proposition 4, ºd is conditionally represented: There exist a vNM utility function υ d : ∆(Z) → R such that p ºd{e} q iff υ d (p(e)) ≥ υ d (q(e)) whenever e ∈ κd . If E ⊆ F , say that pE weakly dominates qE at d if, ∀e ∈ E, υ d (pE (e)) ≥ υ d (qE (e)), with strict inequality for some f ∈ E. Say that ºd is

Belief operators

41

admissible on E if E is non-empty and p Âd q whenever pE weakly dominates qE at d. The following connection between admissibility on subsets and the infinitely-more-likely relation is important for relating the two kinds of accessibility relations derived from preferences below; the one kind is based on the infinitely-more-likely relation, while the other is based on admissibility on subsets. Write ¬E for F \E.

Proposition 6 Let E 6= ∅ and ¬E 6= ∅. ºd is admissible on E iff e ∈ E and f ∈ ¬E imply e Àd f . An epistemic model. In a semantic formulation of belief operators one can, following Aumann (1999), start with an information partition of F , and then assume that the decision maker, for each element of the partition, is endowed with a probability distribution that is concentrated on this element of the partition. Since all states within one element of the partition are indistinguishable, they are assigned the same probability distribution, which however differ from the probability distributions assigned to states outside this element. In particular, probability distributions assigned to two states in different elements of the partition have disjoint supports. Hence, in Aumann’s (1999) formulation, the decision maker’s probability distribution depends on in which element of the information partition the true state is. This is consistent with the approach chosen here, where the probability distribution—or more generally, the preferences—of the decision maker will be different for states in different elements of the information partition, and be the same for all states within the same element. However, in line with our subjective perspective, we will construct the information partition from the preferences of the decision maker, so that each element of the partition is defined as a maximal set of states where the decision maker’s preferences are the same, having the interpretation that states within this set are indistinguishable. Moreover, Aumann’s (1999) assumption that the probability distribution is concentrated within the corresponding element of the partition will in our framework be captured by the property that all states outside (and possibly some states inside) the element are deemed subjectively impossible. Thus, for each d ∈ F , let τ d := {e ∈ F | p ºe q iff p ºd q} be the set of states that are subjectively distinguishable, and write d ≈ e if e ∈ τ d . Note that ≈ is a reflexive, transitive, and symmetric binary relation; i.e., ≈ is an equivalence relation that partitions F into equivalence classes (or “types”).

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Moreover, κd denotes the set of states that are subjectively possible (i.e., not Savage-null) at d. In line with the above discussion, assume that, for each d ∈ F , κd ⊆ τ d . This assumption will ensure that the preference-based operators satisfy positive and negative introspection; it corresponds to “being aware of one’s own type”. Refer to the collection {ºd | d ∈ F } as an epistemic model for the decision maker. In view of Axiom 6, it holds that p ºdφ q ⇔ p ºd q whenever κd ⊆ φ ⊂ F ; in particular, p ºdτd q ⇔ p ºd q. The interpretation is the decision maker’s preferences at d are not changed by ruling out states that he can distinguish from the true state at d. Hence, we can adopt an interim perspective where the decision maker has already become aware of his own preferences (his own “type”); in particular, the decision maker’s unconditional preferences are not obtained by conditioning “ex ante preferences” on his type. Accessibility relation of epistemic priority. Consider the following definition of the accessibility relation Q.

Definition 2 dQe (“d does not have higher epistemic priority than e”) if (1) d ≈ e, (2) e is not Savage-null at d, and (3) d is not deemed infinitely more likely than e at d. Proposition 7 The relation Q is serial,3 transitive, and satisfies forward linearity4 and quasi-backward linearity.5 A vector of nested accessibility relations. Consider the collection of all sets E satisfying that ºd is admissible on E. Since ºd is admissible on κd , it follows that the collection is non-empty as it is contains κd . Also, since no e ∈ E is Savage-null at d if ºd is admissible on E, it follows that any set in this collection is a subset of κd . Finally, since e Àd f implies that f Àd e does not hold, it follows from Proposition 6 that E 0 ⊆ E 00 or E 00 ⊆ E 0 if ºd is admissible on both E 0 and E 00 , implying that the sets in the collection are nested. Hence, there exists a vector of nested sets, (ρd1 , . . . , ρdLd ), on which ºd is admissible, satisfying: ∅ 6= ρd1 ⊂ · · · ⊂ ρd` ⊂ · · · ⊂ ρdLd = κd ⊆ τ d (where ⊂ denotes ⊆ and 6=). 3 ∀d,

∃e such that dQe. and dQf imply eQf or f Qe. 5 If ∃d0 ∈ F such that d0 Qe, then dQf and eQf imply dQe or eQd. 4 dQe

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If we assume that ºd satisfies not only Axiom 10 but also Axiom 1, so that, as reported in Proposition 2, ºd is represented by υ d and an LPS, λd = (µd1 , . . . , µdLd )—i.e., a sequence of Ld levels of subjective probability distributions—then (ρd1 , . . . , ρdLd ) can in an obvious way be derived from the supports of these probability distributions: ∀` ∈ {1, . . . , Ld },

ρd` =

[` k=1

suppµdk .

McLennan (1989a) develops an ordering of κd that is related to (ρd1 , . . . , ρdLd ) in a context where a system of conditional probabilities is taken as primitive. In a similar context, van Fraassen (1976) and Arl´o-Costa and Parith (2003) propose a concept of (belief/probability) cores that correspond to the sets ρd1 , . . . , ρdLd . Grove (1988) spheres and Spohn’s (1988) ordinal conditional functions are also related to these sets. For d ∈ F with Ld < L := maxe∈F Le , let ρdLd = ρd` = κd for ` ∈ {Ld + 1, . . . , L}. The collection of sets, {ρd` | d ∈ F }, defines an accessibility relation, R` .

Definition 3 dR` e (“at d, e is deemed possible at the epistemic level `”) if e ∈ ρd` . Proposition 8 The vector of relations, (R1 , . . . , RL ), has the following properties: For each ` ∈ {1, . . . , L}, R` is serial, transitive, and Euclidean.6 For each ` ∈ {1, . . . , L − 1}, (i) dR` e implies dR`+1 e and (ii) (∃f such that dR`+1 f and eR`+1 f ) implies (∃f 0 such that dR` f 0 and eR` f 0 ). The correspondence between Q and (R1 , . . . , RL ). That d is not Savage-null at d can be interpreted as d being deemed subjectively possible (at some epistemic level) at any state in the same equivalence class. By part (i) of the following result, d being not Savage-null at d has two equivalent representations in terms of accessibility relations: dQd and dRL d. Likewise, e Àd d can be interpreted as e having higher epistemic priority than d. By part (ii) of the following result, e Àd d have two equivalent representations: (dQe and not eQd) and (∃` ∈ {1, . . . , L} such that dR` e and not eR` d). Thus, both Q and (R1 , . . . , RL ) capture ‘subjective possibility’ and ‘epistemic priority’ as implied by the preferences of the preference system.

6 dR e `

and dR` f imply eR` f .

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Proposition 9 (i) dQd iff dRL d. (ii) (dQe and not eQd) iff (∃` ∈ {1, . . . , L} such that dR` e and not eR` d). If Axiom 4 is substituted for Axiom 40 —so that the conditional Archimedean property is strengthened to the Archimedean property—then e being deemed infinitely more likely than f at d implies that f is Savagenull. Hence, L = 1, and by Definitions 2 and 3, Q = R1 . Hence, we are left with a unique serial, transitive, and Euclidean accessibility relation if preferences are continuous.

4.2

Defining and characterizing belief operators

In line with the basic structure illustrated in Figure 4.1, we now use the accessibility relations of Section 4.1 to define and characterize belief operators. Defining certain, conditional, and robust belief. Consider the accessibility relation of epistemic priority, Q, having the properties of Proposition 7. In Asheim and Søvik (2003) we show how equivalence classes can be derived from Q with the properties of Proposition 7, implying that Q with such properties suffices for defining the belief operators. In particular, we show that the set of states that are subjective indistinguishable at d is given by τ d = {e ∈ F | ∃f ∈ F such that dQf and eQf } , and the set of states that are deemed subjectively possible at d equals κd = {e ∈ τ d | ∃f ∈ F such that f Qe} = {e ∈ τ d | eQe} , where κd 6= ∅ since Q is serial, and where the last equality follows since, by quasi-backward linearity, eQe if f Qe. Define ‘certain belief’ as follows.

Definition 4 At d the decision maker certainly believes E if d ∈ KE, where KE := {e ∈ F |κe ⊆ E}. Hence, at d an event E is certainly believed if the complement is deemed subjectively impossible at d. This coincides with what Morris (1997) calls ‘Savage-belief’. ‘Conditional belief’ is defined conditionally on sets that are subjectively possible at any state; i.e., sets in the following collection: \ Φ := Φd , where ∀d ∈ F, Φd = {φ ∈ 2F \{∅}| κd ∩ φ 6= ∅} . d∈F

45

Belief operators

Hence, a non-empty set φ is not in Φ if and only if there exists d ∈ F such that κd ∩ φ = ∅. Note that F ∈ Φ and, ∀φ ∈ Φ, ∅ 6= φ ⊆ F . Since every φ ∈ Φ is subjectively possible at any state, it follows that, ∀φ ∈ Φ, β d (φ) := {e ∈ τ d ∩ φ|∀f ∈ τ d ∩ φ, f Qe} is non-empty, as demonstrated by the following lemma.

Lemma 3 If κd ∩ φ 6= ∅, then ∃e ∈ τ d ∩ φ such that ∀f ∈ τ d ∩ φ, f Qe. Define ‘conditional belief’ as follows.

Definition 5 At d the decision maker believes E conditional on φ if d ∈ B(φ)E, where B(φ)E := {e ∈ F |β e (φ) ⊆ E}. Hence, at d an event E is believed conditional on φ if E contains any state in τ d ∩ φ with at least as high epistemic priority as any other state in τ d ∩ φ. This way of defining conditional belief is in the tradition of, e.g., Grove (1988), Boutilier (1994), and Lamerre and Shoham (1994). Let ΦE be the collection of subjectively possible events φ having the property that E is subjectively possible conditional on φ whenever E is subjectively possible: \ ΦE := ΦdE , where ∀d ∈ F, d∈F

ΦdE := {φ ∈ Φd | E ∩ κd ∩ φ 6= ∅ if E ∩ κd 6= ∅ } . Hence, a non-empty set φ is not in ΦE if and only if (1) there exists d ∈ F such that κd ∩ φ = ∅ or (2) there exists d ∈ F such that E ∩ κd 6= ∅ and E ∩ κd ∩ φ = ∅. Note that ΦE is a subset of Φ that satisfies F ∈ ΦE ; hence, ∅ 6= ΦE ⊆ Φ. Define ‘robust belief’ as follows.

Definition 6 At d the decision maker robustly believes E if d ∈ B0 E, where B0 E := ∩φ∈ΦE B(φ)E. Hence, at d an event E is robustly believed in the following sense: E is believed conditional on any event φ that does not make E subjectively impossible. Indeed, B 0 coincides with what Stalnaker (1998) calls ‘absolutely robust belief’ when we specialize to his setting where Q is also reflexive. The relation between this belief operator and the operators ‘full belief’, ‘assumption’, and ‘strong belief’, introduced by Asheim and Dufwenberg (2003a), Brandenburger and Keisler (2002), and Battigalli and Siniscalchi (2002), respectively, will be discussed at the end of this section as well as in Section 4.4.

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Characterizing certain, conditional, and robust belief. Consider the vector of nested accessibility relations (R1 , . . . , RL ) having the properties of Proposition 8 and being related to Q as in Proposition 9. In Asheim and Søvik (2003) we first derive (R1 , . . . , RL ) from Q and then show how (R1 , . . . , RL ) characterizes the belief operators. In particular, it holds for any ` ∈ {1, . . . , L} that τ d = {e ∈ F | ∃f ∈ F such that dR` f and eR` f } , and ρd` = {e ∈ F | dR` e} . Furthermore, κd = {e ∈ τ d | eRL e} = {e ∈ F | dRL e} . The latter observations yield a characterization of certain belief.

Proposition 10 KE = {d ∈ F | ρdL ⊆ E}. Proposition 10 entails that certain belief as defined in Definition 4 corresponds to what Arl´o-Costa and Parith (2003) call ‘full belief’. Furthermore, by the next result, (unconditional) belief, B(F ), corresponds to what van Fraassen (1995) calls ‘full belief’.

Proposition 11 ∀φ ∈ Φ, B(φ)E = {d ∈ F | ∃` ∈ {1, . . . , L} such that ∅ 6= ρd` ∩ φ ⊆ E}. Finally, by Proposition 9(ii) and the following result, E is robustly believed iff any subjectively possible state in E has higher epistemic priority than any state in the same equivalence class outside E.

Proposition 12 B0 E = {d ∈ F | ∃` ∈ {1, . . . , L} such that ρd` = E ∩ κd }. Asheim and Dufwenberg (2003a) say that an event A is ‘fully believed’ at a if the preferences at a are admissible on the set of states in A that are deemed subjectively possible at a. It follows from Proposition 12 that this coincides with robust belief as defined in Definition 6.

4.3

Properties of belief operators

The present section presents some properties of certain, conditional, and robust belief operators. We do not seek to establish sound and complete axiomatic systems for these operators; this should, however, be

47

Belief operators

standard for the certain and conditional belief operators, while harder to establish for the robust belief operator. Rather, our main goal is to show how the non-monotonic (and thus poorly behaved) robust belief operator is bounded by the two KD45 operators certain and conditional belief. While the results certain belief and conditional belief are included as a background for the results on robust belief, the latter findings in combination with the results of Sections 4.2 and 4.4 shed light on the non-standard notions of belief recently used in epistemic analyses of games. Properties of certain and conditional belief. Note that certain belief implies conditional belief since, by Definitions 4 and 5, β d (φ) ⊆ κd ∩ φ.

Proposition 13 For any φ ∈ Φ, KE ⊆ B(φ)E. Furthermore, combined with Proposition 13 the following result implies that both operators K and B(φ) correspond to KD45 systems.

Proposition 14 For any φ ∈ Φ, the following properties hold: KE ∩ KE 0 = K(E ∩ E 0 )

B(φ)E ∩ B(φ)E 0 = B(φ)(E ∩ E 0 )

KF = F

B(φ)∅ = ∅

KE ⊆ KKE

B(φ)E ⊆ KB(φ)E

¬KE ⊆ K(¬KE)

¬B(φ)E ⊆ K(¬B(φ)E).

Note that K∅ = ∅, B(φ)F = F , B(φ)E ⊆ B(φ)B(φ)E and ¬B(φ)E ⊆ B(φ)(¬B(φ)E) follow from Proposition 14 since KE ⊆ B(φ)E. Since an event can be certainly believed even though the true state is an element of the complement of the event, it follows that neither certain belief nor conditional belief satisfies the truth axiom (i.e. KE ⊆ E and B(φ)E ⊆ E need not hold). Belief revision. Conditional belief satisfies the usual properties for belief revision as given by Stalnaker (1998); see also Alchourr´ on et al. d (1985). To show this we must define the set, β , that determines the decision maker’s unconditional belief at the state d: β d := {e ∈ τ d | ∀f ∈ τ d , f Qe} , i.e. β d = β d (F ). Then the following result can be established.

Proposition 15 1 β d (φ) ⊆ φ.

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2 If β d ∩ φ 6= ∅, then β d (φ) = β d ∩ φ. 3 If φ ∈ Φ, then β d (φ) 6= ∅. 4 If β d (φ) ∩ φ0 6= ∅, then β d (φ ∩ φ0 ) = β d (φ) ∩ φ0 . Properties of robust belief. It is easy to show that certain belief implies robust belief, which in turn implies (unconditional) belief.

Proposition 16 KE ⊆ B0 E ⊆ B(F )E. Even though robust belief is thus bounded by two KD45 operators, robust belief is not itself a KD45 operator.

Proposition 17 The following properties hold: B0 E ∩ B0 E 0 ⊆ B0 (E ∩ E 0 ) B0 E ⊆ KB0 E ¬B0 E ⊆ K(¬B0 E). Note that B0 ∅ = ∅, B0 F = F , B0 E ⊆ B0 B0 E and ¬B0 E ⊆ B0 (¬B0 E) follow from Propositions 14 and 17 since KE ⊆ B0 E ⊆ B(F )E. However, even though the operator B 0 satisfies B 0 E ⊆ ¬B 0 ¬E as well as positive and negative introspection, it does not satisfy monotonicity since E ⊆ E 0 does not imply B 0 E ⊆ B 0 E 0 . To see this let ρd1 = {d} and ρd2 = κd = {d, e, f } for some d ∈ F . Now let E = {d} and E 0 = {d, e}. Clearly, E ⊆ E 0 , and since ρd1 = E ∩ κd we have d ∈ B 0 E. However, since neither ρd1 = E 0 ∩ κd nor ρd2 = E 0 ∩ κd , d ∈ / B0E0.

4.4

Relation to other non-monotonic operators

The purpose of this section is to show how robust belief corresponds to the ‘assumption’ operator of Brandenburger and Keisler (2002) and is related to the ‘strong belief’ operator of Battigalli and Siniscalchi (2002). The ‘assumption’ operator. Brandenburger and Keisler (2002) consider an epistemic model which is more general than the one that we consider in Section 4.1, since the set of states need not be finite, and is more special than ours, since, for all d ∈ F , Axioms 10 , 11, and 40 are strengthened to Axioms 1 and 400 , so that completeness and the partitional Archimedean property are substituted for conditional completeness, partitional priority, and the conditional Archimedean property.

Belief operators

49

Within our setting with a finite set of states, F , it now follows, as reported in Proposition 3, that ºd is represented by υ d and an LCPS, λd = (µd1 , . . . , µdLd )—i.e., a sequence of Ld levels of non-overlapping subjective probability distributions. Hence, ∀` ∈ {1, . . . , Ld }, suppµd` = π`d , where (π1d , . . . , πLd d ) is a partition of κd . In their Appendix B, Brandenburger and Keisler (2002) employ an LCPS to represent preferences in their setting with an infinite set of states. Provided that completeness and the partitional Archimedean property are satisfied, Brandenburger and Keisler (2002) introduce the following belief operator in their Definition B1; see also Brandenburger and Friedenberg (2003).

Definition 7 (Brandenburger and Keisler, 2002) At d the decision maker assumes E if ºdE is nontrivial and p ÂdE q implies p Âd q. Proposition 18 Assume that ºd satisfies Axioms 1 and 4 00 (in addition to the assumptions made in Section 4.1). Then E is assumed at d iff d ∈ B 0 E. Proposition 18 shows that the ‘assumption’ operator coincides with robust belief (and thus with Stalnaker’s ‘absolutely robust belief’) under completeness and the partitional Archimedean property. However, if the partitional Archimedean property is weakened to the conditional Archimedean property, then this equivalence is not obtained. To see this, let κd = {d, e, f }, and let the preferences ºd , in addition to the properties listed in Section 4.1, also satisfy completeness. It then follows from Proposition 2 that ºa is represented by υ a and a LPS—i.e., a sequence of subjective probability distributions with possibly overlapping supports. Consider the example provided by Blume et al. (1991a) in their Section 5 of a two-level LPS, where the primary probability distribution, µd1 , is given by µd1 (d) = 1/2 and µd1 (e) = 1/2, and the secondary probability distribution, µd2 , used to resolve ties, is given by µd2 (d) = 1/2 and µd2 (f ) = 1/2. Consider the acts p and q, where υ d (p(d)) = 2, υ d (p(e)) = 0, and υ d (p(f )) = 0, and where υ d (q(d)) = 1, υ d (q(e)) = 1, and υ d (q(f )) = 2. Even though ºd is admissible on {d, e}, and thus {d, e} is robustly believed at d, it follows that {d, e} is not ‘assumed’ at d since pÂd{d,e} q while p≺d q . Brandenburger and Keisler (2002) do not indicate that their definition —as stated in Definition 7—should be used outside the realm of preferences that satisfy the partitional Archimedean property. Hence, our

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definition of robust belief—combined with the characterization result of Proposition 12 and its interpretation in term of admissibility—yields a preference-based generalization of Brandenburger and Keisler (2002) operator (in our setting with a finite set of states) to preferences that need only satisfy the properties of Section 4.1. The ‘strong belief ’ operator. In the setting of extensive form games, Battigalli and Siniscalchi (2002) have suggested a non-monotonic ‘strong belief’ operator. We now show how their ‘strong belief’ operator is related to robust belief, and thereby, to ‘absolutely robust belief’ of Stalnaker (1998), ‘full belief’ of Asheim and Dufwenberg (2003a), and ‘assumption’ of Brandenburger and Keisler (2002). Battigalli and Siniscalchi (2002) base their ‘strong belief’ operator on a conditional belief operator derived from an epistemic model where, at each state d ∈ F , the decision maker is endowed with a system of conditional preferences {ºdφ | φ ∈ Φd } (with, as before, Φd denoting {φ ∈ 2F \{∅}| κd ∩ φ 6= ∅}). However, Battigalli and Siniscalchi (2002) assume that, if the true state is d, then the decision maker’s system of conditional preferences is represented by υ d and a CPS {µdφ | φ ∈ Φd }. Since a CPS does not satisfy conditionality as specified by Axiom 6, we must embed their conditional belief operator in the framework of the present chapter. We can do so using Corollary 1 of Chapter 3. One the one hand, Battigalli and Siniscalchi (2002) and Ben-Porath (1997) define ‘conditional belief with probability one’ in the following way: At d the decision maker believes E conditional on φ ∈ Φ if suppµdφ ⊆ E, where {µdφ | φ ∈ Φd } is a CPS on F with support κd . On the other hand, according to Definition 5 of the present chapter, at d the decision maker believes E conditional on φ ∈ Φ if β d (φ) ⊆ E. If, however, Axioms 10 , 40 , and 11 are strengthened to Axioms 1 and 400 , so that by Proposition 3 ºd is represented by υ d and an LCPS, λd = (µd1 , . . . , µdLd ), on F with support κd , then Lemma 14 of Appendix A implies that β d (φ) = suppµd` ∩ φ, where ` := min{k| suppµdk ∩ φ 6= ∅}. Hence, by Corollary 1, ‘conditional belief with probability one’ as defined by Battigalli and Siniscalchi (2002) and Ben-Porath (1997) is isomorphic to the conditional belief operator B(φ) derived from an epistemic model satisfying the assumptions of Section 4.1 of the present chapter. Given that the conditional belief operator of Battigalli and Siniscalchi (2002) thus coincides with the B(φ) operator of the present paper, we

Belief operators

51

can define their ‘strong belief’ operator as follows: Let ΦH (⊆ Φ) be some non-empty subcollection of the collection of subsets that are subjectively possible at any state; e.g., in an extensive game ΦH may consist of the subsets that correspond to subgames. Then ΦH ∩ ΦE is the collection of subsets φ satisfying φ ∈ ΦH and having the property that E is subjectively possible conditional on φ whenever E is subjectively possible.

Definition 8 (Battigalli and Siniscalchi, 2002) At d the deciT sion maker strongly believes E if d ∈ ΦH ∩ΦE B(φ). Hence, at d an event E is strongly believed if E is robustly believed in the following sense: E is believed conditional on any subset φ in ΦH that does not make E subjectively impossible. Since ΦE ⊇ ΦH ∩ ΦE ⊇ {F }, it follows that the ‘strong belief’ operator is bounded by the robust belief and (unconditional) belief operators.

Proposition 19 If d ∈ B 0 (E), then E is strongly believed at d. If E is strongly believed at d, then d ∈ B(F )E. As suggested by Battigalli and Bonanno (1999), the ‘strong belief’ operator may also be defined w.r.t. other subcollections of Φ than the collection of subsets that correspond to subgames, and may be seen as a generalization of robust belief by not necessarily requiring belief to be “absolutely robust” in the sense of Stalnaker (1998). However, provided that F is included, Proposition 19 still holds. In any case, the ‘strong belief’ operator shares the properties of robust belief: Also ‘strong belief’ satisfies the properties of Proposition 17, but is not monotonic.

Chapter 5 BASIC CHARACTERIZATIONS

In this chapter we present characterizations of basic game-theoretic concepts. After presenting the concept of an epistemic model of a strategic game form in Section 5.1, we turn to the characterizations of Nash equilibrium and rationalizability in Section 5.2 and characterizations of (strategic form) perfect equilibrium and permissibility in the final Section 5.3. The characterizations of Nash equilibrium and rationalizability will be done by means of the event that each player has preferences that are consistent with the game and the preferences of the opponent. Likewise, the characterizations of (strategic form) perfect equilibrium and permissibility will be done by means of the event that each player has preferences that are admissibly consistent with the game and the preferences of the opponent. Hence, the chapter illustrates the ‘consistent preferences’ approach and set the stage for the analysis of subsequent chapters. Note that the results of this chapter are variants of results that can be found in the literature. In particular, the characterizations of Nash equilibrium and (strategic form) perfect equilibrium are variants of Propositions 3 and 4 of Blume et al. (1991b).

5.1

Epistemic modeling of strategic games

The purpose of this section is to present a framework for strategic games where each player is modeled as a decision maker under uncertainty. The analysis builds on the two previous chapters and introduces

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the concept of an epistemic model for a strategic game form. In this chapter preferences are assumed to be complete, an assumption that will be relaxed in Chapter 6. A strategic game form. Denote by Si player i’s finite set of pure strategies, and let z : S → Z map strategy profiles into outcomes, where S = S1 × S2 is the set of strategy profiles and Z is the finite set of outcomes. Then (S1 , S2 , z) is a finite strategic two-player game form. An epistemic model. For each player i, any of i’s strategies is an act from strategy choices of his opponent j to outcomes. The uncertainty faced by a player i in a strategic game form concerns (a) j’s strategy choice, (b) j’s preferences over acts from i’s strategy choices to outcomes, and so on (cf. the discussion in Section 1.3). A type of a player i corresponds to (a) preferences over acts from j’s strategy choices, (b) preferences over acts from j’s preferences over acts from i’s strategy choices, and so on. For any player i, i’s decision is to choose one of his own strategies. As the player is not uncertain of his own choice, the player’s preferences over acts from his own strategy choices is not relevant and can be ignored. Hence, in line with the discussion in Section 1.3, consider an implicit model—with a finite type set Ti for each player i—where the preferences of a player corresponds to the player’s type, and where the preferences of the player are over acts from the opponent’s strategy-type pairs to outcomes. If we let each player be aware of his own type (as we will assume throughout), this leads to an epistemic model where the state space of player i is Ti × Sj × Tj , and where, for each ti ∈ Ti , {ti } × Sj × Tj constitutes an equivalence class, being the set of states that are indistinguishable for player i at ti , and a non-empty subset of {ti } × Sj × Tj , κti , is the set of states that player i deems subjectively possible at ti .

Definition 9 An epistemic model for the finite strategic two-player game form (S1 , S2 , z) is a structure (S1 , T1 , S2 , T2 ) , where, for each type ti of any player i, ti corresponds to a system of conditional preferences on the collection of sets of acts from elements of Φti := {φ ⊆ Ti × Sj × Tj | κti ∩ φ 6= ∅}

Basic characterizations

55

to ∆(Z), where κti is a non-empty subset of {ti } × Sj × Tj . An implicit model with a finite set of types for each player, as considered throughout this book, does not allow for ‘preference-completeness’, where, for each player i, there exists some type of i for any feasible preferences that i may have.1 Still, even a finite implicit model gives rise to infinite hierarchies of preferences, and – in effect – we assume that each player as a decision maker is able to represent his subjective hierarchy of preferences by means of a finite implicit model. Then, at the true profile of types, the two players’ subjective hierarchies can be embedded in a single implicit model that includes the types of the two players that are needed to represent each player’s hierarchy. Such a construction can fruitfully be used to analyze a wide range of game-theoretic concepts, as will be demonstrated throughout this book. However, when embedding the two player’s subjective hierarchies into a single implicit model, it is illegitimate to require that player i deems the true type of his opponent j subjectively possible. Rather, we cannot rule out that, at the true type profile, player j’s true type is not needed to represent player i’s subjective hierarchy of preferences; this is particularly relevant for the analysis of non-equilibrium game-theoretic concepts. Hence, when applying finite implicit models for interactive analysis of games, it is important to allow—as we do in the framework of the present text—the decision maker to hold objectively possible opponent preferences as subjectively impossible. Throughout this book we will consider two different kinds of epistemic models that differ according to the kind of assumption imposed on the set of conditional preferences that ti determines. For the present chapter, as well Chapters 8, 9, and 10, we will make the following assumption.

Assumption 1 For each ti of any player i, (a) ºtφi satisfies Axioms 1, 2, and 4 0 if ∅ 6= φ ⊆ Ti ×Sj ×Tj , and Axiom 3 if and only if φ ∈ Φti , (b) the system of conditional preferences {ºtφi | φ ∈ Φti } satisfies Axioms 5, 6 0 , and 16 , and (c) there exists a non-empty subset of opponent types, Tjti , such that κti = {ti } × Sj × Tjti .

1 ‘Preference-completeness’

is needed for the interactive epistemic analyses of, e.g., Brandenburger and Keisler (2002) and Battigalli and Siniscalchi (2002), but not for the analysis presented in this book. Brandenburger and Keisler (1999) show that there need not exist a ‘preference-complete’ interactive epistemic model when preferences are not representable by subjective probabilities, implying that ‘preference-completeness’ may be inconsistent with the analysis of Chapters 6, 7, 11, and 12, where Axiom 1 is not imposed.

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In this assumption, Tjti is the non-empty set of opponent types that player i deems subjectively possible at ti . The assumption explicitly allows for preferences over acts from subsets of Ti × Sj × Tj , φ, where projSj φ may be a strict subset of Sj . This accommodates the analysis of extensive game concepts in Chapters 8 and 9 and will permit the concepts in Tables 2.1 and 2.2 to be treated in a common framework. Write ºti for player i’s preferences conditional on being of type ti ; i.e., for ºtφi when φ = {ti } × Sj × Tj . We will refer to ºti as player i’s unconditional preferences at ti . Under Assumption 1 it follows from Proposition 5 that, for each type ti of any player i, i’s system of conditional preferences at ti can be represented by a vNM utility function υiti : ∆(Z) → R and an SCLP (λti , `ti ) on Ti × Sj × Tj with support κti = {ti } × Sj × Tjti . Throughout, we will adopt an interim perspective, where player i has already become aware of his own type. This ¡ entails that we can ¢ w.l.o.g. assume that, for any φ ∈ Φti , `(φ) = ` φ ∩ ({ti } × Sj × Tj ) . The interpretation is player i’s preferences at ti are not changed by ruling out states that i can distinguish from the true state at ti . Consequently, for expositional simplicity we choose to let the SCLP (λti , `ti ) be defined on Sj × Tj with support Sj × Tjti Preferences over strategies. It follows from the above assumptions that, for each type ti of any player i, player i’s unconditional preferences at ti , ºti , are a complete and transitive binary relation on the set of acts from Sj × Tj to ∆(Z) that is represented by a vNM utility function υiti and an LPS λt`i = (µt1i , . . . , µt`i ), where ` = `(Sj × Tj ). Since each pure strategy si ∈ Si is a function that assigns the deterministic outcome z(si , sj ) to any (sj , tj ) ∈ Sj × Tj and is thus an act from Sj × Tj to ∆(Z), we have that ºti determines complete and transitive preferences on i’s set of pure strategies, Si . Player i’s choice set at ti , Siti , is player i’s set of rational pure strategies at ti : Siti := {si ∈ Si | ∀s0i ∈ Si , si ºti s0i } . Since ºti is complete and transitive and satisfies objective independence, and Si is finite, it follows that the choice set Siti is non-empty, and that the set of rational mixed strategies equals ∆(Siti ). A strategic game. Let, for each i, ui : S → R be a vNM utility function that assigns payoff to any strategy profile. Then G = (S1 , S2 , u1 , u2 )

57

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is a finite strategic two-player game. Assume that, for each i, there exist s = (s1 , s2 ), s0 = (s01 , s02 ) ∈ S such that ui (s) > ui (s0 ). The event that i plays the game G is given by [ui ] := {(t1 , t2 ) ∈ T1 ×T2 | υiti ◦z is a positive affine transformation of ui } , while [u] := [u1 ] ∩ [u2 ] is the event that both players play G. Denote by pi , qi ∈ ∆(Si ) mixed strategies for player i, and let Sj0 (⊆ Sj ) be a non-empty set of opponent strategies. Say that pi strongly dominates qi on Sj0 if, ∀sj ∈ Sj0 , ui (pi , sj ) > ui (qi , sj ). Say that qi is strongly dominated on Sj0 if there exists pi ∈ ∆(Si ) such that pi strongly dominated qi on Sj0 . Say that pi weakly dominates qi on Sj0 if, ∀sj ∈ Sj0 , ui (pi , sj ) ≥ ui (qi , sj ) with strict inequality for some s0j ∈ Sj0 . Say that qi is weakly dominated on Sj0 if there exists pi ∈ ∆(Si ) such that pi weakly dominated qi on Sj0 . The following two results will be helpful for some of the proofs.

Lemma 4 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. For each i, pi ∈ ∆(Si ) is strongly dominated on Sj0 if and only if there does not exist µ ∈ ∆(Sj0 ) with suppµ ⊆ Sj0 such that, ∀s0i ∈ Si , X

µ(sj )ui (pi , sj ) ≥

X

µ(sj )ui (s0i , sj ) .

sj ∈Sj0

sj ∈Sj0

Proof. Lemma 3 of Pearce (1984).

Lemma 5 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. For each i, pi ∈ ∆(Si ) is weakly dominated on Sj0 if and only if there does not exist µ ∈ ∆(Sj0 ) with suppµ = Sj0 such that, ∀s0i ∈ Si , X sj ∈Sj0

µ(sj )ui (pi , sj ) ≥

X

µ(sj )ui (s0i , sj ) .

sj ∈Sj0

Proof. Lemma 4 of Pearce (1984). Certain belief. In the present chapter, as well in Chapters 8, 9, and 10, we will apply the certain belief operator (cf. Definition 4 of Chapter 4) for events that are subsets of the set of type profiles, T1 × T2 . In Assumption 1 we allow for the possibility that each player deems some opponent types subjectively impossible, corresponding to an SCLP that does not have full support along the type dimension. Therefore, certain belief (meaning that the complement is subjectively impossible) can be

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derived from the epistemic model and defined for events that are subsets of T1 × T2 . For any E ⊆ T1 × T2 , say that player i certainly believes the event E at ti if ti ∈ projTi Ki E, where Ki E := {(t1 , t2 ) ∈ T1 × T2 | projT1 ×T2 κti = {ti } × Tjti ⊆ E} . Say that there is mutual certain belief of E at (t1 , t2 ) if (t1 , t2 ) ∈ KE, where KE := K1 E ∩ K2 E. Say that there is common certain belief of E at (t1 , t2 ) if (t1 , t2 ) ∈ CKE, where CKE := KE ∩ KKE ∩ KKKE ∩ . . . . As established in Proposition 14, Ki corresponds to a KD45 system. Moreover, the mutual certain belief operator, K, has the following properties, where we write K0 E := E, and for each g ≥ 1, Kg E := KKg−1 E.

Proposition 20 (i) For any E ⊆ T1 ×T2 and all g > 1, Kg E ⊆ Kg−1 E. If E = E 1 ∩ E 2 , where, for each i, E i = projTi E i × Tj , then KE ⊆ E. (ii) For any E ⊆ T1 × T2 , there exists g 0 ≥ 0 such that Kg E = CKE for g ≥ g 0 , implying that CKE = KCKE. Proof. Part (i). If E = E 1 ∩E 2 , where, for each i, E i = projTi E i ×Tj , then KE = K1 E ∩ K2 E ⊆ K1 E 1 ∩ K2 E 2 = E 1 ∩ E 2 = E, establishing the second half of part (i). Since, for any E ⊆ T1 × T2 , KE = K1 E ∩ K2 E, where, for each i, Ki E = projTi Ki E × Tj , the first half of part (i) follows from the result of the second half. Part (ii) is a consequence of part (i) and T1 × T2 being finite.

5.2

Consistency of preferences

In the present section we define the event of consistency of preferences and show how this event can be used to provide characterizations of mixed-strategy Nash equilibrium and mixed rationalizable strategies. Inducing rational choice. In line with the discussion in Section 1.1, and following a tradition from Harsanyi (1973) to Blume et al. (1991b), a mixed strategy will interpreted, not as an object of choice, but as an expression of the beliefs for the other player. Say that the mixed strategy pjti |tj is induced for tj by ti if tj ∈ Tjti and, for all sj ∈ Sj , pjti |tj (sj ) =

µt`i (sj , tj )

µt`i (Sj , tj )

,

P where µt`i (Sj , tj ) := sj ∈Sj µt`i (sj , tj ), and where ` denotes the first level ` of λti for which µt`i (Sj , tj ) > 0. Furthermore, define the set of type profiles for which ti i nduces a r ational mixed strategy for any subjectively

59

Basic characterizations

possible opponent type: ¯ n ¡ 0 ¢o 0 ¯ [iri ] := (t1 , t2 ) ∈ T1 × T2 ¯ ∀t0j ∈ Tjti , pjti |tj ∈ ∆ Sjtj . Write [ir] := [ir1 ] ∩ [ir2 ]. Say that at ti player i’s preferences over his strategies are consistent with the game G = (S1 , S2 , u1 , u2 ) and the preferences of his opponent, if ti ∈ projTi ([ui ] ∩ [iri ]). Refer to [u] ∩ [ir] as the event of consistency. Characterizing Nash equilibrium. In line with the discussion in Section 1.1, we now characterize the concept of a mixed-strategy Nash equilibrium as profiles of induced mixed strategies at a type profile in [u] ∩ [ir] where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible). Before doing so, we define a mixed-strategy Nash equilibrium.

Definition 10 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. A mixed strategy profile p = (p1 , p2 ) is a mixed-strategy Nash equilibrium if for each i, ui (pi , pj ) = max ui (p0i , pj ) . 0 pi

The characterization result—which is a variant of Proposition 3 of Blume et al. (1991b)—can now be stated.

Proposition 21 Consider a finite strategic two-player game G. A profile of mixed strategies p = (p1 , p2 ) is a mixed-strategy Nash equilibrium if and only if there exists an epistemic model with (t1 , t2 ) ∈ [u]∩[ir] such that (1) there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and (2) for each i, pi is induced for ti by tj . Proof. (Only if.) Let (p1 , p2 ) be a mixed-strategy Nash equilibrium. Construct the following epistemic model. Let T1 = {t1 } and T2 = {t2 }. Assume that, for each i, υiti satisfies that υiti ◦ z = ui , the SCLP (λti , `ti ) has the properties that λti = (µt1i , . . . , µtLi ) with support Sj × {tj } satisfies that, ∀sj ∈ Sj , µt1i (sj , tj ) = pj (sj ), and `ti satisfies that `(Sj × {tj }) = 1. Then, it is clear that (t1 , t2 ) ∈ [u], that there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and that, for each i, pi is induced for ti by tj . It

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remains to show that (t1 , t2 ) ∈ [ir], i.e., for each i, pi ∈ ∆(Siti ). Since, by Definition 10, it holds for each i that, ∀s0i ∈ Si , ui (pi , pj ) ≥ ui (s0i , pj ), it follows from the construction of (λti , `ti ) that pi ∈ ∆(Siti ). (If.) Suppose that there exists an epistemic model with (t1 , t2 ) ∈ [u] ∩ [ir] such that there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and, for each i, pi is induced for ti by tj . Then, for each i, ºti is represented by υiti satisfying that υiti ◦z is a positive affine transformation of ui and an LPS λt`i = (µt1i , . . . , µt`i ), where ∀sj ∈ Sj , µt1i (sj , tj ) = pj (sj ), and where ` = `(Sj × Tj ) ≥ 1. Suppose, for some i and p0i ∈ ∆(Si ), ui (pi , pj ) < ui (p0i , pj ). Then there is some si ∈ Si with pi (si ) > 0 and some s0i ∈ Si such that ui (si , pj ) < ui (s0i , pj ), or equivalently X X µt1i (sj , tj )ui (si , sj ) < µt1i (sj , tj )ui (s0i , sj ) . sj

sj

This means that si ∈ / Siti , which, since pi (si ) > 0, contradicts (t1 , t2 ) ∈ [irj ]. Hence, by Definition 10, (p1 , p2 ) is a Nash equilibrium. For the “if” part of Proposition 21, it is sufficient that there is mutual certain belief of the beliefs that each player has about the strategy choice. We do not need the stronger condition that (1) entails. Hence, higher order certain belief plays no role in the characterization, in line with the fundamental insights of Aumann and Brandenburger (1995). Characterizing rationalizability. We now turn to analysis of deductive reasoning in games and present a characterization of (ordinary) rationalizability. Since we are only concerned with two-player games, there is no difference between rationalizability, as defined by Bernheim (1984) and Pearce (1984), and correlated rationalizability, where conjectures are allowed to be correlated. As it follows that rationalizability in two-player games corresponds to IESDS, we use the latter procedure as the primitive definition. For any (∅ = 6 ) X = X1 × X2 ⊆ S1 × S2 , write c˜(X) := c˜1 (X2 ) × c˜2 (X1 ), where c˜i (Xj ) := Si \ {si ∈ Si | ∃pi ∈ ∆(Si ) s.t. pi strongly dominates si on Xj } .

Definition 11 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. Consider the sequence defined by X(0) = S1 × S2 and, ∀g ≥ 1, X(g) = c˜(X(g − 1)). A pure strategy si is said to be rationalizable if \∞ Xi (g) . si ∈ Ri := g=0

A mixed strategy pi is said to be rationalizable if pi is not strongly dominated on Rj .

Basic characterizations

61

While any pure strategy in the support of a rationalizable mixed strategy is itself rationalizable (due to what Pearce calls the pure strategy property), the mixture on a set of rationalizable pure strategies need not be rationalizable. The following lemma is a straightforward implication of Definition 11.

Lemma 6 (i) For each i, Ri 6= ∅. (ii) R = c˜(R). (iii) For each i, si ∈ Ri if and only if there exists X = X1 × X2 with si ∈ Ri such that X ⊆ c˜(X). We next characterize the concept of rationalizable mixed strategies as induced mixed strategies under common certain belief of [u] ∩ [ir].

Proposition 22 A mixed strategy pi for i is rationalizable in a finite strategic two-player game G if and only if there exists an epistemic model with (t1 , t2 ) ∈ CK([u] ∩ [ir]) such that pi is induced for ti by tj . Proof. Part 1: If p∗i is rationalizable, then there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir]) such that p∗i is induced for t∗i by t∗j . Step 1: Construct an epistemic model with T1 × T2 ⊆ CK([u] ∩ [ir]) such that for each si ∈ Ri of any player i, there exists ti ∈ Ti with, si ∈ Siti . Construct an epistemic model with, for each i, a bijection si : Ti → Ri from the set of types to the the set of rationalizable pure strategies. Assume that, for each ti ∈ Ti of any player i, υiti satisfies that (a) υiti ◦ z = ui (so that T1 × T2 ⊆ [u]), and the SCLP (λti , `ti ) on Sj × Tj has the properties that (b) λti = (µt1i , . . . , µtLi ) with support Sj × Tjti satisfies that suppµt11 ∩ (Sj × {tj }) = {(sj (tj ), tj )} for all tj ∈ Tjti (so that, ∀tj ∈ Tjti , piti |tj (sj (tj )) = 1), (c) `ti satisfies `ti (Sj × Tj ) = 1. Property (b) entails that the support of the marginal of µt1i on Sj is included in Rj . By properties (a) and (c) and Lemmas 4 and 6(ii), we can still choose µt1i (and Titi ) so that si (ti ) ∈ Siti . This combined with property (b) means that T1 × T2 ⊆ [ir]. Furthermore, T1 × T2 ⊆ CK([u] ∩ [ir]) since Tjti ⊆ Tj for each ti ∈ Ti of any player i. Since, for each player i, si is onto Ri , it follows that, for each si ∈ Ri of any player i, there exists ti ∈ Ti with si ∈ Siti . ∗ ∗ ∗ Step 2: Add type t∗i to Ti . Assume that υiti satisfies (a) and (λti , `ti ) ∗ ∗ satisfies (b) and (c). Then µ1ti can be chosen so that p∗i ∈ ∆(Siti ).

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Furthermore, (Ti ∪ {t∗i }) × Tj ⊆ [u] ∩ [ir], and since Tjti ⊆ Tj , (Ti ∪ {t∗i }) × Tj ⊆ CK([u] ∩ [ir]). ∗ Step 3: Add type t∗j to Tj . Assume that υjtj satisfies (a) and the SCLP ∗ ∗ ∗ ∗ ∗ (λtj , `tj ) on Si × (Ti ∪ {t∗i }) has the property that λtj = (µ1tj , . . . , µLtj ) ∗ with support Si ×{t∗i } satisfies that, ∀si ∈ Si , µ1tj (si , t∗i ) = p∗i (si ), so that p∗i is induced for t∗i by t∗j . Furthermore, (Ti ∪{t∗i })×(Tj ∪{t∗j }) ⊆ [u]∩[ir], ∗ and since Titj ⊆ Ti ∪{t∗i }, (Ti ∪{t∗i })×(Tj ∪{t∗j }) ⊆ CK([u]∩[ir]). Hence, (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir]) and p∗i is induced for t∗i by t∗j . Part 2: If there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir]) such that p∗i is induced for t∗i by t∗j , then p∗i is rationalizable. Assume that there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ 6= ∅. [ir]) such that p∗i is induced for t∗i by t∗j . In particular, CK([u]∩[ir]) S ti 0 Let, for each i, Ti := projTi CK([u] ∩ [ir]) and Xi := ti ∈T i0 Si . By Proposition 20(ii), for each ti ∈ Ti0 of any player i, ti deems (sj , tj ) subjectively impossible if tj ∈ Tj \Tj0 since CK([u] ∩ [ir]) = KCK([u] ∩ [ir]) ⊆ Ki CK([u] ∩ [ir]), implying Tjti ⊆ Tj0 . By the definitions of [u] and [ir], it follows that, for each ti ∈ Ti0 of any player i, ºti is represented by υiti satisfying that υiti ◦ z is a positive affine transformation of ui and an LPS λt`i = (µt1i , . . . , µt`i ), where ` = `(Sj × Tj ) ≥ 1, and where suppµt1i ⊆ Xj ×Tj . Hence, by Lemma 4, for each ti ∈ Ti0 of any player i, if pi ∈ ∆(Siti ), then no strategy in the support of pi is strongly dominated on Xj , since it follows from pi ∈ ∆(Siti ) and suppµt1i ⊆ Xj × Tj that, ∀si ∈ supppi and ∀s0i ∈ Si , X X t X X t µ1i (sj , tj )ui (s0i , sj ) . µ1i (sj , tj )ui (si , sj ) ≥ sj ∈Xj tj ∈Tj

sj ∈Xj tj ∈Tj

This implies X ⊆ c˜(X), entailing by Lemma 6(iii) that, for each i, Xi ⊆ Ri . Furthermore, since (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir]) and the mixed ∗ strategy induced for t∗i by t∗j , p∗i , satisfies p∗i ∈ ∆(Siti ), it follows that p∗i is not strongly dominated on Xj ⊆ Rj . By Definition 11 this implies that p∗i is a rationalizable mixed strategy.

5.3

Admissible consistency of preferences

We next refine the event of consistency of preferences and show how this leads to characterizations of (strategic form) perfect equilibrium and mixed permissible strategies. Caution. Player i has preference for cautious behavior at ti if he takes into account all opponent strategies for any opponent type that is deemed subjectively possible.

Basic characterizations

63

Throughout this chapter as well as chapters 8, 9, and 10, we assume that Assumption 1 is satisfied so that κti = {ti } × Sj × Tjti . Under Assumption 1 player i is cautious at ti if {ºtφi | ∅ 6= φ ⊆ Φti } satisfies Axiom 6. Because then it follows from Proposition 2 that player i’s unconditional preferences at ti , ºti , are represented by υiti and an LPS λti with support Sj × Tjti . Since thus (sj , tj ) ∈ suppλti for any (sj , tj ) satisfying tj ∈ Tj ti , player i at ti takes into account all opponent strategies for any opponent type that is deemed subjectively possible. Hence, under Assumption 1, we can define the event [caui ] := {(t1 , t2 ) ∈ T1 × T2 | {ºtφi | ∅ 6= φ ⊆ Φti } satisfies Axiom 6 } . In terms of the representation of the system of conditional preferences, 6 φ ⊆ Φti }, by means of a vNM utility function and an SCLP {ºtφi | ∅ = (cf. Proposition 5), caution imposes the additional requirement that for each type ti of any player i the full LPS λti is used to form the conditional beliefs over opponent strategy-type pairs. Formally, if L denotes the number of levels in the LPS λti , then [caui ] = {(t1 , t2 ) ∈ T1 × T2 | `ti (Sj × Tj ) = L} . Since `ti is non-increasing w.r.t. set inclusion, ti ∈ projTi [caui ] implies that `ti (projSj ×Tj φ) = L for all subsets φ of {ti } × Sj × Tj with welldefined conditional beliefs. Since it follows from Assumption 1 that λti has full support on Sj , ti ∈ projTi [caui ] means that i’s choice set at ti never admits a weakly dominated strategy, thereby inducing preference for cautious behavior. Write [cau] := [cau1 ] ∩ [cau2 ]. Say that at ti player i’s preferences over his strategies are admissibly consistent with the game G = (S1 , S2 , u1 , u2 ) and the preferences of his opponent, if ti ∈ projTi ([ui ] ∩ [iri ] ∩ [caui ]). Refer to [u] ∩ [ir] ∩ [cau] as the event of admissible consistency. Characterizing perfect equilibrium. We now characterize the concept of a strategic form (or “trembling-hand”) perfect equilibrium as profiles of induced mixed strategies at a type profile in [u] ∩ [ir] ∩ [cau] where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible). Before doing so, we define a (strategic form) perfect equilibrium.

Definition 12 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. A mixed strategy profile p = (p1 , p2 ) is a (strategic form) perfect equilibrium if there is a sequence (p(n))n∈N of completely mixed strategy

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profiles converging to p such that for each i and every n ∈ N, ui (pi , pj (n)) = max ui (p0i , pj (n)) . 0 pi

The following holds in two-player games.

Lemma 7 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. A mixed strategy profile p = (p1 , p2 ) is a (strategic form) perfect equilibrium if and only if p is a mixed-strategy Nash equilibrium and, for each i, pi is not weakly dominated. Proof. Proposition 248.2 of Osborne and Rubinstein (1994). The characterization result—which is a variant of Proposition 4 of Blume et al. (1991b)—can now be stated.

Proposition 23 Consider a finite strategic two-player game G. A profile of mixed strategies p = (p1 , p2 ) is a (strategic form) perfect equilibrium if and only if there exists an epistemic model with (t1 , t2 ) ∈ [u] ∩ [ir] ∩ [cau] such that (1) there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and (2) for each i, pi is induced for ti by tj . Proof. (Only if.) Let (p1 , p2 ) be a (strategic form) perfect equilibrium. Then, by Lemma 7, (p1 , p2 ) be a mixed-strategy Nash equilibrium and, for each i, pi is not weakly dominated. Construct the following epistemic model. Let T1 = {t1 } and T2 = {t2 }. Assume that, for each i, υiti satisfies that υiti ◦ z = ui , the SCLP (λti , `ti ) has the properties that λti = (µt1i , µt2i ) with support Sj × {tj } has two levels, with the first level chosen so that, ∀sj ∈ Sj , µt1i (sj , tj ) = pj (sj ), and the second level chosen so that suppµt2i = Sj × {tj } and, ∀s0i ∈ Si , X X µt2i (sj , tj )ui (pi , sj ) ≥ µt2i (sj , tj )ui (s0i , sj ) sj

sj

(which is possible by Lemma 5 since pi is not weakly dominated), and `ti satisfies that `ti (Sj × Tj ) = 2. Then, it is clear that (t1 , t2 ) ∈ [u] ∩ [cau], that there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and that, for each i, pi is induced for ti by tj . It remains to show that (t1 , t2 ) ∈ [ir], i.e., for each i, pi ∈ ∆(Siti ). Since, by Lemma 7, it holds for each i that, ∀s0i ∈ Si , ui (pi , pj ) ≥ ui (s0i , pj ), it follows from the construction of (λti , `ti ) that pi ∈ ∆(Siti ).

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(If.) Suppose that there exists an epistemic model with (t1 , t2 ) ∈ [u] ∩ [ir] ∩ [cau] such that there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and, for each i, pi is induced for ti by tj . Then, for each i, ºti is represented by υiti satisfying that υiti ◦z is a positive affine transformation of ui and an LPS λti = (µt1i , . . . , µtLi ), where ∀sj ∈ Sj , µt1i (sj , tj ) = pj (sj ), and where suppλti = Sj × {tj }. Suppose first that (p1 , p2 ) is not a Nash equilibrium; i.e., for some i and p0i ∈ ∆(Si ), ui (pi , pj ) < ui (p0i , pj ). Then there is some si ∈ Si with pi (si ) > 0 and some s0i ∈ Si such that ui (si , pj ) < ui (s0i , pj ), or equivalently X X µt1i (sj , tj )ui (si , sj ) < µt1i (sj , tj )ui (s0i , sj ) . sj

sj

This means that si ∈ / Siti , which, since pi (si ) > 0, contradicts (t1 , t2 ) ∈ [irj ]. Suppose next that, for some i, pi is weakly dominated. Since suppλti = Sj × {tj }, this also implies that si ∈ / Siti for some si ∈ Si with pi (si ) > 0, again contradicting (t1 , t2 ) ∈ [irj ]. Hence, by Lemma 7, (p1 , p2 ) is a (strategic form) perfect equilibrium. As for Proposition 21, higher order certain belief plays no role in this characterization. Characterizing permissibility. We now turn to the non-equilibrium analog to (strategic form) perfect equilibrium, namely the concept of permissibility; cf. B¨orgers (1994) as well as Brandenburger (1992) who coined the term ‘permissibility’. To define the concept of permissible strategies, we use the equivalent Dekel-Fudenberg procedure as the primitive definition. For any (∅ 6=) X = X1 × X2 ⊆ S1 × S2 , write a ˜(X) := a ˜1 (X2 ) × a ˜2 (X1 ), where a ˜i (Xj ) := Si \ {si ∈ Si | ∃pi ∈ ∆(Si ) s.t. pi strongly dominates si on Xj or pi weakly dominates si on Sj } .

Definition 13 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. Consider the sequence defined by X(0) = S1 × S2 and, ∀g ≥ 1, X(g) = a ˜(X(g − 1)). A pure strategy si is said to be permissible if \∞ si ∈ Pi := Xi (g) . g=0

A mixed strategy pi is said to be permissible if pi is not strongly dominated on Pj and not weakly dominated on Sj . While any pure strategy in the support of a permissible mixed strategy is itself permissible, the mixture over a set of permissible pure strategies need not be permissible.

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The following lemma is a straightforward implication of Definition 13.

Lemma 8 (i) For each i, Pi 6= ∅. (ii) P = a ˜(P ). (iii) For each i, si ∈ Pi if and only if there exists X = X1 ×X2 with si ∈ Pi such that X ⊆ a ˜(X). We next characterize the concept of permissible mixed strategies as induced mixed strategies under common certain belief of [u]∩[ir]∩[cau].

Proposition 24 A mixed strategy pi for i is permissible in a finite strategic two-player game G if and only if there exists an epistemic model with (t1 , t2 ) ∈ CK([u] ∩ [ir] ∩ [cau]) such that pi is induced for ti by tj . Proof. Part 1: If p∗i is permissible, then there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir] ∩ [cau]) such that pi is induced for t∗i by t∗j . Step 1: Construct an epistemic model with T1 × T2 ⊆ CK([u] ∩ [ir] ∩ [cau]) such that for each si ∈ Pi of any player i, there exists ti ∈ Ti with, si ∈ Siti . Construct an epistemic model with, for each i, a bijection si : Ti → Pi from the set of types to the the set of permissible pure strategies. Assume that, for each ti ∈ Ti of any player i, υiti satisfies that (a) υiti ◦ z = ui (so that T1 × T2 ⊆ [u]), and the SCLP (λti , `ti ) on Sj × Tj has the properties that (b) λti = (µt1i , µt2i ) with support Sj × Tjti has two levels and satisfies that suppµt11 ∩ (Sj × {tj }) = {(sj (tj ), tj )} for all tj ∈ Tjti (so that, ∀tj ∈ Tjti , piti |tj (sj (tj )) = 1), (c) `ti satisfies `ti (Sj × Tj ) = 2 (so that T1 × T2 ⊆ [cau]). Property (b) entails that the support of the marginal of µt1i on Sj is included in Pj . By properties (a) and (c) and Lemmas 4, 5 and 8(ii), we can still choose µt1i and µt2i (and Ti ti ) so that si (ti ) ∈ Siti . This combined with property (b) means that T1 × T2 ⊆ [ir]. Furthermore, T1 × T2 ⊆ CK([u] ∩ [ir] ∩ [cau]) since Tjti ⊆ Tj for each ti ∈ Ti of any player i. Since, for each player i, si is onto Pi , it follows that, for each si ∈ Pi of any player i, there exists ti ∈ Ti with si ∈ Siti . ∗ ∗ ∗ Step 2: Add type t∗i to Ti . Assume that υiti satisfies (a) and (λti , `ti ) ∗ ∗ satisfies (b) and (c). Then µ1ti and µ2ti can be chosen so that p∗i ∈ ∗ ∆(Siti ). Furthermore, (Ti ∪ {t∗i }) × Tj ⊆ [u] ∩ [ir] ∩ [cau], and since ∗ Tjti ⊆ Tj , (Ti ∪ {t∗i }) × Tj ⊆ CK([u] ∩ [ir] ∩ [cau]). ∗ Step 3: Add type t∗j to Tj . Assume that υjtj satisfies (a) and the SCLP ∗ ∗ ∗ ∗ ∗ (λtj , `tj ) on Si × (Ti ∪ {t∗i }) has the property that λtj = (µ1tj , . . . , µLtj )

67

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with support Si × {t∗i } satisfies that, ∀si ∈ Si , µ1tj (si , t∗i ) = p∗i (si ), so ∗ ∗ that p∗i is induced for t∗i by tj∗ , and `tj satisfies that `tj (Ti ∪ {t∗i }) = L. Furthermore, (Ti ∪ {t∗i }) × (Tj ∪ {t∗j }) ⊆ [u] ∩ [ir] ∩ [cau], and since ∗ Titj ⊆ Ti ∪ {t∗i }, (Ti ∪ {t∗i }) × (Tj ∪ {t∗j }) ⊆ CK([u] ∩ [ir] ∩ [cau]). Hence, (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir] ∩ [cau]) and p∗i is induced for t∗i by t∗j . Part 2: If there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir] ∩ [cau]) such that p∗i is induced for t∗i by t∗j , then p∗i is permissible. Assume that there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir] ∩ [cau]) such that p∗i is induced for t∗i by t∗j . In particular, CK([u] ∩ [ir] ∩ [cau]) 6= ∅. Let, for each i, Ti0 := projTi CK([u] ∩ [ir] ∩ [cau]) and S Xi := ti ∈T i0 Siti . By Proposition 20(ii), for each ti ∈ Ti0 of any player i, ti deems (sj , tj ) subjectively impossible if tj ∈ Tj \Tj0 since CK([u] ∩ [ir] ∩ [cau]) = KCK([u] ∩ [ir] ∩ [cau]) ⊆ Ki CK([u] ∩ [ir] ∩ [cau]), implying Tjti ⊆ Tj0 . By the definitions of [u], [ir], and [cau], it follows that, for each ti ∈ Ti0 of any player i, ºti is represented by υiti satisfying that υiti ◦z is a positive affine transformation of ui and an LPS λti = (µt1i , . . . , µtLi ), and where suppµt1i ⊆ Xj × Tj , and where suppλti = Sj × Tjti . Hence, by Lemma 4, for each ti ∈ Ti0 of any player i, if pi ∈ ∆(Siti ), then no strategy in the support of pi is strongly dominated on Xj , since it follows from pi ∈ ∆(Siti ) and suppµt1i ⊆ Xj × Tj that, ∀si ∈ supppi and ∀s0i ∈ Si , X X t X X t µ1i (sj , tj )ui (s0i , sj ) . µ1i (sj , tj )ui (si , sj ) ≥ sj ∈Xj tj ∈Tj

sj ∈Xj tj ∈Tj

Furthermore, since the projection of λti on Sj has full support, no strategy in the support in pi is weakly dominated on Sj . This implies X⊆a ˜(X), entailing by Lemma 8(iii) that, for each i, Xi ⊆ Pi . Finally, since (t∗1 , t∗2 ) ∈ CK([u] ∩ [ir] ∩ [cau]) and the mixed strategy induced for ∗ t∗i by t∗j , p∗i , satisfies p∗i ∈ ∆(Siti ), it follows that p∗i is not strongly dominated on Xj ⊆ Pj and p∗i is not weakly dominated on Sj . By Definition 13 this implies that p∗i is a permissible mixed strategy.

Chapter 6 RELAXING COMPLETENESS

In the previous chapter, we have presented epistemic characterizations of rationalizability and permissibility. For these non-equilibrium deductive concepts, we have used, respectively, IESDS and the DekelFudenberg procedure (one round of weak elimination followed by iterated strong domination) as the primitive definitions. Neither of these procedures rely on players having subjective probabilities over the strategy choice of the opponent. In contrast, the epistemic characterizations—by relying on Assumption 1—require that players have complete preferences that are representable by means of subjective probabilities. In this chapter we show how rationalizability and permissibility can be epistemically characterized without requiring that players have complete preferences that are representable by means of subjective probabilities. The resulting structure will also be used for the epistemic analysis of backward induction in Chapter 7 and forward induction in Chapter 11. Hence, even though the results of the present chapter may have limited interest in their own right, they set the stage for later analysis.

6.1

Epistemic modeling of strategic games (cont.)

The purpose of this section is to present a framework for strategic games where each player is modeled as a decision maker under uncertainty with preferences that are allowed to be incomplete. An epistemic model. Consider an epistemic model for a finite strategic game form (S1 , S2 , z) as formalized in Definition 9, with a finite type set Ti for each player i, and where the preferences of a player corresponds to the player’s type. Hence, for each type ti of any player i,

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{ti } × Sj × Tj is the set of states that are indistinguishable for player i at ti , a non-empty subset of {ti } × Sj × Tj , κti , is the set of states that player i deems subjectively possible at ti , and ti corresponds to a system of conditional preferences on the collection of sets of acts from subsets of Ti × Sj × Tj whose intersection with κti is non-empty to ∆(Z). However, instead Assumption 1, impose the following assumption, where Φti still denotes {φ ⊆ Ti × Sj × Tj | κti ∩ φ = 6 ∅}.

Assumption 2 For each ti of any player i, (a) ºtφi satisfies Axioms 1 0 , 2, and 4 0 if ∅ = 6 φ ⊆ Ti × Sj × Tj , and Axiom 3 if and only if φ ∈ Φti , and (b) the system of conditional preferences {ºtφi | φ ∈ Φti } satisfies Axioms 5, 6, and 11 . As before, write ºti for player i’s unconditional preferences at ti ; i.e., for ºtφi when φ = {ti } × Sj × Tj . W.l.o.g. we may consider ºti to be preferences over acts from Sj × Tj to ∆(Z) (instead of acts from {ti }×Sj ×Tj to ∆(Z)). Under Assumption 2 it follows from Proposition 4 that, for each ti of any player i, i’s unconditional preferences at ti can be conditionally represented by a vNM utility function υiti : ∆(Z) → R. Conditional representation implies that strong and weak dominance are is well-defined: Let Ej ⊆ Sj × Tj . Say that one act pEj strongly dominates another act qEj at ti if, ∀(sj , tj ) ∈ Ej , υiti (pEj (sj , tj ))) > υiti (qEj (sj , tj ))) . Say that pEj weakly dominates qEj at ti if, ∀(sj , tj ) ∈ Ej , υiti (pEj (sj , tj ))) ≥ υiti (qEj (sj , tj ))) , with strict inequality for some (s0j , t0j ) ∈ Ej . Say that ºti is admissible on {ti }×Ej if Ej is non-empty and p Âti q whenever pEj weakly dominates qEj at ti . Assumption 2 entails that ºti is admissible on κti . Indeed, as shown in Section 4.1, there exists a vector of nested sets, (ρt1i , . . . , ρtLi ), on which ºti is admissible, satisfying: ∅ 6= ρt1i ⊂ · · · ⊂ ρt`i ⊂ · · · ⊂ ρtLi = κti ⊆ {ti } × Sj × Tj (where ⊂ denotes ⊆ and 6=). Preferences over strategies. It follows from the above assumptions that, for each type ti of any player i, player i’s unconditional preferences

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at ti , ºti , is a reflexive and transitive binary relation on acts from Sj ×Tj to ∆(Z) that is conditionally represented by a vNM utility function υiti . Since each mixed strategy pi ∈ ∆(Si ) is a function that assigns the randomized outcome z(pi , sj ) to any (sj , tj ) ∈ Sj × Tj and is thus an act from Sj × Tj to ∆(Z), we have that ºti determines reflexive and transitive preferences on i’s set of mixed strategies, ∆(Si ). Player i’s choice set at ti , Siti , is player i’s set of maximal pure strategies at ti : Siti := {si ∈ Si | @pi ∈ ∆(Si ), pi Âti si } . Hence, a pure strategy, si , is in i’s choice set at ti if there is no mixed strategy that is strictly preferred to si given i’s (possibly incomplete) preferences at ti . If i’s preferences at ti are complete, then @pi ∈ ∆(Si ), pi Âti si is equivalent to ∀s0i ∈ Si , si ºti s0i , and the definition of Siti coincides with the one given in Section 5.1. Since ºti is reflexive and transitive and satisfies objective independence, and Si is finite, it follows that the choice set Siti is non-empty and supports any maximal mixed strategies: If qi ∈ ∆(Si ) and @pi ∈ ∆(Si ) such that pi Âti qi , then qi ∈ ∆(Siti ). On the other hand, with incomplete preferences, it is not the case that all mixed strategies in ∆(Siti ) are maximal. We may have that ∃pi ∈ ∆(Si ) such that pi Âti qi even though qi ∈ ∆(Siti ). As an illustration, consider the case where ºti is defined by p Âti q if and only if pprojS

j ×Tj

κi

weakly dominates qprojS

j ×Tj

κi

at ti ,

and p ∼ti q if and only if υiti (p(sj , tj ))) = υiti (q(sj , tj ))) for all (sj , tj ) ∈ projSj ×Tj κi . Since a mixed strategy qi may be weakly dominated by a pure strategy si that does not weakly dominate any pure strategy in the support of qi , this illustrates the possibility that a non-maximal mixed-strategy qi is supported by maximal pure strategies. The event that player i is rational is defined by [rati ] := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | si ∈ Siti } . A strategic game. As before, G = (S1 , S2 , u1 , u2 ) denotes a finite strategic two-player game, where S = S1 × S2 is the set of strategy profiles and, for each i, ui : S → R is a vNM utility function that assigns payoff to any strategy profile. Assume that, for each i, there exist s = (s1 , s2 ), s0 = (s01 , s02 ) ∈ S such that ui (s) > ui (s0 ). As in Chapter 5—but transferred to S1 × T1 × S2 × T2 space—the event that

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i plays the game G is given by [ui ] := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | υiti ◦ z is a positive affine transformation of ui } , while [u] := [u1 ] ∩ [u2 ] is the event that both players play G. Belief operators. Since Assumptions 2 is compatible with the framework of Chapter 4, we can in line with Section 4.2 define belief operators as follows. For these definitions, say that E ⊆ S1 × T1 × S2 × T2 does not concern player i’s strategy choice if E = Si × projTi ×Sj ×Tj E. If E does not concern player i’s strategy choice, say that player i certainly believes the event E at ti if ti ∈ projTi Ki E, where Ki E := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | κti ⊆ projT1 ×S2 ×T2 E} . If E does not concern the strategy choice of either player, say that there is mutual certain belief of E at (t1 , t2 ) if (t1 , t2 ) ∈ projTi ×T2 KE, where KE := K1 E ∩ K2 E. If E does not concern the strategy choice of either player, say that there is common certain belief of E at (t1 , t2 ) if (t1 , t2 ) ∈ projTi ×T2 CKE, where CKE := KE ∩ KKE ∩ KKKE ∩ . . . . If E does not concern player i’s strategy choice, say that player i (unconditionally) believes the event E at ti if ti ∈ projTi Bi E, where Bi E := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | β ti ⊆ projT1 ×S2 ×T2 E} , and where β ti := ρt1i denotes the smallest set on which ºti is admissible. If E does not concern the strategy choice of either player, say that there is mutual belief of E at (t1 , t2 ) if (t1 , t2 ) ∈ projTi ×T2 BE, where BE := B1 E ∩ B2 E. If E does not concern the strategy choice of either player, say that there is common belief of E at (t1 , t2 ) if (t1 , t2 ) ∈ projTi ×T2 CBE, where CBE := BE ∩ BBE ∩ BBBE ∩ . . . . As established in Proposition 14, Ki and Bi correspond to KD45 systems. Moreover, the mutual certain belief and mutual belief operators, K and B, have the following properties, where we write K0 E := E and B0 E := E, and for each g ≥ 1, Kg E := KKg−1 E and Bg E := BBg−1 E.

Proposition 25 (i) For any E ⊆ T1 ×T2 and all g > 1, Kg E ⊆ Kg−1 E and Bg E ⊆ Bg−1 E. If E = E 1 ∩ E 2 , where, for each i, E i = Si × projTi E i × Sj × Tj , then KE ⊆ E and BE ⊆ E. (ii) For any E ⊆ T1 × T2 , there exist g 0 and g 00 ≥ 0 such that Kg E = CKE for g ≥ g 0 and Bg E = CBE for g ≥ g 00 , implying that CKE = KCKE and CBE = BCBE. Proof. See the proof of Proposition 20.

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6.2

Consistency of preferences (cont.)

In the present section we define the event of consistency of preferences in the case described by Assumption 2, where preferences need not be complete, and use this event to characterize the concept of rationalizable pure strategies. Belief of opponent rationality. In the context of the present chapter, define as follows the event that player i’s preferences over his strategies are consistent with the game G = (S1 , S2 , u1 , u2 ) and the preferences of his opponent: Ci := [ui ] ∩ Bi [ratj ] . Write C := C1 ∩ C2 for the event of consistency. Characterizing rationalizability. We now characterize the concept of rationalizable pure strategies (cf. Definition 11 of Chapter 5) as maximal pure strategies under common certain belief of consistency.

Proposition 26 A pure strategy si for i is rationalizable in a finite strategic two-player game G if and only if there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKC. To prove Proposition 26, it is helpful to establish a variant of Lemma 6. Write, for any (∅ 6=) X = X1 ×X2 ⊆ S1 ×S2 , c(X) := c1 (X2 )×c2 (X1 ), where ci (Xj ) := {si ∈ Si | ∃(∅ 6=) Yj ⊆ Xj such that, ∀pi ∈ ∆(Si ), pi does not weakly dominate si on Yj } .

Lemma 9 (i) R = c(R). (ii) For each i, si ∈ Ri if and only if there exists X = X1 × X2 with si ∈ Ri such that X ⊆ c(X). Proof. In view of Lemma 6, it is sufficient to show that, for any (∅ 6=) Xj ⊆ Sj , ci (Xj ) = c˜i (Xj ). Part 1: ci (Xj ) ⊆ c˜i (Xj ). If si ∈ / c˜i (Xj ), then ∃pi ∈ ∆(Si ) s.t. pi strongly dominates si on Xj . From this it follows that ∀ (∅ 6=) Yj ⊆ Xj , ∃pi ∈ ∆(Si ) s.t. pi weakly dominates si on Yj , implying that si ∈ / ci (Xj ). Part 2: ci (Xj ) ⊇ c˜i (Xj ). If si ∈ c˜i (Xj ), then there does not exist pi ∈ ∆(Si ) s.t. pi strongly dominates si on Xj . Hence, by Lemma 4, there exists a subjective probability distribution µ ∈ ∆(Sj ) with suppµ ⊆ Xj such that si is maximal in ∆(Si ) w.r.t. the preferences represented by the vNM utility function ui and the subjective probability distribution

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µ. Then there does not exist pi ∈ ∆(Si ) s.t. pi weakly dominates si on suppµ (⊆ Xj ), implying that si ∈ ci (Xj ). Proof of Proposition 26. Part 1: If si is rationalizable, then there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKC. It is sufficient to construct a belief system with S1 × T1 × S2 × T2 ⊆ CKC such that, for each si ∈ Ri of any player i, there exists ti ∈ Ti with si ∈ Siti . Construct a belief system with, for each i, a bijection si : Ti → Ri from the set of types to the the set of rationalizable pure strategies. By Lemma 9(i) we have that, for each ti ∈ Ti of any player i, there exists Yjti ⊆ Ri such that there does not exist pi ∈ ∆(Si ) such that pi weakly dominates si (ti ) on Yjti . Determine the set of opponent types that ti deems subjectively possible as follows: Tjti = {tj ∈ Tj | sj (tj ) ∈ Yjti }. Let, for each ti ∈ Ti of any player i, ºti satisfy 1. υiti ◦ z = ui (so that S1 × T1 × S2 × T2 ⊆ [u]), and 2. p Âti q iff pEj weakly dominates qEj for Ej = Ejti := {(sj , tj )|sj = sj (tj ) and tj ∈ Tjti }, which implies that β ti = κti = {ti } × Ejti . By the construction of Ejti , this means that Siti 3 si (ti ) since, for any acts p and q on Sj × Tj satisfying that there exist mixed strategies pi , qi ∈ ∆(Si ) such that, ∀(sj , tj ) ∈ Sj × Tj , p(sj , tj ) = z(pi , sj ) and q(sj , tj ) = z(qi , sj ), p Âti q iff pEj weakly dominates qEj for Ej = Yjti × Tj . This in turn implies, for each ti ∈ Ti any player i, 3. β ti ⊆ projTi ×Sj ×Tj [ratj ] (so that S1 × T1 × S2 × T2 ⊆ Bi [ratj ] ∩ Bj [rati ]). Furthermore, S1 × T1 × S2 × T2 ⊆ CKC since Tjti ⊆ Tj for each ti ∈ Ti of any player i. Since, for each player i, si is onto Ri , it follows that, for each si ∈ Ri of any player i, there exists ti ∈ Ti with si ∈ Siti . ∗ Part 2: If there exists an epistemic model with s∗i ∈ Siti for some (t∗1 , t∗2 ) ∈ projT1 ×T2 CKC, then s∗i is rationalizable. ∗ Assume that there exists an epistemic model with s∗i ∈ Siti for some (t∗1 , t∗2 ) ∈ projT1 ×T2 CKC. InSparticular, CKC 6= ∅. Let, for each i, Ti0 := projTi CKC and Xi := ti ∈T i0 Siti . It is sufficient to show that, for each i, Xi ⊆ Ri . By Proposition 25(ii), for each ti ∈ Ti0 of any player i, β ti ⊆ κti ⊆ {ti } × Sj × Tj0 since CKC = KCKC ⊆ Ki CKC. By the definition of C, it follows that, for each ti ∈ Ti0 of any player i, 1. ºti is conditionally represented by υiti satisfying that υiti ◦ z is a positive affine transformation of ui , and

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2. p Âti q if pEj weakly dominates qEj for Ej = Ejti := projSj ×Tj β ti , where β ti ⊆ projTi ×Sj ×Tj [ratj ]. Write Yjti := projSj Ejti = projSj β ti , and note that β ti ⊆ ({ti }×Sj ×Tj0 )∩ projTi ×Sj ×Tj [ratj ] implies Yjti ⊆ Xi . It follows that, for any acts p and q on Sj × Tj satisfying that there exist mixed strategies pi , qi ∈ ∆(Si ) such that, ∀(sj , tj ) ∈ Sj ×Tj , p(sj , tj ) = z(pi , sj ) and q(sj , tj ) = z(qi , sj ), p Âti q if pEj weakly dominates qEj for Ej = Yjti ×Tj . Hence, if si ∈ Siti , then there does not exist pi ∈ ∆(Si ) such that pi weakly dominates si on Yjti . Since this holds for each ti ∈ Ti0 of any player i, we have that X ⊆ c(X). Hence, Lemma 9(ii) entails that, for each i, Xi ⊆ Ri . Proposition 26 is obtained also if CBC is used instead of CKC.

6.3

Admissible consistency of preferences (cont.)

In the present section we define the event of admissible consistency of preferences in the case considered by Assumption 2, where preferences need not be complete, and use this event to characterize the concept of permissible pure strategies. Caution. As in Section 5.3, player i has preference for cautious behavior at ti if he takes into account all opponent strategies for any opponent type that is deemed subjectively possible. Throughout this chapter, as well as Chapter 7 and 11, we assume that Assumption 2 is satisfied, so that the system of conditional preferences {ºtφi | φ ∈ Φti } satisfies Axiom 6, where Φti denotes {φ ⊆ Ti × Sj × Tj | κti ∩ φ 6= ∅}, and where κti —the set of states that player i deems subjectively possible at ti —satisfies ∅ = 6 κti ⊆ {ti } × Sj × Tj . Hence, Tjti := projTj κti is the set of opponent types that player i deems subjectively possible. Under Assumption 2, player i is cautious at ti if κti = {ti } × Sj × Tjti . Because then, player i at ti takes into account all opponent strategies for any opponent type that is deemed subjectively possible. This means that i’s choice set at ti never admits a weakly dominated strategy, thereby inducing preference for cautious behavior. Hence, under Assumption 2 we can define the event [caui ] := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | ∃Tjti such that κti = {ti } × Sj × Tjti } . Write [cau] := [cau1 ] ∩ [cau2 ]. In the context of the present chapter, define as follows the event that player i’s preferences over his strategies are admissibly consistent with

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the game G = (S1 , S2 , u1 , u2 ) and the preferences of his opponent: Ai := [ui ] ∩ Bi [ratj ] ∩ [caui ] . Write A := A1 ∩ A2 for the event of admissible consistency. Characterizing permissibility. We now characterize the concept of permissible pure strategies (cf. Definition 13 of Chapter 5) as maximal pure strategies under common certain belief of admissible consistency.

Proposition 27 A pure strategy si for i is permissible in a finite strategic two-player game G if and only if there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA. To prove Proposition 27, it is helpful to establish a variant of Lemma 8. Define, for any (∅ 6=) Yj ⊆ Sj , Di (Yj ) := {si ∈ Si | ∃pi ∈ ∆(Si ) such that pi weakly dominates si on Yj or Sj } , and write, for any (∅ 6=) X = X1 × X2 ⊆ S1 × S2 , a(X) := a1 (X2 ) × a2 (X1 ), where ai (Xj ) := {si ∈ Si | ∃(∅ 6=) Yj ⊆ Xj such that si ∈ Si \Di (Yi )} .

Lemma 10 (i) P = a(P ). (ii) For each i, si ∈ Pi if and only if there exists X = X1 × X2 with si ∈ Pi such that X ⊆ a(X). Proof. In view of Lemma 8, it is sufficient to show that, for any (∅ 6=) Xj ⊆ Sj , ai (Xj ) = a ˜i (Xj ). Part 1: ai (Xj ) ⊆ a ˜i (Xj ). If si ∈ / a ˜i (Xj ), then ∃pi ∈ ∆(Si ) s.t. pi strongly dominates si on Xj or pi weakly dominates si on Sj . From this it follows that ∀ (∅ 6=) Yj ⊆ Xj , ∃pi ∈ ∆(Si ) s.t. pi weakly dominates si on Yj or Sj , implying that ∀ (∅ 6=) Yj ⊆ Xj , si ∈ Di (Yj ). This means that si ∈ / ai (Xj ). Part 2: ai (Xj ) ⊇ a ˜i (Xj ). If si ∈ a ˜i (Xj ), then there does not exist pi ∈ ∆(Si ) s.t. pi strongly dominates si on Xj or pi weakly dominates si on Sj . Hence, by Lemmas 4 and 5, there exists an LPS λ = (µ1 , µ2 ) ∈ L∆(Sj ) with suppµ1 ⊆ Xj and suppµ2 = Sj such that si is maximal in ∆(Si ) w.r.t. the preferences represented by the vNM utility function ui and the LPS λ. Then there does not exist pi ∈ ∆(Si ) s.t. pi weakly dominates si on suppµ1 (⊆ Xj ) or suppµ2 (= Sj ), implying that si ∈ / Di (Yj ) for Yj = suppµ1 ⊆ Xj . This means that si ∈ ai (Xj ).

Relaxing completeness

77

Proof of Proposition 27. Part 1: If si is permissible, then there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA. It is sufficient to construct a belief system with S1 × T1 × S2 × T2 ⊆ CKA such that, for each si ∈ Pi of any player i, there exists ti ∈ Ti with si ∈ Siti . Construct a belief system with, for each i, a bijection si : Ti → Pi from the set of types to the the set of permissible pure strategies. By Lemma 10(i) we have that, for each ti ∈ Ti of any player i, there exists Yjti ⊆ Pi such that si (ti ) ∈ Si \Di (Yjti ). Determine the set of opponent types that ti deems subjectively possible as follows: Tjti = {tj ∈ Tj | sj (tj ) ∈ Yjti }. Let, for each ti ∈ Ti of any player i, ºti satisfy 1. υiti ◦ z = ui (so that S1 × T1 × S2 × T2 ⊆ [u]), and 2. p Âti q iff pEj weakly dominates qEj for Ej = Ejti := {(sj , tj )|sj = sj (tj ) and tj ∈ Tjti } or Ej = Sj × Tjti , which implies that β ti = {ti }×Ejti and κti = {ti }×Sj ×Tjti (so that S1 ×T1 ×S2 ×T2 ⊆ [cau]). By the construction of Ejti , this means that Siti = Si \Di (Yjti ) 3 si (ti ) since, for any acts p and q on Sj × Tj satisfying that there exist mixed strategies pi , qi ∈ ∆(Si ) such that, ∀(sj , tj ) ∈ Sj ×Tj , p(sj , tj ) = z(pi , sj ) and q(sj , tj ) = z(qi , sj ), p Âti q iff pEj weakly dominates qEj for Ej = Yjti × Tj or Ej = Sj × Tj . This in turn implies, for each ti ∈ Ti any player i, 3. β ti ⊆ projTi ×Sj ×Tj [ratj ] (so that S1 × T1 × S2 × T2 ⊆ Bi [ratj ] ∩ Bj [rati ]). Furthermore, S1 × T1 × S2 × T2 ⊆ CKA since Tjti ⊆ Tj for each ti ∈ Ti of any player i. Since, for each player i, si is onto Pi , it follows that, for each si ∈ Pi of any player i, there exists ti ∈ Ti with si ∈ Siti . ∗ Part 2: If there exists an epistemic model with s∗i ∈ Siti for some (t∗1 , t∗2 ) ∈ projT1 ×T2 CKA, then s∗i is permissible. ∗ Assume that there exists an epistemic model with s∗i ∈ Siti for some (t∗1 , t∗2 ) ∈ projT1 ×T2 CKA. InSparticular, CKA 6= ∅. Let, for each i, Ti0 := projTi CKA and Xi := ti ∈T i0 Siti . It is sufficient to show that, for each i, Xi ⊆ Pi . By Proposition 25(ii), for each ti ∈ Ti0 of any player i, β ti ⊆ κti ⊆ {ti } × Sj × Tj0 since CKA = KCKA ⊆ Ki CKA. By the definition of A, it follows that, for each ti ∈ Ti0 of any player i, 1. ºti is conditionally represented by υiti satisfying that υiti ◦ z is a positive affine transformation of ui , and

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2. p Âti q if pEj weakly dominates qEj for Ej = Ejti := projSj ×Tj β ti or Ej = Sj × Tjti , where β ti ⊆ projTi ×Sj ×Tj [ratj ]. Write Yjti := projSj Ejti = projSj β ti , and note that β ti ⊆ ({ti }×Sj ×Tj0 )∩ projTi ×Sj ×Tj [ratj ] implies Yjti ⊆ Xi . It follows that, for any acts p and q on Sj × Tj satisfying that there exist mixed strategies pi , qi ∈ ∆(Si ) such that, ∀(sj , tj ) ∈ Sj ×Tj , p(sj , tj ) = z(pi , sj ) and q(sj , tj ) = z(qi , sj ), p Âti q if pEj weakly dominates qEj for Ej = Yjti × Tj or Ej = Sj × Tj . Hence, Siti ⊆ Si \Di (Yjti ). Since this holds for each ti ∈ Ti0 of any player i, we have that X ⊆ a(X). Hence, Lemma 10(ii) entails that, for each i, Xi ⊆ Pi . Proposition 27 is obtained also if CBA is used instead of CKA; this is essentially the corresponding result by Brandenburger (1992). One may argue that the result above is more complicated as it involves two different epistemic operators. Still, it yields the insight that the essential feature in a characterization of the Dekel-Fudenberg procedure is to let irrational opponent choice be deemed subjectively possible. It also turns out to be a useful benchmark for the analysis of backward induction in Section 7.3 where the certain belief operator Ki – rather than the belief operator Bi – must be used for the interactive epistemology (cf. the analysis of Γ5 illustrated in Figure 7.1).

Chapter 7 BACKWARD INDUCTION

In recent years, two influential contributions on backward induction in finite perfect information games have appeared, namely Aumann (1995) and Ben-Porath (1997). These contributions—both of which consider generic perfect information games (where all payoffs are different)— reach opposite conclusions: While Aumann establishes that ‘common knowledge of rationality’ implies that the backward induction outcome is reached, Ben-Porath shows that the backward induction outcome is not the only outcome that is consistent with ‘common certainty of rationality’. The models of Aumann and Ben-Porath are different. One such difference is that Aumann makes use of ‘knowledge’ in the sense of ‘true knowledge’, while Ben-Porath’s analysis is based on ‘certainty’ in the sense of ‘belief with probability one’. Another is that the term ‘rationality’ is used in different senses: Aumann imposes rationality in all subgames, while Ben-Porath assumes rationality initially, in the whole game, only (not after a “surprise” has occurred). The present chapter, which reproduces Asheim (2002), shows how the conclusions of Aumann and Ben-Porath can be captured by imposing requirements on the players within the same general framework. Furthermore, the interpretations of the present analysis correspond closely to the intuitions that Aumann and Ben-Porath convey in their discussions. Hence, the analysis of this chapter may increase our understanding of the differences between the analyses of Aumann and Ben-Porath, and thereby enhance our understanding of the epistemic conditions underlying backward induction. For ease of presentation, the analysis will be limited to two-player games, as in the rest of the book. In this chap-

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ter, this is purely a matter of convenience as everything can directly be generalized to n-player games (with n > 2). Among the large literature on backward induction during the last couple of decades,1 Reny’s (1993) impossibility result is of special importance. Reny associates a player’s ‘rationality’ in an extensive game with perfect (or almost perfect) information with what is called ‘weak sequential rationality’; i.e., that a player chooses rationally in all subgames that are not precluded from being reached by the player’s own strategy. He shows that there exist perfect information games where the event that both players satisfy weak sequential rationality cannot be commonly believed in all subgames. E.g., in the “centipede” game that is illustrated in Figure 2.4, common belief of weak sequential rationality cannot be held in the subgame defined by 2’s decision node. The reason is that if 1 believes that 2 is rational in the subgame, and if 1 believes that 2 believes that 1 will be rational in the subgame defined by 1’s second decision node, then 1 believes that 2 will choose `, implying that only Out is a best response for 1. Then the fact that the subgame defined by 2’s decision node has been reached, contradicts 2’s belief that 1 is rational in the whole game. As a response, Ben-Porath (1997) imposes that common belief of weak sequential rationality is held initially, in the whole game, only. However, backward induction is not implied if weak sequential rationality is commonly believed initially, in the whole game, only. In the “centipede” game of Figure 2.4, the strategies Out and InL for player 1 and ` and r for player 2 are consistent with such common belief, while backward induction implies that down is played at any decision node. In order to obtain an epistemic characterization of backward induction, Aumann (1995) considers ‘sequential rationality’ in the sense that a player chooses rationally in all subgames (see also footnote 3 of this chapter). However, the event that players satisfy sequential rationality is somewhat problematic. If—in the “centipede” game of Figure 2.4—1 believes or knows that 2 chooses `, then only by choosing the strategy OutL will 1 satisfy sequential rationality. However, what does it mean that 1 chooses OutL in the counterfactual event that player 2’s decision node were reached? It is perhaps more natural—as suggested by Stal-

1 Among

contributions that are not otherwise referred to in this chapter are Basu (1990), Bicchieri (1989), Binmore (1987, 1995), Bonanno (1991, 2001), Clausing and Wilks (2000), Dufwenberg and Lind´ en (1996) Feinberg (2004a), Gul (1997), Kaneko (1999), Rabinowicz (1997), and Rosenthal (1981).

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81

naker (1998)—to consider 2’s belief about 1’s subsequent action if 2’s decision node were reached. Since Aumann (1995) assumes knowledge of rational choice in an S5 partition structure, such a question of belief revision cannot be asked within Aumann’s model. By imposing a full support restriction by considering players of types in projT1 ×T2 [cau] (cf. the definition of [cau] in Section 6.3), the present chapter ensures that each player takes all opponent strategies into account, having the structural implication that conditional beliefs are welldefined and the behavioral implication that a rational choice in the whole game is a rational choice in all subgames that are not precluded from being reached by the player’s own strategy. Hence, by this restriction, we may consider ‘rationality’ instead of ‘weak sequential rationality’ (as shown by Lemma 11 and the subsequent text). The main distinguishing feature of the present analysis is, however, to consider the event that a player believes in opponent rationality rather than the event that the player himself chooses rationally. This is of course in line with the ‘consistent preferences’ approach that is the basis for this book. As is shown by Proposition 27 of Chapter 6, permissible pure strategies—strategies surviving the Dekel-Fudenberg procedure, where one round of weak elimination is followed by iterated strong elimination—can be characterized as maximal strategies when there is common certain belief that each player believes initially, in the whole game, that the opponent chooses rationally (‘belief of opponent rationality’). For generic perfect information games, Ben-Porath shows that the set of outcomes consistent with common belief of weak sequential rationality corresponds to the set of outcomes that survives the DekelFudenberg procedure. Hence, maximal strategies when there is common certain belief of ‘belief of opponent rationality’ correspond to outcomes that are promoted by Ben-Porath’s analysis. An extensive game offers choice situations, not only initially, in the whole game, but also in proper subgames. In perfect information games (and, more generally, in multi-stage games) the subgames constitute an exhaustive set of such choice situations. Hence, in perfect information games one can replace ‘belief of opponent rationality’ by ‘belief in each subgame of opponent rationality’: Each player believes in each subgame that his opponent chooses rationally in the subgame. The main results of the present chapter (Propositions 28 and 29 of Section 7.3) show how, for generic perfect information games, common certain belief of ‘belief in each subgame of opponent rationality’ is possible and uniquely deter-

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mines the backward induction outcome. Hence, by substituting ‘belief in each subgame of opponent rationality’ for ‘belief of opponent rationality’, the present analysis provides an alternative route to Aumann’s conclusion, namely that common knowledge (or certain belief) of an appropriate form of (belief of) rationality implies backward induction. This epistemic foundation for backward induction requires common certain belief of ‘belief in each subgame of opponent rationality’, where the term ‘certain belief’ is being used in the sense that an event is certainly believed if the complement is subjectively impossible. As shown by a counterexample in Section 7.3, the characterization does not obtain if instead common belief is applied.2 Furthermore, the event of which there is common certain belief — namely ‘belief in each subgame of opponent rationality’ — cannot be further restricted by taking the intersection with the event of ‘rationality’. The reason is that the full support restriction (i.e., that players are of types in projT1 ×T2 [cau]) is inconsistent with certain belief of opponent ‘rationality’, as the latter prevents a player from taking into account irrational opponent choices and rules out a well-defined theory of belief revision.

7.1

Epistemic modeling of extensive games

The purpose of this section is to present a framework for extensive games of almost perfect information where each player is modeled as a decision maker under uncertainty, with preferences that are allowed to be incomplete. An extensive game form. Inspired by Dubey and Kaneko (1984) and Chapter 6 of Osborne and Rubinstein (1994), a finite extensive twoperson game form of almost perfect information with M − 1 stages can be described as follows. The set of histories is determined inductively: The set of histories at the beginning of the first stage 1 is H 1 = {∅}. Let H m denote the set of histories at the beginning of stage m. At h ∈ H m let, for each player i, i’s action set be denoted Ai (h), where i is inactive at h if Ai (h) is a singleton. Write A(h) := A1 (h) × A2 (h). Define the set of histories at the beginning of stage m + 1 by H m+1 := {(h, a) |hS ∈ H m and a ∈ A(h)}. This concludes the induction. Denote −1 m the set of subgames and denote by Z := H M the set by H := M m=1 H of outcomes.

2 For

definitions of the certain belief operator Ki and the belief operator Bi in the current context, see Section 6.1.

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A pure strategy for player i is a function si that assigns an action in Ai (h) to any h ∈ H. Denote by Si player i’s finite set of pure strategies, and let z : S → Z map strategy profiles into outcomes, where S := S1 × S2 is the set of strategy profiles.3 Then (S1 , S2 , z) is the corresponding finite strategic two-person game form. For any h ∈ H ∪ Z, let S(h) = S1 (h) × S2 (h) denote the set of strategy profiles that are consistent with h being reached. Note that S(∅) = S. For any h, h0 ∈ H ∪ Z, h (weakly) precedes h0 if and only if S(h) ⊇ S(h0 ). If si ∈ Si and h ∈ H, let si |h denote the strategy in Si (h) having the following properties: (1) at subgames preceding h, si |h determines the unique action leading to h, and (2) at all other subgames, si |h coincides with si . Epistemic modeling. Since the extensive game form determines a finite strategic game form, we may represent the strategic interaction by means of an epistemic model as defined by Definition 9 of Chapter 5. Since backward induction is a procedure—like IESDS and the DekelFudenberg procedure—that does not rely on subjective probabilities, the analysis will allow for incomplete preferences. Hence, the epistemic model is combined with Assumption 2 of Chapter 6. In this respect the present analysis follows Aumann (1995) who presents a characterization of backward induction where subjective probabilities play no role. Conditional preferences over strategies. Write ºthi for player i’s preferences at ti conditional on subgame h ∈ H being reached; i.e., for ºtφi when φ = {ti } × Sj (h) × Tj . W.l.o.g. we may consider ºthi to be preferences over acts from Sj (h) × Tj to ∆(Z) (instead of acts from {ti } × Sj (h) × Tj to ∆(Z)). Denote by H ti := {h ∈ H| κti ∩ ({ti } × Sj (h) × Tj ) 6= ∅} the set of subgames that i deems subjectively possible at ti . Under Assumption 2 it follows from Proposition 4 that, for each ti of any player i and all h ∈ H ti , i’s conditional preferences at ti in subgame h can be conditionally represented by a vNM utility function υiti : ∆(Z) → R that does not depend on h. 3 A pure strategy s ∈ S can be viewed as an act on S that assigns z(s , s ) ∈ Z to any i i j i j sj ∈ Sj . The set of pure strategies Si is partitioned into equivalent classes of acts since a pure strategy si also determines actions in subgames which si prevents from being reached. Each such equivalent class corresponds to a plan of action, in the sense of Rubinstein (1991). As there is no need here to differentiate between identical acts in the present analysis, the concept of a plan of action would have sufficed.

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Hence, for each type ti of any player i, player i’s conditional preferences at ti in subgame h, ºthi , is a reflexive and transitive binary relation on acts from Sj (h) × Tj to ∆(Z) that is conditionally represented by a vNM utility function υiti if h ∈ H ti . Since each mixed strategy pi ∈ ∆(Si (h)) is a function that assigns the randomized outcome z(pi , sj ) to any (sj , tj ) ∈ Sj (h) × Tj and is thus an act from Sj (h) × Tj to ∆(Z), we have that ºthi determines reflexive and transitive preferences on i’s set of mixed strategies, ∆(Si ). Player i’s choice function at ti is a function Siti (·) that assigns to every h ∈ H player i’s set of maximal pure strategies at ti in subgame h: Siti (h) := {si ∈ Si (h)| @pi ∈ ∆(Si (h)), pi Âthi si } . Hence, a pure strategy, si , is in the set determined by i’s choice function at ti in subgame h if there is no mixed strategy in ∆(Si (h)) that is strictly preferred to si given i’s (possibly incomplete) conditional preferences at ti in subgame h. Refer to Siti (h) as player i’s choice set at ti in subgame h, and write Siti = Siti (∅), thereby following the notation of Chapter 6. Since ºthi is reflexive and transitive and satisfies objective independence, and Si (h) is finite, it follows that the choice set Siti (h) is nonempty and supports any maximal mixed strategies: If qi ∈ ∆(Si (h)) and @pi ∈ ∆(Si (h)) such that pi Âthi qi , then qi ∈ ∆(Siti (h)). By the following lemma, if si is maximal at ti in subgame h, then si is maximal at ti in any later subgame that si is consistent with.

Lemma 11 If si ∈ Siti (h), then si ∈ Siti (h0 ) for any h0 ∈ H with si ∈ Si (h0 ) ⊆ Si (h). Proof. The proof of this lemma is based on the concept of a ‘strategically independent set’ due to Mailath et al. (1993). The set S 0 ⊆ S is strategically independent for player i in a strategic game G = (S1 , S2 , u1 , u2 ) if S 0 = S10 × S20 and, ∀si , s0i ∈ Si0 , ∃s00i ∈ Si0 such that ui (s00i , sj ) = ui (s0i , sj ) for all sj ∈ Sj0 and ui (s00i , sj ) = ui (si , sj ) for all sj ∈ Sj \Sj0 . It follows from Mailath et al. (Definitions 2 and 3 and the ‘if’ part of Theorem 1) that S(h) is strategically independent for i for any subgame h in a finite extensive game of almost perfect information, and this does not depend on the vNM utility function that assigns payoff to any outcome. The argument is based on the property that ∃s00i ∈ Si (h) such that z(s00i , sj ) = z(s0i , sj ) for all sj ∈ Sj (h) and z(s00i , sj ) = z(si , sj ) for all sj ∈ Sj \Sj (h). The point is that i’s decision conditional on j choosing a strategy consistent with h and i’s decision conditional on j choosing a strategy inconsistent with h can be made independently.

85

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Suppose that si is not a maximal strategy at ti in the subgame h0 . Then there exists s0i ∈ Si (h0 ) such that s0i Âthi si . As noted above, S(h0 ) is strategically independent for i. Hence, ∃s00i ∈ Si (h0 ) such that z(s00i , sj ) = z(s0i , sj ) for all sj ∈ Sj (h0 ) and z(s00i , sj ) = z(si , sj ) for all sj ∈ Sj \Sj (h0 ). By Assumption 2 this implies that s00i Âthi si , which contradicts that si is most preferred at ti in the subgame h. The event that player i is rational in subgame h is defined by [rati (h)] := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | si ∈ Siti (h)} . Write [rati ] = [rati (∅)], thereby following the notation of Chapter 6. The imposition of a full support restriction by considering players of types in projT1 ×T2 [cau] (cf. the definition of [cau] in Section 6.3) has the structural implication that, for all h, the conditional preferences, ºthi , are nontrivial. Moreover, by Lemma 11 it has the behavioral implication that any choice si that is rational in h is also rational in any later subgame that si is consistent with. This means that ‘rationality’ implies ‘weak sequential rationality’. In fact, ºthi is admissible on {ti } × Sj (h) × Tjti (cf. Section 6.1), implying that any strategy that is weakly dominated in h cannot be rational in h. Thus, preference for cautious behavior is induced. However, in the context of generic perfect information games (cf. Section 7.2 of the present chapter) such admissibility has no cutting power beyond ensuring that ‘rationality’ implies ‘weak sequential rationality’; see, e.g., Lemmas 1.1 and 1.2 of Ben-Porath (1997). Hence, in the class of games considered in our main results it is of no consequence to use ‘rationality’ combined with full support rather than ‘weak sequential rationality’. An extensive game. Consider an extensive game form, and let, for each i, υi : Z → R be a vNM utility function that assigns a payoff to any outcome. Then the pair of the extensive game form and the vNM utility functions (υ1 , υ2 ) is a finite extensive two-player game of almost perfect information, Γ. Let G = (S1 , S2 , u1 , u2 ) be the corresponding finite strategic game, where for each i, the vNM utility function ui : S → R is defined by ui = υi ◦ z (i.e., ui (s) = υi (z(s)) for any s = (s1 , s2 ) ∈ S). Assume that, for each i, there exist s = (s1 , s2 ), s0 = (s01 , s02 ) ∈ S such that ui (s) > ui (s0 ). As before, the event that i plays the game G is given by [ui ] := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | υiti ◦ z is a positive affine transformation of ui } ,

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while [u] := [u1 ] ∩ [u2 ] is the event that both players play G. Conditional belief. As before, say that E ⊆ S1 × T1 × S2 × T2 does not concern player i’s strategy choice if E = Si × projTi ×Sj ×Sj E. If E does not concern player i’s strategy choice and h is deemed subjectively possible by i at ti (i.e., h ∈ H ti ), say that player i at ti believes the event E conditional on subgame h if ti ∈ projTi Bi (h)E, where Bi (h)E := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | ∃` ∈ {1, . . . , L} such that ∅ 6= ρt`i ∩ (Ti × Sj (h) × Tj ) ⊆ projTi ×Sj ×Tj E} , and (ρt1i , . . . , ρtLi ) is the vector of nested sets on which ºti is admissible. By writing, for each h ∈ H ti , β ti (h) := ρt`i ∩ (Ti × Sj (h) × Tj ), where ` := min{k ∈ {1, . . . , L}| ρtki ∩ (Ti × Sj (h) × Tj ) 6= ∅}, we have that Bi (h)E = {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | β ti (h) ⊆ projTi ×Sj ×Tj E} . It follows from the analysis of Chapter 4 that, for each h ∈ H ti , ºthi is admissible on β ti (h), and there is no smaller subset of {ti } × Sj (h) × Tj on which ºthi is admissible.4 The collection of sets {β ti (h)| h ∈ H ti } is a system of conditional filter generating sets as defined in Section 5 of Brandenburger (1998). Although completeness of preferences is not imposed under Assumption 2, ºti may encode more information about i’s preferences at ti that what is recoverable from such a system of conditional filter generating sets. It follows from the full support restriction imposed by considering players of types in projT1 ×T2 [cau] (cf. the definition of [cau] in Section 6.3) that ºti has full support on Sj , implying in turn that H ti = H and, at ti , i’s belief conditional on the subgame h is “well-defined” (in the sense that the non-empty set βiti (h) is uniquely determined) for all h ∈ H. Hence, a “well-defined” belief conditional on h is implied by full support alone; it does not require that h is actually being reached. This means that a requirement on i’s belief conditional on the subgame h is a requirement on the preferences (the type) of player i only; it does not impose that i makes a strategy choice consistent with h. Since the conditional belief operator is used only for objectively knowable events that are subjectively possible, we do not consider hypothetical events. Hence, hypothetical epistemic operators of the kind developed by Samet (1996) are not needed in the present framework. 4 The

existence of such a smaller subset would contradict Propositions 6 and 9(ii).

Backward induction

7.2

87

Initial belief of opponent rationality

A finite extensive game is ... of perfect information if, at any h ∈ H, there exists at most one player that has a non-singleton action set. ... generic if, for each i, υi (z) 6= υi (z 0 ) whenever z and z 0 are different outcomes. Generic extensive games of perfect information have a unique subgameperfect equilibrium. Moreover, in such games the procedure of backward induction yields in any subgame the unique subgame-perfect equilibrium outcome. If s∗ denotes the unique subgame-perfect equilibrium, then, for any subgame h, z(s∗ |h ) is the backward induction outcome in the subgame h, and S(z(s∗ |h )) is the set of strategy vectors consistent with the backward induction outcome in the subgame h. Both Aumann (1995) and Ben-Porath (1997) analyze generic extensive games of perfect information. As already pointed out, while Aumann establishes that common (true) knowledge of (sequential) rationality5 implies that the backward induction outcome is reached, Ben-Porath shows that the backward induction outcome is not the only outcome that is consistent with common belief (in the whole game) of (weak sequential) rationality. The purpose of the present section is to interpret the analysis of Ben-Porath by applying Proposition 27 to the class of generic perfect information games. Applying admissible consistency to extensive games. Recall that the event of admissible consistency is defined as A := A1 ∩ A2 , where Ai := [ui ] ∩ Bi [ratj ] ∩ [caui ] . Again note that a full support restriction is imposed by considering players of types in projT1 ×T2 [cau], ensuring that each player takes all opponent strategies into account. In Proposition 27 of Chapter 6 we have established that the concept of permissible pure strategies can be characterized as maximal pure strategies under common certain belief of admissible consistency. Recall also that permissible strategies (cf. Definition 13 of Chapter 5) correspond to 5 Aumann

(1995) uses the term substantive rationality, meaning that for all histories h, if a player were to reach h, then the player would choose rationally at h. See Aumann (1995, pp. 14–16) and Aumann (1998) as well as Halpern (2001) and Stalnaker (1998, Section 5).

88

CONSISTENT PREFERENCES

strategies surviving the Dekel-Fudenberg procedure, where one round of weak elimination is followed by iterated strong elimination. In the context of generic perfect information games, Ben-Porath (1997) establishes through his Theorem 1 that the set of outcomes consistent with common belief (initially, in the whole game) of (weak sequential) rationality corresponds to the set of outcomes that survive the Dekel-Fudenberg procedure. Hence, by Proposition 27, maximal strategies when there is common certain belief of admissible consistency correspond to the outcomes promoted by Ben-Porath’s analysis. An example. To illustrate how common certain belief of admissible consistency is consistent with outcomes other than the unique backward induction outcome, consider the strategic game G03 , with corresponding extensive form Γ03 ; i.e., the “centipede” game illustrated in Figure 2.4. Here, backward induction implies that down is being played at any decision node. Let T1 = {t01 , t001 } and T2 = {t02 , t002 }. Assume that the preferences of each type ti of any player i are represented by a vNM utility function υiti satisfying υiti ◦ z = ui and a 2-level LPS on Sj × Tj . In Table 7.1, the first numbers in the parentheses express primary probability distributions, while the second numbers express secondary probability distributions. The strategies OutL and OutR are merged as their relative likelihood does not matter; see footnote 3. Note that all types are in projT1 ×T2 [cau], implying that players take all opponent strategies into account. With these 2-level LPSs each type’s preferences over the player’s own strategies are given by 0

0

Out Ât1 InL Ât1 InR 00 00 InL Ât1 Out Ât1 InR 0 ` Ât2 r 00 r Ât2 ` It is easy to check that both players satisfy ‘belief of opponent rationality’ at each of their types; e.g., both t02 and t002 assign positive (primary) probability to an opponent strategy-type pair only if it is a maximal strategy for the opponent type (i.e., Out in the case of t01 and InL in the case of t001 ). Thus, S1 × T1 × S2 × T2 ⊆ A. Since, for each ti ∈ Ti of any player i, κti ⊆ {ti } × Sj × Tj , it follows that S1 × T1 × S2 × T2 ⊆ CKA. Hence, preferences consistent with common certain belief of admissible consistency need not reflect backward induction since InL and r are maximal strategies. Note that, conditional on player 2’s decision node being reached (i.e. 1 choosing InL or InR), player 2 at t02 updates her beliefs about the type

89

Backward induction

An epistemic model for G03 with corresponding extensive form Γ03 .

Table 7.1.

t01 :

t02 :

` r

Out InL InR

0 00 ¡ 4 t27 ¢ ¡ t21 ¢ 10 ¢ ¢ ¡ 0, ¡ 5 , 10 1 1 1 , 0, 10 5 10

t001 :

0 ¡ 1t11 ¢ ¡ 2 , 14 ¢ ¡0, 81 ¢ 0, 8

t002 :

00

¡ t11 ¢ 8¢ ¡ 0, 1 1 , 2 ¡ 14 ¢ 0, 8

` r

0 00 ¡ 3 t25 ¢ ¡ t21 ¢ 10 ¢ ¡ 5 , 10 ¢ ¡ 0, 2 3 1 0, 10 , 5 10 0

Out InL InR

¡ t11 ¢ ¡1, 12 ¢ ¡0, 41 ¢ 0, 4

t001 (0, 0) (0, 0) (0, 0)

of player 1 and assigns (primary) probability one to player 1 being of type t001 . Consequently, the conditional belief of player 2 at t02 assigns (primary) probability one to player 1 choosing InL. Player 2 at t002 , on the other hand, does not admit the possibility that 1 is of another type than t01 . Since the choice of In at 1’s first decision node is not rational for player 1 at t01 , there is no restriction concerning the conditional belief of player 2 at t002 about the choice at 1’s second decision node. In the terminology of Ben-Porath, a “surprise” has occurred. Subsequent to such a surprise, a player need not believe that the opponent chooses rationally among his remaining strategies.

7.3

Belief in each subgame of opponent rationality

A simultaneous game offers only one choice situation. Hence, for a game in this class, it seems reasonable that belief of opponent rationality is held in the whole game only, as formalized by the requirement ‘belief of opponent rationality’. An extensive game with a nontrivial dynamic structure, however, offers such choice situations, not only initially, in the whole game, but also in proper subgames. Moreover, for extensive games of almost perfect information, the subgames constitute an exhaustive set of such choice situations. This motivates imposing belief in each subgame of opponent rationality. Hence, consider the event that i believes conditional on subgame h ∈ H ti that j is rational in h: Bi (h)[ratj (h)] = {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | 0

(s0j , t0j ) ∈ projSj ×Tj β ti (h) implies s0j ∈ Sjtj (h)} ,

90

CONSISTENT PREFERENCES

ti = H whenever t ∈ proj [cau ], it follows that, if t ∈ Since £¡ HT i¢ i i T¤i projTi h∈H ti Bi (h)[ratj (h)] ∩ [caui ] , then at ti player i believes conditional on any subgame h that j is rational in h. In other words, ³\ ´ B (h)[rat (h)] ∩ [caui ] i j t h∈H

i

is the event that player i believes in each subgame h that the opponent j is rational in h.6 Consider a finite extensive two-player game Γ of almost perfect information with corresponding strategic game G. Say that at ti player i’s preferences over his strategies are admissibly subgame consistent with the game Γ and the preferences of his opponent if ti ∈ projTi A∗i , where ³\ ´ A∗i := [ui ] ∩ B (h)[rat (h)] ∩ [caui ] . i j t h∈H

i

Refer to A∗ := A∗1 ∩ A∗2 as the event of admissible subgame consistency. This definition of admissible subgame consistency can be applied to any finite extensive game of almost perfect information. However, in order to relate to Aumann’s (1995) Theorems A and B, the following analysis is concerned with generic perfect information games. The example revisited. In the belief system of Table 7.1, player 2 at type t002 does not satisfy ‘belief in each subgame of opponent rationality’. By ‘belief in each subgame of opponent rationality’, player 2 must believe, conditional on the subgame defined by 2’s decision node, that 1 chooses his maximal strategy, InL, in the subgame. This means that player 2 prefers ` to r, implying that player 1 must prefer Out to InL if he satisfies ‘belief in each subgame of opponent rationality’. Thus, common certain belief of admissible subgame consistency entails that any types of players 1 and 2 have the preferences Out Ât1 InL Ât1 InR ` Ât2 r

6 Note

that the requirement of such ‘belief in each subgame of opponent rationality’ allows a player to update his belief about the type of his opponent. Hence, there is no assumption of ‘epistemic independence’ between different agents in the sense of Stalnaker (1998); cf. the remark after the proof of Proposition 28 as well as Section 7.4. Still, the requirement can be considered a non-inductive analog to ‘forward knowledge of rationality’ as defined by Balkenborg and Winter (1997), and it is related to the requirement in Section 5 of Samet (1996) that each player hypothesizes that if h were reached, then the opponent would behave rationally at h.

Backward induction

91

respectively, meaning that if a player chooses a maximal strategy in a subgame, then his choice is made in accordance with backward induction. Demonstrating that this conclusion holds in general for generic perfect information games constitutes the main results of the present chapter. Main results. In analogy with Aumann’s (1995) Theorems A and B, it is established that ... any vector of maximal strategies in a subgame of a generic perfect information game, in a state where there is common certain belief of admissible subgame consistency, leads to the backward induction outcome in the subgame (Proposition 28). Hence, by substituting T B t h∈H i i (h)[ratj (h)] for Bi [ratj ], the present analysis yields support to Aumann’s conclusion, namely that if there is common knowledge (or certain belief) of an appropriate form of (belief of) rationality, then backward induction results. ... for any generic perfect information game, common certain belief of admissible subgame consistency is possible (Proposition 29). Hence, the result of Proposition 28 is not empty.

Proposition 28 Consider a finite generic extensive two-player game of perfect information Γ with corresponding strategic game G. If, for some epistemic model, (t1 , t2 ) ∈ projT1 ×T2 CKA∗ , then, for each h ∈ H, S1t1 (h) × S2t2 (h) ⊆ S(z(s∗ |h )), where s∗ denotes the unique subgameperfect equilibrium. Proof. In view of properties of the certain belief operator (cf. Proposition 25(ii)), it suffices to show for any g = 0, . . . , M − 2 that S1t1 (h) × S2t2 (h) ⊆ S(z(s∗ |h )) for any h ∈ H M −1−g if there exists an epistemic model with (t1 , t2 ) ∈ projT1 ×T2 Kg A∗ . This is established by induction. (g = 0) Let h ∈ H M −1 . First, consider j with a singleton action set at h. Then trivially Sjtj (h) = Sj (h) = Sj (z(s∗ |h )). Now, consider i with a non-singleton action set at h; since Γ has perfect information, there is at most one such i. Let ti ∈ projTi K0 A∗ = projTi A∗ . Then it follows that Siti (h) = Si (z(s∗ |h )) since Γ is generic and A∗ ⊆ [ui ] ∩ [caui ]. (g = 1, . . . , M − 2) Suppose that it has been established for g 0 = 0 0, . . . , g − 1 that S1t1 (h0 ) × S2t2 (h0 ) ⊆ S(z(s∗ |h0 )) for any h0 ∈ H M −1−g 0 if there exists an epistemic model with (t1 , t2 ) ∈ projT1 ×T2 K g A∗ . Let h ∈ H M −1−g . Part 1. Consider j with a singleton action set at h. Let tj ∈ projTj Kg−1 A∗ . Then Sj (h) = Sj (h, a) and, by Lemma 11 and the premise, Sjtj (h) ⊆ Sj tj (h, a) ⊆ Sj (z(s∗ |(h,a) )) if a is a feasible action

92

CONSISTENT PREFERENCES

vector at h. This implies that \ Sjtj (h) ⊆ Sj (z(s∗ |(h,a) )) ⊆ Sj (z(s∗ |h )) . a

tj (h),

Hence, if sj ∈ Sj then sj is consistent with the backward induction outcome in any subgame (h, a) immediately succeeding h. Part 2. Consider i with a non-singleton action set at h; since Γ has perfect g ∗ information, there is at most one such i. Let Ti K A . The T ti ∈ proj t ∗ j preceding argument implies that Sj (h) ⊆ a Sj (z(s |(h,a) )) whenever tj ∈ Tjti since ti ∈ projTi Kg A∗ ⊆ projTi Ki Kg−1 A∗ . Let si ∈ Si (h) be a strategy that differs from s∗i |h by assigning a different action at h (i.e., z(si , s∗j |h ) 6= z(s∗ |h ) and si (h0 ) = s∗i |h (h0 ) whenever Si (h) ⊃ Si (h0 )). Let p and q be acts on Sj × Tj satisfying that, ∀(sj , tj ) ∈ Sj × Tj , p(sj , tj ) = z(s∗i , sj ) and q(sj , tj ) = z(si , sj ). Then, p∩a Sj (z(p|(h,a) ))×Tj

strongly dominates q∩a Sj (z(p|(h,a) ))×Tj

by backward induction since Γ is generic and ti ∈ projTi Kg A∗ ⊆ [ui ]. T Since Sjtj (h) ⊆ a Sj (z(s∗ |(h,a) )) whenever tj ∈ Tj ti , it follows that, ∀tj ∈ Tjti , pSjtj (h)×{tj }

strongly dominates qSjtj (h)×{tj } ,

and, thus, ti ∈ projTi Kg A∗ ⊆ Bi (h)[ratj (h)] ∩ [caui ] implies that p Âthi q . It has thereby been established that si ∈ Si (h)\Siti (h) if si differs from backward induction only by the action taken at h. However, the premise that Siti (h, a) ⊆ Si (z(s∗ |(h,a) )) if a is a feasible action vector at h, it follows that any si ∈ Siti (h) is consistent with the backward induction outcome in the subgame (h, (si (h), aj )) immediately succeeding h when i plays the action si (h) at h (since si ∈ Si (h, (sj (h), aj )) and, by Lemma 11, si ∈ Siti (h, (si (h), aj )). Hence, Siti (h) ⊆ Si (z(p|h )). It follows from the proof of Proposition 28 that, for a generic perfect information game with M −1 stages, it is sufficient with M −2 order mutual certain belief of admissible subgame consistency in order to obtain backward induction. Hence, KM −2 A∗ can be substituted for CKA∗ . Backward induction will not be obtained, however, if CBA∗ is substituted for CKA∗ . This can be shown by considering a counter-example that builds on the four-legged centipede game of Figure 7.1 and the epistemic model of Table 7.2. In the table the preferences of each type ti of any player i are represented by a vNM utility function υiti satisfying

93

Backward induction

1c Out 2 0

In

Figure 7.1.

Table 7.2. t01 : ` r`0 rr 0 t02 : Out InL InR

2s ` 1 3

1s

r

2s

R

L 4 2

`0 3 5

6 4

r0

Γ5 (a four-legged “centipede” game).

An epistemic model for Γ5 .

t02 4 7 , , 5 10 1 0, 10 , 0, 0,




7 12
1 12
1 12

t01 1 1 , , 2 3 0, 16 , 0, 0,




1 4
1 8
1 8

t001 :

t002 1 0, 10 , 1 1 , , 5 10 0, 0,

t001 0, 16 , 1 1 , , 2 3 0, 0,




1 12
1 12
1 12




1 8
1 4
1 8

` r`0 rr0

t000 1 (0, 0, 0) (0, 0, 0) (0, 0, 0)

t02 3 5 , , 5 10 1 0, 10 , 0, 0,




5 12
1 12
1 12

t002 : Out InL InR

t002 1 0, 10 , 2 3 , , 5 10 0, 0,

t000 1 :


1 12
3 12
1 12

t01
1, 21 , 13
0, 41 , 16
0, 0, 61

` r`0 rr 0

t001 (0, 0, 0) (0, 0, 0) (0, 0, 0)

t02
1 10
1 10
3 10

t002
1 10
1 10
3 10

t000 1
1 0, 0, 12
1 0, 0, 12
1 1 0, 4 , 6

υiti ◦ z = ui and a 1 or 3-level LPS on Sj × Tj , where T1 = {t01 , t001 , t000 1} 0 00 and T2 = {t2 , t2 }. While all types are in projT1 ×T2 [cau], implying that players take all opponent strategies into account, inspection shows that A∗ = S1 ×{t01 , t001 }×S2 ×{t02 , t002 }, since player 1 at t000 1 does not satisfy ‘belief in each subgame of opponent rationality’. Furthermore, each player i believes at t0i or t00i that the opponent is of a type in {t0j , t00j }. This implies that CBA∗ = A∗ . Since InL is the maximal strategy for 1 at t001 and r`0 is the maximal strategy for 2 at t002 , it follows that preferences consistent with common belief of admissible subgame consistency need not reflect backward induction. However, 2 does not certainly believe at t002 that the ∗ ∗ 0 00 0 opponent is not of type t000 1 . Therefore, KA = A = S × {t1 , t1 } × {t2 }, while KKA∗ = ∅. Hence, preferences that yield maximal strategies in contradiction with backward induction are not consistent with common certain belief of admissible subgame consistency. The example shows that ti ∈ projTi A∗i is consistent with player i at ti updating his beliefs about the preferences of his opponent conditional on a subgame being reached. I.e., 1 at t01 assigns initially, in the whole, (primary) probability 45 to 2 being of type t02 with preferences ` Â r`0 Â rr0 ,

94

CONSISTENT PREFERENCES

while in the subgame defined by 1’s second decision node 1 at t01 assigns (primary) probability one to 2 being of type t002 with preferences r`0 Â ` ∼ rr. This shows that Stalnaker’s (1998) assumption of ‘epistemic independence’ is not made; a player is in principle allowed to learn about the type of his opponent on the basis of previous play. However, in an epistemic model with CKA∗ 6= ∅, ti ∈ projTi CKA∗ implies that 1 certainly believes at ti that 2 is of a type with preferences ` Â r`0 Â rr0 . In other words, if there is common certain belief of admissible subgame consistency, there is essentially nothing to learn about the opponent.

Proposition 29 For any finite generic two-player extensive game of perfect information Γ with corresponding strategic game G, there exists a belief system for G with CKA∗ 6= ∅. Proof. Construct an epistemic model with one type of each player: T1 = {t1 } and T2 = {t2 }. Write, for each player j, ∀m ∈ {1, . . . , M − 1}, Sj∗ m := {s∗j |h | h ∈ H m } and, Sj∗ M := Sj . Let, for each player i, λti = (µt1i , . . . , µtMi ) ∈ L∆(Sj × {tj }) satisfy the following requirement: ∀m ∈ {1, . . . , M }, suppµtmi = Sj∗ m × {tj }. By letting ºti be represented by a vNM utility function υiti satisfying υiti ◦ z = ui and the LPS λti , then (1) [ui ] ∩ [caui ] = S1 × T1 × S2 × T2 . Let, ∀h ∈ H, λti |h = (µ01ti , . . . µ0Mti|h ) denote the conditional of λti on Sj (h)×Tj . By the properties of a subgame-perfect equilibrium, ∀h ∈ H, µ01ti (s∗j |h , tj ) = 1 ti ∗ ∗ tj and T si |h ∈ Si (h). Hence, since likewise sj |h ∈ Sj (h), we have that (2) h∈H ti Bi (h)[ratj (h)] = S1 × T1 × S2 × T2 . As (1) and (2) hold for both players, it follows that CKA∗ = A∗ = S1 × T1 × S2 × T2 6= ∅.

7.4

Discussion

In this section we first interpret our analysis in view of Aumann (1995) and then present a discussion of the relationship to Battigalli (1996a). Adding belief revision to Aumann’s analysis. Consider a generic perfect information game. Say that a player’s preferences (at a given type) are in accordance with backward induction if, in any subgame, a strategy is a rational choice only if it is consistent with the backward induction outcome. Using this terminology, Proposition 28 can be restated as follows: Under common certain belief of admissible subgame consistency, players are of types with preferences that are in accordance with backward induction. Furthermore, common certain belief of admissible subgame consistency implies that each player deems it subjectively

Backward induction

95

impossible that the opponent is of a type with preferences not in accordance with backward induction. However, since admissible subgame consistency is imposed on preferences, reaching 2’s decision node and 1’s second decision node in the centipede game of Figure 2.4 does not contradict common certain belief of admissible subgame consistency. Of course, these decision nodes will not be reached if players choose rationally. But that players satisfy ‘belief in each subgame of opponent rationality’ is not a requirement concerning whether their own choice is rational; rather, it means that they believe (with probability one) in any subgame that their opponent will choose rationally. Combined with the assumption that all types are in projT1 ×T2 [cau], which entails that each player deems any opponent strategy subjectively possible, this means that belief revision is well-defined. Hence, on the one hand, we capture the spirit of a conclusion that can be drawn from Aumann’s (1995) analysis, namely that when being made subject to epistemic modeling backward induction corresponds to each player having knowledge (or being certain) of some essential feature of the opponent. In Aumann’s case, each player deems it impossible— under common (true) knowledge of (sequential) rationality—that the opponent makes an action inconsistent with backward induction. The analogous result in the present case is that each player deems it subjectivly impossible—under common certain belief of admissible subgame consistency—that the opponent has preferences not in accordance with backward induction. On the other hand, we are still able to present an explicit analysis of how players revise their beliefs about the opponent’s subsequent choice if surprising actions were to be made. As noted in the introduction to this chapter, this fundamental issue of belief revision cannot formally be raised within Aumann’s framework. Stalnaker (1998) argues—contrary to statements made by Aumann (1995, Section 5f)—that an assumption of belief revision is implicit in Aumann’s motivation, namely that information about different agents of the opponent is treated as epistemically independent. In the reformulation by Halpern (2001),7 this means that in a state “closest” to the current state when a player learns that the opponent has not followed

7 See

Halpern (2001) for an instructive discussion of the differences between Aumann (1995) and Stalnaker (1998), as well as how these relate to Samet (1996).

96

CONSISTENT PREFERENCES

her strategy, he believes that the opponent will follow her strategy in the remaining subgame. There is no assumption of ‘epistemic independence’ in the current interpretation of Aumann’s result. Instead, we have changed statements ‘about opponents’ from being concerned with strategy choice to being related to preferences. While it is desirable when modeling backward induction to have an explicit theory of revision of beliefs about opponent choice, a theory of revision of beliefs about opponent preferences is inconsistent with maintaining both (a) that preferences are necessarily revealed from choice, and (b) that there is common certain belief of the game being played (i.e., consider the case where Ai (∅) is non-singleton, and ai ∈ Ai (∅) ends the game and leads to an outcome that is preferred by i to any other outcome). Here we have kept the assumption that there is common certain belief of the game, meaning that the game is of ‘complete information’, while requiring only conditional belief in each subgame of opponent rationality, meaning that irrational opponent choices—although being probability zero events—are not subjectively impossible. We have shown how common certain belief of admissible subgame consistency implies that each player deems it impossible that the opponent has preferences not in accordance with backward induction and thus interprets any deviation from the backward induction path as the opponent not having made a rational choice. In this way we present a model that combines a result that resembles Aumann (1995) by associating backward induction with certainty about opponent type, with an analysis that unlike Aumann’s yields a theory of belief revision about opponent choice. Rationality orderings. The constructive proof of Proposition 29 shows how common certain belief of admissible subgame rationality may lead player i at ti to have preferences over i’s strategies that are represented by a vNM utility function υiti satisfying υiti ◦ z = ui and an LPS λti = (µt1i , ..., µtLi ) ∈ L∆(Sj × Tj ) with more than two levels of subjective probability distributions (i.e., L > 2). E.g., in the “centipede” game of Figure 2.4, common certain belief of admissible subgame consistency implies that player 2 at any type t2 has preferences that can be represented by υ2t2 satisfying υ2t2 ◦ z = u2 and λt2 = (µt12 , µt22 , µt32 ) satisfying projS1 suppµt12 = {Out}, projS1 suppµt22 = {Out, InL}, and projS1 suppµt32 = S1 . One may interpret projSj suppµt1i to be j’s “most rational” strategies,

97

Backward induction

S projSj suppµtLi \ k
k
projSj suppµtki

´

corresponds closely to what Battigalli (1996a) calls a rationality ordering for j. However, the present construction of such a rationality ordering differs from the one proposed by Battigalli. This difference is along two dimensions: 1 Battigalli considers best responses in reachable subgames only (see his Definition 2.1), while here belief of opponent rationality is held in all subgames (cf., ‘belief in each subgame of opponent rationality’). 2 Battigalli considers best responses given beliefs where opponent strategies that are less than “most rational” are given positive probability, while here each player always believes that the opponent chooses rationally. This difference has the following consequences: Although Battigalli’s construction of rationality orderings also yields the backward induction outcome in any generic perfect information game, his proof (cf. Battigalli, 1997) is not tied to the backward induction procedure. Battigalli’s construction promotes the forward induction outcome (InL, `) in the “battle-of-the-sexes with an outside option” game illustrated in Figure 2.6. This conclusion is not reached in the present analysis since there is no choice situation in which 1 under all circumstances will have a particular preference between his “battle-ofthe-sexes” strategies.8 This also indicates how the epistemic foundation for the backward induction procedure offered here differs from the epistemic foundation for backward (and forward) induction outcomes provided by Battigalli and Siniscalchi (2002). 8 Chapter

11 will, following Asheim and Dufwenberg (2003a), demonstrate how the concept of admissible consistency can be strengthened so that the forward induction outcome is promoted in the “battle-of-the-sexes with an outside option” game.

Chapter 8 SEQUENTIALITY

One major problem in the theory of extensive games is the following: How should a player react when he finds himself at an information set that contradicts his previous belief about the opponent’s strategy choice? Different approaches have been proposed to this problem. As mentioned in Chapter 2, Ben-Porath (1997) and Reny (1992) have formulated rationalizability and equilibrium notions based on weak sequentiality, in which a player is allowed to believe, in this situation, that his opponent will no longer choose rationally. Battigalli and Siniscalchi (2002) have shown that Pearce’s (1984) extensive form rationalizability can be characterized by assuming that a player, in such a situation, should look for the highest degree of “strategic sophistication” that is compatible with the event of reaching this information set, and stick to this degree until it is contradicted later on in the game. Perea (2002, 2003) suggests that the player, in such a situation, may revise his conjecture about the opponent’s utility function in order to rationalize her “surprising” move, while maintaining common belief of rational choice at all information sets. The most prominent position, however, is that the player should still believe that his opponent will choose rationally in the remainder of the game; this underlies concepts that promote backward induction. Such concepts will be presented in this and the next chapter, which reproduce joint work with Andr´es Perea, cf. Asheim and Perea (2004). We define sequential rationalizability by imposing common certain belief of the event that each player believes that the opponent chooses rationally at all her information sets. Since this is a non-equilibrium concepts, each player need not be certain of the beliefs that the oppo-

100

CONSISTENT PREFERENCES

nent has about the player’s own action choice. However, by assuming that each player is certain of the beliefs that the opponent has about the player’s own action choice, we obtain an epistemic characterization of the corresponding equilibrium concept: sequential equilibrium. When applied to generic games with perfect information, sequential rationalizability yields the backward induction procedure. As elsewhere, to avoid the issue of whether (and if so, how) each player’s beliefs about the action choice of his opponents are stochastically independent, all analysis is limited to two-player games. The assumption is essential in the present context where a behavior strategy of a player will be interpreted as an expression of the belief of his opponent. For the above mentioned definitions and characterizations, we must describe what a player believes both conditional on reaching his own information sets (to evaluate his rationality) and conditional on his opponent reaching her information sets (to determine his beliefs about her choices). Hence, we must specify a system of conditional beliefs for each player. For reasons given in Section 3.1, this will be done by means of our concept of a system of conditional lexicographic probabilities (SCLP) as defined in Definition 1 and characterized in Proposition 5. We embed the notion of an SCLP in an epistemic model, as defined by Definition 9 of Chapter 5, by invoking Assumption 1. For each type ti of any player i, ti is described by an SCLP, inducing a behavior strategy for each opponent type tj that is deemed subjectively possible by ti . The event that ‘player i believes that the opponent j chooses rationally at each information set’ can then be defined as the event where player i is of a type ti that, for each subjectively possible opponent type tj , induces a behavior strategy which is sequentially rational given tj ’s own SCLP. The characterization of sequential equilibrium reported in Proposition 30 is included in order to motivate the analogous non-equilibrium concept, namely sequential rationalizability. The result may, however, be of interest in its own right and in comparison with other such epistemic characterizations; see, e.g., Theorem 2 of Feinberg (2004b). The concept of sequential rationalizability as stated in Definition 15 is related to various other concepts proposed in the literature. Already in Bernheim (1984) there are suggestions concerning how to define nonequilibrium concepts that involve rational choice at all information sets. By requiring rationalizability in every subgame, Bernheim defines the concept of subgame rationalizability—which coincides with our definition of sequential rationalizability for games of almost perfect information—

Sequentiality

101

but no epistemic characterization is offered. On p. 1022 Bernheim claims that it is possible to define a concept of sequential rationalizability, but does not indicate how this can be done. After related work by Greenberg (1996), sequential rationalizability was finally defined by Dekel et al. (1999, 2002), whose concept coincides with ours in our two-player setting. Our definition of quasi-perfect rationalizability is new. Dekel et al. (1999) and Greenberg et al. (2003) consider also extensive game concepts that lie between equilibrium and rationalizability; such concepts will not be considered here.

8.1

Epistemic modeling of extensive games (cont.)

The purpose of this section is to present a framework for a general class of extensive games where each player is modeled as a decision maker under uncertainty with complete preferences. An extensive game form. Consider a finite extensive two-player game form without chance moves. Assume that the extensive game form satisfies perfect recall. Denote by Hi the finite collection of information sets controlled by player i. For every information set h ∈ Hi , let A(h) be the set of actions available at h. A pure strategy for player i is a function si which assigns to every information set h ∈ Hi some action si (h) ∈ A(h). Denote by Si the set of pure strategies for player i, where, in the subsequent analysis, there is no need to differentiate between pure strategies in Si that differ only at non-reachable information sets. Write S = S1 × S2 , denote by Z the set of outcomes (or terminal nodes), and let z : S → Z map strategy profiles into terminal nodes. Then (S1 , S2 , z) is the corresponding finite strategic two-player game form. For any h ∈ H1 ∪ H2 , let Si (h) be the set of strategies si for which there is some strategy sj such that (si , sj ) reaches h. For any h and any node x ∈ h, denote by S(x) = S1 (x) × S2 (x) the S set of pure strategy profiles for which x is reached, and write S(h) := x∈h S(x). By perfect recall, it holds that S(h) = S1 (h) × S2 (h) for all information sets h. For any h, h0 ∈ Hi , h (weakly) precedes h0 if and only if S(h) ⊇ S(h0 ). For any h ∈ Hi and a ∈ A(h), write Si (h, a) := {si ∈ Si (h)|si (h) = a}. A behavior strategy for player i is a function σi that assigns to every h ∈ Hi some randomization σi (h) ∈ ∆(A(h)) on the set of available actions. If h ∈ Hi , denote by σi |h the behavior strategy with the following properties: (1) at player i information sets preceding h, σi |h determines with probability one the unique action leading to h, and (2) at all other

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player i information sets, σi |h coincides with σi . Say that σi is outcomeequivalent to a mixed strategy pi (∈ ∆(Si )) if, for any sj ∈ Sj , σi and pi induce the same probability distribution over terminal nodes. For any h ∈ Hi , σi |h is outcome-equivalent to some pi ∈ ∆(Si (h)). Epistemic modeling. Since the extensive game form determines a finite strategic game form, we may represent the strategic interaction by means of an epistemic model as defined by Definition 9 of Chapter 5. Since a behavior strategy of a player will be interpreted as an expression of the belief of his opponent, it is essential that the analysis assumes complete preferences. Hence, the epistemic model is combined with Assumption 1 of Chapter 5. Under Assumption 1 it follows from Proposition 5 that, for each type ti of any player i, i’s system of conditional preferences at ti can be represented by a vNM utility function υiti : ∆(Z) → R and an SCLP (λti , `ti ), which for expositional simplicity is defined on Sj × Tj with support Sj × Tjti (instead of being defined on Ti × Sj × Tj with support κti = {ti } × Sj × Tjti ). Hence, writing ºthi for player i’s preferences at ti conditional on player i information set h ∈ Hi being reached, we consider w.l.o.g. ºthi to be preferences over acts from Sj (h) × Tj to ∆(Z) (instead of acts from {ti } × Sj (h) × Tj to ∆(Z)). Conditional preferences over strategies. It follows that, for each ti of any player i and all h ∈ Hi , i’s conditional preferences at ti in subgame h can be represented by the vNM utility function υiti : ∆(Z) → R that does not depend on h, and an LPS i λt`(S | = (µ01 , . . . µ0`(Sj (h)×Tj )|Sj (h)×Tj ) j (h)×Tj ) Sj (h)×Tj

derived from the SCLP (λti , `ti ) on Sj × Tj with support Sj × Tjti . Recall from Assumption 1 that player i deems an opponent strategytype pair (sj , tj ) subjectively possible at ti if and only if sj ∈ Sj and tj ∈ Tj ti . This means that conditional preferences are non-trivial for an event Ej (⊆ Sj × Tj ) if and only if Ej ∩ (Sj × Tjti ) 6= ∅. Note that {Sj (h)×Tj | h ∈ Hi } is the set of events that are objectively observable by i. Hence, conditional preferences are always non-trivial for such events since, for any h ∈ Hi , (Sj (h) × Tj ) ∩ (Sj × Tjti ) = (Sj (h) × Tjti ) 6= ∅. Since, for all h ∈ Hi , each pure strategy si ∈ Si (h) is a function that assigns the deterministic outcome z(si , sj ) to any (sj , tj ) ∈ Sj (h) × Tj , it follows that si ∈ Si (h) is an act from Sj (h) × Tj to ∆(Z), and we have that ºthi determines complete and transitive preferences on i’s set of pure strategies, Si (h), conditional on h ∈ Hi being reached.

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Player i’s choice function at ti is a function Siti (·) that assigns to every h ∈ Hi player i’s set of rational pure strategies at ti conditional on h ∈ Hi : Siti (h) := {si ∈ Si (h)| ∀s0i ∈ Si (h), si ºthi s0i } . Refer to Siti (h) as player i’s choice set at ti conditional on player i information set h, and write Siti = Siti (∅), thereby following the notation of Chapter 5. Since ºthi is complete and transitive and satisfies objective independence, and Si (h) is finite, it follows that the choice set Siti (h) is nonempty, and that the set of rational mixed strategies equals ∆(Siti (h)). Note that Lemma 11 does not hold under Assumption 1, unless also caution is imposed (cf. Section 9.1). The assumption that player i’s system of conditional preferences at ti is representable by means of an SCLP where the set of subjectively possible opponent types equals Sj × Tj ti has the structural implication that, for all h ∈ Hi , the conditional preferences, ºthi , are nontrivial, even without imposing caution. However, representation by means of an SCLP does not have the behavioral implication that any choice si that is rational conditional on h is also rational in any later player i information set that si is consistent with. This means that ‘rationality’ does not imply ‘weak sequential rationality’ if caution is not imposed. An extensive game. As in Chapter 7, a finite extensive two-player game consists of the pair of the extensive game form and the vNM utility functions (υ1 , υ2 ), with G = (S1 , S2 , u1 , u2 ) denoting the corresponding finite strategic game, where for each i, the vNM utility function ui : S → R is defined by ui = υi ◦ z. As before—but transferred to T1 × T2 space—the event that i plays the game G is given by [ui ] := {(t1 , t2 ) ∈ T1 ×T2 |υiti ◦z is a positive affine transformation of ui }, while [u] := [u1 ] ∩ [u2 ] is the event that both players play G. Certain belief. As in Chapter 5, say for any E ⊆ T1 × T2 that player i certainly believes the event E at ti if ti ∈ projTi Ki E, where Ki E := {(t1 , t2 ) ∈ T1 × T2 | projT1 ×T2 κti = {ti } × Tjti ⊆ E} . Say that there is mutual certain belief of E at (t1 , t2 ) if (t1 , t2 ) ∈ KE, where KE := K1 E ∩ K2 E. Say that there is common certain belief of E at (t1 , t2 ) if (t1 , t2 ) ∈ CKE, where CKE := KE ∩ KKE ∩ KKKE ∩ . . . .

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8.2

Sequential consistency

In this section, we use the epistemic model of the previous section— based on the concept of an SCLP—to formalize the requirement that each player believes that the opponent chooses rationally at each of her information sets, given her preferences at these information sets. This enables us to characterize sequential equilibrium (Kreps and Wilson, 1982), and to define sequential rationalizability as a non-equilibrium analog to the concept of Kreps and Wilson (1982). Inducing sequential rationality. In our setting a behavior strategy is not an object of choice, but an expression of the system of beliefs of the other player. Say that the behavior strategy σj ti |tj is induced for tj by ti if tj ∈ Tjti and, for all h ∈ Hj and a ∈ A(h), σjti |tj (h)(a) :=

µt`i (Sj (h, a), tj ) µt`i (Sj (h), tj )

,

where ` is the first level ` of λti for which µt`i (Sj (h), tj ) > 0, implying that µt`i restricted to Sj (h) × {tj } is proportional to the top level probability distribution of the LPS that describes ti ’s conditional P ti ti belief on Sj (h) × {t }. Here, µ (S (h), t ) := j j sj ∈Sj (h) µ` (sj , tj ) and ` Pj ti ti µ` (Sj (h, a), tj ) := sj ∈Sj (h,a) µ` (sj , tj ). Say that the behavior strategy σi is sequentially rational for i at ti if, ∀h ∈ Hi , σi |h is outcome-equivalent to some mixed strategy in ∆(Siti (h)). Define the event that player i is of a type that i nduces a sequentially r ational behavior strategy for any opponent type that is deemed subjectively possible: [isri ] := {(t1 , t2 ) ∈ T1 × T2 | ∀t0j ∈ Tjti , 0

σj ti |tj is sequentially rational for t0j } . Write [isr] := [isr1 ] ∩ [isr2 ] for the event where both players are of such a type. Say that at ti player i’s preferences over his strategies are sequentially consistent with the game Γ and the preferences of his opponent, if ti ∈ projTi ([ui ] ∩ [isri ]). Refer to [u] ∩ [isr] as the event of sequential consistency. Note that the behavior strategy induced for t0j by ti specifies i’s belief revision policy at ti about the behavior of t0j , as it defines probability

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distributions also at player j information sets that are unreachable given i’s initial belief at ti about t0j ’s behavior. Hence, if ti ∈ projTi [isri ], then player i believes at ti that each subjectively possible opponent type t0j chooses rationally also at player j information sets that contradict ti ’s initial belief about the behavior of t0j . The above observation explains why we can characterize a sequential equilibrium as a profile of induced behavior strategies at a type profile in [isr] where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible). Characterizing sequential equilibrium. We first define sequential equilibrium. Player i’s beliefs over past opponent actions at i’s information sets is a function βi that to any h ∈ Hi assigns a probability distribution over the nodes in h. An assessment (σ, β) = ((σ1 , σ2 ), (β1 , β2 )), consisting of a pair of behavior strategies and a pair of beliefs, is consistent if there is a sequence (σ(n), β(n))n∈N of assessments converging to (σ, β) such that for every n, σ(n) is completely mixed and β(n) is induced by σ(n) using Bayes’ rule. If σi and σj are any behavior strategies for i and j, and βi are the beliefs of i, then let, for each h ∈ Hi , ui (σi , σj ; βi )|h denote i’s expected payoff conditional on h, given the belief βi (h), and given that future behavior is determined by σi and σj .

Definition 14 An assessment (σ, β) = ((σ1 , σ2 ), (β1 , β2 )) is a sequential equilibrium if it is consistent and it satisfies that for each i and every h ∈ Hi , ui (σi , σj ; βi )|h = max ui (σi0 , σj ; βi )|h . 0 σi

The characterization result can now be stated; it is proven in Appendix B.

Proposition 30 Consider a finite extensive two-player game Γ. A profile of behavior strategies σ = (σ1 , σ2 ) can be extended to a sequential equilibrium if and only if there exists an epistemic model with (t1 , t2 ) ∈ [u] ∩ [isr] such that (1) there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and (2) for each i, σi is induced for ti by tj . For the “if” part, it is sufficient that there is mutual certain belief of the beliefs that each player has about the action choice of his opponent at each of her information sets. We do not need the stronger condition that (1) entails. Hence, higher order certain belief plays no role in the characterization, in line with the fundamental insights of Aumann and Brandenburger (1995).

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Defining sequential rationalizability. We next define the concept of sequentially rationalizable behavior strategies as induced behavior strategies under common certain belief of [isr].

Definition 15 A behavior strategy σi for i is sequentially rationalizable in a finite extensive two-player game Γ if there exists an epistemic model with (t1 , t2 ) ∈ CK([u] ∩ [isr]) such that σi is induced for ti by tj . It follows from Proposition 30 that a behavior strategy is sequentially rationalizable if it is part of a profile of behavior strategies that can be extended to a sequential equilibrium. Since a sequential equilibrium always exists, we obtain as an immediate consequence that sequentially rationalizable behavior strategies always exist. For the concept of sequential rationalizability—as indeed, throughout the book—we restrict our attention to games with two players. A natural question which arises is whether, and if so how, the present analysis can be extended to the case of three or more players. In order to illustrate the potential difficulties of such an extension, consider a three player game in which player 3 has an information set h with two nodes, x and y, where x is preceded by the player 1 action a and the player 2 action c, and y is preceded by the player 1 action b and the player 2 action d. Suppose that player 3 views b and c as suboptimal choices, and hence player 3 deems a infinitely more likely than b, and deems d infinitely more likely than c. Then, player 3’s LPS at h over player 1’s strategy choice and player 3’s LPS at h over player 2’s strategy choice do not provide sufficient information to derive player 3’s relative likelihoods attached to nodes x and y, and these relative likelihoods are crucial to assess player 3’s rational behavior at h. Hence, in addition to the two LPSs mentioned above, we need another aggregated LPS for player 3 at h over his opponents’ collective strategy profiles. The key problem would then be what restrictions to impose upon the connection between the LPSs over individual strategies on the one hand and the aggregate LPS over strategy profiles on the other hand. Both classes of LPSs are needed, since the former are crucial in order to evaluate the beliefs about rationality of individual players, and the latter are needed in order to determine the conditional preferences of each player, as shown above. This issue is closely related to the problem of how to characterize consistency of assessments in algebraic terms, without the use of sequences; cf. McLennan (1989a, 1989b), Battigalli (1996b), Kohlberg and Reny (1997), and Perea et al. (1997). In these papers, the consistency requirement for assessments has been characterized by

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means of conditional probability systems, relative probability systems and lexicographic probability systems, satisfying some appropriate additional conditions. Perea et al. (1997), for instance, use a refinement of LPS in which, at every information set, not only an LPS over the available actions is defined, but moreover the relative likelihood level between actions is “quantified” by an additional parameter, whenever one action is deemed infinitely more likely than the other. This additional parameter makes it possible to derive a unique aggregate LPS over action profiles (and hence also over strategy profiles). A similar approach can be found in Govindan and Klumpp (2002). Such an approach could possibly be useful when extending our analysis of, e.g., sequentiality to the case of more than two players. For the moment, we leave this issue for future research.

8.3

Weak sequential consistency

In the previous section we have shown how imposing that each player believes that the opponent chooses rationally at all her information sets can be used to characterize sequential equilibrium and define sequential rationalizability. Table 2.2 suggests the following claim: Imposing that each player believes that the opponent chooses rationally only at her reachable information sets can be used to characterize the notion of weak sequential rationalizability, due to Ben-Porath (1997) and coined ‘weak extensive form rationalizablity’ by Battigalli and Bonanno (1999). In this section we verify this claim and shed light on the difference between sequentiality and weak sequentiality. Inducing weak sequential rationality. Recall from Chapter 5 that the mixed strategy pjti |tj is induced for tj by ti if tj ∈ Tjti and, for all sj ∈ Sj , µti (sj , tj ) pjti |tj (sj ) = t`i , µ` (Sj , tj ) where ` is the first level ` of λti for which µt`i (Sj , tj ) > 0. Say that a mixed strategy pi is weak sequentially rational for i at ti if, ∀h ∈ Hi s.t. supppi ∩ Si (h) 6= ∅, supppi ∩ Si (h) ⊆ Siti (h), and define the event that player i is of a type that i nduces a w eakly sequentially r ational mixed strategy for any opponent type that is deemed subjectively possible: [iwri ] := {(t1 , t2 ) ∈ T1 × T2 | ∀t0j ∈ Tjti , 0

pj ti |tj is weak sequentially rational for t0j } .

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Write [iwr] := [iwr1 ] ∩ [iwr2 ]. Say that at ti player i’s preferences over his strategies are weak sequentially consistent with the game Γ and the preferences of his opponent, if ti ∈ projTi ([ui ] ∩ [iwri ]). Refer to [u] ∩ [iwr] as the event of weak sequential consistency. Note that the mixed strategy induced for t0j by ti may be interpreted as i’s initial belief at ti about the behavior of t0j . In contrast to the behavior strategy induced for t0j by ti , as defined in the previous section, the induced mixed strategy gives no information about how i at ti revises his belief about the behavior of t0j at player j information sets that are unreachable given i’s initial belief at ti about t0j ’s behavior. Hence, if ti ∈ projTi [iwri ], then player i believes at ti that each subjectively possible opponent type t0j chooses rationally at player j information sets that do not contradict i’s initial belief at ti about the behavior of t0j . However, and this is the crucial difference when compared to the case where ti ∈ projTi [isri ]: ti ∈ projTi [iwri ] entails no restriction on how i at ti revises his beliefs about t0j ’s behavior conditional on t0j reaching “surprising” information sets. The above observation explains why weak sequentially rationalizable mixed strategies can be shown to correspond to induced mixed strategies under common certain belief of [u] ∩ [iwr]. Characterizing weak sequential rationalizability. We first define weak sequential rationalizability. Since weak sequential rationalizability in two-player games corresponds to iterated elimination of strategies that are strongly dominated at some reachable information set, we use the latter procedure as the primitive definition. For any (∅ 6=) X = X1 × X2 ⊆ S, write ˜b(X) := ˜b1 (X2 ) × ˜b2 (X1 ), where ˜bi (Xj ) := Si \ {si ∈ Si | ∃pi ∈ ∆(Si ) s.t. pi strongly dominates si on Xj or ∃h ∈ Hi with Si (h) 3 si and qi ∈ ∆(Si (h)) s.t. qi strongly dominates si on Sj (h)} . If pi is a mixed strategy and h ∈ Hi satisfies that supppi ∩ Si (h) 6= ∅, then let pi |h be defined by ( pi (si ) if si ∈ Si (h) pi (Si (h)) pi |h (si ) = 0 otherwise .

Definition 16 Let Γ be a finite extensive two-player game. Consider the sequence defined by X(0) = S1 × S2 and, ∀g ≥ 1, X(g) = ˜b(X(g − 1)). A pure strategy si is said to be weak sequentially rational-

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izable if si ∈ Wi :=

\∞ g=0

Xi (g) .

A mixed strategy pi is said to be weak sequentially rationalizable if pi is not strongly dominated on Wj and there does not exist h ∈ Hi with supppi ∩ Si (h) 6= ∅ such that pi |h is strongly dominated on Sj (h). While any pure strategy in the support of a weak sequentially rationalizable mixed strategy is itself weak sequentially rationalizable, the mixture over a set of weak sequentially rationalizable pure strategies need not be weak sequentially rationalizable. The following lemma is a straightforward implication of Definition 11.

Lemma 12 (i) For each i, Wi 6= ∅. (ii) W = ˜b(W ). (iii) For each i, si ∈ Wi if and only if there exists X = X1 × X2 with si ∈ Wi such that X ⊆ ˜b(X). We next characterize the concept of weak sequentially rationalizable mixed strategies as induced mixed strategies under common certain belief of [u] ∩ [iwr].

Proposition 31 A mixed strategy pi for i is weak sequentially rationalizable in a finite extensive two-player game Γ if and only if there exists an epistemic model with (t1 , t2 ) ∈ CK([u] ∩ [iwr]) such that pi is induced for ti by tj . Proof. Part 1: If p∗i is weak sequentially rationalizable, then there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [iwr]) such that p∗i is induced for t∗i by t∗j . Step 1: Construct an epistemic model with T1 × T2 ⊆ CK([u] ∩ [iwr]) such that for each si ∈ Wi of any player i, there exists ti ∈ Ti with, si ∈ Siti . Construct an epistemic model with, for each i, a bijection si : Ti → Wi from the set of types to the the set of weak sequentially rationalizable pure strategies. Assume that, for each ti ∈ Ti of any player i, υiti satisfies that (a) υiti ◦ z = ui (so that T1 × T2 ⊆ [u]), and the SCLP (λti , `ti ) on Sj × Tj has the properties that (b) λti = (µt1i , . . . , µtLi ) with support Sj × Tjti satisfies that suppµt11 ∩ (Sj × {tj }) = {(sj (tj ), tj )} for all tj ∈ Tjti (so that, ∀tj ∈ Tjti , piti |tj (sj (tj )) = 1), (c) ∀Ej ∈ Sj ×Tj such that Ej ∩(Sj × Tjti ) 6= ∅, `ti (Ej ) = min{`| suppλt`i 6= ∅} (so that, by Corollary 1, the SCLP corresponds to a CPS).

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Property (b) entails that the support of the marginal of µt1i on Sj is included in Wj . By properties (a) and (c) and Lemmas 4 and 12(ii), we can still choose µt1i (and Titi ) so that si (ti ) ∈ Siti . Since information sets correspond to strategically independent sets (cf. the discussion in connection with Lemmas 11 and 13) we have that, ∀h ∈ Hi s.t. Si (h) 3 si (ti ) and suppµt1i ∩Sj (h) 6= ∅, si (ti ) ∈ Siti (h), while, ∀h ∈ Hi s.t. Si (h) 3 si (ti ) and suppµt1i ∩Sj (h) = ∅, si (ti ) ∈ Siti (h) by choosing the lower levels of λti appropriately (again invoking properties (a) and (c) and Lemmas 4 and 12(ii)). This combined with property (b) means that T1 ×T2 ⊆ [iwr]. Furthermore, T1 × T2 ⊆ CK([u] ∩ [iwr]) since Tjti ⊆ Tj for each ti ∈ Ti of any player i. Since, for each player i, si is onto Wi , it follows that, for each si ∈ Wi of any player i, there exists ti ∈ Ti with si ∈ Siti . ∗ ∗ ∗ Step 2: Add type t∗i to Ti . Assume that υiti satisfies (a) and (λti , `ti ) ∗ ∗ satisfies (b) and (c). Then µ1ti can be chosen so that p∗i ∈ ∆(Siti ), and ∗ consequently, ∀h ∈ Hi s.t. suppp∗i ∩ Si (h) 6= ∅ and suppµ1ti ∩ Sj (h) 6= ∅, ∗ suppp∗i ∩ Si (h) ⊆ Siti (h), while, ∀h ∈ Hi s.t. suppp∗i ∩ Si (h) 6= ∅ and ∗ ∗ suppµ1ti ∩ Sj (h) = ∅, suppp∗i ∩ Si (h) ⊆ Siti (h) by choosing the lower ∗ levels of λti appropriately. Furthermore, (Ti ∪ {t∗i }) × Tj ⊆ [u] ∩ [iwr], ∗ and since Tjti ⊆ Tj , (Ti ∪ {t∗i }) × Tj ⊆ CK([u] ∩ [iwr]). ∗ Step 3: Add type t∗j to Tj . Assume that υjtj satisfies (a) and the SCLP ∗ ∗ ∗ ∗ ∗ (λtj , `tj ) on Si × (Ti ∪ {t∗i }) has the property that λtj = (µ1tj , . . . , µLtj ) ∗ with support Si ×{t∗i } satisfies that, ∀si ∈ Si , µ1tj (si , t∗i ) = p∗i (si ), so that p∗i is induced for t∗i by t∗j . Furthermore, (Ti ∪ {t∗i }) × (Tj ∪ {t∗j }) ⊆ [u] ∩ ∗ [iwr], and since Titj ⊆ Ti ∪{t∗i }, (Ti ∪{t∗i })×(Tj ∪{t∗j }) ⊆ CK([u]∩[iwr]). Hence, (t∗1 , t∗2 ) ∈ CK([u] ∩ [iwr]) and pi is induced for t∗i by t∗j . Part 2: If there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [iwr]) such that p∗i is induced for t∗i by t∗j , then p∗i is weak sequentially rationalizable. Assume that there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [iwr]) such that p∗i is induced for t∗i by t∗j . In particular, CK([u]∩[iwr]) 6= ∅. Let, for each i, Ti0 := projTi CK([u] ∩ [iwr]) and [ Xi := {si ∈ Si |∀h ∈ Hi s.t. Si (h) 3 si , si ∈ Siti (h)} . 0 ti ∈Ti

By Proposition 20(ii), for each ti ∈ Ti0 of any player i, ti deems (sj , tj ) subjectively impossible if tj ∈ Tj \Tj0 since CK([u] ∩ [iwr]) = KCK([u] ∩ [iwr]) ⊆ Ki CK([u] ∩ [iwr]), implying Tjti ⊆ Tj0 . By the definitions of [u] and [iwr], it follows that, for each ti ∈ Ti0 of any player i, ºti is represented by υiti satisfying that υiti ◦z is a positive affine transformation of ui and an LPS λt`i = (µt1i , . . . , µt`i ), where ` = `(Sj × Tj ) ≥ 1, and where suppµt1i ⊆ Xj × Tj . Hence, by Lemma 4, for each ti ∈ Ti0 of any

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2c d 1 1

f

1s D 0 0

Figure 8.1.

F

3 3

f d 1, 1 0, 0 D F 1, 1 3, 3

Γ6 and its strategic form.

player i, if pi ∈ ∆(Si ) satisfies that, ∀h ∈ Hi s.t. supppi ∩ Si (h) 6= ∅, supppi ∩ Si (h) ⊆ Siti (h), then no strategy in the support of pi is strongly dominated on Xj , since then pi ∈ ∆(Siti ), and it follows from pi ∈ ∆(Siti ) and suppµt1i ⊆ Xj × Tj that, ∀si ∈ supppi and ∀s0i ∈ Si , X X t X X t µ1i (sj , tj )ui (s0i , sj ) , µ1i (sj , tj )ui (si , sj ) ≥ sj ∈Xj tj ∈Tj

sj ∈Xj tj ∈Tj

∀h ∈ Hi s.t. supppi ∩ Si (h) 6= ∅, no strategy in supppi ∩ Si (h) is strongly dominated on Sj (h) since supppi ∩ Si (h) ⊆ Siti (h). This implies X ⊆ ˜b(X), entailing by Lemma 12(iii) that, for each i, Xi ⊆ Wi . Furthermore, since (t∗1 , t∗2 ) ∈ CK([u] ∩ [iwr]) and the mixed strategy induced for t∗i by t∗j , p∗i , satisfies that, ∀h ∈ Hi s.t. suppp∗i ∩ Si (h) 6= ∅, suppp∗i ∩ Si (h) ⊆ Siti (h), it follows that p∗i is not strongly dominated on Xj ⊆ Wj and there does not exist h ∈ Hi with suppp∗i ∩ Si (h) 6= ∅ such that pi |h is strongly dominated on Sj (h). By Definition 16 this implies that p∗i is a weak sequentially rationalizable mixed strategy. The following observation (which is stated without proof) can now be used to establish the relationships between the rationalizability concepts on the lower row of Table 2.2.

Proposition 32 For any epistemic model and for each player i, [isri ] ⊆ [iwri ] ⊆ [iri ] . Since [iwri ] ⊆ [iri ], Propositions 31 and 22 entail that weak sequential rationalizability refines (ordinary) rationalizability, and since [isri ] ⊆ [iwri ], Definition 15 and Proposition 31 entail that sequential rationalizability refines weak sequential rationalizability. That the two latter inclusions can be strict, is illustrated by Γ6 and Γ06 of Figures 8.1 and 8.2, respectively. In Γ6 rationalizability does not have any bite, while weak

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1c D 2 2

F

2s d 1 1

Figure 8.2.

f

1s D 0 0

F

3 3

f d D 2, 2 2, 2 FD 1, 1 0, 0 FF 1, 1 3, 3

Γ06 and its pure strategy reduced strategic form.

sequential rationalizability promotes that player 1 plays F and player 2 plays f . In Γ06 —introduced by Reny (1992, Figure 1)—weak sequential rationalizability only precludes the play of D at 1’s second decision node. This can be established by applying the Dekel-Fudenberg procedure (i.e., one round of weak elimination followed by iterated strong elimination) which eliminates a strategy if and only if it is not permissible. Since all terminal nodes yield different payoffs, weak sequential rationalizability leads to the same conclusion.1 However, only the play of F at both of 1’s decision nodes and the play of f at 2’s single decision node are sequentially rationalizable. This follows from Proposition 33 of the next section, showing that the latter concepts imply the backward induction procedure. Extensive form rationalizability (EFR), cf. Pearce (1984) as well as Battigalli (1997) and Battigalli and Siniscalchi (2002), is an iterative deletion procedure where, at any information set reached by a remaining strategy, any deleted strategy is deemed infinitely less likely than some remaining strategy. Even though EFR only requires players to choose rationally at reachable information sets and preference for cautious behavior is not imposed, EFR is different from weak sequential rationalizability. Unlike all concepts in Tables 2, EFR yields forward induction in common examples like the “battle-of-the-sexes with an outside option” game, see Figure 2.6.2 EFR also leads to the backward induction out-

1 To

see how the characterization in Proposition 31 of weak sequential rationalizability is consistent with (D, d) in Γ06 , let T1 = {t1 } with λt1 = ((1, 0), (0, 1)) (assigning probabilities to (d, t2 ) and (f, t2 ) respectively), and T2 = {t2 } with λt2 = ((1, 0, 0), (0, 1, 0), (0, 0, 1)) (assigning probabilities to (D, t1 ), (F D, t1 ), and (F F, t1 ) respectively). Then, independently of how `t1 and `t2 are specified, (t1 , t2 ) ∈ CK([u] ∩ [iwr]), and, for each i, pi is induced for ti by tj , where p1 (D) = 1 and p2 (d) = 1. 2 By strengthening permissibility, Asheim and Dufwenberg (2003a) define a rationalizability concept, fully permissible sets, which is different from those of Table 2.2 as well as EFR, as it yields forward induction, but does not always promote backward induction. This concept will be presented in Chapters 11 and 12.

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come. However, unlike sequential rationalibility, EFR need not promote the backward induction procedure.

8.4

Relation to backward induction

The following result shows how sequential rationalizability implies the backward induction procedure in perfect information games. A finite extensive game Γ, as introduced in Section 8.1, is of perfect information if, at any information set h ∈ H1 ∪ H2 , h = {x}; i.e., h contains only one node. It is generic if, for each i, υi (z) 6= υi (z 0 ) whenever z and z 0 are different outcomes. A generic extensive game of perfect information has a unique subgame-perfect equilibrium in pure strategies. Moreover, in such games the backward induction procedure yields in any subgame the unique subgame-perfect equilibrium outcome.

Proposition 33 Consider a finite generic extensive two-player game of perfect information Γ. If there exists an epistemic model with (t1 , t2 ) ∈ CK([u]∩[isr]) and, for each i, σi is induced for ti by tj , then σ = (σ1 , σ2 ) is the subgame-perfect equilibrium. Proof. In a perfect information game, the action a ∈ A(h) taken at the information set h determines the immediate succeeding information set, which can thus be denoted (h, a). Also, any information set h ∈ H1 ∪ H2 determines a subgame. Set H −1 = Z (i.e. the set of terminal nodes) and determine H g for g ≥ 0 by induction: h ∈ H g if and only if h satisfies 0

max{g 0 | ∃h0 ∈ H g and a ∈ A(h) such that h0 = (h, a)} = g − 1 . In words, h ∈ H g if and only if g is the maximal number of decision nodes between h and a terminal node in the subgame determined by h. If σ is a profile of behavior strategies and h ∈ H1 ∪ H2 , denote by σ|h the strategy profile with the following properties: (1) at information sets preceding h, σ|h determines with probability one the unique action leading to h, and (2) at all other information sets, σ|h coincides with σ. Say that σ 0 is outcome-equivalent to σ 00 if σ 0 and σ 00 induce the same probability distribution over terminal nodes. In view of properties of the certain belief operator (cf. Proposition 20 0 of Chapter 5), it is sufficient to show for any g = 0, . . . , max{g 0 |H g 6= ∅} that if there exists an epistemic model with (t1 , t2 ) ∈ Kg ([u] ∩ [isr]) and, for each i, σi is induced for ti by tj , then, ∀h ∈ H g , σ|h is outcome-

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equivalent to σ ∗ |h , where σ ∗ = (σ1∗ , σ2∗ ) denotes the subgame-perfect equilibrium. This is established by induction. (g = 0) Let (t1 , t2 ) ∈ K0 ([u] ∩ [isr]) = [u] ∩ [isr] and, for each i, σi be induced for ti by tj . Let h ∈ H 0 and assume w.l.o.g. that h ∈ Hi . Since (t1 , t2 ) ∈ [ui ] ∩ [isrj ] and j takes no action at h, σ|h is outcome equivalent to σ ∗ |h . 0 (g = 1, . . . , max{g 0 |H g 6= ∅}) Suppose that it has been established for g 0 = 0, . . . , g − 1 that if there exists an epistemic model with (t1 , t2 ) ∈ 0 0 Kg ([u] ∩ [isr]) and, for each i, σi is induced for ti by tj , then, ∀h0 ∈ H g , σ|h0 is outcome-equivalent to σ ∗ |h0 . Let (t1 , t2 ) ∈ Kg ([u] ∩ [isr]) and, for each i, σi be induced for ti by tj . Let h ∈ H g and assume w.l.o.g. that h ∈ Hi . Since (t1 , t2 ) ∈ Ki Kg−1 [isr], it follows from the premise of the inductive step that ti ’s SCLP (λti , `ti ) satisfies, ∀t0j ∈ Tjti , ∀h0 ∈ Hj succeeding h, and ∀a0 ∈ A(h0 ), µt`i (Sj (h0 , a0 ), t0j ) µt`i (Sj (h0 ), t0j )

= σj∗ (h0 )(a0 ) ,

where ` is the first level ` of λti for which µt`i (Sj (h0 ), t0j ) > 0. Since Γ is generic, σi is sequentially rational for ti only if σi (h) = σi∗ (h). Since (t1 , t2 ) ∈ [ui ] ∩ [isrj ] and j takes no action at h, it follows from the premise that σ|h is outcome-equivalent to σ ∗ |h . Since sequentially rationalizable strategies always exist, there is an epistemic model with (t1 , t2 ) ∈ CK([u] ∩ [isr]), implying that the result of Proposition 33 is not empty.

Chapter 9 QUASI-PERFECTNESS

In Chapter 5 we saw how the characterizations of Nash equilibrium and rationalizability lead to characterizations of (strategic form) perfect equilibrium and permissibility by adding preference for cautious behavior. In this chapter we show that the characterization of sequential equilibrium leads to a characterization of quasi-perfect equilibrium by adding caution. The concept of a quasi-perfect equilibrium, proposed by van Damme (1984), differs from Selten’s (1975) extensive form perfect equilibrium by the property that, at each information set, the player taking an action ignores the possibility of his own future mistakes. So, parallelling Chapter 8, we define quasi-perfect rationalizability by imposing common certain belief of the event that each player has preference for cautious behavior (i.e., at every information set, one strategy is preferred to another if the former weakly dominates the latter) and believes that the opponent chooses rationally at all her information sets. Moreover, by assuming that each player is certain of the beliefs that the opponent has about the player’s own action choice, we obtain an epistemic characterization of the corresponding equilibrium concept: quasi-perfect equilibrium. Since quasi-perfect rationalizability refines sequential rationalizability, it follows from Proposition 33 that also the former concept yields the backward induction procedure. By embedding the notion of an SCLP in an epistemic model with a set of epistemic types for each player, we are able to model quasi-perfectness as a special case of sequentiality. For each type ti of any player i, ti is described by an SCLP, which under the event that “player i believes that the opponent j chooses rationally at each information set” induces,

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for each opponent type tj that is deemed subjectively possible by ti , a behavior strategy which is sequentially rational given tj ’s own SCLP. An SCLP ensures well-defined conditional beliefs representing nontrivial conditional preferences, while allowing for flexibility w.r.t. whether to assume preference for cautious behavior. Preference for cautious behavior, as needed for quasi-perfect rationalizability, is obtained by imposing the following additional requirement on ti ’s SCLP for each conditioning event: If an opponent strategy-type pair (sj , tj ) is compatible with the event and tj is deemed subjectively possible by ti , then (sj , tj ) is in the support the LPS that represents type ti ’s conditional preferences. This chapter’s definition of quasi-perfect rationalizability was proposed by Asheim and Perea (2004).

9.1

Quasi-perfect consistency

In this section, we add preference for cautious behavior to the analysis of Chapter 8. This enables us to characterize quasi-perfect equilibrium (van Damme, 1984), and define quasi-perfect rationalizability as a non-equilibrium analog to the concept of van Damme (1984). The epistemic modeling is identical to the one given in Section 8.1; hence, this will not be recapitulated here. Caution. Under Assumption 1 it follows from Proposition 5 that, for each type ti of any player i, i’s system of conditional preferences at ti can be represented by a vNM utility function υiti : ∆(Z) → R and an SCLP (λti , `ti ) on Sj × Tj with support Sj × Tjti . Recall from Section 5.3 that caution imposes the additional requirement that for each type ti of any player i the full LPS λti is used to form the conditional beliefs over opponent strategy-type pairs. Formally, if L denotes the number of levels in the LPS λti , then [caui ] = {(t1 , t2 ) ∈ T1 × T2 | `ti (Sj × Tj ) = L} . Since `ti is non-increasing w.r.t. set inclusion, ti ∈ projTi [caui ] implies that `ti (projSj ×Tj φ) = L for all subsets φ of {ti } × Sj × Tj with welldefined conditional beliefs. Since it follows from Assumption 1 that λti has full support on Sj , ti ∈ projTi [caui ] means that i’s choice function at ti never admits a weakly dominated strategy, thereby inducing preference for cautious behavior. As before, write [cau] := [cau1 ] ∩ [cau2 ].

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Say that at ti player i’s preferences over his strategies are quasiperfectly consistent with the game Γ and the preferences of his opponent, if ti ∈ projTi ([ui ] ∩ [isri ] ∩ [caui ]). Refer to [u] ∩ [isr] ∩ [cau] as the event of quasi-perfect consistency. Characterizing quasi-perfect equilibrium. We now characterize the concept of a quasi-perfect equilibrium as profiles of induced behavior strategies at a type profile in [u] ∩ [isr] ∩ [cau] where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible). To state the definition of quasi-perfect equilibrium, we need some preliminary definitions. Define the concepts of a behavior representation of a mixed strategy and the mixed representation of a behavior strategy in the standard way, cf., e.g., p. 159 of Myerson (1991). If a behavior strategy σj and a mixed strategy pj are both completely mixed, and σj is a behavior representation of pj or pj is the mixed representation of σj , then, ∀h ∈ Hj , ∀a ∈ A(h), σj (h)(a) =

pj (Sj (h, a)) . pj (Sj (h))

If σi is any behavior strategy for i and σj is a completely mixed behavior strategy for j, then abuse notation slightly by writing, for each h ∈ Hi , ui (σi , σj )|h := ui (pi , pj |h ) , where pi is outcome-equivalent to σi |h and pj is the mixed representation of σj .

Definition 17 A behavior strategy profile σ = (σ1 , σ2 ) is a quasiperfect equilibrium if there is a sequence (σ(n))n∈N of completely mixed behavior strategy profiles converging to σ such that for each i and every n ∈ N and h ∈ Hi , ui (σi , σj (n))|h = max ui (σi0 , σj (n))|h . 0 σi

The characterization result can now be stated; it is proven in Appendix B.

Proposition 34 Consider a finite extensive two-player game Γ. A profile of behavior strategies σ = (σ1 , σ2 ) is a quasi-perfect equilibrium if and only if there exists an epistemic model with (t1 , t2 ) ∈ [u] ∩ [isr] ∩ [cau] such that (1) there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and (2) for each i, σi is induced for ti by tj .

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As for Proposition 31, higher order certain belief plays no role in this characterization. Defining quasi-perfect rationalizability. We next define the concept of quasi-perfectly rationalizable behavior strategies as induced behavior strategies under common certain belief of [u] ∩ [isr] ∩ [cau].

Definition 18 A behavior strategy σi for i is quasi-perfectly rationalizable in a finite extensive two-player game Γ if there exists an epistemic model with (t1 , t2 ) ∈ CK([u] ∩ [isr] ∩ [cau]) such that σi is induced for ti by tj . It follows from Proposition 34 that a behavior strategy is quasi-perfectly rationalizable if it is part of a quasi-perfect equilibrium. Since a quasiperfect equilibrium always exists, we obtain as an immediate consequence that quasi-perfectly rationalizable behavior strategies always exist. Propositions 30 and 34 imply the well-known result that every quasiperfect equilibrium can be extended to a sequential equilibrium, while Definitions 15 and 18 imply that the set of quasi-perfectly rationalizable strategies is included in the set of sequentially rationalizable strategies. To illustrate that this inclusion can be strict, consider Γ4 of Figure 3.1. Both concepts predict that player 2 plays d with probability one. However, only quasi-perfect rationalizability predicts that player 1 plays D with probability one. Preferring D to F amounts to preference for cautious behavior since by choosing D player 1 avoids the risk that player 2 may choose f . Since quasi-perfect rationalizability is thus a refinement of sequential rationalizability, it follows from Proposition 33 that quasi-perfect rationalizability implies the backward induction procedure in perfect information games.

9.2

Relating rationalizability concepts

The following result helps establishing some of the remaining relationships between the rationalizability concepts of Table 2.2.

Proposition 35 For any epistemic model and for each player i, [iri ] ∩ Ki [cauj ] ⊆ [iwri ] . To prove Proposition 35 we need the following lemma.

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Lemma 13 If ti ∈ projTi Ki [cauj ], then, for each tj ∈ Tj ti and any h ∈ Hj , sj ∈ Sj (h)\Sjtj (h) implies that there exists s0j ∈ Sj (h) such that s0j Âtj sj . Proof. As for Lemma 11 the proof of this lemma is based on the concept of a strategically independent set due to Mailath et al. (1993). It follows from Mailath et al. (Definitions 2 and 3 and the ‘if’ part of Thm. 1) that S(h) is strategically independent for j at any player j information set h in a finite extensive game, and this does not depend on the vNM utility function that assigns payoff to any outcome. If ti ∈ projTi Ki [cauj ], then the following holds for each tj ∈ Tj ti : Player j’s system of conditional preferences at tj satisfies Axiom 6 (Conditionality). Suppose sj ∈ Sj (h)\Sjtj (h). Then there exists s0j ∈ Sj (h) t such that s0j Âhj sj . As noted above, S(h) is a strategically independent set for j. Hence, s0j can be chosen such that z(s0j , si ) = z(sj , si ) for all si ∈ Si \Si (h). By Axiom 6 (Conditionality), this implies s0j Âtj sj . Proof of Proposition 35. Consider any epistemic model with ti ∈ projTi ([iri ] ∩ Ki [cauj ]) . Suppose ti ∈ / projTi [iwri ]; i.e., there exist tj ∈ Tjti and h ∈ Hj such that pjti |tj (sj ) > 0 for some sj ∈ Sj (h)\Sjtj (h). Since ti ∈ projTi Ki [cauj ], it follows from Lemma 13 that ∃s0j ∈ Sj (h) s.t. s0j Âtj sj . Hence, pjti |tj ∈ / ∆(Sjtj ) , contradicting ti ∈ projTi [iri ]. This shows that ti ∈ projTi [iwri ]. Since [iri ] ∩ Ki [cauj ] ⊆ [iwri ], the cell in Table 2.2 to the left of ‘permissibility’ is not applicable, and permissibility refines weak sequential rationalizability. Figure 3.1 shows that the inclusion can be strict: Permissibility, but not weak sequential rationalizability, precludes that player 1 plays F in Γ4 . Since [isri ] ⊆ [iri ], Definition 18 and Proposition 24 entail that quasiperfect rationalizability refines permissibility. That the latter inclusion can be strict is illustrated by Γ06 of Figure 8.2. Since this is a generic extensive game, imposing preference for cautious behavior has no bite, and the difference between permissibility and quasi-perfect rationalizability corresponds to the difference between weak sequential rationalizability and sequential rationalizability, as discussed in Section 8.3.

Chapter 10 PROPERNESS

Most contributions on the relation between common knowledge/belief of rationality and backward induction in perfect information games perform the analysis in the extensive form of the game. Indeed, the analyses in Chapters 7 and 8 of this book are examples of this. An exception to this rule is Schuhmacher (1999) who—based on Myerson’s (1978) concept of a proper equilibrium, but without making equilibrium assumptions—defines the concept of proper rationalizability in the strategic form and shows that proper rationalizable play leads to backward induction. Schuhmacher defines the concept of ε-proper rationalizability by assuming that players make mistakes, but where more costly mistakes are made with a much smaller probability than less costly ones. A properly rationalizable strategy can then be defined as the limit of a sequence of ε-properly rationalizable strategies as ε goes to zero. For a given ε, Schuhmacher offers an epistemic foundation for ε-proper rationalizability. However, this does not provide an epistemic foundation for the limiting concept, i.e. proper rationalizability. It is one purpose of the present chapter, which reproduces Asheim (2001), to establish how proper rationalizability can be given an epistemic characterization in strategic two-player games, within an epistemic model where preferences are represented by a vNM utility function and an SCLP (i.e., an epistemic model satisfying Assumption 1 of Chapter 5). Blume et al. (1991b) characterize proper equilibrium as a property of preferences. When doing so they represent a player’s preferences

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by a vNM utility function and an LPS, whereby the player may deem one opponent strategy to be infinitely more likely than another while still taking the latter strategy into account. In two-player games, their characterization of proper equilibrium can be described by the following two properties. 1 Each player is certain of the preferences of his opponent, 2 Each player’s preferences satisfies that the player takes all opponent strategies into account (‘caution’) and that the player deems one opponent strategy to be infinitely more likely than another if the opponent prefers the one to the other (‘respect of opponent preferences’). The present characterization of proper rationalizability in two-player games drops property 1, which is an equilibrium assumption; instead it will be assumed that there is common certain belief of property 2, which will be referred to as proper consistency. Since, in the present framework, a player is not certain of the preferences of his opponent, player i’s preferences must be defined on acts from Sj × Tj , where Sj denotes the set of opponent strategies and Tj denotes the set of opponent types. Under Assumption 1, each type of player i corresponds to a vNM utility function and an SCLP on Sj × Tj . As before, a player i has preference for cautious behavior at ti if he takes into account all strategies of any opponent type that is deemed subjectively possible. Moreover, a player i is said to respect opponent preferences at ti if, for any opponent type that is deemed subjectively possible, he deems one strategy of the opponent type to be infinitely more likely than another if the opponent type prefers the one to the other. At ti player i’s preferences are said to be properly consistent with the game and the preferences of his opponent if at ti i both has preference for cautious behavior and respects opponent preferences. Hence, the present analysis follows the ‘consistent preferences’ approach by imposing requirements on the preferences of players rather than their choice. In this chapter it is first shown (in Proposition 36) how the event of proper consistency combined with mutual certain belief of the type profile can be used to characterize the concept of proper equilibrium. It is then established (in Proposition 37) that common certain belief of proper consistency corresponds to Schuhmacher’s (1999) concept of proper rationalizability. Furthermore, by relating ‘respect of preferences’ to ‘inducement of sequential rationality’ in Proposition 38, it follows by comparing Proposition 37 with Proposition 33 of Chapter 8 that only strategies leading to the backward induction outcome are properly ra-

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c r ` 1, 1 1, 1 1, 0 U M 1, 1 2, 2 2, 2 D 0, 1 2, 2 3, 3 Figure 10.1.

G7 , illustrating common certain belief of proper consistency.

tionalizable in the strategic form of a generic perfect information game. Thus, Schuhmacher’s Theorem 2 (which shows that the backward induction outcome obtains with “high” probability for any given “small” ε) is strengthened, and an epistemic foundation for the backward induction procedure, as an alternative to Aumann’s (1995) and others, is provided. Lastly, it is illustrated through an example how proper rationalizability can be used to test the robustness of inductive procedures.

10.1

An illustration

The symmetric game of Figure 10.1 is an example where common certain belief of proper consistency is sufficient to determine completely each player’s preferences over his or her own strategies. The game is due to Blume et al. (1991b, Figure 1). In this game, caution implies that player 1 prefers M to U since M weakly dominates U . Likewise, player 2 prefers c to `. Since 1 respects the preferences of 2 and, in addition, certainly believes that 2 has preference for cautious behavior, it follows that 1 deems c infinitely more likely than `. This in turn implies that 1 prefers D to U . Likewise, since 2 respects the preferences of 1 and, in addition, certainly believes that 1 has preference for cautious behavior, it follows that 2 prefers r to `. As a consequence, since 1 respects the preferences of 2, certainly believes that 2 respects the preferences of 1, and certainly believes that 2 certainly believes that 1 has preference for cautious behavior, it follows that 1 deems r infinitely more likely than `. Consequently, 1 prefers D to M . A symmetric reasoning entails that 2 prefers r to c. Hence, if there is common certain belief of proper consistency, it follows that the players’ preferences over their own strategies are given by 1’s preferences: D  M  U 2’s preferences: r  c  ` . The facts that D is the unique most preferred strategy for 1 and r is the unique most preferred strategy for 2 mean that only D and r are properly

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rationalizable; cf. Proposition 37 of Section 10.2. By Proposition 36 of the same section, it then follows that the pure strategy profile (D, r) is the unique proper equilibrium, which can easily be checked. However, note that in the argument above, each player obtains certainty about the preferences of his opponent through deductive reasoning; i.e. such certainty is not assumed as in the concept of proper equilibrium. The concept of proper rationalizability yields a strict refinement of (ordinary) rationalizability (cf. Definition 11 of Chapter 5). All strategies for both players are rationalizable, which is implied by the fact that, in addition to (D, r), the pure strategy profiles (U, `) and (M, c) are also Nash equilibria. The concept of proper rationalizability yields even a strict refinement when compared permissibility (cf. Definition 13 of Chapter 5), corresponding to the Dekel-Fudenberg procedure, where one round of weak elimination followed by iterated strong elimination. When the Dekel-Fudenberg procedure is employed, only U is eliminated for 1, and only ` is eliminated for 2, reflecting that also the pure strategy profile (M, c) is a strategic form perfect equilibrium. It is a general result that proper rationalizability refines the Dekel-Fudenberg procedure; this follows from Section 10.3 as well as Theorem 4 of Herings and Vannetelbosch (1999).

10.2

Proper consistency

In this section, we add respect for opponent preferences to the analysis of Chapter 5. This enables us to characterize proper equilibrium (Myerson, 1978), and proper rationalizability (Schuhmacher, 1999). The epistemic modeling is identical to the one given in Section 5.1; hence, this will not be recapitulated here. Respect of opponent preferences. Player i respects the preferences of his opponent at ti if the following holds for any opponent type that is deemed subjectively possible: Player i deems one opponent strategy of the opponent type to be infinitely more likely than another if the opponent type prefers the one to the other. To capture this, define the event [respi ] := {(t1 , t2 ) ∈ T1 × T2 | (sj , t0j ) Àti (s0j , t0j ) 0

whenever t0j ∈ Titi and sj ºtj s0j } , where the notation Àti means “infinitely more likely at ti ”, as defined in Section 3.2.

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Write [resp] := [resp1 ] ∩ [resp2 ]. Say that at ti player i’s preferences over his strategies are properly consistent with the game G = (S1 , S2 , u1 , u2 ) and the preferences of his opponent, if ti ∈ projTi ([ui ] ∩ [respi ] ∩ [caui ]). Refer to [u]∩[resp]∩[cau] as the event of proper consistency. Characterizing proper equilibrium. We now characterize the concept of a proper equilibrium as profiles of induced mixed strategies at a type profile in [u] ∩ [resp] ∩ [cau] where there is mutual certain belief of the type profile (i.e., for each player, only the true opponent type is deemed subjectively possible). Before doing so, we define a proper equilibrium.

Definition 19 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. A completely mixed strategy profile p = (p1 , p2 ) is a ε-proper equilibrium if, for each i, εpi (si ) ≥ pi (s0i ) whenever ui (si , pj ) > ui (s0i , pj ) . A mixed strategy profile p = (p1 , p2 ) is a proper equilibrium if there is a sequence (p(n))n∈N of ε(n)-proper equilibria converging to p, where ε(n) → 0 as n → ∞. The characterization result—which is a variant of Proposition 5 of Blume et al. (1991b)—can now be stated. For this result, recall from Sections 5.2 and 8.3 that the mixed strategy pjti |tj is induced for tj by ti if tj ∈ Tjti and, for all sj ∈ Sj , pjti |tj (sj ) =

µt`i (sj , tj )

µt`i (Sj , tj )

,

where ` is the first level ` of λti for which µt`i (Sj , tj ) > 0.

Proposition 36 Consider a finite strategic two-player game G. A profile of mixed strategies p = (p1 , p2 ) is a proper equilibrium if and only if there exists an epistemic model with (t1 , t2 ) ∈ [u] ∩ [resp] ∩ [cau] such that (1) there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and (2) for each i, pi is induced for ti by tj . The proof is contained in Appendix B. As for similar earlier results, higher order certain belief plays no role in this characterization. Characterizing proper rationalizability. We now turn to the non-equilibrium analog to proper equilibrium, namely the concept of proper rationalizability; cf. Schuhmacher (1999). To define the concept of properly rationalizable strategies, we must introduce the following

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variant of an epistemic model, with a mixed strategy piti being associated to each type ti of any player i, where piti is completely mixed.

Definition 20 An ∗-epistemic model for the finite strategic two-player game form (S1 , S2 , z) is a structure (S1 , T1 , S2 , T2 ) , where, for each type ti of any player i, ti corresponds to (1) mixed strategy piti , where supppiti = Si , and (2) a system of conditional preferences on the collection of sets of acts from elements of Φti := {φ ⊆ Ti × Sj × Tj | κti ∩ φ 6= ∅} to ∆(Z), where κti is a non-empty subset of {ti } × Sj × Tj . Moreover, Schuhmacher (1999) in effect makes the following assumption.

Assumption 3 For each ti of any player i, (a) ºtφi satisfies Axioms 1, 2, and 4 if ∅ 6= φ ⊆ Ti × Sj × Tj , and Axiom 3 if and only if φ ∈ Φti , (b) the system of conditional preferences {ºtφi | φ ∈ Φti } satisfies Axioms 5 and 6, and (c) there exists a non-empty subset of opponent types, Tjti , such that κti = {ti } × Sj × Tjti . Under Assumption 3 it follows from Proposition 1 that, for each type ti of any player i, i’s system of conditional preferences at ti can be represented by a vNM utility function υiti : ∆(Z) → R and a subjective probability distribution µti which for expositional simplicity is defined on Sj ×Tj with support Sj ×Tjti (instead of being defined on Ti ×Sj ×Tj with support κti = {ti } × Sj × Tjti ). Hence, as before we consider w.l.o.g. i’s unconditional preferences at ti , ºti , to be preferences over acts from Sj × Tj to ∆(Z) (instead of acts from {ti } × Sj × Tj to ∆(Z)). The combination of κti having full support on Si and Axiom 6 (Conditionality) being satisfied means that all opponent strategies are taken into account for any opponent type that is deemed subjectively possible, something that is reflected by µti having full support on Sj . Hence, preference for cautious behavior need not be explicitly imposed. Rather, following Schuhmacher (1999) we consider the following events. First, define the set of type profiles for which ti , for any subjectively possible opponent type, ind uces that type’s mixed strategy: ¯ n o 0 0 ¯ [indi ] := (t1 , t2 ) ∈ T1 × T2 ¯ ∀t0j ∈ Tjti , pjti |tj = pjtj . Write [ind] := [ind1 ]∩[ind2 ]. Furthermore, define the set of type profiles for which ti , according to his mixed strategy piti , plays a pure strategy

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with much greater probability than another if player i at ti prefers the former to the latter: ¯ n ¯ [ε-prop tremi ] := (t1 , t2 ) ∈ T1 × T2 ¯ o εpiti (si ) ≥ piti (si ) whenever si ºti s0i . If ti ∈ projTi [ε-prop tremi ], then player i is said to satisfy the ε-proper trembling condition at ti . Schuhmacher’s (1999) definition of ε-proper rationalizability can now be formally stated.

Definition 21 (Schuhmacher, 1999) A mixed strategy pi for i is εproperly rationalizable in a finite strategic two-player game G if there exists an ∗-epistemic model with piti = pi for some ti ∈ projTi CK([u] ∩ [ind] ∩ [ε-prop trem]). A mixed strategy pi for i is properly rationalizable if there exists a sequence (pi (n))n∈N of ε(n)-properly rationalizable strategies converging to pi , where ²(n) → 0 as n → ∞. We next characterize the concept of properly rationalizable strategies as induced mixed strategies under common certain belief of [u] ∩ [resp] ∩ [cau]. The result is proven in Appendix B.

Proposition 37 A mixed strategy pi for i is properly rationalizable in a finite strategic two-player game G if and only if there exists an epistemic model with (t1 , t2 ) ∈ CK([u] ∩ [resp] ∩ [cau]) such that pi is induced for ti by tj . It follows from Propositions 36 and 37 that any mixed strategy is properly rationalizable if it is part of a proper equilibrium. Since a proper equilibrium always exists, we obtain as an immediate consequence that properly rationalizable strategies always exist.

10.3

Relating rationalizability concepts (cont.)

As shown by van Damme (1984), any proper equilibrium in the strategic form corresponds to a quasi-perfect equilibrium in the extensive form. The following result shows, by Propositions 34 and 36, this relationship between the equilibrium concepts and establishes, by Definition 18 and Proposition 37, the corresponding relationship between the rationalizability concepts. Furthermore, it means that the two cells in Table 2.2 to the left of ‘proper rationalizability’ are not applicable.

Proposition 38 For any epistemic model and for each player i, [respi ] ∩ Ki [cauj ] ⊆ [isri ] .

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Proof. Consider any epistemic model with ti ∈ projTi ([respi ] ∩ Ki [cauj ]) . Suppose ti ∈ / projTi [isri ]; i.e., there exist tj ∈ Tj ti and h ∈ Hj such that σjti |tj |h is outcome equivalent to pj , where pj (sj ) > 0 for some sj ∈ Sj (h)\Sjtj (h). Since ti ∈ projTi Ki [cauj ], it follows from Lemma 13 that ∃s0j ∈ Sj (h) s.t. s0j Âtj sj . Since ti ∈ projTi [respi ], this means that ∃s0j ∈ Sj (h) s.t. (s0j , tj ) Àti (sj , tj ). Furthermore, pj (sj ) > 0 implies µt`i (sj , tj ) > 0, where ` is the first level ` of λti for which µt`i (Sj (h), tj ) > 0. Since then ` is also the first level ` of λti for which µt`i ({sj , s0j }, tj ) > 0, this contradicts (s0j , tj ) Àti (sj , tj ) and shows that ti ∈ projTi [isri ]. Since proper rationalizability is thus a refinement of quasi-perfect rationalizability, which in turn is a refinement of sequential rationalizability, it follows from Proposition 33 that proper rationalizability implies the backward induction procedure in perfect information games. E.g., in the “centipede” game illustrated in Γ03 of Figure 2.4, common certain belief of proper consistency implies that the players’ preferences over their own strategies are given by 1’s preferences: Out  InL  OutR 2’s preferences: `  r . This property of proper rationalizability has been discussed by both Schuhmacher (1999) and Asheim (2001). From the proof of Proposition 1 in Mailath et al. (1997) one can conjecture that quasi-perfect rationalizability in every extensive form corresponding to a given strategic game coincides with proper rationalizability in that game. However, for any given extensive form the set of proper rationalizable strategies can be a strict subset of the set of quasi-perfect rationalizable strategies, as illustrated by Γ02 of Figure 2.5. Here, quasi-perfect rationalizability only precludes the play of InR with positive probability. However, since InL strongly dominates InR, it follows that 2 prefers ` to r if she respects 1’s preferences. Hence, only ` with probability one is properly rationalizable for 2, which implies that only InL with probability one is properly rationalizable for 1.

10.4

Induction in a betting game

The games G7 (of Figure 10.1), Γ03 (of Figure 2.4), and Γ02 (of Figure 2.5) have in common that the properly rationalizable strategies coincide with those surviving iterated (maximal) elimination of weakly domi-

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Properness

a -9 9

b 6 -6

c -3 3

1/3

1/3

1/3

Player 1 Player 2

Figure 10.2.

A betting game.

nated strategies (IEWDS). In the present section it will be shown that this conclusion does not hold in general. Rather, the concept of proper rationalizability can be used to test the robustness of IEWDS and other inductive procedures. Figure 10.2 illustrates a simplified version of a betting game introduced by Sonsino et al. (2000) for the purpose of experimental study; Søvik (2001) has subsequently repeated their experiment in alternative designs. The two players consider to bet and have a common and uniform prior over the states that determine the outcome of the bet. If the state is a, then 1 looses 9 and 2 wins 9 if betting occurs. If the state is b, then 1 wins 6 and 2 looses 6 if betting occurs. Finally, if the state is c, then 1 looses 3 and 2 wins 3 if betting occurs. Player 1 is informed of whether the state of the bet is equal to a or in the set {b, c}. Player 2 is informed of whether the state of the bet is in the set {a, b} or equal to c. As a function of their information, each player can announce to accept the bet or not. For player 1 the strategy YN means to accept the bet if informed of a and not to accept the bet if informed of {b, c}, etc. For player 2 the strategy yn means to accept the bet if informed of {a, b} and not to accept the bet if informed of c, etc. Betting occurs if and only if both players have accepted the bet. This yields the strategic game of Figure 10.3. An inductive procedure. If player 2 naively believes that player 1 is equally likely to accept the bet when informed of a as when informed of {b, c}, then 2 will wish to accept the bet when informed of {a, b}. However, the following, seemingly intuitive, inductive procedure appears to indicate that 2 should never accept the bet if informed of {a, b}: Player 1 should not accept the bet when informed of a since he cannot win by doing so. This eliminates his strategies YY and YN. Player 2, realizing this, should never accept the bet when informed of {a, b}, since—as long as 1 never accepts the bet when informed of a—she cannot win by doing so. This eliminates her strategies yy and yn. This in turn means

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YY YN NY NN Figure 10.3.

yy yn ny -2, 2 -1, 1 -1, 1 -3, 3 -3, 3 0, 0 1, -1 2, -2 -1, 1 0, 0 0, 0 0, 0

nn 0, 0 0, 0 0, 0 0, 0

The strategic form of the betting game.

that player 1, realizing this, should never accept the bet when informed of {b, c}, since—as long as 2 never accepts the bet when informed of {a, b}—he cannot win by doing so. This eliminates his strategy NY. This inductive argument corresponds to IEWDS, except that the latter procedure eliminates 2’s strategies yn and nn in the first round. The argument seems to imply that player 2 should never accept the bet if informed of {a, b} and that player 1 should never accept the bet if informed of {b, c}. Is this a robust conclusion? Proper rationalizability in the betting game. The strategic game of Figure 10.3 has a set of Nash equilibria that includes the pure strategy profiles (NN, ny) and (NN, nn), and a set of (strategic form) perfect equilibria that includes the pure strategy profile (N N, ny). However, there is a unique proper equilibrium where player 1 plays NN with probability one, and where player 2 mixes between yy with probability 1/5 and ny with probability 4/5. It is instructive to see why the pure strategy profile (NN, ny) is not a proper equilibrium. If 1 assigns probability one to 2 playing ny, then he prefers YN to NY (since the more serious mistake to avoid is to accept the bet when being informed of {b, c}). However, if 2 respects 1’s preferences and certainly believes that 1 prefers YN to NY, then she will herself prefer yy to ny, undermining (NN, ny) as a proper equilibrium. The mixture between yy and ny in the proper equilibrium is constructed so that 1 is indifferent between YN and NY. Since any mixed strategy is properly rationalizable if it is part of a proper equilibrium, it follows that both yy and yn are properly rationalizable pure strategies for 2. Moreover, if 1 certainly believes that 2 is of a type with only yy as a most preferred strategy, then NY is a most preferred strategy for 1, implying that NY in addition to NN is a properly rationalizable strategy for 1. That these strategies are in fact properly rationalizable is verified by the epistemic model of Ta-

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Table 10.1.

t01

t02

An epistemic model for the betting game.

yy yn ny nn

t02 (0, 0, 1, 0) (0, 0, 0, 1) (1, 0, 0, 0) (0, 1, 0, 0)

t002 (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0)

t001

YY YN NY NN

t01 (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0)

t001 (0, 0, 1, 0) (0, 0, 0, 1) (1, 0, 0, 0) (0, 1, 0, 0)

t002

yy yn ny nn

t02 (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0)

t002 (1, 0, 0, 0) (0, 1, 0, 0) (0, 0, 1, 0) (0, 0, 0, 1)

YY YN NY NN

t01 (0, 0, 0, 1) (0, 1, 0, 0) (0, 0, 1, 0) (1, 0, 0, 0)

t001 (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0)

ble 10.1. In the table the preferences of any player i at each type ti are represented by a vNM utility function υiti satisfying υiti ◦ z = ui and a 4-level LPS on Sj × {t0j , t00j }, with the first numbers in the parantheses expressing primary probability distributions, the second numbers expressing secondary probability distributions, etc. It can be checked that {t01 , t001 } × {t02 , t002 } ⊆ [u] ∩ [resp] ∩ [cau], which in turn implies {t01 , t001 } × {t02 , t002 } ⊆ CK([u] ∩ [resp] ∩ [cau]) since, for each ti ∈ Ti of any player i, Tjti ⊆ {t0j , t00j }. Since each type’s preferences over his/her own strategies are given by 0

0

0

N N Ât1 Y N Ât1 N Y Ât1 Y Y 00 00 00 N Y Ât1 N N Ât1 Y Y Ât1 Y N 0 0 0 ny Ât2 nn Ât2 yy Ât2 yn 00 00 00 yy Ât2 yn Ât2 ny Ât2 nn , it follows that NY and NN are properly rationalizable for player 1 and yy and ny are properly rationalizable for player 2. Note that YY and YN for player 1 and yn and nn for player 2 cannot be properly rationalizable since these strategies are weakly dominated and, thus, cannot be most preferred strategies for cautious players. The lesson to be learned from this analysis is that is not obvious that deductive reasoning should lead players to refrain from accepting the bet in the betting game. The experiments by Sonsino et al. (2000) and Søvik (2001) show that some subjects do in fact accept the bet in a slightly more complicated version of this game. By comparison to

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Propositions 33 and 38, the analysis can be used to support the argument that backward induction in generic perfect information games is more convincing than the inductive procedure for the betting game discussed above.

Chapter 11 CAPTURING FORWARD INDUCTION THROUGH FULL PERMISSIBILITY

The procedure of iterated (maximal) elimination of weakly dominated strategies (IEWDS) has a long history and some intuitive appeal, yet it is not as easy to interpret as iterated elimination of strongly dominated strategies (IESDS). IESDS is known to be equivalent to common belief of rational choice; cf. Tan and Werlang (1988) as well as Propositions 22 and 26 of this book. IEWDS would appear simply to add a requirement of admissibility, i.e., that one strategy should be preferred to another if the former weakly dominates the latter on a set of strategies that the opponent “may choose”. However, numerous authors—in particular, Samuelson (1992)—have noted that it is not clear that we can interpret IEWDS this way. To see this, consider the following two examples. The left-hand side of Figure 2.6 shows G01 , the pure strategy reduced strategic form of the “battle-of-the-sexes with an outside option” game. Here IEWDS works by eliminating InR, r, and Out, leading to the forward induction outcome (InL, `). This prediction appears consistent: if 2 believes that 1 will choose InL, then she will prefer ` to r as 2’s preference over her strategies depends only on the relative likelihood of InL and InR. The situation is different in G8 of Figure 11.1, where IEWDS works by eliminating D, r, and M , leading to (U, `). Since 2 is indifferent at the predicted outcome, we must here appeal to admissibility on a superset of {U }, namely {U, M }, to justify the statement that 2 must play L. However, it is not clear that this is reasonable. Admissibility on {U, M } means that 2’s preferences respect weak dominance on this set and implies that M is deemed infinitely more likely than D (in the sense of Blume et al., 1991a, Definition 5.1; see also Chapter 3). However, why

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r ` 1, 1 1, 1 U M 0, 1 2, 0 D 1, 0 0, 1

Figure 11.1.

G8 , illustrating that IEWDS may be problematic.

should 2 deem M more likely than D? If 2 believes that 1 believes in the prediction that 2 plays ` (as IEWDS suggests), then it seems odd to assume that 2 believes that 1 considers D to be a less attractive choice than M . A sense in which D is “less rational” than M is simply that it was eliminated first. This hardly seems a justification for insisting on the belief that D is much less likely than M . Still, Stahl (1995) has shown that IEWDS effectively assumes this: a strategy survives IEWDS if and only if it is a best response to a belief where one strategy is infinitely less likely than another if the former is eliminated at an earlier round than the latter. Thus, IEWDS adds extraneous and hard-to-justify restrictions on beliefs, and may not appear to correspond to the most natural formalization of deductive reasoning under admissibility. So what does? Reproducing joint work with Martin Dufwenberg, cf. Asheim and Dufwenberg (2003a), this chapter presents the concept of ‘fully permissible sets’ as an answer. In G01 this concept agrees with the prediction of IEWDS, as seems natural. The procedure leading to this prediction is quite different, though, as is its interpretation. In G8 , however, full permissibility predicts that 1’s set of rational choices is either {U } or {U, M }, while 2’s set of rational choices is either {`} or {`, r}. This has interesting implications. If 2 is certain that 1’s set is {U }, then—absent extraneous restrictions on beliefs—one cannot conclude that 2 prefers ` to r or vice versa. On the other hand, if 2 considers it possible that 1’s set is {U, M }, then ` weakly dominates r on this set and justifies {`} as 2’s set of rational choices. Similarly, one can justify that U is preferred to M if and only if 1 considers it impossible that 2’s set is {`, r}. Thus, full permissibility tells a consistent story of deductive reasoning under admissibility, without adding extraneous restrictions on beliefs. This chapter is organized as follows. Section 11.1 illustrates the key features of the requirement—called ’full admissible consistency’—that is imposed on players to arrive at full permissibility. Section 11.2 formally defines the concept of fully permissible sets through an algorithm that

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135

eliminates strategy sets under full admissible consistency. General existence as well as other properties are shown. Section 11.3 establishes epistemic conditions for the concept of fully permissible sets, and checks that these conditions are indeed needed and thereby relates full permissibility to other concepts. Section 11.4 investigates examples, showing how forward induction is promoted and how multiple fully permissible sets may arise. Section 11.5 compares our epistemic conditions to those provided in related literature. As elsewhere in this book, the analysis will be limited to two-player games. In this chapter (and the next), this is for ease of presentation, as everything can essentially be generalized to n-player games (with n > 2).

11.1

Illustrating the key features

Our modeling captures three key features: 1 Caution. A player should prefer one strategy to another if the former weakly dominates the latter. Such admissibility of a player’s preferences on the set of all opponent strategies is defended, e.g., in Chapter 13 of Luce and Raiffa (1957) and is implicit in procedures that start out by eliminating all weakly dominated strategies. 2 Robust belief of opponent rationality. A player should deem any opponent strategy that is a rational choice infinitely more likely than any opponent strategy not having this property. This is equivalent to preferring one strategy to another if the former weakly dominates the latter on the set of rational choices for the opponent. Such admissibility of a player’s preferences on a particular subset of opponent strategies is an ingredient of the analyses of weak dominance by Samuelson (1992) and B¨orgers and Samuelson (1992), and is essentially satisfied by ‘extensive form rationalizability’ (EFR; cf. Pearce, 1984 and Battigalli, 1996a, 1997) and IEWDS. 3 No extraneous restrictions on beliefs. A player should prefer one strategy to another only if the former weakly dominates the latter on the set of all opponent strategies or on the set of rational choices for the opponent. Such equal treatment of opponent strategies that are all rational—or all irrational—have in principle been argued by Samuelson (1992, p. 311), Gul (1997), and Mariotti (1997). These features are combined as follows. A player’s preferences over his own strategies leads to a choice set (i.e., a set of maximal pure strategies; cf. Section 6.1). A player’s preferences is said to be fully

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r ` 1, 1 1, 1 U M 1, 1 1, 0 D 1, 0 0, 1

Figure 11.2.

G9 , illustrating the key features of full admissible consistency.

admissibly consistent with the game and the preferences of his opponent if one strategy is preferred to another if and only if the former weakly dominates the latter on the set of all opponent strategies, or on the union of the choice sets that are deemed possible for the opponent. A subset of strategies is a fully permissible set if and only if it can be a choice set when there is common certain belief of full admissible consistency. Hence, the analysis yields a solution concept that determines a collection of choice sets for each player. This collection can be found via a simple algorithm, introduced in the next section. We use G9 of Fig. 11.2 to illustrate the consequences of imposing ‘caution’ and ‘robust belief of opponent rationality’. Since ‘caution’ means that each player takes all opponent strategies into account, it follows that player 1’s preferences over his strategies will be U ∼ M  D (where ∼ and  denote indifference and preference, respectively). Player 1 must prefer each of the strategies U and M to the strategy D, because the former strategies weakly dominate D. Hence, U and M are maximal, implying that 1’s choice set is {U, M }. The requirement of ‘robust belief in opponent rationality’ comes into effect when considering the preferences of player 2. Suppose that 2 certainly believes that 1 is cautious and therefore (as indicated above) certainly believes that {U, M } is 1’s choice set. Our assumption that 2 has robust belief of 1’s rationality captures that 2 deems each element of {U, M } infinitely more likely than D. Thus, 2’s preferences respect weak dominance on 1’s choice set {U, M }, regardless of what happens if 1 chooses D. Hence, 2’s preferences over her strategies will be `  r. Summing up, we get to the following solution for G9 : 1’s preferences: U ∼ M  D 2’s preferences: `  r

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137

Hence, {U, M } and {`} are the players’ fully permissible sets. The third feature of full admissible consistency—‘no extraneous restrictions on beliefs’—means in G9 that 2 does not assess the relative likelihood of 1’s maximal strategies U and M . This does not have any bearing on the analysis of G9 , but is essential for capturing forward induction in G01 of Figure 2.6. In this case the issue is not whether a player assesses the relative likelihood of different maximal strategies, but rather whether a player assesses the relative likelihood of different non-maximal strategies. To see the significance in G01 , assume that 1 deems r infinitely more likely than `, while 2 deems Out infinitely more likely than InR and InR infinitely more likely than InL. Then the players rank their strategies as follows: 1’s preferences: Out  InR  InL 2’s preferences: r  ` Both ‘caution’ and ‘robust belief of opponent rationality’ are satisfied and still the forward induction outcome (InL, `) is not promoted. However, the requirement of ‘no extraneous restrictions on beliefs’ is not satisfied since the preferences of 2 introduce extraneous restrictions on beliefs by deeming one of 1’s non-maximal strategies, InR, infinitely more likely than another non-maximal strategy, InL. When we return to G01 in Sections 11.4 and 11.5, we show how the additional imposition of ‘no extraneous restrictions on beliefs’ leads to (InL, `) in this game. Several concepts with natural epistemic foundations fail to match these predictions in G01 and G9 . In the case of rationalizability—cf. Bernheim (1984) and Pearce (1984)—this is perhaps not so surprising since this concept in two-player games corresponds to IESDS. It can be understood as a consequence of common belief of rational choice without imposing caution, so there is no guarantee that a player prefers one strategy to another if the former weakly dominates the latter. In G9 , for example, all strategies are rationalizable. It is more surprising that the concept of ‘permissibility’ does not match our solution of G9 . Permissibility can be given rigorous epistemic foundations in models with cautious players—cf. B¨orgers (1994) and Brandenburger (1992), who coined the term ‘permissible’; see also Ben-Porath (1997) and Gul (1997) as well as Propositions 24 and 27 of this book. In these models players take into account all opponent strategies, while assigning more weight to a subset of those deemed to

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be rational choices. As noted earlier, permissibility corresponds to the Dekel-Fudenberg procedure where one round of elimination of all weakly dominated strategies is followed by iterated elimination of strongly dominated strategies. In G9 , this means that 1 cannot choose his weakly dominated strategy D. However, while 2 prefers ` to r in our solution, permissibility allows that 2 chooses r. To exemplify using Brandenburger’s (1992) approach, this will be the case if 2 deems U to be infinitely more likely than D which in turn is deemed infinitely more likely than M . The problem is that ‘robust belief of opponent rationality’ is not satisfied: Player 2 deems D more likely than M even though M is in 1’s choice set, while D is not. In Section 11.3 we establish in Proposition 40 that the concept of fully permissible sets refines the Dekel-Fudenberg procedure.

11.2

IECFA and fully permissible sets

We present in this section an algorithm—‘iterated elimination of choice sets under full admissible consistency’ (IECFA)—leading to the concept of ‘fully permissible sets’. This concept will in turn be given an epistemic characterization in Section 11.3 by imposing common certain belief of full admissible consistency. We present the algorithm before the epistemic characterization for different reasons: IECFA is fairly accessible. By defining it early, we can apply it early, and offer early indications of the nature of the solution concept we wish to promote. By defining IECFA, we point to a parallel to the concepts of rationalizable strategies and permissible strategies. These concepts are motivated by epistemic assumptions, but turn out to be identical in 2-player games to the set of strategies surviving simple algorithms: respectively, IESDS and the Dekel-Fudenberg procedure. Just like IESDS and the Dekel-Fudenberg procedure, IECFA is easier to use than the corresponding epistemic characterizations. The algorithm should be handy for applied economists, independently of the foundational issues discussed in Section 11.3. IESDS and the Dekel-Fudenberg procedure iteratively eliminate dominated strategies. In the corresponding epistemic models, these strategies in turn cannot be rational choices, cannot be rational choices given that other players do not use strategies that cannot be rational choices, etc.

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IECFA is also an elimination procedure. However, the interpretation of the basic item thrown out is not that of a strategy that cannot be a rational choice, but rather that of a set of strategies that cannot be a choice set for any preferences that are in a given sense consistent with the preferences of the opponent. The specific kind of consistency involved in IECFA—which will be defined in Section 11.3 and referred to as ‘full admissible consistency’—requires that a player’s preferences are characterized by the properties of ‘caution’, ‘robust belief of opponent rationality’ and ‘no extraneous restrictions on beliefs’. Thus, IECFA does not start with each player’s strategy set and then iteratively eliminates strategies. Rather, IECFA starts with each player’s collection of non-empty subsets of his strategy set and then iteratively eliminates subsets from this collection. Definition. Consider a finite strategic two-player game G = (S1 , S2 , u1 , u2 ), and recall the following notation from Chapter 6: For any (∅ 6=) Yj ⊆ Sj , Di (Yj ) := {si ∈ Si | ∃pi ∈ ∆(Si ) such that pi weakly dominates si on Yj or Sj } . Interpret Yj as the set of strategies that player i deems to be the set of rational choices for his opponent. Let i’s choice set be equal to Si \Di (Yj ), entailing that i’s choice set consists of pure strategies that are not weakly dominated by any mixed strategy on Yj or Sj . In Section 11.3 we show how this corresponds to a set of maximal strategies given the player’s preferences over his own strategies. Let Σ = Σ1 × Σ2 , where Σi := 2Si \{∅} denotes the collection of nonempty subsets of Si . Write σi (∈ Σi ) for a subset of pure strategies. For any (∅ 6=) Ξ = Ξ1 × Ξ2 ⊆ Σ, write α(Ξ) := α1 (Ξ2 ) × α2 (Ξ1 ), where αi (Ξj ) := {σi ∈ Σi |∃(∅ 6=) Ψj ⊆ Ξj s.t. σi = Si \Di (∪σj0 ∈Ψj σj0 )} . Hence, αi (Ξj ) is the collection of strategy subsets that can be choice sets for player i if he associates Yj —the set of rational choices for his opponent—with the union of the strategy subsets in a non-empty subcollection of Ξj . We can now define the concept of a fully permissible set.

Definition 22 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. Consider the sequence defined by Ξ(0) = Σ and, ∀g ≥ 1, Ξ(g) = α(Ξ(g − 1)). A non-empty strategy set σi is said to be fully permissible if

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CONSISTENT PREFERENCES

σi ∈

\∞ g=0

Ξi (g) .

Let Π = Π1 × Π2 denote the collection of profiles of fully permissible sets. Since ∅ 6= αi (Ξ0j ) ⊆ αi (Ξ00j ) ⊆ αi (Σj ) whenever ∅ 6= Ξ0j ⊆ Ξ00j ⊆ Σj and since the game is finite, Ξ(g) is a monotone sequence that converges to Π in a finite number of iterations. IECFA is the procedure that in round g eliminates sets in Ξ(g − 1)\Ξ(g) as possible choice sets. As defined in Definition 22, IECFA eliminates maximally in each round in the sense that, ∀g ≥ 1, Ξ(g) = α(Ξ(g − 1)). However, it follows from the monotonicity of αi that any non-maximal procedure, where ∃g ≥ 1 such that Ξ(g − 1) ⊃ Ξ(g) ⊃ α(Ξ(g − 1)), will also converge to Π. A strategy subset survives elimination round g if it can be a choice set when the set of rational choices for his opponent is associated with the union of some (or all) of opponent sets that have survived the procedure up till round g − 1. A fully permissible set is a set that survives in this way for any g. The analysis of Section 11.3 justifies that strategy subsets that this algorithm has not eliminated by round g be interpreted as choice sets compatible with g − 1 order of mutual certain belief of full admissible consistency. Applications. We illustrate IECFA by applying it. Consider G9 of Figure 11.2. We get: Ξ(0) = Σ1 × Σ2 Ξ(1) = {{U, M }} × Σ2 Π = Ξ(2) = {{U, M }} × {{`}} . Independently of Y2 , S1 \D1 (Y2 ) = {U, M }, so for 1 only {U, M } survives the first elimination round, while S2 \D2 ({U, M }) = {`}, S2 \D2 ({D}) = {r} and S2 \D2 ({U }) = {`, r}, so that no elimination is possible for player 2. However, in the second round only {`} survives since ` weakly dominates r on {U, M }, implying that S2 \D2 ({U, M }) = {`}. Next, consider G01 of Figure 2.6. Applying IECFA we get: Ξ(0) = Σ1 × Σ2 Ξ(1) = {{Out}, {InL}, {Out, InL}} × Σ2 Ξ(2) = {{Out}, {InL}, {Out, InL}} × {{`}, {`, r}} Ξ(3) = {{InL}, {Out, InL}} × {{`}, {`, r}} Ξ(4) = {{InL}, {Out, InL}} × {{`}} Π = Ξ(5) = {{InL}} × {{`}} .

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141

Again the algorithm yields a unique fully permissible set for each player. Finally, apply IECFA to G8 of Figure 11.1: Ξ(0) = Σ1 × Σ2 Ξ(1) = {{U }, {M }, {U, M }} × Σ2 Ξ(2) = {{U }, {M }, {U, M }} × {{`}, {`, r}} Π = Ξ(3) = {{U }, {U, M }} × {{`}, {`, r}} . Here we are left with two fully permissible sets for each player. There is no further elimination, as {U } = S1 \D1 ({`}), {U, M } = S1 \D1 ({`, r}), {`} = S2 \D2 ({U, M }), and {`, r} = S2 \D2 ({U }). The elimination process for G01 and G8 is explained and interpreted in Section 11.4. Results. The following proposition characterizes the strategy subsets that survive IECFA and thus are fully permissible, and is a straightforward implication of Definition 22 (keeping in mind that Σ is finite and, for each i, αi is monotone).

Proposition 39 (i) For each i, Πi 6= ∅. (ii) Π = α(Π). (iii) For each i, σi ∈ Πi if and only if there exists Ξ = Ξ1 × Ξ2 with σi ∈ Ξi such that Ξ ⊆ α(Ξ). Proposition 39(i) shows existence, but not uniqueness, of each player’s fully permissible set(s). In addition to G2 , games with multiple strict Nash equilibria illustrate the possibility of such multiplicity; by Proposition 39(iii) any strict Nash equilibrium corresponds to a profile of fully permissible sets. Proposition 39(ii) means that Π is a fixed point in terms of a collection of profiles of strategy sets as illustrated by G2 above. By Proposition 39(iii) it is the largest such fixed point. We close this section by recording some connections between IECFA on the one hand, and IESDS, the Dekel-Fudenberg procedure (i.e., permissibility), and IEWDS on the other. First, we note through the following Proposition 40 that IECFA has more bite than the Dekel-Fudenberg procedure. Both G1 and G3 illustrate that this refinement may be strict.

Proposition 40 A pure strategy si is permissible if there exists a fully permissible set σi such that si ∈ σi . Proof. Using Proposition 39(ii), the definitions of α(·) (given above) and a(·) (given in Chapter 6) imply, for each i, Pi0 := ∪σi ∈Πi σi = ∪σi ∈αi (Πj ) σi ⊆ ai (Pj0 ) .

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UU UD DU DD

Figure 11.3.

` 1, 1 1, 1 0, 1 0, 0

c 1, 1 0, 1 0, 0 0, 1

r 0, 0 1, 0 2, 0 0, 2

G10 , illustrating the relation between IECFA and IEWDS.

Since P 0 ⊆ a(P 0 ) implies P 0 ⊆ P , by Lemma 10(iii), it follows that, for each i, ∪σi ∈Πi σi ⊆ Pi . It is a corollary that IECFA has also more cutting power than IESDS. However, neither of IECFA and IEWDS has more bite than the other, as demonstrated by the game G10 of Fig. 11.3. It is straightforward to verify that UU and UD for player 1 and ` for player 2 survive IEWDS, while {UU} for 1 and {`, c} for 2 survive IECFA and are thus the fully permissible sets, as shown below: Ξ(0) = Σ1 × Σ2 Ξ(1) = {{UU}, {DU}, {UU, UD}, {UU, DU}, {UD, DU}, {UU, UD, DU}} × {{`}, {r}, {`, c}, {`, r}, {c, r}, {`, c, r}} Ξ(2) = {{UU}, {DU}, {UU, UD}, {UU, DU}, {UD, DU}, {UU, UD, DU}} × {{`}, {`, c}} Ξ(3) = {{UU}, {UU, UD}} × {{`}, {`, c}} Ξ(4) = {{UU}, {UU, UD}} × {{`, c}} Π = Ξ(5) = {{UU}} × {{`, c}} . Strategy UD survives IEWDS but does not appear in any fully permissible set. Strategy c appears in a fully permissible set but does not survive IEWDS.

11.3

Full admissible consistency

When justifying rationalizable and permissible strategies through epistemic conditions, players are usually modeled as decision makers under uncertainty. Tan and Werlang (1988) characterize rationalizable strategies by common belief (with probability one) of the event that each player chooses a maximal strategy given preferences that are represented by a subjective probability distribution. Hence, preferences are both complete and continuous (cf. Proposition 1). Brandenburger (1992) characterizes permissible strategies by common belief (with pri-

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143

mary probability one) of the event that each player chooses a maximal strategy given preferences that are represented by an LPS with full support on the set of opponent strategies (cf. Proposition 2). Hence, preferences are still complete, but not continuous due to the full support requirement. Since preferences are complete and representable by a probability distribution or an LPS, these epistemic justifications differ significantly from the corresponding algorithms, IESDS and the DekelFudenberg procedure, neither of which makes reference to subjective probabilities.1 When doing analogously for fully permissible sets, not only must continuity of preferences be relaxed to allow for ‘caution’ and ‘robust belief of opponent rationality’, as discussed in Section 11.1. One must also relax completeness of preferences to accommodate ‘no extraneous restrictions on beliefs’, which is a requirement of minimal completeness and implies that preferences are expressed solely in terms of admissibility on nested sets. Hence, preferences are not in general representable by subjective probabilities (except through treating incomplete preferences as a set of complete preferences; cf. Aumann, 1962; Bewley, 1986). This means that epistemic operators must be derived directly from the underlying preferences—as observed by Morris (1997) and explored further in Chapter 4 of this book—since there is no probability distribution or LPS that represents the preferences. It also entails that the resulting characterization, given in Proposition 41, must be closely related to the algorithm used in the definition of fully permissible sets. There is another fundamental difference. When characterizing rationalizable and permissible strategies within the ‘rational choice’ approach, the event that is made subject to interactive epistemology is defined by requiring that each player’s strategy choice is an element of his choice set (i.e. his set of maximal strategies) given his belief about the opponent’s strategy choice.2 In contrast, in the characterization of Proposition 40, the event that is made subject to interactive epistemology is defined by imposing requirements on how each player’s choice set is related to his belief about the opponent’s choice set. Since a player’s choice set equals the set of maximal strategies given the ranking that the player has over his strategies, the imposed requirements relate a player’s ranking over

1 However,

as shown by Propositions 26 and 27 of this book, epistemic characterization of rationalizability and permissibility can be provided without using subjective probabilities. 2 As illustrated in Chapters 5 and 6 of this book, it is also possible to characterize rationalizable and permissible strategies within the ‘consistent preferences’ approach.

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his strategies to the opponent’s ranking. Hence, fully permissible sets are characterized within the ‘consistent preferences’ approach. The epistemic modeling is identical to the one given in Section 6.1; hence, this will not be recapitulated here. Recall, however, that κti (⊆ {ti }×Sj ×Tj ) denotes the set of states that player i deems subjectively possible at ti , that β ti (⊆ κti ) denotes the smallest set of states on which player i’s preferences at ti , ºti , are admissible, and that Assumption 2 is imposed so that preferences are conditionally represented by a vNM utility function (cf. Proposition 4). Characterizing full permissibility. To characterize the concept of fully permissible sets, consider for each i, ¯ 0i [ratj ] := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | B β ti = (projTi ×Sj ×Tj [ratj ]) ∩ κti , and p Âti q only if pEj weakly dominates qEj for Ej = projSj ×Tj β ti or Ej = projSj ×Tj κti } , Define as follows the event that player i’s preferences over his strategies are fully admissibly consistent with the game G = (S1 , S2 , u1 , u2 ) and the preferences of his opponent: ¯ 0i [ratj ] ∩ [caui ] . A¯0i := [ui ] ∩ B Write A¯0 := A¯01 ∩ A¯02 for the event of full admissible consistency.

Proposition 41 A strategy set σi for i is fully permissible in a finite strategic two-player game G if and only if there exists an epistemic model with σi = Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA¯0 . Proof. Part 1: If σi is fully permissible, then there exists an epistemic model with σi = Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA¯0 . It is sufficient to construct a belief system with S1 × T1 × S2 × T2 ⊆ CKA¯0 such that, for each σi ∈ Πi of any player i, there exists ti ∈ Ti with σi = Siti . Construct a belief system with, for each i, a bijection σ i : Ti → Πi from the set of types to the the collection of fully permissible sets. By Proposition 39(ii) we have that, for each ti ∈ Ti of any player i, there exists Ψjti ⊆ Πj such that σ i (ti ) = Si \Di (Yjti ), where Yjti := {sj ∈ Sj | ∃σj ∈ Ψjti s.t. sj ∈ σj }. Determine the set of opponent types that ti deems subjectively possible as follows: Tjti = {tj ∈ Tj | σ j (tj ) ∈ Ψjti }. Let, for each ti ∈ Ti of any player i, ºti satisfy 1. υiti ◦ z = ui (so that S1 × T1 × S2 × T2 ⊆ [u]), and

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2. p Âti q iff pEj weakly dominates qEj for Ej = Ejti := {(sj , tj )|sj ∈ σ j (tj ) and tj ∈ Tjti } or Ej = Sj × Tjti , which implies that β ti = {ti }×Ejti and κti = {ti }×Sj ×Tjti (so that S1 ×T1 ×S2 ×T2 ⊆ [cau]). By the construction of Ejti , this means that Siti = Si \Di (Yjti ) = σ i (ti ) since, for any acts p and q on Sj × Tj satisfying that there exist mixed strategies pi , qi ∈ ∆(Si ) such that, ∀(sj , tj ) ∈ Sj ×Tj , p(sj , tj ) = z(pi , sj ) and q(sj , tj ) = z(qi , sj ), p Âti q iff pEj weakly dominates qEj for Ej = Yjti × Tj or Ej = Sj × Tj . This in turn implies, for each ti ∈ Ti any player i, 3. β ti = (projTi ×Sj ×Tj [ratj ]) ∩ κti (so that, in combination with 2., S1 × ¯ 0i [ratj ] ∩ B ¯ 0j [rati ]). T1 × S2 × T2 ⊆ B Furthermore, S1 × T1 × S2 × T2 ⊆ CKA¯0 since Tjti ⊆ Tj for each ti ∈ Ti of any player i. Since, for each player i, σ i is onto Πi , it follows that, for each σi ∈ Πi of any player i, there exists ti ∈ Ti with σi = Siti . ∗ Part 2: If there exists an epistemic model with σi∗ = Siti for some (t∗1 , t∗2 ) ∈ projT1 ×T2 CKA¯0 , then σi∗ is fully permissible. ∗ Assume that there exists an epistemic model with σi∗ = Siti for some (t∗1 , t∗2 ) ∈ projT1 ×T2 CKA¯0 . In particular, CKA¯0 6= ∅. Let, for each i, Ti0 := projTi CKA¯0 and Ξi := {Siti | ti ∈ Ti0 }. It is sufficient to show that, for each i, Ξi ⊆ Πi . By Proposition 25(ii), for each ti ∈ Ti0 of any player i, β ti ⊆ κti ⊆ {ti } × Sj × Tj0 since CKA¯0 = KCKA¯0 ⊆ Ki CKA¯0 . By the definition of A¯0 , it follows that, for each ti ∈ Ti0 of any player i, 1. ºti is conditionally represented by υiti satisfying that υiti ◦ z is a positive affine transformation of ui , and 2. p Âti q iff pEj weakly dominates qEj for Ej = Ejti := projSj ×Tj β ti or Ej = Sj × Tjti , where β ti = (projTi ×Sj ×Tj [ratj ]) ∩ κti . Write Ψjti := {Sj tj | tj ∈ Tj ti } and Yjti := {sj ∈ Sj | ∃σj ∈ Ψjti s.t. sj ∈ σj }, and note that κti ⊆ {ti } × Sj × Tj0 implies Ψjti ⊆ Ξj . It follows that, for any acts p and q on Sj × Tj satisfying that there exist mixed strategies pi , qi ∈ ∆(Si ) such that, ∀(sj , tj ) ∈ Sj × Tj , p(sj , tj ) = z(pi , sj ) and q(sj , tj ) = z(qi , sj ), p Âti q iff pEj weakly dominates qEj for Ej = Yjti × Tj or Ej = Sj × Tj . Hence, Siti = Si \Di (Yjti ). Since this holds for each ti ∈ Ti0 of any player i, we have that Ξ ⊆ α(Ξ). Hence, Proposition 39(iii) entails that, for each i, Ξi ⊆ Πi . Interpretation. We now show how the event used to characterize fully permissible sets—full admissible consistency—can be interpreted in terms of the requirements of ‘caution’, ‘robust belief of opponent ratio-

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nality’, and ‘no extraneous restrictions on beliefs’. Following a common procedure of the axiomatic method, this will in turn be used to verify that these requirements are indeed needed for the characterization in Proposition 41 by investigating the consequences of relaxing one requirement at a time. These exercises contribute to the understanding of fully permissible sets by showing that the concept is related to properly rationalizable, permissible, and rationalizable pure strategies in the following manner: When allowing extraneous restrictions on beliefs, we open for any properly rationalizable pure strategy, implying that forward induction is no longer promoted in G01 of Figure 2.6.3 When weakening ‘robust belief of opponent rationality’ to ‘belief of opponent rationality’, we characterize the concept of permissible pure strategies independently of whether a requirement of ‘no extraneous restrictions on beliefs’ is retained. When removing ‘caution’, we characterize the concept of rationalizable pure strategies independently of whether extraneous restrictions on beliefs are allowed and robust belief of opponent rationality is weakened. Since it is clear that [cau] = [cau1 ] ∩ [cau2 ] corresponds to caution ¯ 0 [rat2 ]∩ B ¯ 0 [rat1 ] into ‘robust belief (cf. Section 6.3), it remains to split B 1 2 of opponent rationality’ and ‘no extraneous restrictions on beliefs’. To state the condition of ‘robust belief of opponent rationality’ we need to recall the robust belief operator as defined and characterized in Chapter 4. Since Assumption 2 is compatible with the framework of Chapter 4, we can in line with Section 4.2 define robust belief as follows. If E does not concern player i’s strategy choice (i.e., E = Si × projTi ×Sj ×Sj E), say that player i robustly believes the event E at ti if ti ∈ projTi B0i E, where B0i E := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | ∃` ∈ {1, . . . , L} s.t. ρt`i = projT1 ×S2 ×T2 E ∩ κti } ,

3 To

relax ‘no extraneous restrictions on beliefs’ we need an epistemic model—as the one introduced in Section 6.1—that is versatile enough to allow for preferences that are more complete than being determined by admissibility on two nested sets.

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147

and where (ρt1i , . . . , ρtLi ) is the profile of nested sets on which ºti is admissible, and which satisfies: ∅ 6= β ti = ρt1i ⊂ · · · ⊂ ρt`i ⊂ · · · ⊂ ρtLi = κti ⊆ {ti } × Sj × Tj (where ⊂ denotes ⊆ and 6=). If ti ∈ projTi B0i [ratj ], then i robustly believes at ti that j is rational. By Proposition 6 this means that any (sj , tj ) that is deemed subjectively possible and where sj is a rational choice by j at tj is considered infinitely more likely than any (s0j , t0j ) where s0j is not a rational choice by j at t0j . ¯ 0 [ratj ] entails that β ti = (projT ×S ×T [ratj ]) ∩ κti , it As ti ∈ projTi B i i j j ¯ 0 [ratj ] ⊆ B0 [ratj ]. Hence, relative to B0 [rat2 ] ∩ B0 [rat1 ], follows that B 1 2 i i ¯ 0 [rat2 ]∩ B ¯ 0 [rat1 ] is obtained by imposing minimal completeness, which B 1 2 in this context yields the requirement of ‘no extraneous restrictions on beliefs’. As established in Section 4.3, robust belief B0i is a non-monotone operator which is bounded by the two KD45 operators, namely belief Bi and certain belief Ki . Furthermore, as shown in Chapter 4, the robust belief operator coincides with the notions of ‘absolutely robust belief’, as introduced by Stalnaker (1998), and ‘assumption’, as proposed by Brandenburger and Keisler (2002), and is closely related to the concept of ‘strong belief’, as used by Battigalli and Siniscalchi (2002). However, in contrast to the use of non-monotonic operators in these contributions, our non-monotonic operator B0i is used only to interpret ‘full admissible consistency’, while the KD45 operator Ki is used for the interactive epistemology. The importance of this will be discussed in Section 11.5. There we also comment on how the present requirement of ‘no extraneous restrictions on beliefs’ is related to Brandenburger and Keisler’s and Battigalli and Siniscalchi’s use of a ‘preference-complete’ epistemic model. Allowing extraneous restrictions on beliefs. In view of the previous discussion, we allow extraneous restrictions on beliefs by replacing, ¯ 0 [ratj ] by B0 [ratj ]. Hence, let for each i, for each i, B i

i

A0i := [ui ] ∩ B0i [ratj ] ∩ [caui ] . The following result is proven in Appendix C and shows that any properly rationalizable pure strategy is consistent with common certain belief of A0 := A01 ∩ A02 .

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Proposition 42 Consider a finite strategic two-player game G. If a pure strategy si for i is properly rationalizable, then there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA0 . Note that both Out and r are properly rationalizable pure strategies (and, indeed (Out, r) is a proper equilibrium) in G01 , the ‘battle-of-thesexes-with-an-outside-option’ game of Figure 2.6, while neither Out nor r is consistent with common certain belief of full admissible consistency. This demonstrates that ‘no extraneous restrictions on beliefs’ is needed for the characterization in Proposition 41 of the concept of fully permissible sets, which in G01 promotes only the forward induction outcome (InL, `) (cf. the analysis of G01 in Sections 11.2 and 11.4). Weakening robust belief of opponent rationality. By applying the belief operator Bi , as defined in Section 6.1, we can weaken B01 [rat2 ]∩ B02 [rat1 ] (i.e., robust belief of opponent rationality) to B1 [rat2 ]∩B2 [rat1 ] ¯ 0 [rat2 ] (i.e., belief of opponent rationality). Moreover, we can weaken B 1 ¯ 0 [rat1 ] to B ¯ 1 [rat2 ] ∩ B ¯ 2 [rat1 ], where for each i, ∩B 2 ¯ i [ratj ] := {(s1 , t1 , s2 , t2 ) ∈ S1 × T1 × S2 × T2 | B β ti ⊆ (projTi ×Sj ×Tj [ratj ]), and p Âti q only if pEj weakly dominates qEj for Ej = projSj ×Tj β ti or Ej = projSj ×Tj κti } , ¯ 1 [rat2 ] ∩ B ¯ 2 [rat1 ] is obtained by imRelative to B1 [rat2 ] ∩ B2 [rat1 ], B posing minimal completeness, which in the context of belief of opponent rationality yields the requirement of ‘no extraneous restrictions on beliefs’. To impose ‘caution’ and ‘belief of opponent rationality’, recall from Section 6.3 that A = A1 ∩ A2 is the event of admissible consistency where, for each i, Ai = [ui ] ∩ Bi [ratj ] ∩ [caui ] , To add ‘no extraneous restrictions on beliefs’, consider for each i, ¯ i [ratj ] ∩ [caui ] , A¯i := [ui ] ∩ B and write A¯ := A¯1 ∩ A¯2 . Since A¯ ⊆ A, the following proposition implies that permissibility (i.e., the Dekel-Fudenberg procedure; see Definition 13) is characterized if ‘robust belief of opponent rationality’ is weakened to ‘belief of opponent rationality’, independently of whether a requirement of ‘no extraneous restrictions on beliefs’ is retained. This result, which is a strengthening of Proposition 27 and is proven in Appendix

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149

C, shows that ‘robust belief of opponent rationality’ is needed for the characterization in Proposition 41 of the concept of fully permissible sets.

Proposition 43 Consider a finite strategic two-player game G. If a pure strategy si for i is permissible, then there exists an epistemic model ¯ A pure strategy si for i with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA. is permissible if there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA. Removing caution. Recall from Section 6.2 that C = C1 ∩ C2 is the event of consistency where, for each i, Ci = [ui ] ∩ Bi [ratj ] . To add ‘no extraneous restrictions on beliefs’ and ‘robust belief of opponent rationality’, consider for each i, ¯ 0i [ratj ] , C¯i0 := [ui ] ∩ B and write C¯ 0 := C¯10 ∩ C¯20 . Since C¯ 0 ⊆ C, the following strengthening of Proposition 25 means that the removal of ‘caution’ leads to a characterization of rationalizability (i.e., IESDS; see Definition 11), independently of whether extraneous restrictions on beliefs are allowed and robust belief of opponent rationality is weakened. Thus, ‘caution’ is necessary for the characterization in Proposition 41.

Proposition 44 Consider a finite strategic two-player game G. If a pure strategy si for i is rationalizable, then there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKC¯ 0 . A pure strategy si for i is rationalizable if there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKC. Also the proof of this result is contained in Appendix C.

11.4

Investigating examples

The present section illustrates the concept of fully permissible sets by returning to the previously discussed games G01 and G8 . Here, G01 will serve to show how our concept captures aspects of forward induction, while G8 will be used to interpret the occurrence of multiple fully permissible sets. The two examples will be used to shed light on the differences between, on the one hand, the approach suggested here and, on the other hand,

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IEWDS as characterized by Stahl (1995): A strategy survives IEWDS if and only if it is a best response to a belief where one strategy is infinitely less likely than another if the former is eliminated at an earlier round than the latter.4 Forward induction. Reconsider G01 Figure 2.6, and apply our algorithm IECFA to this “battle-of-the-sexes with an outside option” game. Since InR is a dominated strategy, InR cannot be an element of 1’s choice set. This does not imply, as in the procedure of IEWDS (given Stahl’s, 1995, characterization), that 2 deems InL infinitely more likely than InR. However, 2 certainly believes that only {Out}, {InL} and {Out, InL} are candidates for 1’s choice set. This excludes {r} as 2’s choice set, since {r} is 2’s choice set only if 2 deems {InR} or {Out, InR} possible. This in turn means that 1 certainly believes that only {`} and {`, r} are candidates for 2’s choice set, implying that {Out} cannot be 1’s choice set. Certainly believing that only {InL} and {Out, InL} are candidates for 1’s choice set does imply that 2 deems InL infinitely more likely than InR. Hence, 2’s choice set is {`} and, therefore, 1’s choice set {InL}. Thus, the forward induction outcome (InL, `) is promoted. To show how common certain belief of the event A¯0 is consistent with the fully permissible sets {InL} and {`}—and thus illustrate Proposition 41—consider an epistemic model with only one type of each player; i.e., T1 × T2 = {t1 } × {t2 }. Let, for each i, ºti satisfy that υiti ◦ z = ui . Also, let β t1 = {t1 } × {`} × {t2 } κt1 = {t1 } × S2 × {t2 } t 2 β = {t2 } × {InL} × {t1 } κt2 = {t2 } × S1 × {t1 } . Finally, let for each i, p Âti q if and only if pEj weakly dominates qEj for Ej = projSj ×Tj β ti or Ej = projSj ×Tj κti . Then S1t1 = {InL}

S2t2 = {`} .

Inspection will verify that CK A¯0 = A¯0 = S1 × T1 × S2 × T2 . Multiple fully permissible sets. Let us also return to G8 of Figure 11.1, where IEWDS eliminates D in the first round, r in the second round, and M in the third round, so that U and ` survive. Stahl’s (1995) characterization of IEWDS entails that 2 deems each of U and M infinitely more likely than D. Hence, the procedure forces 2 to deem 4 Cf.

Brandenburger and Keisler (2002, Theorem 1) as well as Battigalli (1996a) and Rajan (1998). See also Bicchieri and Schulte (1997), who give conceptually related interpretations of IEWDS.

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M infinitely more likely than D for the sole reason that D is eliminated before M , even though both M and D are eventually eliminated by the procedure. Applying our algorithm IECFA yields the following result. Since D is a weakly dominated strategy, D cannot be an element of 1’s choice set. Hence, 2 certainly believes that only {U }, {M } and {U, M } are candidates for 1’s choice set. This excludes {r} as 2’s choice set, since {r} is 2’s choice set only if 2 deems {D} or {U, D} possible. This in turn means that 1 certainly believes that only {`} and {`, r} are candidates for 2’s choice set, implying that {M } cannot be 1’s choice set. There is no further elimination. This means that 1’s collection of fully permissible sets is {{U }, {U, M }} and 2’s collection of fully permissible sets is {`}, {`, r}}. Thus, common certain belief of full admissible consistency implies that 2 deems U infinitely more likely than D since U (respectively, D) is an element of any (respectively, no) fully permissible set for 1. However, whether 2 deems M infinitely more likely than D depends on the type of player 2. To show how common certain belief of the event A¯0 is consistent with the collections of fully permissible sets {{U }, {U, M }} and {{`}, {`, r}}— and thus illustrate Proposition 41 also in the case of G8 —consider an epistemic model with two types of each player; i.e., T1 × T2 = {t01 , t001 } × {t02 , t002 }. Let, for each type ti of any player i, ºti satisfy that υiti ◦ z = ui . Moreover, let 0

κt1 = {t01 } × S2 × {t02 } 00 κt1 = {t001 } × S2 × T2

0

κt2 = {t02 } × S1 × T1 00 κt2 = {t002 } × S1 × {t01 } .

0

β t1 = {t01 } × {`} × {t02 } 00 β t1 = {t001 } × {(`, t02 ), (`, t002 ), (r, t002 )}

0

β t2 = {t02 } × {(U, t01 ), (U, t001 ), (M, t001 )} 00 β t2 = {t002 } × {U } × {t01 }

Finally, let for each type ti of any player i, p Âti q if and only if pEj weakly dominates qEj for Ej = projSj ×Tj β ti or Ej = projSj ×Tj κti . Then t0

S11 = {U }

t00

S11 = {U, M }

t0

S22 = {`}

t00

S22 = {`, r} .

Inspection will verify that CK A¯0 = A¯0 = S1 × T1 × S2 × T2 Our analysis of G8 allows a player to deem an opponent choice set to be subjectively impossible even when it is the true choice set of the opponent. E.g., at (t01 , t002 ), player 1 deems it subjectively impossible that player 2’s choice set is {`, r} even though this is the true choice set of player 2. Likewise, at (t001 , t002 ), player 2 deems it subjectively impossible that player 1’s choice set is {U, M } even though this is the true choice set

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of player 1. This is an unavoidable feature of this game as there exists no pair of non-empty strategy subsets (Y1 , Y2 ) such that Y1 = S1 \D1 (Y2 ) and Y2 = S2 \D2 (Y1 ). It implies that under full admissible consistency we cannot have in G8 that each player is certain of the true choice set of the opponent. Multiplicity of fully permissible sets arises also in the strategic form of certain extensive games in which the application of backward induction is controversial, e.g. the ‘centipede’ game Γ03 illustrated in Figure 2.4. For more on this, see Chapter 12 where the concept of fully permissible sets is used to analyze extensive games.

11.5

Related literature

It is instructive to explain how our analysis differs from the epistemic foundations of IEWDS and EFR provided by Brandenburger and Keisler (2002) (BK) and Battigalli and Siniscalchi (2002) (BS), respectively. It is of minor importance for the comparison that EFR makes use of the extensive form, while the present analysis is performed in the strategic form. The reason is that, by ‘caution’, a rational choice in the whole game implies a rational choice at all information sets that are not precluded from being reached by the player’s own strategy (cf. Lemma 11). To capture forward induction players must essentially deem any opponent strategy that is a rational choice infinitely more likely than any opponent strategy not having this property. An analysis incorporating this feature must involve a non-monotonic epistemic operator, which is called robust belief in the present analysis (cf. Section 11.3), while the corresponding operators are called ‘assumption’ and ‘strong belief’ by BK and BS, respectively (see Chapter 4 for an analysis of the relationship between these non-monotonic operators). We use robust belief only to define the event that the preferences of each player is ‘fully admissibly consistent’ with the preferences of his opponent, while the monotonic certain belief operator is used for the interactive epistemology: each player certainly believes (in the sense of deeming the complement subjectively impossible) that the preferences of his opponent are fully admissibly consistent, each player certainly believes that his opponent certainly believes that he himself has preferences that are fully admissibly consistent, and so on. As the examples of Section 11.4 illustrate, it is here a central question what opponent types (choice sets) a player deems subjectively

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possible. Consequently, the certain belief operator is appropriate for the interactive epistemology. In contrast, BK and BS use their non-monotonic operators for the interactive epistemology. In the process of defining higher order beliefs both BK and BS impose that lower order beliefs are maintained. This is precisely how BK obtain Stahl’s (1995) characterization which—e.g., in G8 of Figure 11.1—seems to correspond to extraneous and hard-tojustify restrictions on beliefs. Stahl’s characterization provides an interpretation of IEWDS where strategies eliminated in the first round are completely irrational, while strategies eliminated in later rounds are at intermediate degrees of rationality. Likewise, Battigalli (1996a) has shown how EFR corresponds to the ‘best rationalization principle’, entailing that some opponent strategies are neither completely rational nor completely irrational. The present analysis, in contrast, differentiates only between whether a strategy is maximal (i.e., a rational choice) or not. As the examples of Section 11.4 illustrate, although a strategy that is weakly dominated on the set of all opponent strategies is a “stupid” choice, it need not be “more stupid” than any remaining admissible strategy, as this depends on the interactive analysis of the game. The fact that a non-monotonic epistemic operator is involved when capturing forward induction also means that the analysis must ensure that all rational choices for the opponent are included in the epistemic model. BK and BS ensure this by employing ‘preference-complete’ epistemic models, where all possible epistemic types of each player are represented. Instead, the present analysis achieves this by requiring ‘no extraneous restrictions on beliefs’, meaning that the preferences are minimally complete (cf. Section 11.3). Since an ordinary monotonic operator is used for the interactive epistemology, there is no more need for a ‘preference-complete’ epistemic model here than in usual epistemic analyses of rationalizability and permissibility. Our paper has a predecessor in Samuelson (1992), who also presents an epistemic analysis of admissibility that leads to a collection of sets for each player, called a ‘generalized consistent pair’. Samuelson requires that a player’s choice set equals the set of strategies that are not weakly dominated on the union of choice sets that are deemed possible for the opponent; this implies our requirements of ‘robust belief of opponent rationality’ and ‘no extraneous restrictions on beliefs’ (cf. Samuelson, 1992, p. 311). However, he does not require that each player deems no

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opponent strategy impossible, as implied by our requirement of ‘caution’. Hence, his analysis does not yield {{U, M }} × {{`}} in G9 of Figure 11.2. Furthermore, he defines possibility relative to a knowledge operator that satisfies the truth axiom, while our analysis—as illustrated by the discussion of G8 in Section 11.4—allows a player to deem an opponent choice set to be subjectively impossible even when it is the true choice set of the opponent. This explains why we in contrast to Samuelson obtain general existence (cf. Proposition 39(i)). If each player is certain of the true choice set of the opponent, one obtains a ‘consistent pair’ as defined by B¨orgers and Samuelson (1992), a concept that need not exist even when a generalized consistent pair exists. Ewerhart (1998) modifies the concept of a consistent pair by adding ‘caution’. However, since he allows extraneous restrictions on beliefs to ensure general existence, his concept of a ‘modified consistent pair’ does not promote forward induction in G01 . A ‘self-admissible set’ in the terminology of Brandenburger and Friedenberg (2003) is a Cartesian product of strategy subsets, where each player’s subset consists of strategies that weakly dominated neither on the subset of opponent strategies nor on the set of all opponent strategies. Also Brandenburger and Friedenberg allow extraneous restrictions on beliefs. Hence, ‘modified consistent pairs’ and ‘self-admissible sets’ need not correspond to profiles of fully permissible sets. However, if there is a unique fully permissible set for each player, then the pair constitutes both a ‘modified consistent pair’ and a ‘self-admissible set’. Basu and Weibull’s (1991) ‘tight curb* set’ is another variant of a consistent pair that ensures existence without yielding forward induction in G01 , as they impose ‘caution’ but weaken ‘robust belief of opponent rationality’ to ‘belief of opponent rationality’. In particular, the set of permissible strategy profiles is ‘tight curb*’. ‘Caution’ and ‘robust belief of opponent rationality’ are admissibility requirements on the preferences of players, thus positioning the analysis of the present chapter in the ‘consistent preferences’ approach. Moreover, by imposing ‘no extraneous restrictions on beliefs’ as a requirement of minimal completeness, preferences are not in general representable by subjective probabilities, thus showing the usefulness of an analysis that relax completeness. 5

5 By

not employing subjective probabilities, the analysis is related to the filter model of beliefs presented by Brandenburger (1997, 1998).

Chapter 12 APPLYING FULL PERMISSIBILITY TO EXTENSIVE GAMES

In many economic contexts decision makers interact and take actions that extend through time. A bargaining party makes an offer, which is observed by the adversary, and accepted, rejected or followed by a counter-offer. Firms competing in markets choose prices, levels of advertisement, or investments with the intent of thereby influencing the future behavior of competitors. One could add many examples. The standard economic model for analyzing such situations is that of an extensive game. Reproducing joint work with Martin Dufwenberg, cf. Asheim and Dufwenberg (2003b), this chapter revisits a question that was already posed in Chapters 7–10: What happens in an extensive game if players reason deductively by trying to figure out one another’s moves? We have in Asheim and Dufwenberg (2003a), incorporated in Chapter 11 of this book, proposed a model for deductive reasoning leading to the concept of ‘fully permissible sets’, which can be applied to many strategic situations. In the present chapter we argue that the model is appropriate for analyzing extensive games and we apply it to several such games.

12.1

Motivation

There is already a literature exploring the implications of deductive reasoning in extensive games, but the answers provided differ and the issue is controversial. Much of the excitement concerns whether or not deductive reasoning implies backward induction in games where that principle is applicable. We next discuss this issue, since it provides a useful backdrop against which to motivate our own approach.

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1c D 1 0

F

2s d 0 2

Figure 12.1.

f

1s D 3 0

F

0 3

f d D 1, 0 1, 0 FD 0, 2 3, 0 FF 0, 2 0, 3

Γ11 and its pure strategy reduced strategic form.

Consider the 3-stage “take-it-or-leave-it”, introduced by Reny (1993) (a version of Rosenthal’s, 1981, “centipede” game, see Γ03 of Figure 2.4), and shown in Figure 12.1 together with its pure strategy reduced strategic form.1 What would 2 do in Γ11 if called upon to play? Backward induction implies that 2 would choose d, which is consistent with the following idea: 2 chooses d because she “figures out” that 1 would choose D at the last node. Many models of deductive reasoning support this story, starting with Bernheim’s concept of ‘subgame rationalizability’ and Pearce’s concept of ‘extensive form rationalizability’ (EFR). More recently, Battigalli and Siniscalchi (2002) provide a rigorous epistemic foundation for EFR, while Chapters 7–10 of this book epistemically model rationalizability concepts that resemble ‘subgame rationalizability’. However, showing that backward induction can be given some kind of underpinning does not imply that the underpinning is convincing. Indeed, skepticism concerning backward induction can be expressed by means of Γ11 . Suppose that each player believes the opponent will play in accordance with backward induction; i.e., 1 believes that 2 chooses d if asked to play, and 2 believes that 1 plays D at his initial note. Then player 1 prefers playing D to any of his two other strategies FD and FF. Moreover, if 2 is certain that 1 believes that 2 chooses d if she were asked to play, then 2 realizes that 1 has not chosen in accordance with his preferences if she after all is asked to play. Why then should 2 believe that 1 will make any particular choice between his two less preferred strategies, FD and FF, at his last node? So why then should 2 prefer d to f ? This kind of perspective on the “take-it-or-leave-it” game is much inspired by the approach proposed by Ben-Porath (1997), where similar

1 We

need not consider what players plan to do at decision nodes that their own strategy precludes them from reaching (cf. Section 12.2).

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objections against backward inductive reasoning are raised. We shall discuss his contribution in some detail, since the key features of our approach can be appreciated via a comparison to his model. Applied to Γ11 , Ben-Porath’s model captures the following intuition: Each player has an initial belief about the opponent’s behavior. If this belief is contradicted by the play (a “surprise” occurs) he may subsequently entertain any belief consistent with the path of play. The only restriction imposed on updated beliefs is Bayes’ rule. In Γ11 , Ben-Porath’s model allows player 2 to make any choice. In particular, 2 may choose f if she initially believes with probability one that player 1 will choose D, and conditionally on D not being chosen assigns sufficient probability on FF. This entails that if 2 initially believes that 1 will comply with backward induction, then 2 need not follow backward induction herself. In Γ11 , our analysis captures much the same intuition as Ben-Porath’s approach, and it has equal cutting power in this game. However, it yields a more structured solution as it is concerned with what strategy subsets that are deemed to be the set of rational choices for each player. While agreeing with Ben-Porath that deductive reasoning may lead to each of D and FD being rational for 1 and each of d and f being rational for 2, our concept of full permissibility predicts that 1’s set of rational choices is either {D} or {D, F }, and 2’s set of rational choices is either {d} or {d, f }. This has appealing features. If 2 is certain that 1’s set is {D}, then—unless 2 has an assessment of the relative likelihood of 1’s less preferred strategies FD and FF —one cannot conclude that 2 prefers d to f or vice versa; this justifies {d, f } as 2’s set of rational choices. On the other hand, if 2 considers it possible that 1’s set is {D, FD}, then d weakly dominates f on this set and justifies {d} as 2’s set of rational choices. Similarly, one can justify that D is preferred to FD if and only if 1 considers it impossible that 2’s set is {d, f }. This additional structure is important for the analysis of Γ06 , illustrated in Figure 8.2. This game is due to Reny (1992, Figure 1) and has appeared in many contributions. Suppose in this game that each player believes the opponent will play in accordance with backward induction by choosing FF and f respectively. Then both players will prefer FF and f to any alternative strategy. Moreover, as will be shown in Section 12.3, our analysis implies that {FF} and {f } are the unique sets of rational choices. Ben-Porath’s approach, by contrast, does not have such cutting power in Γ06 , as it entails that deductive reasoning may lead to each of the strategies D and FF being rational for 1 and each of the strategies d and

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f being rational for 2. The intuition for why the strategies D and d are admitted is as follows: D is 1’s unique best strategy if he believes with probability one that 2 plays d. Player 1 is justified in this belief in the sense that d is 2’s best strategy if she initially believes with probability one that 1 will choose D, and if called upon to play 2 revises this belief so as to believe with sufficiently high probability (e.g., probability one) that 1 is using FD. This belief revision is consistent with Bayes’ rule, and so is acceptable. Ben-Porath’s approach is a very important contribution to the literature, since it is a natural next step if one accepts the above critique of backward induction. Yet we shall argue below that it is too permissive, using Γ06 as an illustration. Assume that 1 deems d infinitely more likely than f , while 2 deems D infinitely more likely than FD and FD infinitely more likely than FF. Then the players rank their strategies as follows: 1’s preferences: D  FF  FD 2’s preferences: d  f This is in fact precisely the justification of the strategies D and d given above when applying Ben-Porath’s approach to Γ06 . Here, ‘caution’ is satisfied since all opponent strategies are taken into account; in particular, FF is preferred to FD as the former strategy weakly dominates the latter. Moreover, ‘robust belief of opponent rationality’ is satisfied since each player deems the opponent’s maximal strategy infinitely more likely that any non-maximal strategy. However, the requirement of ‘no extraneous restrictions on beliefs’, as described in Chapter 11, is not satisfied since the preferences of 2 introduce extraneous restrictions on beliefs by deeming one of 1’s non-maximal strategies, FD, infinitely more likely than another non-maximal strategy, FF. When we return to G06 in Section 12.3, we show how the additional imposition of ‘no extraneous restrictions on beliefs’ means that deductive reasoning leads to the conclusion that {FF} and {f } are the players’ choice sets in this game. As established in Chapter 11, our concept of fully permissible sets is characterized by ‘caution’, ‘robust belief of opponent rationality’, and ‘no extraneous restrictions on beliefs’. In Section 12.2 we prove results that justify the claim that interesting implications of deductive reasoning in a given extensive game can be derived by applying this concept to the strategic form of that game. Sections 12.3 and 12.4 are concerned with such applications, with the aim of showing how our solution concept gives new and economically relevant insights into the implications of deductive reasoning in extensive

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games. The material is organized around two central themes: backward and forward induction. Other support for forward induction through the concept of EFR and the procedure of IEWDS precludes outcomes in conflict with backward induction; see, e.g., Battigalli (1997). In contrast, we will show how the concept of fully permissible sets promotes forward induction in the “battle-of-the-sexes with an outside option” and “burning money” games as well as an economic application from organization theory, while not insisting on the backward induction outcome in games (like Γ11 and the 3-period prisoners’ dilemma) where earlier contributions, like Basu (1990), Reny (1993) and others, have argued on theoretical grounds that this is problematic. Still, we will show that the backward induction outcome is obtained in Γ06 , and that our concept has considerable bite in the 3-period “prisoners’ dilemma” game. Lastly, in Section 12.5 we compare our approach to related work.

12.2

Justifying extensive form application

The concept of fully permissible sets, presented and epistemically characterized in Chapter 11 of this book, is designed to analyze the implications of deductive reasoning in strategic form games. In this chapter, we propose that this concept can be fruitfully applied for analyzing any extensive game through its strategic form. In fact, we propose that it is legitimate to confine attention to the game’s pure strategy reduced strategic form (cf. Definition 23 below), which is computationally more convenient. In this section we prove two results which, taken together, justify such applications. An extensive game. A finite extensive two-player game Γ (without chance moves) includes a set of terminal nodes Z and, for each player i, a vNM utility function υi : Z → R that assigns a payoff to any outcome. For each player i, there is a finite collection of information sets Hi , with a finite set of actions A(h) being associated with each h ∈ Hi . A pure strategy for player i is a function si that to any h ∈ Hi assigns an action in A(h). Let Si denote player i’s finite set of pure strategies, and let S = S1 × S2 . As before, write si (∈ Si ) for pure strategies and pi and qi (∈ ∆(Si )) for mixed strategies. Define ui : S → R by ui = υi ◦ z, and refer to G = (S1 , S2 , u1 , u2 ) as the strategic form of the extensive game Γ. For any h ∈ H1 ∪ H2 , let S(h) = S1 (h) × S2 (h) denote the set of strategy profiles that are consistent with h being reached. Weak sequential rationality. Consider any strategy that is maximal given preferences that satisfy that one strategy is preferred to an-

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other if and only if the one weakly dominates the other on Yj — the set of strategies that player i deems to be the set of rational choices for his opponent — or Sj — the set of all opponent strategies. Hence, the strategy is maximal at the outset of a corresponding extensive game. Corollary 2 makes the observation that this strategy is still maximal when the preferences have been updated upon reaching any information set that the choice of this strategy does not preclude. Assume that player i’s preferences over his own strategies satisfy that pi is preferred to qi if and only if pi weakly dominates qi on Yj or Sj . Let, for any h ∈ Hi , Yj (h) := Yj ∩ Sj (h) denote the set of strategies in Yj that are consistent with the information set h being reached. If pi , qi (∈ ∆(Si (h))), then i’s preferences conditional on the information set h ∈ Hi being reached satisfy that pi is preferred to qi if and only if pi weakly dominates qi on Yj (h) or Sj (h) (where it follows from the definition that weak dominance on Yj (h) is not possible if Yj (h) = ∅). Furthermore, i’s choice set conditional on h ∈ Hi , SiYj (h), is given by SiYj (h) := Si (h) \ {si ∈ Si (h)| ∃xi ∈ ∆(Si (h)) s.t. xi weakly dominates si on Yj (h) or Sj (h)} . Write SiYj := SiYj (∅) (= Si \Di (Yj ) in earlier notation). By the result below, if si is maximal at the outset of an extensive game, then it is also maximal at later information sets for i that si does not preclude.

Corollary 2 Let (∅ = 6 ) Yj ⊆ Sj . If si ∈ SiYj , then si ∈ SiYj (h) for any h ∈ Hi with Si (h) 3 si . Proof. This follow from Lemma 11 by letting i’s preferences (at ti ) on i’s set of mixed strategies satisfy that pi is preferred to qi if and only if pi weakly dominates qi on Yj or Sj . By the assumption of ‘caution’, each player i takes into account the possibility of reaching any information set for i that the player’s own strategy does not preclude from being reached. Hence, ‘rationality’ implies ‘weak sequential rationality’; i.e., that a player chooses rationally at all information sets that are not precluded from being reached by the player’s own strategy. Reduced strategic form. It follows from Proposition 45 below that it is in fact sufficient to consider the pure strategy reduced strategic form when deriving the fully permissible sets of the game. The following definition is needed.

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Definition 23 Let G = (S1 , S2 , u1 , u2 ) be a finite strategic two-player game. The pure strategies si and s0i (∈ Si ) are equivalent if for each player k, uk (s0i , sj ) = uk (si , sj ) for all sj ∈ Sj . The pure strategy reduced strategic form (PRSF) of G is obtained by letting, for each player i, each class of equivalent pure strategies be represented by exactly one pure strategy. Since the maximality of one of two equivalent strategies implies that the other is maximal as well, the following observation holds: If si and s0i are equivalent and σi is a fully permissible set for i, then si ∈ σi if and only if s0i ∈ σi . To see this formally, note that if si ∈ σi for some fully permissible set σi , then, by Proposition 39(ii), there exists (∅ 6=) Ψj ⊆ Πj such that si ∈ σi = SiYj for Yj = ∪σj0 ∈Ψj σj0 . Since si and s0i are equivalent, s0i ∈ SiYj = σi . This observation explains why the following result can be established. ˜ = (S˜1 , S˜2 , u Proposition 45 Let G ˜1 , u ˜2 ) be a finite strategic two-player 0 game where si and si are two equivalent strategies for i. Consider G = (S1 , S2 , u1 , u2 ) where Si = S˜i \{s0i } and Sj = S˜j for j 6= i, and where, for each player k, uk is the restriction of u ˜k to S = S1 × S2 . Let, for each ˜ player k, Πk (Πk ) denote the collection of fully permissible sets for k in ˜ Then Πi is obtained from Π ˜ i by removing s0 from any σ ˜i G (G). ˜i ∈ Π i ˜ j. with si ∈ σ ˜i , while, for j 6= i, Πj = Π Proof. By Proposition 39(iii) it suffices to show that ˜ ⊆ α(Ξ) ˜ for G, ˜ then Ξ ⊆ α(Ξ) for G, where Ξi is obtained from 1 If Ξ ˜ ˜ i with si ∈ σ Ξi by removing s0i from any σ ˜i ∈ Ξ ˜i , while, for j 6= i, ˜ Ξj = Ξj . ˜ ⊆ α(Ξ) ˜ for G, ˜ where Ξ ˜ i is obtained from 2 If Ξ ⊆ α(Ξ) for G, then Ξ 0 ˜ j = Ξj . Ξi by adding si to any σi ∈ Ξi with si ∈ σi , while, for j 6= i, Ξ ˜ ⊆ α(Ξ). ˜ By the observation preceding Proposition Part 1. Assume Ξ 0 ˜ 45, if σ ˜i ∈ Πi , then si ∈ σ ˜i if and only if si ∈ σ ˜i . Pick any player k ˜ and any σ ˜k ∈ Πk . Let ` denote the other player. By the definition of ˜ ˜` ⊆ Π ˜ ` such that σ αk (·), there exists (∅ 6=) Ψ ˜k = SkY` for Y˜` = ∪σ`0 ∈Ψ˜ ` σ`0 . ˜ i with si ∈ σ Construct Ψi by removing s0i from any σ ˜i ∈ Ψ ˜i and replace ˜ ˜ ˜ Si by Si , while, for j 6= i, Ψj = Ψj and Sj = Sj . Let Y` = ∪σ`0 ∈Ψ` σ`0 . ˜k \{s0k } if k = i Then it follows from the definition of SkY` that SkY` = σ ˜k if k 6= i. Since, for each player k, (∅ 6=) Ψk ⊆ Ξk , we have and SkY` = σ that Ξ ⊆ α(Ξ). Part 2 is shown similarly.

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Proposition 45 means that the PRSF is sufficient for analyzing common certain belief of full admissible consistency, which is the epistemic foundation for the concept of fully permissible sets. Consequently, in the strategic form of an extensive game, it is unnecessary to specify actions at information sets that a strategy precludes from being reached. Hence, instead of fully specified strategies, it is sufficient to consider what Rubinstein (1991) calls plans of action. For a generic extensive game, the set of plans of action is identical to the strategy set in the PRSF. In the following two sections we apply the concept of fully permissible sets to extensive games. We organize the discussion around two themes: backward and forward induction. Motivated by Corollary 2 and Proposition 45, we analyze each extensive game via its PRSF (cf. Definition 23), given in conjunction to the extensive form. In each example, each plan of action that appears in the underlying extensive game corresponds to a distinct strategy in the PRSF.

12.3

Backward induction

Does deductive reasoning in extensive games imply backward induction? In this section we show that the answer provided by the concept of fully permissible sets is “sometimes, but not always”. ‘Sometimes.’ There are many games where Ben-Porath’s approach does not capture backward induction while our approach does (and the converse is not true). Ben-Porath (1997) assumes ‘initial common certainty of rationality’ in extensive games of perfect information. As discussed in Chapter 7 he proves that in generic games (with no payoff ties at terminal nodes for any player) the outcomes consistent with that assumption coincide with those that survive the Dekel-Fudenberg procedure (where one round of elimination of all weakly dominated strategies is followed by iterated elimination of strongly dominated strategies). It is a general result that the concept of fully permissible sets refines the Dekel-Fudenberg procedure (cf. Proposition 40). Game Γ06 of Figure 8.2 shows that the refinement may be strict even for generic extensive games with perfect information, and indeed that fully permissible sets may respect backward induction where Ben-Porath’s solution does not. The strategies surviving the Dekel-Fudenberg procedure, and thus consistent with ‘initial common certainty of rationality’, are D and FF for player 1 and d and f for player 2. In Section 12.2 we gave an intuition for

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why the strategies D and d are possible. This is, however, at odds with the implications of common certain belief of full admissible consistency. Applying IECFA to the PRSF of Γ06 of Figure 8.2 yields: Ξ(0) = Σ1 × Σ2 Ξ(1) = {{D}, {FF}, {D, FF}} × Σ2 Ξ(2) = {{D}, {FF}, {D, FF}} × {{f }, {d, f }} Ξ(3) = {{FF}, {D, FF}} × {{f }, {d, f }} Ξ(4) = {{FF}, {D, FF}} × {{f }} Π = Ξ(5) = {{FF}} × {{f }} Interpretation: Ξ(1): ‘Caution’ implies that FD cannot be a maximal strategy (i.e., an element of a choice set) for 1 since it is weakly dominated (in fact, even strongly dominated). Ξ(2): Player 2 certainly believes that only {D}, {FF} and {D, FF} are candidates for 1’s choice set. By ‘robust belief of opponent rationality’ and ‘no extraneous restrictions on beliefs’ this excludes {d} as 2’s choice set, since d weakly dominates f only on {FD} or {D, FD}. Ξ(3): 1 certainly believes that only {f } and {f, d} are candidates for 2’s choice set. By ‘robust belief of opponent rationality’ and ‘no extraneous restrictions on beliefs’ this excludes {D} as 1’s choice set, since D weakly dominates FD and FF only on {d}. Ξ(4): Player 2 certainly believes that only {FF} and {D, FF} are candidates for 1’s choice set. By ‘robust belief of opponent rationality’ this implies that 2’s choice set is {f } since f weakly dominates d on both {FF} and {D, FF}. Ξ(5): 1 certainly believes that 2’s choice set is {f }. By ‘robust belief of opponent rationality’ this implies that {FF} is 1’s choice set since FF weakly dominates D on {f }. No further elimination of choice sets is possible, so {FF} and {f } are the respective players’ unique fully permissible sets. ‘Not always.’ While fully permissible sets capture backward induction in Γ06 and other games, the concept does not capture backward induction in certain games where the procedure has been considered controversial.2 The background for the controversy is the following paradoxical aspect: Why should a player believe that an opponent’s future play will satisfy backward induction if the opponent’s previous play is incompatible with backward induction? A prototypical game for cast-

2 See

discussion and references in Chapter 7 of the present text.

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ing doubt on backward induction is the “take-it-or-leave-it” game Γ11 of Figure 12.1, which we next analyze in detail. Applying IECFA to the PRSF of Γ11 of Figure 12.1 yields: Ξ(0) = Σ1 × Σ2 Ξ(1) = {{D}, {FD}, {D, FD}} × Σ2 Ξ(2) = {{D}, {FD}, {D, FD}} × {{d}, {d, f }} Π = Ξ(3) = {{D}, {D, FD}} × {{d}, {d, f }} Interpretation: Ξ(1): FF cannot be a maximal strategy for 1 since it is strongly dominated. Ξ(2): Player 2 certainly believes that only {D}, {FD} and {D, FD} are candidates for 1’s choice set. This excludes {f } as 2’s choice set since {f } is 2’s choice set only if 2 deems {FF} or {FD, FF} subjectively possible. Ξ(3): 1 certainly believes that only {d} and {d, f } are candidates for 2’s choice set, implying that {FD} cannot be 1’s choice set. No further elimination of choice sets is possible and the collection of profiles of fully permissible sets is as specified. Note that backward induction is not implied. To illustrate why, we focus on player 2 and explain why {d, f } may be a choice set for her. Player 2 certainly believes that 1’s choice set is {D} or {D, FD}. This leaves room for two basic cases. First, suppose 2 deems {D, FD} subjectively possible. Then {d} must be her choice set, since she must consider it infinitely more likely that 1 uses FD than that he uses FF. Second, and more interestingly, suppose 2 does not deem {D, FD} subjectively possible. Then conditional on 2’s node being reached 2 certainly believes that 1 is not choosing a maximal strategy. As player 2 does not assess the relative likelihood of strategies that are not maximal (cf. the requirement of ‘no extraneous restrictions on beliefs’), {d, f } is her choice set in this case. Even in the case where 2 deems {D} to be the only subjectively possible choice set for 1, she still considers it subjectively possible that 1 may choose one of his non-maximal strategies FD and FF (cf. the requirement of ‘caution’), although each of these strategies is in this case deemed infinitely less likely than the unique maximal strategy D. Applied to (the PRSF of) Γ11 , our concept permits two fully permissible sets for each player. How can this multiplicity of fully permissible sets be interpreted? The following interpretation corresponds to the underlying formalism: The concept of fully permissible sets, when applied to Γ11 , allows for two different types of each player. Consider player 2. Either she may consider that {D, FD} is a subjectively possible choice set for 1, in which case her choice set will be {d} so that she complies

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with backward induction. Or she may consider {D} to be the only subjectively possible choice set for 1, in which case 2’s choice set is {d, f }. Intuitively, if 2 is certain that 1 is a backward inducter, then 2 need not be a backward inducter herself! In this game, our model captures an intuition that is very similar to that of Ben-Porath’s model. Reny (1993) defines a class of “belief consistent” games, and argues on epistemic grounds that backward induction is problematic only for games that are not in this class. It is interesting to note that the game where our concept of fully permissible sets differs from Ben-Porath’s analysis by promoting backward induction, Γ06 , is belief-consistent. In contrast, the game where the present concept coincides with his by not yielding backward induction, Γ11 , is not belief-consistent. There are examples of games that are not belief consistent, where full permissibility still implies backward induction, meaning that belief consistency is not necessary for this conclusion. It is, however, an as-of-yet unproven conjecture that belief consistency is sufficient for the concept of fully permissible sets to promote backward induction. We now compare our results to the very different findings of Aumann (1995), cf. also Section 5 of Stalnaker (1998) as well as Chapter 7 of this book. In Aumann’s model, where it is crucial to specify full strategies (rather than plans of actions), common knowledge of rational choice implies in Γ11 that all strategies for 1 but DD (where he takes a payoff of 1 at his first node and a payoff of 3 at his last node) are impossible. Hence, it is impossible for 1 to play FD or FF and thereby ask 2 to play. However, in the counterfactual event that 2 is asked to play, she optimizes as if player 1 at his last node follows his only possible strategy DD, implying that it is impossible for 2 to choose f (cf. Aumann’s Sections 4b, 5b, and 5c). Thus, in Aumann’s analysis, if there is common knowledge of rational choice, then each player chooses the backward induction strategy. By contrast, in our analysis player 2 being asked to play is seen to be incompatible with 1 playing DD or DF. For the determination of 2’s preference over her strategies it is the relative likelihood of FD versus FF that is important to her. As seen above, this assessment depends on whether she deems {D, FD} as a possible candidate for 1’s choice set. Prisoners’ dilemma. We close this section by considering a finitely repeated “prisoners’ dilemma” game. Such a game does not have perfect information, but it can still be solved by backward induction to find the unique subgame perfect equilibrium (no one cooperates in the last period, given this no one cooperates in the penultimate period, etc.).

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T sN 1 N V s1 E sN 1 sRT 1 sRV 1 sRE 1

Figure 12.2.

T sN 2 7, 7 8, 4 8, 4 5, 5 6, 2 6, 2

V sN 2 4, 8 5, 5 5, 5 8, 4 9, 1 9, 1

E sN 2 4, 8 5, 5 5, 5 5, 5 6, 2 6, 2

sRT 2 5, 5 4, 8 5, 5 3, 3 2, 6 3, 3

sRV 2 2, 6 1, 9 2, 6 6, 2 5, 5 6, 2

sRE 2 2, 6 1, 9 2, 6 3, 3 2, 6 3, 3

Reduced form of Γ12 (a 3-period “prisoners’ dilemma” game).

This solution has been taken to be counterintuitive; cf, e.g. Pettit and Sugden (1989). We consider the case of a 3-period “prisoners’ dilemma” game (Γ12 ) and show that, again, the concept of fully permissible sets does not capture backward induction. However, the fully permissible sets nevertheless have considerable cutting power. Our solution refines the Dekel-Fudenberg procedure and generates some special “structure” on the choice sets that survive. The payoffs of the stage game are given as follows, using Aumann’s (1987b, pp. 468–9) description: Each player decides whether he will receive 1 (defect) or the other will receive 3 (cooperate). There is no discounting. Hence, the action defect is strongly dominant in the stage game, but still, each player is willing to cooperate in one stage if this induces the other player to cooperate instead of defect in the next stage. It follows from Proposition 45 that we need only consider what Rubinstein (1991) calls plans of action. There are six plans of actions for each player that survive the DekelFudenberg procedure. In any of these, a player always defects in the 3rd stage, and does not always cooperate in the 2nd stage. The six plans T N V , sN E , sRT , sRV and of actions for each player i are denoted sN i ,si i i i sRE , where N denotes that i is nice in the sense of cooperating in the 1st i stage, where R denotes that i is rude in the sense of defecting in the 1st stage, where T denotes that i plays tit-for-tat in the sense of cooperating in the 2nd stage if and only j 6= i has cooperated in the 1st stage, where V denotes that i plays inverse tit-for-tat in the sense of defecting in the 2nd stage if and only if j 6= i has cooperated in the 1st stage, and where E denotes that i is exploitive in the sense of defecting in the 2nd stage independently of what j 6= i has played in the 1st stage. The strategic form after elimination of all other plans of actions is given in Figure

Applying full permissibility to extensive games

167

12.2. Note that none of these plans of actions are weakly dominated in the full strategic form. Proposition 40 shows that any fully permissible set is a subset of the set of strategies surviving the Dekel-Fudenberg procedure. Hence, only subsets of T NV E RT RV RE {sN , sN i , si i , si , si , si } can be i’s choice set under common certain belief of full admissible consistency. Furthermore, under common certain belief of full admissible consistency, we have for each player i that T must also contain sN E , since sN T is any choice set that contains sN i i i E is a maximal strategy, a maximal strategy only if sN i V must also contain sN E , since sN V any choice set that contains sN i i i N is a maximal strategy only if si E is a maximal strategy, RT is any choice set that contains sRT must also contain sRE i i , since si RE a maximal strategy only if si is a maximal strategy, RV is any choice set that contains sRV must also contain sRE i i , since si a maximal strategy only if sRE is a maximal strategy, i

Given that the choice set of the opponent satisfies these conditions, this implies that E is included in i’s choice set, only the following sets are candiif sN i T N E RT RE N V , sN E , sRV , sRE }, dates for i’s choice set: {sN i , si , si , si }, {si i i i N E RE N E or {si , si }. The reason is that si is a maximal strategy only T if i considers it subjectively possible that j’s choice set contains sN j E RT (and hence, sRE ). (and hence, sN j ) or sj j NE if sRE i , but not si , is included in i’s choice set, only the followRE RV RE ing sets are candidates for i’s choice set: {sRT i , si }, {si , si }, or RE is a maximal strategy only if i con{sRE i }. The reason is that si V NE siders it subjectively possible that j’s choice set contains sN j , sj , RV RE sj , or sj .

This in turn implies that V or sRV since any candidate for j’s i’s choice set does not contain sN i i E is preferred to sN V and choice set contains sRE , implying that sN j i i RE RV si is preferred to si .

Hence, the only candidates for i’s choice set under common certain beT N E RT RE N E RE lief of full admissible consistency are {sN i , si , si , si }, {si , si }, RE RE {sRT i , si }, and {si }. Moreover, it follows from Proposition 39(iii) that all these sets are indeed fully permissible since

168

CONSISTENT PREFERENCES

T N E RT RE RT RE {sN i , si , si , si } is i’s choice set if he deems {sj , sj }, but not N E RE N T N E RT RE {sj , sj } and {sj , sj , sj , sj }, as possible candidates for j’s choice set, E RE N T N E RT RE {sN i , si } is i’s choice set if he deems {sj , sj , sj , sj } as a possible candidate for j’s choice set, RE RE {sRT i , si } is i’s choice set if he deems {sj } as the only possible candidate for j’s choice set, N E RE RT RE {sRE i } is i’s choice set if he deems {sj , sj }, but not {sj , sj } RT RE N E N T and {sj , sj , sj , sj }, as possible candidates for j’s choice set.

While play in accordance with strategies surviving the Dekel-Fudenberg procedure does not provide any prediction other than both players defecting in the 3rd stage, the concept of fully permissible sets has more bite. In particular, a player cooperates in the 2nd stage only if the opponent has cooperated in the 1st stage. This implies that only the following paths can be realized if players choose strategies in fully permissible sets: ((cooperate, cooperate), (cooperate, cooperate), (defect, defect)) ((cooperate, cooperate), (cooperate, defect), (defect, defect)) and vice versa ((cooperate, defect), (defect, cooperate), (defect, defect)) and vice versa ((cooperate, cooperate), (defect, defect), (defect, defect)) ((cooperate, defect), (defect, defect), (defect, defect)) and vice versa ((defect, defect), (defect, defect), (defect, defect)). That the path ((cooperate, defect), (cooperate, defect), (defect, defect)) or vice versa cannot be realized if players choose strategies in fully permissible sets can be interpreted as an indication that the present analysis seems to produce some element of reciprocity in the 3-period “prisoners’ dilemma” game.

12.4

Forward induction

In Chapter 11 we have already seen how the concept of fully permissible sets promotes the forward induction outcome, (InL, `), in the PRSF of the “battle-of-the-sexes with an outside option” game Γ01 , illustrated in Figure 2.6. In this section we first investigate whether this conclusion carries over to two other variants of the “battle-of-the-sexes” game, before testing the concept of fully permissible sets in an economic application.

Applying full permissibility to extensive games

NU ND BU BD Figure 12.3.

169

`` `r r` rr 3, 1 3, 1 0, 0 0, 0 0, 0 0, 0 1, 3 1, 3 2, 1 -1, 0 2, 1 -1, 0 -1, 0 0, 3 -1, 0 0, 3

G13 (the pure strategy reduced strategic form of “burning money”).

The “battle-of-the-sexes” game with variations. Consider first the “burning money” game due to van Damme (1989) and Ben-Porath and Dekel (1992). Game G13 of Figure 12.3 is the PRSF of a “battleof-the-sexes” game with the addition that 1 can publicly destroy 1 unit of payoff before the “battle-of-the-sexes” game starts. BU (NU ) is the strategy where 1 burns (does not burn), and then plays U , etc., while `r is the strategy where 2 responds with ` conditional on 1 not burning and r conditional on 1 burning, etc. The forward induction outcome (supported e.g. by IEWDS) involves implementation of 1’s preferred “battleof-the-sexes” outcome, with no payoff being burnt. One might be skeptical to the use of IEWDS in the “burning money” game, because it effectively requires 2 to infer that BU is infinitely more likely than BD based on the sole premise that BD is eliminated before BU, even though all strategies involving burning (i.e. both BU and BD) are eventually eliminated by the procedure. On the basis of this premise such an inference seems at best to be questionable. As shown in Table 12.1, the application of our algorithm IECFA yields a sequence of iterations where at no stage need 2 deem BU infinitely more likely than BD, since {NU} is always included as a candidate for 1’s choice set. The procedure uniquely determines {NU} as 1’s fully permissible set and {``, `r} as 2’s fully permissible set. Even though the forward induction outcome is obtained, 2 does not have any assessment concerning the relative likelihood of opponent strategies conditional on burning; hence, she need not interpret burning as a signal that 1 will play according with his preferred “battle-of-the-sexes” outcome.3

3 Also

Battigalli (1991), Asheim (1994), and Dufwenberg (1994), as well as Hurkens (1996) in a different context, argue that (NU, `r) in addition to (NU, ``) is viable in “burning money”.

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CONSISTENT PREFERENCES

Table 12.1.

Applying IECFA to “burning money”.

Ξ(0) = Σ1 × Σ2 Ξ(1) = {{NU}, {ND}, {BU}, {NU, ND}, {ND, BU}, {NU, BU}, {NU, ND, BU}} × Σ2 Ξ(2) = {{NU}, {ND}, {BU}, {NU, ND}, {ND, BU}, {NU, BU}, {NU, ND, BU}} × {{``}, {r`}, {``, `r}, {r`, rr}, {``, r`}, {``, `r, r`, rr}} Ξ(3) = {{NU}, {BU}, {ND, BU}, {NU, BU}, {NU, ND, BU}} × {{``}, {r`}, {``, `r}, {r`, rr}, {``, r`}, {``, `r, r`, rr}} Ξ(4) = {{NU}, {BU}, {ND, BU}, {NU, BU}, {NU, ND, BU}} × {{``}, {r`}, {``, `r}, {``, r`}} Ξ(5) = {{NU}, {BU}, {NU, BU}} × {{``}, {r`}, {``, r`}, {``, r`}} Ξ(6) = {{NU}, {BU}, {NU, BU}} × {{``}, {``, `r}, {``, r`}} Ξ(7) = {{NU}, {NU, BU}} × {{``}, {``, `r}, {``, r`}} Ξ(8) = {{NU}, {NU, BU}} × {{``}, {``, `r}} Ξ(9) = {{NU}} × {{``}, {``, `r}} Π = Ξ(10) = {{NU}} × {{``, `r}}

We next turn now to a game introduced by Dekel and Fudenberg (1990) (cf. their Figure 7.1) and discussed by Hammond (1993), and which is reproduced here as Γ001 of Figure 12.4. It is a modification of Γ01 which introduces an “extra outside option” for player 2. In this game there may seem to be a tension between forward and backward induction: For player 2 not to choose out may seem to suggest that 2 “signals” that she seeks a payoff of at least 3/2, in contrast to the payoff of 1 that she gets when the subgame structured like Γ01 is considered in isolation (as seen in the analysis of Γ01 ). However, this intuition is not quite supported by the concept of fully permissible sets. Applying our algorithm IECFA to the PRSF of Γ001 yields: Ξ(0) = Σ1 × Σ2 Ξ(1) = {{Out}, {InL}, {Out, InL}} × {{out}, {inr}, {out, in`}, {out, inr}, {in`, inr}, {out, in`, inb}} Ξ(2) = {{Out}, {InL}, {Out, InL}} × {{out}, {out, in`}, {in`, inr}} Ξ(3) = {{InL}, {Out, InL}} × {{out}, {out, in`}, {in`, inr}} Π = Ξ(4) = {{InL}, {Out, InL}} × {{out}, {out, in`}} . The only way for Out to be a maximal strategy for player 1 is that he deems {out} as the only subjectively possible candidate for 2’s choice

171

Applying full permissibility to extensive games 3 2 3 2

2 2

out

2c

Out

s1 ¡@

in InL ¡

¡ s ¡ `¡@ r ¡ @

3 1

0 0

Figure 12.4.

@ InR @ 2 @s ` ¡ @r ¡ @

0 0

1 3

out in` 3 Out 2 2, 2 3 InL 2 3, 1 3 InR 2 0, 0 3 2, 3 2, 3 2,

inr 2, 2 0, 0 1, 3

Γ001 and its pure strategy reduced strategic form.

set, in which case 1’s choice set is {Out, InL}. Else {InL} is 1’s choice set. Furthermore, 2 can have a choice set different from {out} only if she deems {Out, InL} as a subjectively possible candidate for 1’s choice set. Intuitively this means that if 2’s choice set differs from {out} (i.e., equals {out, in`}), then she deems it subjectively possible that 1 considers it subjectively impossible that in` is a maximal strategy for 2. Since it is only under such circumstances that in` is a maximal element for 2, perhaps this strategy is better thought of in terms of “strategic manipulation” than in terms of “forward induction”. Note that the concept of fully permissible sets has more bite than the Dekel-Fudenberg procedure; in addition to the strategies appearing in fully permissible sets also inr survives the Dekel-Fudenberg procedure. An economic application. Finally, we apply the concept of fully permissible sets to an economic model from organization theory. Schotter (2000) discusses in his Chapter 8 incentives schemes for firms and the moral hazard problems that may plague them. “Revenue-sharing contracts”, for example, often invite free-riding behavior by the workers, and so lead to inefficient outcomes. However, Schotter points to “forcing contracts”—incentive schemes of a kind introduced by Holmstr¨om (1982)—as a possible remedy: Each worker is paid a bonus if and only if the collective of workers achieve a certain level of total production. If incentives are set right, then there is a symmetric and efficient Nash equilibrium in which each worker exerts a substantial effort. Each worker avoids shirking because he feels that his role is “pivotal”, believing that any reduction in effort leads to a loss of the bonus.

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CONSISTENT PREFERENCES

1c ¡@

Out ¡ s ¡

¡

out ¡ @ in ¡ @ w w w w

@ In @ 2 @s out ¡ @ in ¡ @s1 w ¡@ w ¡ @ H S ¡ @ 2 s ¡ @s s¡ @ h s¡@h ¡ @ ¡ @

0 0

Figure 12.5.

0 −c

−c 0

b−c b−c

out ins inh Out w, w w, w w, w InS w, w 0, 0 0, -c InH w, w -c, 0 b-c,b-c

Γ14 and its pure strategy reduced strategic form.

However, forcing contracts are often problematic in that there typically exists a Nash equilibrium in which no worker exerts any effort at all. How serious is this problem? Schotter offers the following argument in support of the forcing-contract (p. 302): “While the no-work equilibrium for the forcing-contract game does indeed exist, it is unlikely that we will ever see this equilibrium occur. If workers actually accept such a contract and agree to work under its terms, we must conclude that they intend to exert the necessary effort and that they expect their coworkers to do the same. Otherwise, they would be better off obtaining a job elsewhere at their opportunity wage and not wasting their time pretending that they will work hard.” Schotter appeals to intuition, but his argument has a forward induction flavor to it. We now show how the concept of fully permissible sets lends support. Consider the following situation involving a forcing contract: A firm needs two workers to operate. The workers simultaneously choose shirking at zero cost of effort, or high effort at cost c > 0. They get a bonus b > c if and only if both workers choose high effort. As indicated above, this situation can be modeled as a game with two Nash equilibria (S, s) and (H, h), where (H, h) Pareto-dominates (S, s). However, let this game be a subgame of a larger game. In line with Schotter’s intuitive discussion, add a preceding stage where each worker simultaneously decides whether to indicate willingness to join the firm with the forcing contract, or to work elsewhere at opportunity wage w, 0 < w < b − c. The firm with the forcing contract is established if and only if both workers indicate willingness to join it.

Applying full permissibility to extensive games

173

This situation is depicted by the extensive game Γ14 . Again, we analyze the PRSF (cf. Figure 12.5). Application of IECFA yields: Ξ(0) = Σ1 × Σ2 Ξ(1) = {{Out}, {InH}, {Out, InH}} × {{out}, {inh}, {out, inh}} Ξ(2) = {{InH}, {Out, InH}} × {{inh}, {out, inh}} Π = Ξ(3) = {{InH}} × {{inh}} . Interpretation: Ξ(1): Shirking cannot be a maximal strategy for either worker since it is weakly dominated. Ξ(2): This excludes the possibility that a worker’s choice set contains only the outside option. Ξ(3): Since each worker certainly believes that hard work is, while shirking is not, an element of the opponent’s choice set, it follows that each worker deems it infinitely more likely that the opponent chooses hard work rather than shirking. This means that, for each worker, only hard work is in his choice set, a conclusion that supports Schotter’s argument.

12.5

Concluding remarks

In this final chapter of the book we have explored the implications of the concept of fully permissible sets in extensive games. In Chapter 11 we have already seen—based on Asheim and Dufwenberg (2003a)—that this concept can be characterized as choice sets under common certain belief of full admissible consistency. Full admissible consistency consists of the requirements ‘caution’, ‘robust belief of opponent rationality’, and ‘no extraneous restrictions on beliefs’, and entails that one strategy is preferred to another if and only if the former weakly dominates the latter on the union of the choice sets that are deemed possible for the opponent, or on the set of all opponent strategies. The requirement of ‘robust belief of opponent rationality’ is concerned with strategy choices of the opponent only initially, in the whole game, not with choices among the remaining available strategies at each and every information set. To illustrate this point, look back at Γ11 and consider a type of player 2 who deems {D} as the only subjectively possible choice set for 1. Conditional on 2’s node being reached it is clear that 1 cannot be choosing a strategy that is maximal given his preferences. Conditional on 2’s node being reached, the modeling of the

174

CONSISTENT PREFERENCES

current chapter imposes no constraint on 2’s assessment of likelihood concerning which non-maximal strategy FF or FD that 1 has chosen. This crucially presumes that 2 assesses the likelihood of different strategies as chosen by player 1 initially, in the whole game. It is possible to model players being concerned with opponent choices at all information sets. In Γ11 this would amount to the following when player 2 is of a type who deems {D} as the only possible choice set for 1: Conditional on 2’s node being reached she realizes that 1 cannot be choosing a strategy which is maximal given his preferences. Still, 2 considers it infinitely more likely that 1 at his last node chooses a strategy that is maximal among his remaining available strategies given his conditional preferences at that node. In Section 12.2 we argued with Ben-Porath (1997) that this is not necessarily reasonable, a view which permeates the working hypotheses on which the current chapter in grounded. Yet, research on the basis of this alternative approach is illuminating and worthwhile. Indeed, Chapters 7–9 of this book have reproduced the epistemic models of Asheim (2002) and Asheim and Perea (2004) where each player believes that his opponent chooses rationally at all information sets. The former model yields an analysis that is related to Bernheim’s (1984) subgame rationalizability, while the latter model demonstrates how it—in accordance with Bernheim’s conjecture—is possible to define sequential rationalizability. Moreover, Chapter 10 has considered the closely related strategic form analyses of Schuhmacher (1999) and Asheim (2001) that define and characterize proper rationalizability as a non-equilibrium analog to Myerson’s (1978) proper equilibrium. Analysis that goes in this alternative direction promotes concepts that imply backward induction without yielding forward induction. Thus, they lead to implications that are significantly different from those of the current final chapter, where forward induction is promoted without insisting on backward induction in all games. The tension between these two approaches to extensive games cannot be resolved by formal epistemic analysis alone. It is worth noting, though, that the analysis—independently of this issue—makes use of the ‘consistent preferences’ approach to deductive reasoning in games.

Appendix A Proofs of results in Chapter 4

Proof of Proposition 6. Only if. Assume that ºd is admissible on E. Let e ∈ E and f ∈ ¬E. It now follows directly that e is not Savage-null at ` and that p Âd{e} q implies p Âd{e,f } q. If. Assume that e ∈ E and f ∈ ¬E imply e Àd f . Let p and q satisfy that pE weakly dominates qE at d. Then there exists e0 ∈ E such that υ d (p(e0 )) > υ d (q(e0 )). Write ¬A = {f1 , . . . , fn }. Let, for m ∈ {0, . . . , n},

pm (d0 ) =

8 0 n+1−m > > n+1 p(d ) + > > q(d ) > > > : 0 0

p(d )

m q(d0 ) n+1

if d0 = e0 if d0 ∈ E\e0 if d0 = fm0 and m0 ∈ {1, . . . , m} if d0 = fm0 and m0 ∈ {m + 1, . . . , n}.

Then p = p0 , pm−1 Âd pm for all m = {1, . . . , n} (since e ∈ E and f ∈ ¬E imply that e Àd f ), and p(n) Âd q (since p(n) weakly dominates q at d with υ d (pn (e0 )) > υ d (q(e0 ))). By transitivity of ºe , it follows that p Âd q. Proof of Proposition 7. (Q serial.) If d is Savage-null at d, then there exists e ∈ τ d such that e is not Savage-null at d since ºd is nontrivial. Clearly, d is not infinitely more likely than e at d, and dQe. If d is not Savage-null at d, then dQd since d is not infinitely more likely than itself at d. (Q transitive.) We must show that dQe and eQf imply dQf . Clearly, dQe and eQf imply d ≈ e ≈ f , and that f is not Savage-null at d. It remains to be shown that d Àd f does not hold if dQe and eQf . Suppose to the contrary that d Àd f . It suffices to show that dQe contradicts eQf . Since f is not Savage-null at d ≈ e, e Àd f is needed to contradict eQf . This follows from Axiom 11 because dQe entails that d Àd e does not hold. (Q satisfies forward linearity.) We must show that dQe and dQf imply eQf or f Qe. From dQe and dQf it follows that d ≈ e ≈ f and that both e and f are not Savage-null at e ≈ f . Since e Àe f and f Àf e cannot both hold, we have that eQf or f Qe. (Q satisfies quasi-backward linearity.) We must show that dQf and eQf imply dQe or eQd if ∃d0 ∈ F such that d0 Qe. From dQf and eQf it follows that d ≈ e ≈ f ,

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CONSISTENT PREFERENCES

while d0 Qe implies that e is not Savage-null at d0 ≈ d ≈ e. If d is Savage-null at d, then d Àd e cannot hold, implying that dQe. If d is not Savage-null at d ≈ e, then d Àd e and e Àe d cannot both hold, implying that dQe or eQd. Proof of Proposition 8. (R` serial.) For all d ∈ F , ρd` 6= ∅. (R` transitive.) We must show that dR` e and eR` f imply dR` f . Since dR` e implies that d ≈ e, we have that ρd` = ρe` . Now, eR` f (i.e., f ∈ ρe` ) implies dR` f (i.e., f ∈ ρd` ). (R` Euclidean.) We must show that dR` e and dR` f imply eR` f . Since dR` e implies that d ≈ e, we have that ρd` = ρe` . Now, dR` f (i.e., f ∈ ρd` ) implies eR` f (i.e., f ∈ ρe` ). (dR` e implies dR`+1 e.) This follows from the property that ρd` ⊆ ρd`+1 . (∃f such that dR`+1 f and eR`+1 f ) implies (∃f 0 such that dR` f 0 and eR` f 0 ). Since dR`+1 f implies that d ≈ f and eR`+1 f implies that e ≈ f , we have that d ≈ e and ρd` = ρe` . By the non-emptiness of this set, ∃f 0 such that dR` f 0 and eR` f 0 . Proof of Proposition 9. (i) (dQd is equivalent to d being not Savage-null at d.) If dQd, then it follows directly from Definition 2 that d is not Savage-null at d. If d is not Savage-null at d, then by Definition 2 it follows that dQd since d ≈ d and not d Àd d. (dRL d is equivalent to d being not Savage-null at d.) By Definition 3, dRL d iff d ∈ ρdL = κd , which directly establishes the result. (ii) Only if. Assume that dQe and not eQd. From dQe it follows that d ≈ e and e is not Savage-null at d, i.e. e ∈ κd (⊆ τ d ). Consider E := {e0 ∈ F | eQe0 }. Clearly, e ∈ E ⊆ κd (⊆ τ d ) and d ∈ τ d \E 6= ∅. If e0 ∈ E and f ∈ τ d \E, then not e0 Qf , since otherwise it would follow from eQf 0 and the transitivity of Q that eQf , thereby contradicting f ∈ / E. If, on the one hand, f ∈ κd \E, then e0 Àd f since f is not Savage-null at d ≈ e0 and e0 Qf does not hold. If, on the other hand, f ∈ / κd , then 0 d 0 0 e À f since f is Savage-null at d and e is not. Hence, e ∈ E and f ∈ ¬E imply e0 Àd f . By Proposition 6, ºd is admissible on E, entailing that ∃` ∈ {1, . . . , L} such that ρd` = E. By Definition 3, dR` e and not eR` d since e ∈ E and d ∈ τ d \E. If. Assume that ∃` ∈ {1, . . . , L} such that dR` e and not eR` d. From dR` e it follows that d ≈ e and e ∈ ρd` (⊆ κd ); in particular, e is not Savage-null at d. Since eR` d does not hold, however, d ∈ / ρe` = ρd` . By construction, ºd is admissible on ρd` , and it now follows from Proposition 6 that e Àd d. Furthermore, e Àd d implies that d Àd e does not hold. Hence, dQe since d ≈ e, e is not Savage-null at d and d Àd e does not hold, while not eQd since e Àd d. Proof of Lemma 3. Since κd = {e ∈ τ d |eQe}, it follows that ∃e1 ∈ τ d ∩ φ such that e1 Qe1 if κd ∩ φ 6= ∅. Either, ∀f ∈ τ d ∩ φ, f Qe1 – in which case we are through – or not. In the latter case, ∃e2 ∈ τ d ∩ φ such that e2 Qe1 does not hold. Since e1 , e2 ∈ τ d , ∃e02 ∈ τ d such that e1 Qe02 and e2 Qe02 . Since e1 Qe1 and not e2 Qe1 it now follows from quasi-backward linearity that e1 Qe2 . Moreover, not e2 Qe1 implies e2 6= e1 . Either ∀f ∈ τ d ∩ φ, f Qe2 – in which case we are through – or not. In the latter case we can, by repeating the above argument and invoking transitivity, show the existence of some e3 ∈ τ d ∩ φ such that e1 Qe3 , e2 Qe3 , and e3 6= e1 , e2 . Since τ d ∩ φ is finite, this algorithm converges to some e satisfying, ∀f ∈ τ d ∩ φ, f Qe. To prove Proposition 11 it suffices to show the following lemma.

Appendix A: Proofs of results in Chapter 4

177

Lemma 14 If φ ∈ Φ, then β d (φ) = ρd` ∩ φ, where ` := min{k ∈ {1, . . . , L}| ρdk ∩ φ 6= ∅}. Proof. (β d (φ) ⊆ ρd` ∩ φ) Assume that (τ d ∩ φ)\ρd` 6= ∅. Let e ∈ (τ d ∩ φ)\ρd` . Since ρd` ∩ φ 6= ∅, ∃f ∈ ρd` ∩ φ. Then, by Definition 3 eR` f and not f R` e, which by Proposition 9(ii) implies eQf and not f Qe. Hence, e ∈ (τ d ∩ φ)\β d (φ), and ρd` ∩ φ = (τ d ∩ φ) ∩ ρd` ⊇ (τ d ∩ φ) ∩ β d (φ) = β d (φ). Assume then that (τ d ∩ φ)\ρd` = ∅. In this case, ρd` ∩ φ = (τ d ∩ φ) ∩ ρd` = τ d ∩ φ ⊇ β d (φ). (ρd` ∩ φ ⊆ β d (φ)) Let e ∈ ρd` ∩ φ. If f ∈ ρd` ∩ φ, then f RL f since ρd` ⊆ ρdL , and f Qf by Proposition 9(i). Since e, f ∈ τ d and f Qf , it follows by quasi-backward linearity of Q that f Qf or eQf . However, since by construction, ∀k ∈ {1, . . . ` − 1}, ρdk ∩ φ = ∅, there is no k ∈ {1, . . . ` − 1} such that f Rk e and not eRk f or vice versa, and Proposition 9(ii) implies that both f Qe and eQf must hold. In particular, f Qe. If, on the other hand, f ∈ (τ d ∩ φ)\ρd` , then by Definition 3 f R` e and not eR` f , implying by Proposition 9(ii) that f Qe. Thus, ∀f ∈ τ d ∩ φ, f Qe, and e ∈ β d (φ) follows. Proof of Proposition 12. Recall that B0 E := ∩φ∈ΦE B(φ)E, where ΦE := d ∩d∈F ΦE is non-empty and defined by, ∀d ∈ F , ΦdE := {φ ∈ Φd | E ∩ κd ∩ φ 6= ∅ if E ∩ κd 6= ∅ }. (If ∃` ∈ {1, . . . , L} such that ρd` = E ∩ κd , then d ∈ B0 E.) Let ρd` = E ∩ κd and consider any φ ∈ ΦE . We must show that d ∈ B(φ)E. By the definition of ΦE , E ∩ κd ∩ φ 6= ∅ since φ ∈ ΦE and E ∩ κd = ρd` 6= ∅. Since ρd` ∩ φ = E ∩ κd ∩ φ, it follows that ∅ 6= ρd` ∩ φ ⊆ E, so by Proposition 11, d ∈ B(φ)E. (If d ∈ B0 E, then ∃` ∈ {1, . . . , L} such that ρd` = E ∩ κd .) Let d ∈ B0 E; i.e., ∀φ ∈ ΦE , d ∈ B(φ)E. We first show that ρd1 ⊆ E. Consider some φ0 ∈ ΦE satisfying τ d ∩ φ0 = (E ∩ τ d ) ∪ ρd1 . Since d ∈ B(φ0 )E, ∃k ∈ {1, . . . , L} such that ∅ 6= ρdk ∩ φ0 = ρdk ∩ (E ∪ ρd1 ) ⊆ E. Since ρd1 ⊆ ρdk , ρd1 ⊆ E. Let ` = max{k|ρdk ⊆ E}. If ` = L, then ρd` = κd , and ρd` ⊆ E implies ρd` = E ∩ κd . If ` < L, then, since ρd` ⊂ ρdL = κd , ρd` = ρd` ∩ κd ⊆ E ∩ κd . To show that ρd` = E ∩ κd also in this case, suppose instead that (E ∩ κd )\ρd` 6= ∅, and consider some φ00 ∈ ΦE satisfying τ d ∩φ00 = ((E∩κd )∪ρd`+1 )\ρd` . Since, ∀k ∈ {1, .., `}, ρdk ⊆ ρd` , it follows from ρd` ∩φ00 = ∅ that, ∀k ∈ {1, .., `}, ρdk ∩ φ00 = ∅. Since by construction, ρd` ⊆ E, while ρd`+1 ⊆ E does not hold, ρd`+1 ∩ φ00 = ρd`+1 \ρd` is not included in E. Since ρd1 ⊂ · · · ⊂ ρdL , there is no k ∈ {0, . . . , L} such that ∅ 6= ρdk ∩ φ00 ⊆ E, contradicting by Proposition 11 that d ∈ B(φ00 )E. Hence, ρd` = E ∩ κd . Proof of Proposition 14. (KE ∩ KE 0 = K(E ∩ E 0 )) To prove KE ∩ KE 0 ⊆ K(E ∩ E 0 ), let d ∈ KE and d ∈ KE 0 . Then, by Definition 4, κd ⊆ E and κd ⊆ E 0 and hence, κd ⊆ E ∩ E 0 , implying that d ∈ K(E ∩ E 0 ). To prove KE ∩ KE 0 ⊇ K(E ∩ E 0 ), let d ∈ K(E ∩ E 0 ). Then κd ⊆ E ∩ E 0 and hence, κd ⊆ E and κd ⊆ E 0 , implying that d ∈ KE and d ∈ KE 0 . (B(φ)E ∩ B(φ)E 0 = B(φ)(E ∩ E 0 )) Using Defintion 5 the proof of conjunction for B(φ) is identical to the one for K except that β d (φ) is substituted for κd . (KF = F ) KF ⊆ F is obvious. That KF ⊇ F follows from Definition 4 since, ∀d ∈ F , κd ⊆ τ d ⊆ F . (B(φ)∅ = ∅) This follows from Definition 5 since, ∀d ∈ F , β d (φ) 6= ∅, implying that there exists no d ∈ F such that β d (φ) ⊆ ∅.

178

CONSISTENT PREFERENCES

(KE ⊆ KKE) Let d ∈ KE. By Definition 4, d ∈ KE is equivalent to κd ⊆ E. Since ∀e ∈ τ d , κe = κd , it follows that τ d ⊆ KE. Hence, κd ⊆ τ d ⊆ KE, implying by Definition 4 that d ∈ KKE. (B(φ)E ⊆ KB(φ)E) Let d ∈ B(φ)E. By Definition 5, d ∈ B(φ)E is equivalent to β d (φ) ⊆ E. Since ∀e ∈ τ d , β e (φ) = β d (φ), it follows that τ d ⊆ B(φ)E. Hence, κd ⊆ τ d ⊆ B(φ)E, implying by Definition 4 that d ∈ KB(φ)E. (¬KE ⊆ K(¬KE)) Let d ∈ ¬KE. By Definition 4, d ∈ ¬KE is equivalent to κd ⊆ E not holding. Since ∀e ∈ τ d , κe = κd , it follows that τ d ⊆ ¬KE. Hence, κd ⊆ τ d ⊆ ¬KE, implying by Definition 4 that d ∈ K(¬KE). (¬B(φ)E ⊆ K(¬B(φ)E)) Let d ∈ ¬B(φ)E. By Definition 5, d ∈ ¬B(φ)E is equivalent to β d (φ) ⊆ E not holding. Since ∀e ∈ τ d , β e (φ) = β d (φ), it follows that τ d ⊆ ¬B(φ)E. Hence, κd ⊆ τ d ⊆ ¬B(φ)E, implying by Definition 4 that d ∈ K(¬B(φ)E). Proof of Proposition 15. (1.) β d (φ) ⊆ φ follows by definition since, ∀e ∈ β d (φ), e ∈ φ. (2.) By Definitions 2 and 3 and Proposition 9, β d = ρd1 . Hence, β d ∩ φ 6= ∅ implies d ρ1 ∩ φ 6= ∅ and min{`|ρd` ∩ φ 6= ∅} = 1. By Lemma 14, β d (φ) = ρd1 ∩ φ = β d ∩ φ. (3.) This follows directly from Lemma 3, since φ ∈ Φ implies that, ∀d ∈ F , κd ∩ φ 6= ∅. (4.) Let β d (φ) ∩ φ0 6= ∅. By Lemma 14, β d (φ) = ρd` ∩ φ 6= ∅ where ` := min{k|ρdk ∩ φ 6= ∅}. Likewise, β d (φ ∩ φ0 ) = ρd`0 ∩ φ ∩ φ0 , where `0 := min{k|ρdk ∩ φ ∩ φ0 6= ∅}. It suffices to show that ` = `0 . Obviously, ` ≤ `0 . However, ∅ 6= β d (φ) ∩ φ0 = (ρd` ∩ φ) ∩ φ0 = ρd` ∩ φ ∩ φ0 implies that `0 ≥ `. Proof of Proposition 16. That KE ⊆ B0 E follows from Definition 4 and Propositions 9 and 12 since κd ⊆ E implies that ρdL = κd = κd ∩ E. That B0 E ⊆ B(F )E follows from Definition 6 since F ∈ ΦE . Proof of Proposition 17. (B0 E ∩ B0 E 0 ⊆ B0 (E ∩ E 0 )) Let d ∈ B0 E and d ∈ B0 E 0 . Then, by Proposition 12, there exist k such that ρd` = E ∩ κd and k0 such that ρd`0 = E 0 ∩ κd . Since ρd1 ⊂ · · · ⊂ ρdL , either ρd` ⊆ ρd`0 or ρd` ⊇ ρd`0 , or equivalently, E ∩ κd ⊆ E 0 ∩ κd or E ∩ κd ⊇ E 0 ∩ κd . Hence, either ρd` = E ∩ κd = E ∩ E 0 ∩ κd or ρd`0 = E 0 ∩ κd = E ∩ E 0 ∩ κd , implying by Proposition 12 that a ∈ B0 (E ∩ E 0 ). (B0 E ⊆ KB0 E) Let d ∈ B0 E. By Proposition 12, d ∈ B0 E is equivalent to ∃` ∈ {1, . . . , L} such that ρd` = E ∩ κd . Since ∀e ∈ τ d , ρe` = ρd` and κe = κd , it follows that τ d ⊆ B0 E. Hence, κd ⊆ τ d ⊆ B0 E, implying by Definition 4 that d ∈ KB0 E. (¬B0 E ⊆ K(¬B0 E)) Let d ∈ ¬B0 E. By Proposition 12, d ∈ ¬B0 E is equivalent to there not existing k ∈ {1, . . . , L} such that ρd` = E ∩ κd . Since ∀e ∈ τ d , ρe` = ρd` and κe = κd , it follows that τ d ⊆ ¬B0 E. Hence, κd ⊆ τ d ⊆ ¬B0 E, implying by Definition 4 that d ∈ K(¬B0 E). To prove Proposition 18 the following lemma is helpful.

Lemma 15 Assume that ºd satisfies Axioms 1 and 4 00 (in addition to the assumptions made in Section 4.1), and let `, `0 ∈ {1, . . . , Ld } satisfy ` < `0 . Then pÂdπd q ` implies pÂdπd ∪πd q. `

`0

Proof. This follows from Proposition 3.

179

Appendix A: Proofs of results in Chapter 4

Proof of Proposition 18. (If E is assumed at d, then d ∈ B 0 E.) Let E be assumed at d. Then it follows that ºdE is nontrivial; hence, E ∩ κd 6= ∅. Assume that pE∩κd weakly dominates qE∩κd at d. Since E ∩ κd 6= ∅, we have that p ÂdE q. Hence, it follows from the premise (viz., that E is assumed at d) that p Âd q. This shows that ºd is admissible on E ∩ κd , and, by Proposition 12, d ∈ B 0 E. (If d ∈ B 0 E, then E is assumed at d.) Let d ∈ B 0 E, so by Proposition 12 ºd is admissible on E ∩ κd (6= ∅). Hence, by Proposition 6, e ∈ E ∩ κd and f ∈ ¬(E ∩ κd ) implies e Àd f . By Axiom 400 this in turn implies that ∃` such that E ∩ κd =

[` k=1

πkd ,

since the first property of Axiom 400 – the Archimedean property of ºd within each partitional element – rules out that e and f are in the same element of the partition d d {π1d , . . . , πL d } if e À f . d Assume that p ÂE q. Then p ÂdE∩κd q, and, by the above argument, p ÂdS`

k=1

d πk

q.

By completeness and the partitional Archimedean property, Lemma 15 entails that ∃`0 ∈ {1, . . . , `} such that pÂdπd q and, ∀k ∈ {1, . . . , `0 − 1}, p∼dπd q . `0

d

d d since ∪L k=1 πk d

k

d

ÂdE

By Lemma 15, p  q = κ . Hence, p q implies p Âd q. Moreover, d ºE is nontrivial since E ∩ κ 6= ∅, and it follows from Definition 7 that E is assumed at d.

Appendix B Proofs of results in Chapters 8–10

For the proofs of Propositions 30, 34, 36, and 37 we need two results from Blume et al. (1991b). To state these results, introduce the following notation. Let λ = (µ1 , ..., µL ) be an LPS on a finite set F and let r = (r1 , ..., rL−1 ) ∈ (0, 1)L−1 . Then, rλ denotes the probability distribution on F given by the nested convex combination (1 − r1 )µ1 + r1 [(1 − r2 )µ2 + r2 [(1 − r3 )µ3 + r3 [. . . ] . . . ]] . The first is a restatement of Proposition 2 in Blume et al. (1991b).

Lemma 16 Let (x(n))n∈N be a sequence of probability distributions on a finite set

F . Then, there exists a subsequence x(m) of (x(n))n∈N , an LPS λ = (µ1 , ..., µL ), and a sequence r(m) of vectors in (0, 1)L−1 converging to zero such that x(m) = r(m)λ for all m. The second is a variant of Proposition 1 in Blume et al. (1991b).

Lemma 17 Consider a type ti of player i whose preferences over acts on Sj × Tj

are represented by υiti — with υiti ◦ z = ui — and λti = (µt1i , . . . , µtLi ) ∈ L∆(Sj × Tj ). Then, for every sequence (r(n))n∈N in (0, 1)L−1 converging to zero there is an n0 such that, ∀si , s0i ∈ Si , si Âti s0i if and only if XX sj

(r(m)λti )(sj , tj )ui (si , sj ) >

tj

XX sj

(r(m)λti )(sj , tj )ui (s0i , sj )

tj

for all n ≥ n0 . Proof. Suppose that si Âti s0i . Then, there is some ` ∈ {1, ..., L} such that XX sj

µtki (sj , tj )ui (si , sj ) =

XX sj

tj

µtki (sj , tj )ui (s0i , sj )

(B.1)

µt`i (sj , tj )ui (s0i , sj ).

(B.2)

tj

for all k < ` and XX sj

tj

µt`i (sj , tj )ui (si , sj ) >

XX sj

tj

182

CONSISTENT PREFERENCES

Let (r(n))n∈N be a sequence in (0, 1)L−1 converging to zero. By (B.1) and (B.2), XX

(r(n)λti )(sj , tj )ui (si , sj ) >

sj

tj

XX sj

(r(n)λti )(sj , tj )ui (s0i , sj )

tj

if n is large enough. Since Si is finite, this is true if n is large enough for any si , s0i ∈ Si satisfying si Âti s0i . The other direction follows from the proof of Proposition 1 in Blume et al. (1991b). For the proofs of Propositions 30 and 34 we need the following definitions. Let the LPS λi = (µi1 , . . . , µiL ) ∈ L∆(Sj ) have full support on Sj . Say that the behavior strategy σj is induced by λi if for all h ∈ Hj and a ∈ A(h), σj (h)(a) :=

µi` (Sj (h, a)) , µi` (Sj (h))

where ` = min{k| supp λik ∩ Sj (h) 6= ∅}. Moreover, say that player i’s beliefs over past opponent actions βi are induced by λi if for all h ∈ Hi and x ∈ h, βi (h)(x) :=

µi` (Sj (x)) , µi` (Sj (h))

where ` = min{k| supp λik ∩ Sj (h) 6= ∅}. Proof of Proposition 30. (Only if.) Let (σ, β) be a sequential equilibrium. Then (σ, β) is consistent and hence there is a sequence (σ(n))n∈N of completely mixed behavior strategy profiles converging to σ such that the sequence (β(n))n∈N of induced belief systems converges to β. For each i and all n, let pi (n) ∈ ∆(Si ) be the mixed representation of σi (n). By Lemma 16, the sequence (pj (n))n∈N of probability distributions on Sj contains a subsequence pj (m) such that we can find an LPS λi = (µi1 , . . . , µiL ) with full support on Sj and a sequence of vectors r(m) ∈ (0, 1)L−1 converging to zero with pj (m) = r(m)λi for all m. W.l.o.g., we assume that pj (n) = r(n)λi for all n ∈ N. We first show that λi induces the behavior strategy σj . Let σ ˜j be the behavior strategy induced by λi . By definition, ∀h ∈ Hj , ∀a ∈ A(h), σ ˜j (h)(a)

= =

µi` (Sj (h, a)) (r(n)λi )(Sj (h, a)) = lim i n→∞ (r(n)λi )(Sj (h)) µ` (Sj (h)) pj (n)(Sj (h, a)) = lim σj (n)(h)(a) = σj (h)(a) , lim n→∞ n→∞ pj (n)(Sj (h))

where ` = min{k| supp λik ∩ Sj (h) 6= ∅}. For the fourth equation we used the fact that pj (n) is the mixed representation of σj (n). Hence, for each i, λi induces σj . We then show that λi induces the beliefs βi . Let β˜i be player i’s beliefs over past opponent actions induced by λi . By definition, ∀h ∈ Hi , ∀x ∈ h, β˜i (h)(x)

= =

r(n)λi (Sj (x)) µi` (Sj (x)) = lim i n→∞ r(n)λi (Sj (h)) µ` (Sj (h)) pj (n)(Sj (x)) = lim βi (n)(h)(x) = βi (h)(x), lim n→∞ n→∞ pj (n)(Sj (h))

Appendix B: Proofs of results in Chapters 8–10

183

where ` = min{k| supp λik ∩ Sj (h) 6= ∅}. For the fourth equality we used the facts that pj (n) is the mixed representation of σj (n) and βi (n) is induced by σj (n). Hence, for each i, λi induces βi . We now define the following epistemic model. Let T1 = {t1 } and T2 = {t2 }. Let, for each i, υiti satisfy υiti ◦ z = ui , and (λti , `ti ) be the SCLP with support Sj ×{tj }, where (1) λti coincides with the LPS λi constructed above, and (2) `ti (Ej ) = min{`| supp λt`i ∩ Ej 6= ∅} for all (∅ 6=) Ej ⊆ Sj × {tj }. Then, it is clear that (t1 , t2 ) ∈ [u], there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and for each i, σi is induced for ti by tj . It remains to show that (t1 , t2 ) ∈ [isr]. For this, it is sufficient to show, for each i, that σi is sequentially rational for ti . Suppose not. By the choice of `ti , it then follows that there is some information set h ∈ Hi and some mixed strategy pi ∈ ∆(Si (h)) that is outcome-equivalent to σi |h such that there exist si ∈ Si (h) with pi (si ) > 0 and s0i ∈ Si (h) having the property that ui (si , µt`i |Sj (h) ) < ui (s0i , µt`i |Sj (h) ) , where ` = min{k| supp λtki ∩ (Sj (h) × {tj }) 6= ∅} and µt`i |Sj (h) ∈ ∆(Sj (h)) is the conditional probability distribution on Sj (h) induced by µt`i . Recall that µt`i is the `-th level of the LPS λti . Since the beliefs βi and the behavior strategy σj are induced by λi , it follows that ui (si , µt`i |Sj (h) ) = ui (si , σj ; βi )|h and ui (s0i , µt`i |Sj (h) ) = ui (s0i , σj ; βi )|h and hence ui (si , σj ; βi )|h < ui (s0i , σj ; βi )|h , which is a contradiction to the fact that (σ, β) is sequentially rational. (If ) Suppose that there is an epistemic model with (t1 , t2 ) ∈ [u] ∩ [isr] such that there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and for each i, σi is induced for ti by tj . We show that σ = (σ1 , σ2 ) can be extended to a sequential equilibrium. For each i, let λi = (µi1 , . . . , µiL ) ∈ L∆(Sj ) be the LPS coinciding with λti , and let βi be player i’s beliefs over past opponent choices induced by λi . Write β = (β1 , β2 ). We first show that (σ, β) is consistent. Choose sequences (r(n))n∈N in (0, 1)L−1 converging to zero and let the sequences (pj (n))n∈N of mixed strategies be given by pj (n) = r(n)λi for all n. Since λi has full support on Sj for every n, pj (n) is completely mixed. For every n, let σj (n) be a behavior representation of pj (n) and let βi (n) be the beliefs induced by σj (n). We show that (σj (n))n∈N converges to σj and that (βi (n))n∈N converges to βi , which imply consistency of (σ, β). Note that the inducement of σj by ti depends on λti through, for each h ∈ Hj , µt`i , where ` = min{k| supp λtki ∩ (Sj (h) × {tj }) 6= ∅}. This implies that σj is induced by λi . Since σj (n) is a behavior representation of pj (n) and σj is induced by λi , we have, ∀h ∈ Hj , ∀a ∈ A(h), lim σj (n)(h)(a)

n→∞

= =

lim

n→∞

pj (n)(Sj (h, a)) r(n)λi (Sj (h, a)) = lim n→∞ r(n)λi (Sj (h)) pj (n)(Sj (h))

µi` (Sj (h, a)) = σj (h)(a), µi` (Sj (h))

where ` = min{k| supp λik ∩ Sj (h) 6= ∅}. Hence, (σj (n))n∈N converges to σj .

184

CONSISTENT PREFERENCES

Since βi (n) is induced by σj (n) and σj (n) is a behavior representation of pj (n), and furthermore, βi is induced by λi , we have, ∀h ∈ Hi , ∀x ∈ h, lim βi (n)(h)(x)

n→∞

= =

lim

n→∞

pj (n)(Sj (x)) r(n)λi (Sj (x)) = lim n→∞ r(n)λi (Sj (h)) pj (n)(Sj (h))

µi` (Sj (x)) = βi (h)(x), µi` (Sj (h))

where ` = min{k| suppλik ∩ Sj (h) 6= ∅}. Hence, (βi (n))n∈N converges to βi . This establishes that (σ, β) is consistent. It remains to show that for each i and ∀h ∈ Hi , ui (σi0 , σj ; βi )|h . ui (σi , σj ; βi )|h = max 0 σi

Suppose not. Then, ui (σi , σj ; βi )|h < ui (σi0 , σj ; βi )|h for some h ∈ Hi and some σi0 . Let pi ∈ ∆(Si (h)) be outcome-equivalent to σi |h . Then, there is some si ∈ Si (h) with pi (si ) > 0 and some s0i ∈ Si (h) such that ui (si , σj ; βi )|h < ui (s0i , σj ; βi )|h . Since the beliefs βi and the behavior strategy σj are induced by λi , it follows (using the notation that has been introduced in the ‘only if’ part of this proof) that ui (si , σj ; βi )|h = ui (si , µt`i |Sj (h) ) and ui (s0i , σj ; βi )|h = ui (s0i , µt`i |Sj (h) )|h and hence ui (si , µt`i |Sj (h) ) < ui (s0i , µt`i |Sj (h) ), which contradicts the fact that σi is sequentially rational for ti . This completes the proof of this proposition. Proof of Proposition 34. (Only if.) Let (σ1 , σ2 ) be a quasi-perfect equilibrium. By definition, there is a sequence (σ(n))n∈N of completely mixed behavior strategy profiles converging to σ such that for each i and every n ∈ N and h ∈ Hi , ui (σi , σj (n))|h = max ui (σi0 , σj (n))|h . 0 σi

For each j and every n, let pj (n) be the mixed representation of σj (n). By Lemma 16, the sequence (pj (n))n∈N of probability distributions on Sj contains a subsequence pj (m) such that we can find an LPS λi = (µi1 , . . . , µiL ) with full support on Sj and a sequence of vectors r(m) ∈ (0, 1)L−1 converging to zero with pj (m) = r(m)λi for all m. W.l.o.g., we assume that pj (n) = r(n)λi for all n ∈ N. By the same argument as in the proof of Proposition 30, it follows that λi induces the behavior strategy σj . Now, we define an epistemic model as follows. Let T1 = {t1 } and T2 = {t2 }. Let, for each i, υiti satisfy υiti ◦ z = ui , and (λti , `ti ) be the SCLP with support Sj × {tj }, where (1) λti coincides with the LPS λi constructed above, and (2) `ti (Sj × {tj }) = L. Then, it is clear that (t1 , t2 ) ∈ [u], there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and for each i, σi is induced for ti by tj . It remains to show that (t1 , t2 ) ∈ [isr] ∩ [cau].

185

Appendix B: Proofs of results in Chapters 8–10

Since, obviously, (t1 , t2 ) ∈ [cau], it suffices to show, for each i, that σi is sequentially rational for ti . Fix a player i and let h ∈ Hi be given. Let pi (∈ ∆(Si (h))) be outcomeequivalent to σi |h and let pj (n) be the mixed representation of σj (n). Then, since (σ1 , σ2 ) is a quasi-perfect equilibrium, it follows that ui (pi , pj (n)|h ) =

max

p0i ∈∆(Si (h))

ui (p0i , pj (n)|h )

for all n. Hence, pi (si ) > 0 implies that X sj ∈Sj (h)

X

pj (n)|h (sj )ui (si , sj ) = 0 max

si ∈Si (h)

pj (n)|h (sj )ui (s0i , sj )

(B.3)

sj ∈Sj (h)

for all n. Let λthi be i’s preferences at ti conditional on h. Since ti ∈ projTi [caui ]—so that i’s system of conditional preferences at ti satisfies Axiom 6 (Conditionality)—and pj (n) = r(n)projSj λti for all n, there exist vectors r(n)|h converging to zero such that pj (n)|h = r(n)|h projSj λthi for all n. Together with equation (B.3) we obtain that pi (si ) > 0 implies X

(r(n)|h projSj λthi )(sj )ui (si , sj )

sj ∈Sj (h)

X

= 0 max

si ∈Si (h)

(r(n)|h projSj λthi )(sj )ui (s0i , sj ) .

(B.4)

sj ∈Sj (h)

Siti (h).

We show that pi (si ) > 0 implies si ∈ Suppose that si ∈ Si (h)\Siti (h). Then, ti 0 0 there is some si ∈ Si (h) with si Âh si . By applying Lemma 17 in the case of acts on Sj (h) × {tj }, it follows that r(n)|h has a subsequence r(m)|h for which X

X

(r(m)|h projSj λthi )(sj )ui (s0i , sj ) >

sj ∈Sj (h)

(r(m)|h projSj λthi )(sj )ui (si , sj )

sj ∈Sj (h)

for all m, which is a contradiction to (B.4). Hence, si ∈ Siti (h) whenever pi (si ) > 0, which implies that pi ∈ ∆(Siti (h)). Hence, σi |h is outcome equivalent to some pi ∈ ∆(Siti (h)). This holds for every h ∈ Hi , and hence σi is sequentially rational for ti . (If ) Suppose, there is an epistemic model with (t1 , t2 ) ∈ [u] ∩ [isr] ∩ [cau] such that there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and for both i, σi is induced for ti by tj . We show that (σ1 , σ2 ) is a quasi-perfect equilibrium. For each i, let λi = (µi1 , . . . , µiL ) ∈ L∆(Sj ) be the LPS coinciding with λti . Choose sequences (r(n))n∈N in (0, 1)L−1 converging to zero and let the sequences (pj (n))n∈N of mixed strategies be given by pj (n) = r(n)λi for all n. Since λi has full support on Sj for every n, pj (n) is completely mixed. For every n, let σj (n) be a behavior representation of pj (n). Since λi induces σj , it follows that (σj (n))n∈N converges to σj ; this is shown explicitly under the ‘if’ part of Proposition 30. Hence, to establish that (σ1 , σ2 ) is a quasi-perfect equilibrium, we must show that, for each i and ∀n ∈ N and ∀h ∈ Hi , ui (σi , σj (n))|h = max ui (σi0 , σj (n))|h . (B.5) 0 σi

Fix a player i and an information set h ∈ Hi . Let pi (∈ ∆(Si (h))) be outcomeequivalent to σi |h . Then, equation (B.5) is equivalent to ui (pi , pj (n)|h ) =

max

p0i ∈∆(Si (h))

ui (p0i , pj (n)|h )

186

CONSISTENT PREFERENCES

for all n. Hence, we must show that pi (si ) > 0 implies that X

pj (n)|h (sj )ui (si , sj ) = 0 max

X

si ∈Si (h)

sj ∈Sj (h)

pj (n)|h (sj )ui (s0i , sj )

(B.6)

sj ∈Sj (h)

for all n. In fact, it suffices to show this equation for infinitely many n, since in this case we can choose a subsequence for which the above equation holds, and this would be sufficient to show that (σ1 , σ2 ) is a quasi-perfect equilibrium. Since, by assumption, σi is sequentially rational for ti , σi |h is outcome equivalent to some mixed strategy in ∆(Siti (h)). Hence, pi ∈ ∆(Siti (h)). Let pi (si ) > 0. By construction, si ∈ Siti (h). Suppose that si would not satisfy (B.6) for infinitely many n. Then, there exists some s0i ∈ Si (h) such that X

pj (n)|h (sj )ui (si , sj ) <

sj ∈Sj (h)

X

pj (n)|h (sj )ui (s0i , sj )

sj ∈Sj (h)

for infinitely many n. Assume, w.l.o.g., that it is true for all n. Let λthi be i’s preferences at ti conditional on h. Since ti ∈ projTi [caui ]—so that i’s system of conditional preferences at ti satisfies Axiom 6 (Conditionality)—and pj (n) = r(n)projSj λti for all n, there exist vectors r(n)|h converging to zero such that pj (n)|h = r(n)|h projSj λthi for all n. This implies that X

(r(n)|h projSj λthi )(sj )ui (si , sj ) <

sj ∈Sj (h)

X

(r(n)|h projSj λthi )(sj )ui (s0i , sj )

sj ∈Sj (h)

for all n. By applying Lemma 17 in the case of acts on Sj (h)×{tj }, it follows that i at ti strictly prefers s0i to si conditional on h, which contradicts the fact that si ∈ Siti (h). Hence, pi (si ) > 0 implies (B.6) for infinitely many n, and as a consequence, (σ1 , σ2 ) is a quasi-perfect equilibrium. Proof of Proposition 36. (Only if.) Let (p1 , p2 ) be a proper equilibrium. Then, by Definition 7, there is a sequence (p(n))n∈N of ε(n)-proper equilibria converging to p, where ε(n) → 0 as n → ∞. By the necessity part of Proposition 5 of Blume et al. (1991b), there exists an epistemic model with T1 = {t1 } and T2 = {t2 } where, for each i, υiti satisfies that υiti ◦ z = ui , the SCLP (λti , `ti ) has the properties that λti = (µt1i , . . . , µtLi ) with support Sj × {tj } satisfies that, ∀sj ∈ Sj , µt1i (sj , tj ) = pj (sj ), and `ti satisfies that `ti (Sj × Tj ) = L, such that (t1 , t2 ) ∈ [resp]. This argument involves Lemma 16 (which yields, for each i, the existence of λti with full support on ∆(Sj × {tj }) by means of a subsequence pj (m) of (pj (n))n∈N ) and Lemma 17 (which yields that, for m large enough, i having the conjecture pj (m) leads to the same preferences over i’s strategies as ºti ). The only-if part follows since it is clear that (t1 , t2 ) ∈ [u] ∩ [cau], that there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and that, for each i, pi is induced for ti by tj . (If.) Suppose that there exists an epistemic model with (t1 , t2 ) ∈ [u]∩[resp]∩[cau] such that there is mutual certain belief of {(t1 , t2 )} at (t1 , t2 ), and, for each i, pi is induced for ti by tj . Then, by the sufficiency part of Proposition 5 in Blume et al. (1991b), there exists, for each i, a sequence of completely mixed strategies

Appendix B: Proofs of results in Chapters 8–10

187

(pi (n))n∈ℵ converging to pi , where, for each n, (p1 (n), p2 (n)) is an ε(n)-proper equilibrium and ²(n) → 0 as n → ∞. This argument involves Lemma 17 (which yields, for each j, the existence of (xj (n))n∈N so that, for all n, i having the conjecture pj (n) leads to the same preferences over i’s strategies as ºti ). Proof of Proposition 37. Part 1: If p∗i is properly rationalizable, then there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [resp] ∩ [cau]) such that p∗i is induced for t∗i by t∗j . In the definition of proper rationalizability, g in Kg [²-prop trem] goes to infinity for each ε, and then ε converges to 0. The strategy for the proof of the ‘only if’ part of Proposition 37 is to reverse the order of g and ε, by first noting that ε-proper rationalizability implies ε-proper g-rationalizability for all g, then showing that ε-proper g-rationalizability as ε converges to 0 corresponds to the gth round of a finite algorithm, and finally proving that any mixed strategy surviving all rounds of the algorithm is rational under common certain belief of [u] ∩ [resp] ∩ [cau] in some epistemic model. The algorithm eliminates preference relations on the players’ strategy sets. It is related to, but differs from, Hammond’s (2001) ‘rationalizable dominance relations’, which are recursively constructed by gradually extending a single incomplete binary relation on each player’s strategy set. Say that a mixed strategy pi for i is ε-properly g-rationalizable if there exists an ∗-epistemic model with piti = pi for some ti ∈ projTi Kg ([u] ∩ [ind] ∩ [ε-prop trem]). Since, for all g, CK[ε-prop trem]) ⊆ Kg [ε-prop trem]) , it follows from Definition 21 that if p∗i is an ε-properly rationalizable strategy, then, for all g, there exists an ∗-epistemic model with piti = p∗i for some ti ∈ projTi Kg ([u] ∩ [ind] ∩ [ε-prop trem]). Consequently, if a mixed strategy p∗i for i is ∗-properly rationalizable, then, for all g, there exists a sequence (pi (n))n∈N of ε(n)-properly grationalizable strategies converging to p∗i , where ²(n) → 0 as n → ∞. This means that it is sufficient to show that if p∗i satisfies that, for all g, there exists a sequence (pi (n))n∈N of ε(n)-properly g-rationalizable strategies converging to p∗i and ²(n) → 0 as n → ∞, then p∗i is rational under common certain belief of [u] ∩ [resp] ∩ [cau] in some epistemic model. This will in turn be shown in two steps: 1 If a sequence of ²(n)-properly g-rationalizable strategies converges to p∗i , then p∗i survives the gth round of a finite algorithm. 2 Any mixed strategy surviving all rounds of the algorithm is rational under common certain belief of [u] ∩ [resp] ∩ [cau] in some epistemic model. To construct the algorithm, note that any complete and transitive binary relation on Si can be represented by a vector of sets (Si (1), . . . , Si (L)) (with L ≥ 1) that constitute a partition of Si . The interpretation is that si is preferred or indifferent to s0i if and only if si ∈ Si (`), s0i ∈ Si (`0 ) and ` ≤ `0 . Let, for each i, Σi := 2Si \{∅} be the collection of non-empty subsets of Si and πi

Πi := {πi = (Si (1), . . . , Si (Lπi )) ∈ ΣiL | {Si (1), . . . , Si (Lπi )} is a partition of Si } denote the collection of vectors of sets that constitute a partition of Si . Define the algorithm by, for each i, setting Π−1 = Πi and determining, ∀g ≥ 0, Πgi as follows: i g πi πi = (Si (1), . . . , Si (L )) ∈ Πi if and only if πi ∈ Πi and there exists an LPS λπi ∈ L∆(Sj × Πj ) with suppλπi = Sj × Ππj i for some Πjπi ⊆ Πg−1 , satisfying that j (sj , πj ) À (s0j , πj ) according to ºπi

188

CONSISTENT PREFERENCES

if πj = (Sj (1), . . . , Sj (Lπj )) ∈ Πjπi , sj ∈ Sj (`), s0j ∈ Sj (`0 ) and ` < `0 , and si Âπi s0i if and only if si ∈ Si (`), s0i ∈ Si (`0 ) and ` < `0 , where ºπi is represented by υiπi satisfying υiπi ◦ z = ui and λπi . Write Π := Π1 × Π2 and, ∀g ≥ 0, Πg = Πg1 × Πg2 . Since Π0 ⊆ Π, it follows by induction that, ∀g ≥ 0, Πg ⊆ Πg−1 . Moreover, since the finiteness of S = S1 × S2 implies that Π is finite, it follows that Πg converges to Π∞ in a finite number of rounds. Say that pi survives the gth round of the algorithm if there exists πi = (Si (1), . . . , Si (Lπi )) ∈ Πgi with ∆(Si (1)) 3 pi . Step 1. We first show that p∗i survives the gth round of the algorithm if there exists a sequence (pi (n))n∈N of ε(n)-properly g-rationalizable strategies converging to p∗i , where ε(n) → 0 as n → ∞. Say that the probability distribution µ ∈ ∆(Sj ×Tj ) is an ε-properly g-rationalizable belief for i if there is an ∗-epistemic model with µti = µ for some ti ∈ projTi Kg ([u] ∩ [ind] ∩ [ε-prop trem]). It is sufficient to establish the following result: If πi = (Si (1), . . . , Si (Lπi )) ∈ Πi satisfies that there exists a sequence (µπi (n))n∈ℵ of ε(n)-properly g-rationalizable beliefs for i, where ε(n) → 0 as n → ∞, and where, for all n, XX π XX π µ i (n)(sj , tj )ui (si , sj ) > µ i (n)(sj , tj )ui (s0i , sj ) (B.7) sj

tj

sj

if and only if si ∈ Si (`),

s0i

tj

∈ Si (` ) and ` < ` , then πi ∈ Πgi . 0

0

This result is established by induction. If (µπi (n))n∈N is a sequence of ε(n)-properly g-rationalizable beliefs for i, then, for each n, there exists an ∗-epistemic model with T1 (n)×T2 (n) as the set of type vectors, such that µπi (n) ∈ ∆(Sj × Tj (n)). For the inductive proof we can w.l.o.g. partition Tj (n) into Πj , where πj = (Sj (1), . . . , Sj (Lπj )) ∈ Πj corresponds to the subset of j-types in Tj (n) satisfying that XX si

µtj (n)(si , ti )uj (sj , si ) >

ti

XX si

s0j

0

µtj (n)(si , ti )uj (s0j , si )

ti 0

if and only if sj ∈ Sj (`), ∈ Sj (` ) and ` < ` , since i’s certain belief of j’s ε(n)proper trembling only matters through j-types’ preferences over j’s pure strategies . Hence, we can w.l.o.g. assume that µπi (n) ∈ ∆(Sj × Πj ). (g = 0) Let (µπi (n))n∈N be a sequence of ε(n)-properly 0-rationalizable beliefs for i, where ²(n) → 0 as n → ∞, and where, for all n, (B.7) is satisfied. By Lemma 16, the sequence (µπi (n))n∈N contains a subsequence µπi (m) such that one can find an LPS λπi ∈ L∆(Sj × Πj ) and a sequence of vectors rπi (m) ∈ (0, 1)L−1 (for some L) converging to 0 with µπi (m) = rπi (m)λπi for all m. By Definition 20, suppλπi = Sj × Πjπi for some Πjπi ⊆ Πj . Let ºπi be represented by υiπi satisfying υiπi ◦ z = ui and λπi . Since Definition 20 is the only requirement on (µπi (n))n∈N for g = 0, we may, for each πj ∈ Πjπi , associate πj with (Sj (1), . . . , Si (Lπj )) ∈ Π−1 satisfying that (sj , πj ) À (s0j , πj ) according to ºπi j 0 0 if sj ∈ Sj (`), sj ∈ Sj (` ) and ` < `0 . By Lemma 2, ºπi yields the same preferences on Si as µπi (n) (for any n). Hence, πi ∈ Π0i .

189

Appendix B: Proofs of results in Chapters 8–10

(g > 0) Suppose the result holds for g 0 = 0, . . . , g − 1. Let (µπi (n))n∈N be a sequence of ε(n)-properly g-rationalizable beliefs for i, where ε(n) → 0 as n → ∞, and where, for all n, (B.7) is satisfied. As for g = 0, use Lemma 16 to construct an LPS λπi ∈ L∆(Sj × Πj ), where suppλπi = Sj × Πjπi for some Πjπi ⊆ Πj , and where ºπi is represented by υiπi satisfying υiπi ◦ z = ui and λπi . Since


Kg [²-prop trem] ⊆ Ki [²-prop tremj ] ∩ Kg−1 [²-prop trem] , the induction hypothesis implies that Πjπi ⊆ Πg−1 and (sj , πj ) À (s0j , πj ) according j πi πj πi to º if πj = (Sj (1), . . . , Si (L )) ∈ Πj , sj ∈ Sj (`), s0j ∈ Sj (`0 ) and ` < `0 . By Lemma 17, ºπi yields the same preferences on Si as µπi (n) (for any n). Hence, πi ∈ Πgi . This concludes the induction and thereby Step 1. Step 2. We then show that if a mixed strategy p∗i survives all rounds of the ∗ algorithm, then there exists an epistemic model with p∗i ∈ ∆(Siti ) for some t∗i ∈ projTi CK([u] ∩ [resp] ∩ [cau]). It is sufficient to show that one can construct an epistemic model with T1 × T2 ⊆ CK([u] ∩ [resp] ∩ [cau]) such that, for each i, ∀πi = ti 0 (Si (1), . . . , Si (Lπi )) ∈ Π∞ si if and only if i , there exists ti ∈ Ti satisfying that si  0 0 0 si ∈ Si (`), si ∈ Si (` ) and ` < ` . Construct an epistemic model with, for each i, a ∞ bijection  i : Ti → Π∞ i from the set of types to the collection of vectors in Πi . Since ∃g 0 such that Πg = Π∞ for g ≥ g 0 , it follows from the definition of the algorithm πi (Πg )g≥0 that, for each i, Π∞ i is characterized as follows: πi = (Si (1), . . . , Si (L )) ∈ ∞ Πi if and only if there exists ti ∈ Ti such that  i (ti ) = πi , and an LPS λti = (µt1i , . . . µtLi ) ∈ L∆(Sj × Tj ) with suppλti = Sj × Tjti for some Tjti ⊆ Tj , satisfying for each tj ∈ Tjti that (sj , tj ) À (s0j , tj ) according to ºti if

j (tj ) = (Sj (1), . . . , Sj (Lj (tj ) ), sj ∈ Sj (`), s0j ∈ Sj (`0 ) and ` < `0 , and si Âti s0i

if and only if si ∈ Si (`), s0i ∈ Si (`0 ) and ` < `0 , where υiti satisfies υiti ◦ z = ui and the SCLP (λti , `ti ) has the property that `ti satisfies `ti (Sj × Tj ) = L (so that ºti is represented by υiti and λti ). Consider any πi = (Si (1), . . . , Si (Lπi )) ∈ Π∞ i . By the construction of the type sets, there exists ti ∈ Ti such that  i (ti ) = πi , and si Âti s0i if and only if si ∈ Si (`), s0i ∈ Si (`0 ) and ` < `0 ; in particular, Si (1) = Siti . It remains to be shown that, for each i, T1 × T2 ⊆ [ui ] ∩ [respi ] ∩ [caui ], implying that T1 × T2 ⊆ CK([u] ∩ [resp] ∩ [cau]) since Tjti ⊆ Tj for each ti ∈ Ti of any player i. It is clear that T1 × T2 ⊆ [ui ] ∩ [caui ]. That T1 × T2 ⊆ [respi ] follows from the property that, for any ti ∈ Ti , (sj , tj ) À (s0j , tj ) according to ºti whenever tj ∈ Tjti if sj ∈ Sj (`), s0j ∈ Sj (`0 ) and ` < `0 , while sj Âtj s0j if and only if sj ∈ Sj (`), s0j ∈ Sj (`0 ) and ` < `0 (where  j (tj ) = (Sj (1), . . . , Sj (Lj (tj ) )). This concludes Step 2. ∗ In the construction in Step 2, let t∗i ∈ Ti satisfy that p∗i ∈ ∆(Siti ). To conclude Part 1 of the proof of Proposition 37, add type t∗j to Tj having the property that p∗i is ∗ ∗ induced for t∗i by t∗j . Assume that υjtj satisfies υjtj ◦z = uj and the SCLP (λti , `ti ) on ∗ ∗ ∗ ∗ Si × Tj with support Si × {ti } has the property that λtj = (µ1tj , . . . , µLtj ) satisfies, ∗ t∗ ∗ ∗ t t ∀si ∈ Si , µ1 j (si , ti ) = pi (si ) and ` i satisfies ` i (Sj × Tj ) = L (so that ºtj is ∗ t∗ t represented by υj j and λ j ). Furthermore, assume that ∗

(si , t∗i ) À (s0i , t∗i ) according to ºtj

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CONSISTENT PREFERENCES

∗ if  i (t∗i ) = (Si (1), . . . , Si (Li (ti ) )), si ∈ Si (`), s0i ∈ Si (`0 ) and ` < `0 . Then t∗j ∈ ∗ projTj ([uj ] ∩ [respj ] ∩ [cauj ]), and since Titj ⊆ Ti , Ti × (Tj ∪ {t∗j }) ⊆ CK([u] ∩ [resp] ∩ [cau]). Hence, (t∗1 , t∗2 ) ∈ CK([u] ∩ [resp] ∩ [cau]) and p∗i is induced for t∗i for t∗j . Part 2: If there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [resp] ∩ [cau]) such that p∗1 is induced for t∗1 by t∗2 , then p∗1 is properly rationalizable. Schuhmacher (1999) considers a set of type profiles T = T1 × T2 , where each type ti of either player i plays a completely mixed strategy piti and has a subjective probability distribution on Sj × Tj , for which the conditional distribution on Sj × {tj } coincides with pjtj whenever the conditional distribution is defined. His formulation implies that all types of a player agrees not only on the preferences but also on the relative likelihood of the strategies for any given opponent type. In contrast, the characterization given in Proposition 37 requires the types of a player only to agree on the preferences of any given opponent type. This difference implies that expanded type sets must be constructed for the ‘if’ part of the proof of Proposition 37. Assume that there exists an epistemic model with (t∗1 , t∗2 ) ∈ CK([u] ∩ [resp] ∩ [cau]) such that p∗1 is induced for t∗1 by t∗2 . In particular, CK([u] ∩ [resp] ∩ [cau]) 6= ∅, ∗ and p∗1 ∈ ∆(S1t1 ) since CK([u] ∩ [resp] ∩ [cau]) ⊆ [resp2 ]. Let, for each i, Ti0 := projTi CK([u] ∩ [resp] ∩ [cau]). Note that, for each ti ∈ Ti0 of any player i, ti deems (sj , tj ) subjectively impossible if tj ∈ Tj \Tj0 since CK([u] ∩ [ir] ∩ [cau]) = KCK([u] ∩ [ir] ∩ [cau]) ⊆ Ki CK([u] ∩ [ir] ∩ [cau]), implying Tjti ⊆ Tj0 . We first construct a sequence, indexed by n, of ∗-epistemic models. By Definition 20 and Assumption 3 this involves, for each n and for each player i, a finite set of types—which we below denote by Ti00 and which will not vary with n—and, for each n, for each i, and for each type τi ∈ Ti00 , a mixed strategy and a probability distribution (pτi i (n), µτi (n)) ∈ ∆(Si ) × ∆(Sj × Tj00 ) that will vary with n. For either player i and each type ti ∈ Ti0 of the original epistemic model, make as many “clones” of ti as there are members of Tj0 : For each i, Ti00 := {τi (ti , tj )| ti ∈ Ti0 and tj ∈ Tj0 }, where τi (ti , tj ) is the “clone” of ti associated with tj . The term “clone” in the above statement reflects that, ∀tj ∈ Tj0 , τi (ti , tj ) is assumed to “share” the preferences of ti in the sense that

1 the set of opponent types that τi (ti , tj ) does not deem subjectively possible, Tjτi (ti ,tj ) , is equal to {τj (t0j , ti )|t0j ∈ Tjti } (⊆ Tj00 since Tjti ⊆ Tj0 ), and 2 the likelihood of (sj , τj (t0j , ti )) according to ºτi (ti ,tj ) is equal to the likelihood of (sj , t0j ) according to ºti . Since Tjτi (ti ,tj ) = {τj (t0j , ti )| t0j ∈ Tjti } is independent of tj , but corresponds to disjoint subsets of Tj00 for different ti ’s, we obtain the following conclusion for any pair of type vectors (t1 , t2 ), (t01 , t02 ) ∈ T10 × T20 : 0

0

Tjτi (ti ,tj ) = Tjτi (ti ,tj ) 0 0 τi (ti ,tj ) Tj ∩ Tjτi (ti ,tj ) = ∅

if if

ti = t0i , ti 6= t0i .

This ends the construction of type sets in the sequence of ∗-epistemic models. Fix a player i and consider any τi ∈ Ti00 . Since CK([u] ∩ [resp] ∩ [cau]) ⊆ [ui ], ºτi can be represented by a vNM utility function υiτi satisfying υiτi ◦ z = ui and an LPS λτi on Sj × Tjτi . Since CK([u] ∩ [resp] ∩ [cau]) ⊆ [caui ], this LPS yields, for each τj ∈ Tjτi , a partition {Ejτi (1), . . . , Ejτi (Lτi )} of Sj × Tjτi , where (sj , τj ) À (s0j , τj ) according to ºτi if and only (sj , τj ) ∈ Ejτi (`), (s0j , τj0 ) ∈ Ejτi (`0 ) and ` < `0 . Since

191

Appendix B: Proofs of results in Chapters 8–10

CK([u] ∩ [resp] ∩ [cau]) ⊆ [respi ], it follows that sj is a most preferred strategy for τj in {s0j ∈ Sj |(s0j , τj ) ∈ Ejτi (`) ∪ · · · ∪ Ejτi (Lτi )} if (sj , τj ) ∈ Ejτi (`). Consider any i and τi ∈ Ti00 . Construct the sequence (µτi (n))n∈N as follows. Choose τi ∀τi ∈ {τi (ti , tj )|tj ∈ Tj0 } one common sequence (rτi (n))n∈N in (0, 1)L −1 converging τi to 0 and let the sequence of probability distributions (µ (n))n∈N be given by µτi (n) = rτi (n)λτi . For all n, suppµτi (n) = Sj × Tjτi . By Lemma 17 (rτi (n))n∈ℵ can be chosen such that, for all n, XX sj

µτi (n)(sj , τj )ui (si , sj ) >

XX sj

τj

µτi (n)(sj , τj )ui (s0i , sj )

τj

if and only if si Âτi s0i . Hence, for all n, the belief µτi (n) leads to the same preferences over i’s strategies as ºτi . This ends the construction of the sequences (µτi (n))n∈N in the sequence of ∗-epistemic models. Consider now the construction of the sequence (pτi i (n))n∈N for any i and τi ∈ Ti00 . There are two cases. Case 1: If there is τj ∈ Tj00 such that τi ∈ Tiτj , implying that Si × {τi } ⊆ suppµτj (n), then let pτi i (n) be determined by pτi i (n)(si ) =

µτj (n)(si , τi ) . µτj (n)(Si , τi )

Moreover, for each n, there exists ²(n) such that, for each player i, the ²(n)-proper trembling condition is satisfied at all such types in Ti00 : Since pτi i (n)(s0i ) µτj (n)(s0i , τi ) = τj → 0 as n → ∞ τi pi (n)(si ) µ (n)(si , τi ) τ

τ

if (si , τi ) ∈ Ei j (`), (s0i , τi ) ∈ Ei j (`0 ) and ` < `0 , and since si is a most preferred τ τ τ strategy for τi in {s0i ∈ Si |(s0i , τi ) ∈ Ei j (`) ∪ · · · ∪ Ei j (Lτj )} if (si , τi ) ∈ Ei j (`), it τi follows that there exists a sequence (² (n))n∈N converging to 0 such that, for all n, ²τi (n)pτi i (n)(si ) ≥ pτi i (n)(s0i ) XX sj

τj

whenever

µτi (n)(sj , τj )ui (si , sj ) >

XX sj

µτi (n)(sj , τj )ui (s0i , sj ) .

τj

Let, for each n, ²(n) := max {²τ1 (n)|∃τ2 ∈ T200 s.t. τ1 ∈ T1τ2 } ∪ {²τ2 (n)|∃τ1 ∈ T100 s.t. τ2 ∈ T2τ1 } . Since the type sets are finite, ²(n) → 0 as n → ∞. Case 2: If there is no τj ∈ Tj00 such that τi ∈ Tiτj , then let pτi i (n) be any mixed strategy having the property that τi satisfies the ²(n)-proper trembling condition given the belief µτi (n). This ends the construction of the sequences (pτi i (n))n∈N in the sequence of ∗-epistemic models. ∗ We then turn to the construction of a sequence (p1τ1 (n))n∈N converging to p∗1 . Add ∗ ∗ type τ1∗ to T100 having the property that µτ1 (n) = µτ1 (t1 ,t2 ) (n) for some t2 ∈ T20 , but ∗ ∗ τ1∗ ∗ τ (t ,t ) where p1 (n) = (1 − n1 )p1 + n1 p1 1 1 2 (n). For all n, we have that the belief µτ1 (n) t∗ leads to the same preferences over 1’s strategies as º i . This in turn implies that i ∗ satisfies the ²(n)-trembling condition at τ1∗ since p∗1 ∈ ∆(C1t1 ). Consider the sequence, indexed by n, of ∗-epistemic models, with T100 ∪ {τ1∗ } as the type set for 1 and T200 as the type set for 2,

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with, for each type τi of any player i, (pτi i (n), µτi (n)) as the sequence of a mixed strategy and a probability distribution, as constructed above. Furthermore, it follows that, for all n, the ²(n)-proper trembling condition is satisfied at all types in T100 ∪ {τ1∗ } and at all types in T200 , where ²(n) → 0 as n → ∞. Hence, for all n, (T100 ∪ {τ1∗ }) × T200 ⊆ CK[²(n)-prop trem] ; ∗

∗

in particular, p1τ1 (n) is ²(n)-properly rationalizable. Moreover, (p1τ1 (n))n∈N converges to p∗1 . By Definition 21, p∗1 is a properly rationalizable strategy.

Appendix C Proofs of results in Chapter 11

Proof of Proposition 42. Assume that the pure strategy si for i is properly rationalizable in a finite strategic two-player game G. Then, there exists an epistemic model satisfying Assumption 1 with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CK([u] ∩ [resp] ∩ [cau]) (this follows from Proposition 37 since CK([u] ∩ [resp] ∩ [cau]) = KCK([u] ∩ [resp] ∩ [cau]) ⊆ Kj CK([u] ∩ [resp] ∩ [cau])). In particular, CK([u] ∩ [resp] ∩ [cau]) 6= ∅. By Proposition 20(ii), for each i, CK([u] ∩ [resp] ∩ [cau]) = KCK([u] ∩ [resp] ∩ [cau]) ⊆ Ki CK([u] ∩ [resp] ∩ [cau]). Hence, we can construct a new epistemic model (S1 , T10 , S2 , T20 ) where, for each i, Ti0 := projTi CK([u] ∩ [resp] ∩ [cau]), as for each ti ∈ Ti0 of any player i, κti = {ti } × Sj × Tjti ⊆ {ti } × Sj × Tj0 . Since T10 × T20 ⊆ [cau], according to the definition of caution given in Section 5.3, it follows that the new epistemic model satisfies Axiom 6 for each ti ∈ Ti0 of any player i. Therefore, the new epistemic model satisfies Assumption 2 with S1 × T10 × S2 × T20 ⊆ [cau] according to the definition of caution given in Section 6.3. Also, S1 × T10 × S2 × T20 ⊆ [u]. It remains to be shown that, for each i, S1 × T10 × S2 × T20 ⊆ B0i [ratj ], since by the fact that κti ⊆ {ti } × Sj × Tj0 for each ti ∈ Ti0 of any player i, we then have an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA0 . Since T10 × T20 ⊆ [resp], we have that, for each ti ∈ Ti0 of any player i, (sj , tj ) Àti whenever tj ∈ Titi and sj ºtj s0j . In particular, for each ti ∈ Ti0 of any player i, (sj , tj ) Àti (s0j , tj ) whenever tj ∈ Titi , sj ∈ Sjtj and s0j ∈ / Sjtj . By Proposition 6 this 0 ti means that, for each ti ∈ Ti of any player i, º is admissible on projT1 ×S2 ×T2 [ratj ] ∩ κti , showing that S1 × T10 × S2 × T20 ⊆ (B01 [rat2 ] ∩ B02 [rat1 ]). (s0j , tj )

Proof of Proposition 43. Part 1: If si is permissible, then there exists an ¯ It is sufficient to epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA. construct a belief system with S1 × T1 × S2 × T2 ⊆ CKA¯ such that, for each si ∈ Pi of any player i, there exists ti ∈ Ti with si ∈ Siti . Construct a belief system with, for each i, a bijection si : Ti → Pi from the set of types to the the set of permissible pure strategies. By Lemma 10(i) we have that, for each ti ∈ Ti of any player i, there exists Yjti ⊆ Pi such that si (ti ) ∈ Si \Di (Yjti ). Determine the set of opponent types

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that ti deems subjectively possible as follows: Tjti = {tj ∈ Tj | sj (tj ) ∈ Yjti }. Let, for each ti ∈ Ti of any player i, ºti satisfy 1. υiti ◦ z = ui (so that S1 × T1 × S2 × T2 ⊆ [u]), and 2. p Âti q iff pEj weakly dominates qEj for Ej = Ejti := {(sj , tj )|sj = sj (tj ) and tj ∈ Tjti } or Ej = Sj × Tjti , which implies that β ti = {ti } × Ejti and κti = {ti } × Sj × Tjti (so that S1 × T1 × S2 × T2 ⊆ [cau]). By the construction of Ejti , this means that Siti = Si \Di (Yjti ) 3 si (ti ) since, for any acts p and q on Sj × Tj satisfying that there exist mixed strategies pi , qi ∈ ∆(Si ) such that, ∀(sj , tj ) ∈ Sj × Tj , p(sj , tj ) = z(pi , sj ) and q(sj , tj ) = z(qi , sj ), p Âti q iff pEj weakly dominates qEj for Ej = Yjti × Tj or Ej = Sj × Tj . This in turn implies, for each ti ∈ Ti any player i, 3. β ti ⊆ projTi ×Sj ×Tj [ratj ] (so that, in combination with 2., S1 × T1 × S2 × T2 ⊆ ¯ i [ratj ] ∩ B ¯ j [rati ]). B Furthermore, S1 × T1 × S2 × T2 ⊆ CKA¯ since Tjti ⊆ Tj for each ti ∈ Ti of any player i. Since, for each player i, si is onto Pi , it follows that, for each si ∈ Pi of any player i, there exists ti ∈ Ti with si ∈ Siti . Part 2: If there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKA, then si is permissible. Part 2 of the proof of Proposition 27. Proof of Proposition 44. Part 1: If si is rationalizable, then there exists an ¯ 0 . It is sufficient to epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKC construct a belief system with S1 × T1 × S2 × T2 ⊆ CKC such that, for each si ∈ Ri of any player i, there exists ti ∈ Ti with si ∈ Siti . Construct a belief system with, for each i, a bijection si : Ti → Ri from the set of types to the the set of rationalizable pure strategies. By Lemma 9(i) we have that, for each ti ∈ Ti of any player i, there exists Yjti ⊆ Ri such that there does not exist pi ∈ ∆(Si ) such that pi weakly dominates si (ti ) on Yjti . Determine the set of opponent types that ti deems subjectively possible as follows: Tjti = {tj ∈ Tj | sj (tj ) ∈ Yjti }. Let, for each ti ∈ Ti of any player i, ºti satisfy 1. υiti ◦ z = ui (so that S1 × T1 × S2 × T2 ⊆ [u]), and 2. p Âti q iff pEj weakly dominates qEj for Ej = Ejti := {(sj , tj )|sj = sj (tj ) and tj ∈ Tjti }, which implies that β ti = κti = {ti } × Ejti . By the construction of Ejti , this means that Siti 3 si (ti ) since, for any acts p and q on Sj × Tj satisfying that there exist mixed strategies pi , qi ∈ ∆(Si ) such that, ∀(sj , tj ) ∈ Sj × Tj , p(sj , tj ) = z(pi , sj ) and q(sj , tj ) = z(qi , sj ), p Âti q iff pEj weakly dominates qEj for Ej = Yjti × Tj . This in turn implies, for each ti ∈ Ti any player i, 3. β ti ⊆ projTi ×Sj ×Tj [ratj ] (so that, in combination with 2., S1 × T1 × S2 × T2 ⊆ ¯ 0i [ratj ] ∩ B ¯ 0j [rati ]). B ¯ 0 since Tjti ⊆ Tj for each ti ∈ Ti of any player Furthermore, S1 × T1 × S2 × T2 ⊆ CKC i. Since, for each player i, si is onto Ri , it follows that, for each si ∈ Ri of any player i, there exists ti ∈ Ti with si ∈ Siti . Part 2: If there exists an epistemic model with si ∈ Siti for some (t1 , t2 ) ∈ projT1 ×T2 CKC, then si is rationalizable. Part 2 of the proof of Proposition 25.

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Index

Accessibility relation, 39–44, 46 Act Anscombe-Aumann act, 8, 22, 26, 32–33, 39–40, 49, 54, 56, 70–71, 74–75, 77–78, 83–84, 92, 102, 122, 126, 145, 181, 185–186, 194 Admissibility, 39, 41–42, 46, 49–50, 70, 72, 85–86, 133–135, 143–144, 147, 153–154, 175–176, 179, 193 Backward induction, 2, 6–7, 10–11, 14–17, 20–21, 23–24, 38, 69, 78–80, 82–83, 87–88, 91–97, 99–100, 112–113, 115, 118, 121, 123, 128, 132, 152, 155–159, 162–166, 170, 174 Belief operators absolutely robust belief, 38, 40, 45, 49–50, 147 assumption, 38, 40, 45, 48–50, 147, 152 certain belief, 19, 39, 44, 46–48, 57–61, 63–66, 72–73, 76, 78, 81–82, 87–88, 90–96, 99, 103, 105–106, 108–109, 113, 115, 117–118, 122–123, 125, 127–128, 136, 138, 140, 147–148, 150–153, 162–163, 167, 173, 183–188 conditional belief, 39–40, 44–45, 47, 50, 86 full belief, 38, 40, 45–46, 50 robust belief, 40, 44–46, 48–51, 135–139, 143, 145–149, 152–154, 158, 163, 173 strong belief, 38, 40, 45, 48, 50–51, 147, 152 Caution, 10, 14, 23, 62–63, 75, 103, 115–116, 123, 135–137, 139, 143, 145–146, 148–149, 152, 154, 158, 160, 163–164, 173, 193 Consistency of preferences (ordinary) consistency, 5, 12, 53, 58–59, 73, 149

admissible consistency, 53, 63, 75–76, 87–88, 97, 148 admissible subgame consistency, 90–96 full admissible consistency, 134–140, 144–145, 147–148, 151–152, 162–163, 167, 173 proper consistency, 122–123, 125, 128 quasi-perfect consistency, 117 sequential consistency, 104 weak sequential consistency, 108 Consistent preferences approach, 1–7, 11–12, 15–17, 21, 53, 81, 144, 154, 174 Epistemic independence, 90, 94, 96 Epistemic model, 3–5, 8–9, 15, 41–42, 48, 50, 53–55, 58–62, 64–67, 69, 73–74, 76–77, 83, 91–92, 94, 100, 102, 104–106, 109–111, 113–115, 117–119, 121, 125–128, 130, 138, 144–145, 147–151, 153, 174, 183–190, 193–194 Epistemic priority, 38–39, 42–46 Equilibrium Nash equilibrium, 2–6, 11–13, 18, 53, 58–60, 64–65, 115, 124, 130, 141, 171–172 perfect equilibrium, 18, 53, 62–65, 115, 124, 130 proper equilibrium, 16, 18–19, 121–122, 124–125, 127, 130, 148, 174, 186 quasi-perfect equilibrium, 18–19, 24, 115–118, 127, 184–186 sequential equilibrium, 18–19, 24, 100, 104–107, 115, 118, 182–183 subgame-perfect equilibrium, 87, 91, 94, 113–114 weak sequential equilibrium, 18 Forward induction, 2, 6–7, 10–11, 17, 21, 24, 38, 69, 97, 112, 133, 135, 137, 146, 148–150, 152–154, 159, 162, 168–172,

202 174 Game extensive game, 6, 14–15, 17, 23, 50–51, 56, 80–85, 87, 89–91, 94, 99, 101–103, 105–106, 108–109, 113, 117–119, 152, 155, 158–160, 162, 173 of perfect information, 20, 79–82, 84–85, 87–92, 94, 97, 100–101, 113, 118, 121, 123, 128, 132, 162 strategic game, 2–4, 7–8, 53–54, 56–57, 59–61, 63–66, 69, 71, 73, 76, 83–85, 88, 90–91, 94, 101–103, 121, 125–130, 139, 144, 148–149, 159, 161, 193 pure strategy reduced strategic form (PRSF), 133, 156, 159–164, 168–170, 173 Inducement (of rationality) of a rational mixed strategy, 5, 58 of a sequentially rational behavior strategy, 104 of a weak sequentially rational mixed strategy, 107 Iterated elimination Dekel-Fudenberg procedure, 13–14, 23–24, 65, 69, 78, 81, 83, 88, 112, 124, 138, 141, 143, 148, 162, 166–168, 171 of choice sets under full admissible consistency (IECFA), 138–142, 150–151, 163–164, 169–170, 173 of strongly dominated strategies (IESDS), 13, 23–24, 60, 69, 83, 133, 137–138, 141–143, 149 of weakly dominated strategies (IEWDS), 14, 16–17, 129–130, 133–135, 141–142, 150, 152–153, 159, 169 No extraneous restrictions on beliefs, 135, 137, 139, 143, 146–149, 153–154, 158, 163–164, 173 Probability system conditional probability system (CPS), 24–25, 34–36, 50, 109

REFERENCES lexicographic conditional probability system (LCPS), 31, 35–36, 49–50 lexicographic probability system (LPS), 24–25, 30–33, 36, 43, 49, 56, 60, 62–63, 65, 67, 76, 88, 93–94, 96, 102, 104, 106–107, 110, 116, 122, 131, 143, 181–185, 187–190 system of conditional lexicographic probabilities (SCLP), 25, 32, 35–36, 56–57, 59, 61–64, 66, 100, 102–104, 109–110, 114–116, 121–122, 183–184, 186, 189 Rational choice approach, 1–3, 6, 11–12, 143 Rationalizability (ordinary) rationalizability, 8, 13, 18, 53, 60, 69, 73, 124, 137–138, 142–143, 146, 149, 153 extensive form rationalizability, 99, 112–113, 135, 152–153, 156, 159 full permissibility, 17, 112, 134–146, 148–152, 154–155, 157–173 permissibility, 13–15, 17–18, 53, 62, 65–67, 69, 75–77, 81, 87, 112, 115, 119, 124, 137–138, 141–143, 146, 148–149, 153–154, 193–194 proper rationalizability, 1, 16, 18–19, 121–125, 127–131, 146–148, 174, 187, 190, 192–193 quasi-perfect rationalizability, 1, 15–16, 18, 24, 101, 115–116, 118–119, 128 sequential rationalizability, 1, 15–16, 18, 99–101, 104, 106–107, 111–115, 118–119, 128, 174 weak sequential rationalizability, 18, 20, 107–112, 119 Strategic manipulation, 171 Strategically independent set, 84–85, 110, 119 Subjective possibility, 38–39, 43

About the Author

Geir B. Asheim is Professor of Economics at the University of Oslo, Norway. In additional to investigating epistemic conditions for gametheoretic solution concepts, he is doing research on questions relating to intergenerational justice.