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k and no a^Pik) such that 6 < y f l for some y > 0 for any k, k>k>kj then let b^Pik) for all k, k>k>k. (We could make such a construction in Step 2. P{k) will still meet the definition of trade proposal, and the trade proposals will still be nested. It will simply be the case that when b G P{k) is matched for M, then there is no trade and so r' = 0 for each / where f^{u) = e' + r^a(u\ d)a. Thus the previous steps of the proof still hold.) First let us treat the situation in which there exist a e Pik) and b ^P{k) such that b^ya for all y 9t 0. We must show that fiu) = e. Case J: There exists A with a{u^^ b) = 0, and such that if A had a{ii^, b) > 0, then there would be at least k agents with aiu'y b) > 0. In this case, it must be that r^ = 0. (See Case 1 of 5.3.1.) It must also be that /^ = 0 for all B^A. Suppose the contrary. Choose p e IR(^^ such that /? a < 0 for all a in all P{k\ (The fact that such a p exists follows from the fact that P{k) is a trade proposal and it contains a and b with d =jfc yfl for all y =?fc 0.) Since x-f^e, there must exist B^A and r^ =5^ 0 such that pt^^O. Then find it such that a(M,fl)<0 for all a in all P{Jc) and t^ is preferred to no trade. It follows that /(M~^'^,w^,M^) = e. Let M'^ concavify both u^ and M^ through e. Then fiu~'^'^,ii^,u^) = e and f(u~^,ii'^)=x. Then B can manipulate at u^ via u^. Case 2: There exists A such that M'(y')<w'(eO for any i&A and for any y^e such that y' < e^ + ^aePik^^a^ f^^ ^^'"^ ^^^ of ^a ^^^ ^^^^ ^d> 0 for each a. Also, if A had aiit^y b)>0 then there would be at least k agents with a(u', b) > 0. Suppose that /^ # 0. Case 1 implies that e is available to y4 so the outcome must be individually rational for A. It also follows from Case 1 that r'^ < — yfe for some y > 0. (Otherwise some A from Case 1 could manipulate.) Thus there exists p e U^^^ such that p • /^ = 0 and /? • a < 0 for all a in P{k). There is no C #y4 such that p't^>Q; otherwise find u with both t^ and H preferred to no trade and aiu, a)<0 for all a in P{k). The outcome if both A and C announced u would be e, and they could manipulate via u^ and u^. Thus all C are such that pt^ = 0. Since this is true for all p' ^U+^ such that p'-t^ = 0 and /?' a < 0, all t^ are coUinear with r^. If there are two groups with nonzero trades in the same direction, then it is possible to .find u with both trades preferred to no trade, and «(«, a) < 0 for all a in P(k). The outcome if both groups announced M would be e and so they could manipulate via u. Thus there must be some group C with /^ = 0. Find « with H preferred to no trade and aiu, a) < 0 for all a in P(k). It follows that f{u~^'^, u^, u^) = e. Let M^ concavify both u^ and u^ through e. Then /(M"'^'^, M"^,M^) = e and fiu~^,ii^)=f{u). Then A can manipulate at u^ via u"^. Thus our supposition was wrong and so f "^ = 0. It then follows from Case 1 and strong nonbossiness that / ( « ) = e. Case 3: There exists A such that u'iy^)
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a(u^yb)>0 and aiu^,b)>0 and so choose u so that a(M,^)>0. Otherwise choose w so that af(w, a) < 0 for all a &P(k). In either case if both B and C announce «, then the outcome is e and they can jointly manipulate / via u^ and u^. Case 4: Not cases 1, 2, or 3. Suppose xi=e. Then there exists some A and /? e !R(^^ such that pt"^ >0 and p-a <0 for all fl ^P(k) for any A:. Then there exists w^ such that M prefers t^ to no trade and w'(y') < M'(e') for / Gy4 for any y ^ e such that y' < e' + I^flet/;k/>(;fe)^a^ ^^^ some set of A^ such that A^ ^ 0 for each a. This contradicts Cases 1, 2, and 3. Thus / ( « ) = e. Finally, let us treat the case where either P(k) = {a} or P(k) = {a, -ya} for some 7 > 0. We show that if there is any trade, then it must be in proportion a. Without loss of generality we can treat the case P(k} = [a] as if it were Pik) = {A, —ya) for some y > 0. (We could make such a construction in Step 2. P(k) will still meet the definition of trade proposal, and the trade proposals will still be nested. It will simply be the case that when -ya e P ( ^ ) is matched for M, then there is no trade (unless k = n/2 and a is also matched) and so /•' = 0 for each i where fKu) = e^ + r^a{u^,a)a. Thus the previous steps of the proof will still hold.) There must exist some groups of agents with a{u\ a) = 0; otherwise some other step applies. If there is only one such group A, then t"^ = 0. (The proof of this parallels Case 1 of 53.1.) It must also be that t^ is in proportion a for all B ¥'A. Suppose the contrary. Then there must exist some B and p G R'^^ such that p't^>0 and pa = 0. Then find u such that a(Uj a) = 0, and u prefers t^ to no trade. It follows that /'(M~^'^,M^,M^) = e' for any i^AUB, Let u^ concavify both u"^ and M'^ through e. Then /'(M~'^'^, M^, U^) = e' for / e B and /(«"^,«^) = A:. Then B can manipulate at u^ via M^. If there are two groups of agents A and B with «(«', a) = 0, then both get the endowment. (Neither t^ nor t"^ can be such that p't> 0 for some p^U^^+ such that /?•« =0; otherwise that group could manipulate from the previous case or if a were matched. Since they can get e by matching the other, it must be that the outcome is e.) We can repeat the previous argmnent to establish that all other groups trade in proportion a. Iteration of this reasoning allows for additional groups B^^A such that aiu^y a) = 0. The proof that there exists r' e [0,1] such that /'(«) = e' + r'a(u\ a)a for all i and that (7) holds is a straightforward extension of the proof of the same fact in Step 5.2 and Step 4. PROOF OF THEOREM 4: First, notice that if / satisfies (4), (5), (6), and (7), then it is strategy-proof, nonbossy and tie-free (see Step 1 of the proof of Theorem 3). We begin by showing that it has an anonymous extension / which is strategy-proof, nonbossy and tie-free. We first identify the M*S at which / is constrained and should be extended. / is constrained at u if u matches a e P{k) for some ky and there exists / such that /*(M) = e' -I- a{u\ a)a, a(u\ a) # a(u'ya)y and there exists w and. « ~ ^ M such that f^^'Xu) == e'^^'^ + r'^'^a(u'^^'\a}a and If-^^O^iu-^oy^a)] > \a(u\a)\ or f''^'\u)-e'^'^=fKu)-e' and there exists j such that sign[a(M'^({>,fl)] = sign[a(M*,fl)] and \r''^^^a(u''^^\a)\ > \a(u\a)\. Define / as follows. If for M, there exists ir and M ~^ M such that / is not constrained at M, then let f'(u) - e^ =f'^^'\u) - e'^^'^ for each i?^ If at M, / is constrained at u for every M ~^ «, then define / by uniform rationing subject to anyone^s trade being no larger than the maximum trade possible when the given a is matched. More formally this is as follows. Notice that u must match a GPik) for some k. If E,a(M',fl)>0, then let f'iu)-e' =a(u',a)a for each i such that a(u'ya)<0 and fKu)-e' = min{X,a(u'ya)}a for each / such that a(M',fl)^0, where A soh^es ^i:a(a',a)>o ^^^(A, ^(M',^)} = E,.aiu\a}
. u and both are unconstrained. Notice that by (6) fKu) = e' ^fa(Jl\a)a and7^^'XM) = e'^^'^ + r'^'>a(M^<'>,fl)a. By the definition of uniform rationing if fKu) - e' ^^f^^^KH) - e'^^^^ for some /, then it must be that for some /either {«(«', a)!
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that M ~^ u~\u'. Then strategy-proofness follows directly from the definition of / and uniform rationing. Next, suppose that / is constrained at u for all v and « such that M ~^ M and / is not constrained atw for some TT and u such that M ~^ u~\u\ Without loss of generality, by the anonymity of / , suppose that IT is the identity. From the definition of / it follows that either /'(M) = e* + a{u\ a)a or that the total trade in direction of JC' - e* at fiu) is at least as large as that at fiu~\u'X and since / is not constrained at u~', M* it follows that agents with the same sign of a(u\ a) as a(w^ a) are simply uniformly rationed the amount of trade available_ from agents trading the other direction, as if a = a. Thus / receives at least as large a trade at fiu) as at f{u~\u^). Finally, suppose that / is not constrained at u for some TT and u such that M ~^ u. This means that either fKu)-e' = r'^^'^a{u''^'\a)a and r^'^Kl or fKu)-e'= a(u'^'\a)a. In the latter case no manipulation is possible, since i is obtaining the most preferred trade in direction a. In the former case, if \a(u\a)\ >J'^^'^a(u'^^'^y a)\ then by (6) and (7) u ~Tr."~'^ "' ^^^^5 ^^ ^^^ ^^^^ f ^"^ ^^ unconstrained so / is unchanged. If \aiu\a)\
We verify t h a t / is nonbossy. Consider M, i, and u' such that /^(w) =/'(«"', MO. First suppose that e^ = f[(u) = f[(u~i^u[X Given that only f s utility function changes, the definition of / implies that e' =f'(u) =fKu~\ u') and that this is true for all permutations of u and (u~\ MO. Then by (7) fiu)^fiu~\«0 and this is true for all permutations of u and u~\u\ If e =fiu) =fiu~\iV) for all permutations, then neither profile is constrained and so e =fiu) ^fiu~\ u'). If e ^fiu) =fiu~\u') for some permutation, which is without loss of generahty, the identity, then trade is in the same proportion a, and the definition o f / and_ the fact that e ' = / ' ( « ) =/'(M"',_MO implies that aiu\a) = aiu'',a) which implies that fiu)=fiu~',u}). Next suppose that e^¥-fKu)=fKu~\u') and thus u and u~\'u} both match the same a ^Pik), If / is constrained at u for all TT and u such that M'^jr" .3"^ / is constrained at u for all TT and u such that u^^u~\u\ then since /'(«) -=-f'iu~\u% it follows from uniform rationing that fiu) =-fiu~'\ MO. If / is not constrained at M for some IT and u such that w ~^ M and / is not constrained at M~',M' according to the same permutation, then by the nonbossiness of / it follows that fiu) =fiu~\ MO. Finally, suppose that / is not constrained at u for some permutation, which is without loss of generahty, the identity I>ermutation, and that / is constrained at u ~\ M'. It must be that aiu\ a) ^ aiu\ a). So first consider \aiu^, a)\ < \aiu', a)|. Since / is not constrained at «, agents with the same sign a as / are simply uniformly rationed the amount agents trading in the other direction are trading,_as if a = a . Since \aiu',a)\ < \aiu',a)\, and fKu)=-f'iu~',u'X it must be that i is rationed at /(M"',MO_and that rationing is uniform. Given that / is not constrained at M, it must be that / ( « ) = / ( « " / , MO. Next, consider \aiu',a)\ > \aiu\a)\. Since \aiu\a)\ > \aiu\a)\ and / is constrained at u~\u\ but not at M, it must be that / gives i a smaller trade at «"',«' than at u. Then it must be that /'(M"', M') = e^ + aiu', a)a, which has a smaller trade than /'(M). This means that fKu~\ «') # / ' ( M ) which is a contradiction. To complete the proof, we show that if / is strategy-proof, nonbossy, and tie-free, and has an anonymous extension^ / which is strategy-proof, nonbossy, and tie-free, then / satisfies (4), (5), (6), and (7). Notice that / is the result of fixed proportion anonymous trading, where A is relaxed to be the set {x e U"^\Zx' = Ee'} and in (6) and (7) a is rewritten to be a. (Mimic the proof of Theorem 3 for this new ^4.) So to satisfy (4) and (5), use the (4) and (5) associated with / . We veriiy (6). If fiu) e ^ , then fiu) =fiu) and so (6) is satisfied. If fiu) ^A, then u matches a ^Pik) for some k. For each i such that aiu\ a) # aiu\ a), let M' be a concavification of u' through JC' =f%u) such that aiu\ a) = «(«', a) and if aiu', a) > 0, then «(«', b)> 0 for each b ^Pik') such that k'
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If fKu) = e' ==fKu~\ u'\ then it follows from the nonbossiness of / that / ( « ) =f{u~\ u'). To verify uniform rationing, consider u and w which match some a ^P(k) for some k, and such that for each / there exists r' e [0,1] and r' e [0,1] such that /'(M) = e' + r^a(u\ d)a and /'(«) = e' + r'a{u\ a)a. Suppose that there exists a permutation ir such that sign[aiu'^^^^,a)] = sign[aiu\a)]y and either |a(M^<'>,fl)| >r'\diu',a)\ and r' < 1, \aiu',a)\ >r'\a(u'^^^\a)\ and f ' < 1, or a(M'^^'^fl) = «(«',fl) for all /. We need to show that f^^'Ku) - e''^'^=fKu) - e' for all /. Concayify u' through /'(M) to M' and M' through /'(M) to M' for each /, so that «(«', a) = a{u\ a) and «(«*, a) = «(«', a). It follows that f{u)=^f{u) and fiu)=fiu)&A and that / ( M ) = / ( M ) and fiu}=fiu)^A. Notice that for sign[a(M^^'\fl)| ¥= sign [«(«',«)], it would have to be that either 0 = aiu^,a) or 0 = a(u'^^'\ a}. For our premise to be satisfied, it would have to be that 0 = «(«', a) = a(M'^^'\ a). Thus sign[a(w^^'\ a)] = sign[a(w',fl)], and either laiu'^^'^.a)] >r'\a(u\a)\ and r'
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any neighborhood of a must intersect cr^w^). Find v^ ^U which has peak in B'{a)naKu^\ and such that there is some b'^Bib) such that vKb')>vHc) for all c ^aKu^)U(THU^X c^B{b)U B'ia)^^ It follows that f{v\u^) ^ B'{a) while f{v\u^)^B{b). Thus uKfHv\iP-))> u\fHv^, M^)), which contradicts the strategy-proofness of / . We have shown that / is strategy-proof over (U U S}) X U. The argument to show that / is strategy-proof over iU U S}) X{UU Sp parallels Cases 1 and 2 above. Q.E.D. LEMMA 13: The restriction offto Sj X Sj can be written as f(u)=mm(a,max
[r'(^y^,M),fc],max [t^(^Af,uyc^,max
[f'(v4},M),/^(y4},M), j ] | ,
where max and min are defined relative to ^ , /' and t^ are strategy-proof tie breaking rules (j and i-fauorable, respectivelyX and a>=b>^c>^d for a, byC,d^ Af. PROOF: This is an exact paraphrasing of Theorem 3 in Barbera and Jackson (1994) (and we refer the reader there to see a proof), given strategy-proofness (Lemma 12) and provided that 5} restricted to Af generates all the preferences which are single peaked over Af. Given any preference which is single peaked over J } , the recipe for constructing a preference in 5 | which generates it is straightforward. Draw a line from the peak x to the origin (0 from t's point of view). From each point a &Ap a>Xy draw a segment connecting a to a point on the line connecting the peak to the origin. Draw these lines so that they are parallel to each other. Then, for the point b which is indifferent to a(x > b) draw a segment from b to the segment between x and the origin, and so that it hits at the same point as the segment from a. (If A^ has gaps, then draw the line to some b in the gap which would be indifferent to a, given the ordenng over remaining points.) These "K" shapes connecting a and b represent the indifference curves of K'e5^. Notice that this is possible, since by the definition of 5} it is not necessary that u' be increasing. Q.E.D. LEMMA 14: Aj- satisfies (8) with a, b,c,d corresponding to those given in Lemma 13. PROOF: Strategy-proofness implies that unanimity (if both agents have the same unique most preferred point out of the range, then it_is selected by / ) is satisfied for u^SJX Sj. This implies that / restricted to Sj X Sj has range Aj. From Lemma 13 we know that (i) holds. Next we verify (ii). Suppose that x ^Ay, x>=b, but that x, a, and b are not coUinear. One of the two agents, say agent 1, has a utility function M* e f/, with a as the most preferred point (or else some a' close to a), which is preferred to b which is preferred to x. Let agent 2 have a single peaked preference u^ with x as peak. By strategy-proofness and the form of Lemma 13, we know that /(M) = y, where x>y. (It is notd>x since then if agent 1 had a single peaked preference with peak at a, the outcome of / by Lemma 13 would be JC, and agent 1 would be better off announcing u^ and getting y. It is not x, since then agent 1 could benefit from the deviating from u^ and announcing a single peaked preference with peak at b, thus obtaining b which is preferred to x under M^) Since the only assumption made about u^ is that it is single peaked at ;t, the above argument still holds if uHa^)>uHy'^X which is a feasible preference. But this contradicts strategy-proofness, since agent 2 can then deviate and announce a single peaked preference with peak at a, thus improving the outcome from y to a. Hence our assumption was wrong. Parallel arguments establish that c :^= JC ^ rf are collinear. Thus (ii) holds. Next we verify (iii). Consider i, the agent defined in Lemma 13. Notice that when u^ e Sf, that by the structure of the expression in Lemma 13, agent / can always obtain any outcome between c and b. Define g as in (iii). We show that all points in Af are "below'* the line segment connecting a and gy relative to Ts preferences. More specifically, there is no point x and_preference u' e S'^ such that 6 ^ jc>- c and «'(«') > uKx') > uW) (where z is the peak of u' over A^). Suppose the contrary. Let y be i's most preferred point between b and c. Let ; have the single peaked preference with peak at c and u^Xa^)>uKy^\ By strategy-proofness and Lemma 13, the outcome of / at M is y. (The point cannot be outside of b and c since then i would wish to manipulate / from some single "^ Without loss of generality, assume that a^b. Notice that since u^ e Sj and uHa^) > u\b^X it must be that if x e o-\u% then either x>^a or b^x. (Otherwise 1 would choose x instead of b from (T\U^X) Given the diagonality of A^ and the fact that o-\u^) U aKu^) does not include any points between a and by such a t;^ can be found.
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peaked preference with peak outside b and c. And since by Lemma 13 / can obtain any point between b and c, it must be his best in this interval.) Now by deviating and announcing a single peaked preference with peak at a,j can change the outcome from y to a, an improvement. Thus our supposition was wrong. Q.E.D. LEMMA 15: fis obtained by the procedure (9) -* (11) over A f. PROOF: Combining Lemmas 13 and 14, we know that this is true on Sj X Sj over Af. This implies that if for some u&U^ agent / ever has a peak in the dictatorial region, then one of agent Vs tops is chosen (according to a strategy-proof tie breaking rule). Otherwise, agent /'s peak over Af lies on one of the segments and behaves as a single peaked preference on the segment, in which case the characterization from Lemma 13 extends. Q.E.D.
REFERENCES ABREU, D . , AND H . MATSUSHIMA (1991): "Virtual Implementation in Iteratively Undominated Strategies IL Incomplete Information," mimeo. AuMANN, R., AND J. DREZE (1986): "Values of Markets with Satiation or Fixed Prices,'' Econometrica, 54, 1271-1318. BARBERA, S., AND M. JACKSON (1994): "A Characterization of Strategy-proof Social Choice Functions for Economies with Pure Public Goods," Social Choice and Welfare, 11, 241-252. BARBERA, S., M . JACKSON, AND A. NEME (1994): "Strategy-Proof Sharing," mimeo. BARBERA, S., AND B. PELEG (1990): "Strategy-Proof Voting Schemes with Continuous Preferences," Social Choice and Welfare, 7, 31-38. BENASSY, J. P. (1993): "Nonclearing Markets: Microeconomic Concepts and Macroeconomic Applications," Journal of Economic Literature, 31, 732-761. BLUME, L., AND D . EASLEY (1990): "Implementation of Walrasian Expectations Equilibria," Journal of Economic Theory, 51, 207-227. DASGUPTA, P.,.P. HAMMOND, AND E . MASKIN (1979): "The Implementation of Social Choice Rules: Some General Results on Incentive Compatibility," Review of Economic Studies, 46, 185-216. GiBBARD, A. (1973): "Manipulation of Voting Schemes: A General Result," Econometrica, 41, 587-601. GuL, F., AND A. POSTLEWAITE (1992): "Asymptotic Efficiency in Large Exchange Economies with Asymmetric Information," Econometrica, 60, 1273-1292. HAGERTY, K., AND W . ROGERSON (1987): "Robust Trading Mechanisms," Journal of Economic Theory, 42, 94-107. HuRwicz, L. (1972): "On Informationally Decentralized Systems," in Decision and Organization, ed. by C. B. McGuire and R. Radner. Amsterdam: North Holland. (1986): "On Informational Decentralization and Efficiency in Resource Allocation Mechanisms" in Studies in Mathematical Economics, ed. by S. Reiter. Cambridge: Mathematics Association of America. HuRwicz, L., E. MASKIN, AND A. PosTLEWArre (1984): "Feasible Implementation of Social Choice Correspondence by Nash Equilibria," Mimeo. HuRwicz, L., AND M. WALKER (1990): "On the Generic Nonoptimality of Dominant-Strategy Allocation Mechanisms: A General Theorem that Includes Pure Exchange Economies," Econometrica, 58, 683-704. JACKSON, M. (1991): "Bayesian Implementation," Econometrica, 59, 461-478. (1992): "Competitive Allocations and Incentive Compatibility," Economics Letters, 40, 299-302. JACKSON, M., AND A. MANELLI (1994): "Approximate Competitive Equilibria in Large Economies," CMSEMS DP 1101, Northwestern University. LEDYARD, J. (1979): "Incentive Compatibility and Incomplete Information" in Aggregation and Revelation of {References, ed. by J.-J. Lalfont. Amsterdam: North Holland. MAS-COLELL, A . , AND X. VIVES (1993): "Implementation in Economies with a Continuum of . Agents," Review of Economic Studies, 60, 613-630. MOORE, J. (1992): "Implementation in Environments with Complete Information," in Advances in Economic Theory: The Proceedings of the Congress of the Econometric Society, ed. by J.-J. Laffont. Cambridge: Cambridge University Press.
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MORENO, D . (1994): "Nonmanipulable Decision Mechanisms For Economic Environments," Social Choice and Welfare, 11, 225-240. PALFREY, T . (1992): "Implementation in Bayesian Equilibrium" in Advances in Economic Theory: The Proceedings of the Congress of the Econometric Society, ed. by J.-J. Laffont. Cambridge: Cambridge University Press. PALFREY, T., AND S. SRIVASTAVA (1987): "On Bayesian Implementable Allocations," The Review of Economic Studies, 54, 193-208. (1989): "Mechanism Design with Incomplete Information: A Solution to the Implementation Problem," Journal of Political Economy, 97, 668-691. - (1993): Bayesian Implementation. Switzerland: Harwood Academic Publishers. PosTLEWAiTE, A., AND D. ScHMEiDLER (1986): "Implementation in Differential Information Economies," Journal of Economic Theory, 39, 14-33. RoBER-re, D. J,, AND A. PosTLEWATTE (1976): "The Incentives for Price Taking Behavior in Large Exchange Economies," Econometrica, 44, 115-127. SATTERTHWArTE, M. (1975): "Strategy-proofness and Arrow's Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Theorems," Journal of Economic Theory, 10, 187-217. SATTERTHWAIIE, M., AND H . SONNENSCHEIN (1981): "Strategy-Proof Allocation Mechanisms at Differentiable Points," Review of Economic Studies, 48, 587-597. SERIZAWA, S. (1993): "Strategy-Proof and Individually Rational Social Choice Functions for Public Good Economies," Mimeo, University of Rochester and Osaka University. SHENKER, S. (1992): "On the Strategy-Proof and Smooth Allocation of Private Goods in Production Economies," mimeo, Xerox Corporation. SPRUMONT, Y . (1991): "The Division Problem with Single Peaked Preferences: A Characterization of the Uniform Allocation Rules," Econometrica, 59, 509-519. THOMSON, W. (1994): "Monotonic and Consistent Solutions to the Problem of Fair Allocation when Preferences are Single-Peaked," Journal of Economic Theory, 63, 219-245. ZHOU, L. (1991): "InelSiciency of Strategy-Proof Allocation Mechanisms in Pure Exchange Economies," Social Choice and Welfare, 8, 247-254.
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13 Marc Dudey on Hugo F. Sonnenschein
I've felt Hugo's influence in so many ways, but here are three examples. First, the way he talked about his subject helped me to beheve it was something I could spend my professional life on. Second, when I expressed an interest in writing a Finance dissertation, Hugo introduced me to his friend Rich Kihlstrom, who met with me once a week at Penn and became an unofficial member of my committee. Third, I can't imagine that I would have studied mathematics (or met my wife, Yan) after leaving Princeton, if not for Hugo. My paper is a revision of a chapter from my Ph.D. thesis. Although Hugo was not on my committee, his lectures and our conversations contributed greatly to my sense of what to look for in a topic. The paper asks if there are noncooperative foundations for inefficiency in dynamic monopoly. It studies a seller who can repeatedly post prices before producing to order. An example wherein all buyers have the same quadratic utility function reveals two possibilities. In one double limit (buyers, then periods), buyers act as future price takers, and the outcome is that of static monopoly. In the other double limit (periods, then buyers), the seller earns the static monopoly profit per buyer, but the outcome is efficient. In the second case, buyers do not act as if future prices are beyond their control, but the seller may not care enough about reopening her market. These possibilities are not related to the "Coase conjecture," which Hugo, Faruk Gul and Robert Wilson proved in their landmark paper on durable good monopoly. However, there is a literature that studies the effect of relaxing Gul, Sonnenschein, and Wilson's assumption that buyers form a continuum. This includes my 1995 and 2003 papers, which adapt ideas from the reprinted article to durable good monopoly, in order to illustrate Coase's intuition for the finite-buyer case. References Dudey, Marc (1984) "Monopoly with Strategic Buyers," in Essays on Monopoly Pricing Strategies, Ph.D. thesis, Princeton University. Dudey, Marc (1995) "On the Foundations of Dynamic Monopoly Theory,'' Journal of Political Economy 103, 893-902. Dudey, Marc (1996) "Dynamic Monopoly with Nondurable Goods," Journal of Economic Theory 70, 470-488.
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13 Marc Dudey on Hugo F. Sonnenschein Dudey, Marc (2003) "The Free Rider Problem in Durable Good Monopoly," prepared for the First International Industrial Organization Conference, Boston MA. Gul, Faruk, Hugo Sonnenschein, and Robert Wilson (1986), "Foundations of Durable Good Monopoly and the Coase Conjecture,'' Journal ofEconomic Theory 39,155-190.
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E>ynamic Monopoly with Nondurable Goods JOURNAL OF ECONOMIC THEORY 70, 470-488 (1996) ARTICLE NO. 0099
Dynamic Monopoly with Nondurable Goods* Marc Dudey Economics Department, Rice University, P.O. Box 1892, Houston, Texas 77251 Received October 14, 1994; revised October 2, 1995
A nondurable good monopolist who posts a single price will generally achieve an inefficient outcome. But is it possible that the monopoHst would achieve efficiency by repeatedly posting prices before dehvery? If buyers recognize the effect of current purchases on future prices, then, under complementary ideal conditions, the answer is yes. On the other hand, traditional concerns about monopoly are viable if the seller bears a small cost per buyer of market reopening. Journal of Economic Literature Classification Numbers: I>42, LI2. © 1996 Academic Press, inc.
I. INTRODUCTION A monopolist who posts a single price for a nondurable good will generally achieve an inefficient outcome. But is it possible that the monopolist would achieve efficiency by repeatedly posting prices before dehvery? This paper shows that, if buyers recognize the effect of current purchases on future prices, then, under complementary ideal conditions, the answer is yes. On the other hand, traditional concerns about monopoly are viable if the seller bears a small cost per buyer of market reopening. It is not difficult to see why repeated price posting before dehvery can generate efficiency gains. Let n be the number of times that the seller opens her market before dehvery. Suppose the seller sets the static monopoly price, P, in the first n — \ periods. Also assume that buyers make no purchases in the first n—2 periods. If buyers then make purchases in the «—1st period, the n\h period price will fall below P. As a result, some buyers will gain, no buyers will lose, and the monopolist will earn more than the static monopoly profit (in dechning to choose P again in the last period, the seller rejects the static monopoly profit). So after the seller sets * Propositions 1 and 2 previously appeared in [15], I am indebted to Dwight Jaffee, Richard Kihlstrom, Pete Kyle, and Joseph Stiglitz for guidance and encouragement. My thanks also go to John Nash, Jr. for suggesting the use of Stirhng's Formula in Proposition 5. Finally, I am grateful to the many individuals who offered helpful comments on previous drafts, and to Rice University and the Sloan Foundation for financial support.
470
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Marc Dudey DYNAMIC MONOPOLY
471
P in the first n—l periods and buyers make no purchases in the first n — 2 periods, one or more buyers should be willing to make strategic n—lst period purchases. The result is a Pareto improvement over the static monopoly solution. Working backward, one finds that a time-consistent sequence of prices for the entire model generates a Pareto improvement over the static outcome. It follows that the dynamic foundations of nondurable good monopoly deserve careful consideration. This paper analyzes these foundations with a parametric example—one in which buyers have the same linear-quadratic utiHty function. My first result sharpens the challenge to static monopoly theory [12]. I show that, not only do the traders strike an intertemporal bargain, but the equilibrium approaches efficiency as the number of trading periods grows large. (A surprising feature of the result is that it does not depend on the number of buyers.*) A second finding is that traditional concerns about monopoly are viable under almost ideal conditions. Analysis of the efficient period limit equiHbrium reveals that, for any number of buyers, the monopolist's profit per buyer is not much greater than the static monopoly profit per buyer. Thus, if the monopolist bears a small cost per buyer of market reopening, she will have no incentive to reopen her market,^ In fact, by increasing the number of buyers, the monopolist's profit per buyer in the efficient period limit equilibrium can be made arbitrarily close to the static monopoly profit per buyer. Thus, the cost per buyer of market reopening that deters the seller from reopening her market can be made arbitrarily small. These findings throw light on the source of inefficiency in nondurable good monopoly. The first result shows that the inefficiency theorem of static monopoly (see [12]) depends on the ad hoc assumption that the seller and buyers stop trading after one period. The second result shows that, while the single period assumption is unjustified under ideal conditions, it may nonetheless be valid in the presence of a minor market imperfection! My results can be compared with the literature stemming from Coase's discussion of durable good monopoly in [10], This literature also deals with the time-inconsistency of static monopoly. However, there are important differences in both assumptions and results. In a durable good monopoly.
' This distinguishes the model from others in which incentives for strategic behavior disappear as the number of traders increases. See, for example, [12, 14, 25, 28]. - I have in mind sunk costs that are borne by the seller prior to the first trading period. Scenarios in which such costs have a natural interpretation are presented in Section IV below. (Costs of this form have sometimes been introduced into two-person bargaining models; see the references on p. 414 of [ 17]).
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deliveries are made at the time of purchase, and buyers receive asymmetric consumption benefits from accelerated purchases. Here, trading and consumption are not mixed together, so consumers are indifferent about the timing of their purchases. Coase claimed that a durable good monopolist would produce an efficient and zero profit outcome by competing with her future self Here, the monopohst does not compete with her future self; in fact, market reopening before delivery makes all traders better off ^ (This difference does not depend on my assumption of a finite trading horizon; see, for example, [5].) The results may also be compared with AUaz and Vila's study of a dynamic game in which duopolists repeatedly choose quantities before delivery [ 1 ]. The period limit equilibrium of their game is also efficient, but it yields zero profits to the sellers. The zero profit result reflects competition between the duopolists. In the present paper, it is a form of cooperation between the seller and buyers that generates the dynamics and these dynamics result in a Pareto improvement."^ The paper is organized as follows. The basic model, which involves no costs of market reopening, is presented and discussed in Sections II and III, respectively. Sections IV and V introduce small costs of market reopening. In Section VI, buyers accurately forecast future prices, but ignore the effect of their current purchases on future prices. Given this mild form of irrationality, market reopening does not result in any efficiency gains. Section VII concludes.
^Coase's conjecture, proved by Stokey [32] and Gul et al [18], is generally viewed as a paradox. As Butz [ 7 ] put it, "[s]ince real-world monopolists do not routinely behave as competitors, either real-world conditions do not mirror those assumed in Coase's illustration or monopolists somehow commit not to behave in this manner." Thus, it is natural to ask how a durable good monopolist might avoid Coasian difficulties. There are many possible answers. For instance, a durable good monopolist might lease her good or limit her own capacity to produce [5, 20, 32]. Alternatively, she could issue a price-protection guarantee or offer a repurchasing agreement [7, 11, 27]. She might also try to develop a reputation for not lowering her price [ 3 ] . Examples of an "anti-Coase conjecture" are studied by Bagnoli et al. [ 4 ] . These authors consider situations in which finitely many buyers with different unit demands occupy singular positions in the market. They find that dynamic trading can increase the durable good monopohst's profit. In [ 1 4 ] , I show how demand replication weakens the anti-Coase conjecture in the context of Bulow's two-period model [ 5 ] . See [8, 30, 33, 34] for more extensive literature reviews. "^ Much of the literature on financial market microstructure has emphasized the effect of imperfect competition on dynamic pricing behavior in various financial markets. For instance, see [6, 16, 22]. Indeed, AUaz and Vila offer their example as a possible explanation of forward trading that does not depend on any trader's desire to hedge against risk. They note that many forward and futures markets are dominated by a small number of producers of the commodity being traded (see [2, 24]).
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IL THE BASIC MODEL A monopolist faces w identical buyers, and there are two goods: the one sold by the monopolist and a numeraire commodity. Trade occurs over the course of n two-stage periods. In the first stage of a period, the monopolist chooses a price for her good. In the second stage, buyers independently and simultaneously decide how much of the monopolized good to buy at the quoted price.^ The monopohsf s marginal cost of seUing a unit of output is constant and, without further loss of generality, equal to zero. Consumption occurs immediately after the final trading period. The final trading period could be determined by a fixed production or harvest date (as in the case of a market for a consumption good) or by the revelation of some information (as in the case of a financial market).^ The following notation is used. Let /?, denote the price set by the monopolist in period / and let rj" denote the total amount of numeraire paid to the monopohst by the end of period /. In addition, define rj to be the total amount of numeraire buyer j holds at the end of period i and define xj to be the total amount of the monopolized good that buyer j has purchased by the end of period /. Let x,- represent the vector [x],..., xf). Traders have perfect information. In the first stage of period 1, the monopohst chooses/^i G [0, 1 ] . In the second stage of period 1, buyer y, who is initially endowed with none of the monopolized good, chooses x{ G [0, 1 ] as a function ofpi. In the first stage of period z> 1, the monopolist chooses Pi as a function ofp^, ^>.,Pi-i and Xj,..., x , _ i . In the second stage of period i, buyer 7 chooses x/ as a function of/?i,...,/?, and Xj,..., x , _ i . The buyers' quantity choices in the second stage of any period are simultaneous.^ The monopolist's utility is her total numeraire profit at the end of n periods. The notation for this is c/"^ = C .
(1)
Every buyer/ seeks to maximize a linear-quadratic payoff that depends on period n holdings:
UJ = ri + xi-(xir/2.
(2)
^ One may suppose that the seller's power to post prices comes from a regulator (as the New York Stock Exchange gives pricing power to its speciaHsts) and that price discrimination is illegal. Of course, the posted offer institution is fairly common in the retail markets of everyday life. ^The absence of discounting means that delayed consumption cannot be a source of inefficiency in the model. ^ Assume that numeraire endowments are large enough to prevent traders from acquiring short positions at any trading date.
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Variable w
Definition number of buyers
n
number of trading periods
Pi
price set by the monopolist in period /
r{
amount of numeraire buyer J holds at the end of period /
x{
amount of durable good that buyer 7 holds at the end of period i
rf
amount of numeraire paid to the monopolist by the end of period /
f/™
monopolist's profit
U-'
buyer fs payoff
Note that, when n equals 1, the problem is equivalent to static monopoly with linear demand. It will be shown in the next section that the linear demand structure is maintained when there are more trading periods. For the reader's convenience, the notation for the model is summarized in Table I.
III.
EQUILIBRIUM
The model described in Section II is a linear-quadratic dynamic game with a dominant player. A backward induction argument applied to such a game yields a subgame perfect equilibrium path in closed form. (See [21,31].) The argument's starting point is the second stage of period n, when buyer j solves
M^x-(xi-xi_,)p„+xi~{xir/2 subject to x^j; G [ 0 , 1]. This formulation comes from (2) and the fact that — (x^ —x{_j)/7„ is the cost of b u y e r / s period n purchases of the monopolized good. As in the static case, the solution is a linear function of /'«6[0,l]: K^^'^-Pn-
(3)
The monopolist's nth period problem is to solve
Max
Y^i-lLK-AiPr.) 279
Marc Dudey DYNAMIC MONOPOLY
subject to p^e[0, solution is
475
1] and (3). The first order condition for an interior
(4) for x„_i G [0, 1]^''. Substituting (3) into (4) yields
Pn=VV2-]
I {\-K-^)l^
(5)
for x„_i G [0, 1]". It now foUows from (3) and (5) that
Z(l-x{_0/w
l - x i = [l/2]
(6)
for x„_i G [0, 1]*^. From (5) and (6), the equilibrium price and quantities of the subgame beginning in period n can be written as a function of x „ _ , and w. Notice that, if no purchases are made in periods 1 through n — \, the expressions for /?„ and x^ in (5) and (6) would be the static monopoly price and quantity. Purchases in periods 1 through n — 1 would lower the 71th period demand and the «th period price, and would raise the ^zth period quantity. Next consider the behavior of the monopolist and buyers in period n—\. The assumption that trader strategies form a subgame perfect equilibrium implies that buyer j will take into account the effect of his period n — 1 purchases on the period n price. However, according to (5), p^ depends on the earlier purchases of all buyers. So in the second stage of period « — 1, buyer 7 chooses x{_^ to maximize
subject to x;i_j G [0, 1], (5) and (6), and taking x^_, as given for Some calculations show that, in equilibrium.
0
if
;7„_,G[0,(2w+l)/(4vi;)]
if
;7_JG((2W+1)/(4M;),1]-
a\\k^j\
(7)
To formulate the seller's period « — 1 problem, we need expressions for x^ and p„ in terms of p„_i. By (5) and (7), p„ = (2w/i2w+\-\)p„_,
1/2
280
if
;>„_,6[0,(2H'+1)/(4>V)]
if
;7„_,6((2w+l)/(4>v),l]
(8)
Dynamic Monopoly with Nondurable Goods 476
MARC DUDEY
and, by (3) and (8), xi=\-(2w/l2w+l])p„_, 1/2
if ;7„_,G[0,(2w+l)/(4w)] if p„_,e{{2w+iy{4wlll
(9)
It follows that the monopolist's period n — 1 problem is to solve Max
I x ^ _ , - X ^{-2 (/'„-,)+
E^i-Exi_,
(;;„),
subject to Pn-\ ^ [0> 1]? (7)> (8), and (9). The first order condition for an interior solution can be written as Z <-X-
Z <-2]-^Pn-X
[d Z <-xldPn-X 7=1
7=1
+ {dpjdp,_,){
X ^ ' « - Z ^^-1
(10) Using (7), (8), (9), and (10) we have that ; ' « - I = [1/(4»V)][(2H^+1)7(2^ + 2)]
(11)
The remaining equilibrium price and quantities can be obtained by combining (7), (8), (9), and (11): [ 1 - ^ ; ; - . ] = [(2H'+1)/(2M; + 2 ) ]
^-[t<-2h
;7„ = (1/2)[(2W+1)/(2M; + 2 ) ]
(12) (13)
^ By rearranging and renaming terms in (10), one can obtain an "inverse elasticity rule" (see [33]) for the period n—\ price. The rule takes the form lPn-\-Pn\IPn-\
=
^l\^n-\\,
where £„_, is the elasticity of (w— l)st period demand. The period 72 price is viewed by the monopolist as the marginal cost of period n — \ sales.
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DYNAMIC MONOPOLY
and [ l - * i ] = ( l / 2 ) [ ( 2 » ' + l ) / ( 2 " ' + 2)]
'- £
(14)
^-l/w
7=1
Repeating the exact same logic and using the initial condition xl = 0 yields the following result. PROPOSITION L
In period i {i=\, ...,n — 1), the monopolist sets a price
of p.= "ll l{2w +
2h-l)/{2w-h2h-2)-i
h= \ n
x(l/2) n
[(2w-h2^-3)/(2>v + 2 ^ - 2 ) ]
and the monopolisfs final period price is Pn = {V2) n
(15)
[(2>v + 2/c-3)/(2w + 2A:-2)].
At the end of period i (/ = 1,..., n — \), buyer fs holding of the monopolized good equals n
x{=l-
n
[(2w + 2k-3)/(2w
+
2k-2)']
k = n — i+l
and buyer fs final period holding of the monopolized good is xi=l-{l/2)
n
(16)
[(2vt/ + 2;t-3)/(2H; + 2 ^ - 2 ) ] .
k = 2
{Recursive formulas are given in an Appendix,) In addition, when (15) and (16) are substituted into (1) and (2), one has an expression for the monopolisfs equilibrium profit per buyer (1/w) n
[(2w + 2/:-3)/(2w + 2 ^ - 2 ) ] ^ [ ( w + « - l ) / 4 ]
k = 2
as well as an expression for buyer fs equilibrium payoff less his endowment (1/2)-[(«-1)/(4M;) + (3/8)]
n /t = 2
282
(2w + 2k-3)/{2w
+
2k-2)
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Dynamic Monopoly with Nondurable Goods 478
MARC DUDEY
Proposition 1 may be used to demonstrate the efficiency of the period hmit equihbrium. PROPOSITION 2. As the number of trading periods grows without bound, the deadweight loss per buyer converges to zero. That is, Hm„_^ ^^ (1 — x^) = 0.^
Proof of Proposition 2. The proof consists of two simple steps. The first step is to rewrite (1 --x{^) using (16). ( l - x 0 - ( l / 2 ) n { l - [ l / ( 2 > i ; + 2z-2)]}
(17)
i= 2
The second step is to show that the right hand side of (17) converges to zero as n goes to infinity. Observe first that 1 - [ 1 / ( 2 M ; + 2 / - 2 ) ] < 1 / [ 1 + 1/(2W + 2 Z - 2 ) ]
(18)
for i^2. From (17) and (18), we have that ( l - x ; : ) < ( l / 2 ) ( ^ l ^ n [ l + V(2H^ + 2 / - 2 ) ] j .
(19)
n [l + l/(2>v + 2 z - 2 ) ] > l + i [l/(2vt; + 2 / - 2 ) ]
(20)
Now,
/=2
i=2
(expand the left hand side). Taken together, (19) and (20) imply i\-xi)<{l/2)(lll^l-\-J: [\/{2w-i-2i-2)-]^y
(21)
Observe that lim„_ ^ XiLi [ V(2>v + 2 / - 2 ) ] can be obtained by deleting a finite number of terms from the series oo
S = {\/2) X (1/4 5=1
Since S diverges, so does lim„_,^ X"=2 [ V(2>v + 2/ —2)]. Hence, both sides of (21) must converge to zero as n grows without bound. ^Propositions I and 2 previously appeared in [15].
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479
The distribution of the efficiency gains is the final section. What I find is that, when the number of buyers is seller's equilibrium profit per buyer is only slightly higher monopoly profit per buyer. As a first step, recall from Proposition 1 the expression equilibrium profit per buyer. Let
topic of this very large, the than the static for the seller's
n
f/"(«,
W)/W = (1/M;)
n
[(2vt; + 2A:-3)/(2w + 2 / c - 2 ) ] ^ [ ( w + « - l ) / 4 ] .
k = 2
(22) A straightforward consequence of (22) is that an extra round of trading increases the monopolist's equilibrium profit per buyer. To see why, note from (22) that the seller's equilibrium profit per buyer with n-\-\ periods is U^{n^-\,w)lw
= {\lw) n
[{2w + 2k-l>)l{2w + 2k-2)Yl{w
+ n)IA\
k = 2
(23) Dividing (23) by (22) gives [(2w + 2 « - l ) / ( 2 w + 2«)]2[(>v + «)/(w + « - l ) ] = [ l - l / ( 2 w + 2 « ) ] 7 [ l - l / ( > v + «)]. Expanding [1 — 1/(2H;H-2«)]^ shows that this fraction, or C/"'(n+1, w)/ U'^{n, w), is greater than one. Notice that this also means that l i m „ _ ^ t/"'(«, vv)/w exists because U'^{n,w)lw is increasing in n and the maximal profit per buyer is bounded. Now, we have from (22) that ir{n, w)lw=[ V^in, w- l ) / ( w - l ) ] [ 2 w / 2 w - 1]^ x[{2n + 2w-3)/{2n x[{n + w-l)/{n
+
2w-2)y
+ w-2)][{w-\)/wl
(24)
Applying (24) recursively yields ir"{n, w)/w=U"'in,
1) n [ 2 / / ( 2 / - 1)]^ f ] [ ( ^ - 1 ) / ' ] i=2
i=2
w
x f ] [(2n + 2 i - 3 ) ] [ ( 2 « + 2 z - 2 ) ] 2
xfl (« + /-!)/(«+ /-2).
284
(25)
Dynamic Monopoly with Nondurable Goods 480
MARC DUDEY
Cancelling some nonsquared terms in the first part of the right side of (25) and taking limits with respect to n yields
lim V^in^wyw^llim n -^ CO
U'"{n,\)'] f] [2i/{2i-l)y
n - > oo
[l/w]
-I
= [ lim f/-(«, I)][l/4] n [2//(2z-l)]Ml/v^]. n -^
CO
. I =
(26)
^ 1
Note that the w-Iimit of n r = i [ 2 / / ( 2 i - 1)]^ [ l / w ] is [lim„^ „/7"(«, 1)] - ' . Hence, hm
lim V^in, w)/w = Mm t/""(«, l ) - ( l / 4 ) - [ lim f/™(n, 1 ) ] - ' .
w - > CO « - > oo
n - > oo
w->oo
It follows that lim
lim L^(n, ii/)/w = 1/4.
w -* CO n —* oo
To sum up, PROPOSITION 3. The seller's equilibrium profit per buyer, given by (22), is strictly increasing in the number of trading periods}^ Increasing the number of buyers does not affect the efficiency of the period limit equilibrium. However, the monopolist's share of the efficiency gains per buyer converges to zero as the number of buyers becomes larger and larger. It is interesting to note that an increase in the number of buyers shifts surplus from the monopolist to the buyers in the efficient limit equilibrium. The idea is that, as the number of buyers increases, the influence of each buyer's current purchases on future prices falls. Thus, at any given price in some period before the last, each buyer has less of an incentive to make strategic purchases (i.e., demand in each period before the last becomes more elastic). In order to stimulate strategic purchases after the number of buyers has increased, the monopolist lowers early period mark-ups. Thus, an increase in the number of buyers lowers mark-ups without affecting the efficiency of the period limit equilibrium. It follows that, in the period limit equilibrium, an increase in the number of buyers results in a transfer of surplus from the seller to the buyers. ^^The expression for each buyer's equilibrium payoff less his endowment is given in Proposition 1. This expression can be used to show that each buyer's equilibrium payoff is increasing in the number of trading periods. The proof of this result is omitted since it is not essential for the remainder.
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481
IV. COSTLY MARKET REOPENING The model of the previous sections represents one benchmark for the study of dynamic monopoly. The analysis of that model was intended to show what is possible under ideal conditions. In this section and the next, I demonstrate that a small market imperfection can have a drastic effect on the traders' ability to achieve an efficient outcome. This imperfection takes the form of a cost of market reopening. The model is the same as that in Section II, except that the seller chooses the number of trading periods and incurs a cost per buyer of market reopening. The seller's choice of the number of trading periods, n, is assumed to be constrained by a positive upper bound, T, The constraint is meant to reflect the fact that trading takes time and that consumption cannot be delayed indefinitely. The per-buyer cost of market reopening will be captured by the function fw-£
^^"'^^ = lo
if
n>\
if « = i.
A simple interpretation of C{n, w) is that it represents the seller's cost of informing buyers that they should arrive at a particular date prior to the "spot market" (i.e., the final period). The parameter £ represents the seller's cost of informing each buyer of forward trading opportunities. The assumption that C(n, w) = 0 if 77 = 1 means that buyers think there are no forward trading opportunities if they do not receive a call from the seller. Another simple interpretation of C{n, w) is based on the following scenario. Buyers can make all purchases on credit by telephone or computer until the final trading period, when they take delivery and pay their debts. Within this context, C(/2, w) may be seen as the cost to the seller of checking each buyer's endowment (i.e., each buyer's ability to cover her position in the last period). If £ is the cost of each credit check, then C(«, w) may be written as above. In this case, the assumption that C{n, w) equals zero if n equals one means that spot market transactions involve cash instead of credit. The main result of the section emphasizes the fragihty of Proposition 2. PROPOSITION 4. Pick any e that is less than the static monopoly profit per buyer. When the number of buyers exceeds some critical value, the subgame perfect equilibrium outcome of the variant game is the static monopoly outcome The critical number of buyers depends on 8 but is independent of T.
[The fact that the critical number of buyers in Proposition 4 depends on 8 but not on T is significant. It shows that the inefficiency result is not a
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Dynamic Monopoly with Nondurable Goods 482
MARC DUDEY
matter of buyers "becoming nonstrategic" as w increases. For any w above the critical level, T can be increased to a point where, in the absence of transaction costs, an approximately efficient outcome would have been achieved.] Proof of Proposition 4. solves
If the monopolist reopens her market, she Max t/"'(«, w) — W'£. n
By Proposition 3, U^{n, >V)/M^ is increasing in n for any w. Hence, if « is greater than one, n must be equal to T, It follows that the monopolist will choose between setting n equal to one and setting n equal to T, That is, she will compare 17^(1, M;) = W/4 with
By Proposition 3, n -^ oo
But Proposition 3 also tells us that l i m „ ^ ^ £/""(«, w)/w can be made arbitrarily close to 1/4 by increasing w. Thus, for sufficiently large w we have
V. MORE ON COSTLY MARKET REOPENING This subsection takes a second look at the model of Section IV. In that section, I showed that, with arbitrarily small costs per buyer of market reopening and a sufficiently large number of buyers, the seller would have no incentive to reopen her market. Here, I will show that, as long as the cost per buyer of market reopening, s, is not too small, the seller has no incentive to reopen her market regardless of the number of buyers. PROPOSITION 5. As long as s is not less than 15% of the total surplus per consumer, the monopolist of the Section IV game will not reopen her market.
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Marc Dudey DYNAMIC MONOPOLY
483
Proof of Proposition 5. The first of three steps is to show that, with one buyer, the monopoHst receives a payoff of I/TT in the Hmit equihbrium. From (22), we have U^{n,\) = nY\
[ ( 2 / - l ) / 2 / ] ^ = n[(2^2)!]7[2^"(n!)^].
According to Stiriing's Formula, ( 7 2 ^ ) x " e - " < x ! < ( y 2 ^ ) x " e - " ( l + l/(4x)). Thus, the monopoHst's profit is bounded above by [ l + l/(8«)]^(l/7r), and below by [ l + l/(4n)]-^(l/7r). Thus, lim„_,<^ U^{n, 1)= I/TT, or approximately 0.3183, is the monopolist's share of the pie. The second step of the proof is to show that the monopolist's share of the efficiency gains per buyer in the limit equilibrium is decreasing in the number of buyers. From (26), we have lim [ ir{n, H; + 1 )/(H; + 1)/[ V^in, \v)lw'\ n -^ CO
= [(2H;-f2)/(2wH-l)]^[l/(vt;+l)]ii; = [4M/^ + 4}V]/[4>V^ + 4 > V 4 - 1 ] < 1 .
So lim„_ ^ f/'"(«, w)/w is decreasing in w. Finally, consider the monopolist's choice of n. If the monopohst does not reopen her market, she earns w/4. If she does reopen her market, she must solve Max U'^{n, w) — w-e subject to n^T. By Proposition 3, the solution is to set n equal to T. Thus, if the monopolist does reopen her market, her profit is U^{ T, >v) — w - s. But
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Dynamic Monopoly with Nondurable Goods 484
MARC DUDEY U'^{T,w)-W'£<
lim f/^(«, w)-W'S
(by Proposition 3)
77 —> OO
< w [ lim {/"^(n, l ) - e ] (since lim U'^{n, w)/w is decreasing in w) = M;[l/;;r-£]
(since lim U^'in, I) =1/71). n—»• 0 0
Thus, the monopohst will not reopen her market if I/TT — £ < l / 4 . This inequality holds if s > 0.0684. Fifteen percent of the surplus per buyer amounts to (0.15)( 1/2) = 0.075. Hence, if s is no less than 15% of the surplus per buyer, market reopening will not occur.
VL A MILD FORM O F IRRATIONALITY In the above analysis, I assumed the maximal degree of trader rationality. Under this assumption, a small market reopening cost was shown to generate the inefficiency of static monopoly. In this section, I will show that a mild departure from complete rationality can also restore the static monopoly outcome. In particular, suppose buyers ignore the effect of their own current purchases on future prices, but accurately forecast future prices both on and off the equilibrium path. (This is the mild form of irrationahty encountered in most of the durable good monopoly literature.) Under this assumption, market reopening prior to delivery may produce no efficiency gains. To understand why, consider a two period version of the setup described in Section II. Suppose that each buyer's expectation of the second period price is PliPi) =
(F IPi
for for
Pi^F p,
where P is the static monopoly price. My claim is that these expectations induce behavior that results in the expectations being fulfilled on and off the equilibrium path. First observe that, given flipi), it is rational for buyers to make no first period purchases iipi^F, Now, if the monopolist sets Pi^F and buyers purchase nothing in the first period, then it will actually be rational for the monopolist to charge P in the second period—in other words, the buyers' expectations will have been fulfilled for any j!7, ^ P .
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Suppose, on the other hand, that the monopoHst were to set/^j below P in the first period. If buyers expect the same p^ in the second period, they will be indifferent about how they divide their demand at p^ between the two periods. If each buyer purchases D'{pi) -p^ + D{p^) in the first period, where Z)( •) is the buyer's static demand function, then the monopolist's second period profit maximization problem will be to solve Maximize [i)(;72)-(i)'(/7i)'J^i+^(;>i))]/^2P2
The first order condition is
D\P2)'P2 + D{p^) =
D\p,)^p,^D{p,y
If D' <0 and D"
Different specifications of trader irrationality can lead to very different outcomes. In fact, the repeated price posting framework of this paper was examined by Josef Hadar [19] under the assumption that buyers are myopic. He showed that, in the period limit, an efficient outcome would be
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achieved. However, unlike the period limit considered in previous sections of this paper, the seller would be able to extract all of the consumer surplus. See also [13].
VII. CONCLUSION The classical discussion on inefficiency in monopoly is due to Cournot [12]. He assumed that the price-posting seller opens her market only once. However, as noted by Coase in [10], the inefficiency of static monopoly would provide the monopolist with an incentive to reopen her market. This paper considers a model of dynamic monopoly in which delivery does not occur until all trading is completed. A key feature of the model is that buyers are permitted to behave as strategic agents. Thus, they take into account the effect of their current purchases on future prices. The assumption that all traders should behave strategically is in accord with von Neumann and Morgenstem's dictum that an economic model should be "formulated, solved and understood for small numbers of participants before an3l:hing can be proved about changes of its character in any limiting case of large numbers" [35]. The parametric example analyzed here shows that dynamics caused by strategic buyer behavior can produce an approximately efficient outcome when the number of trading periods is sufficiently large. Remarkably, the result is independent of the number of buyers. Yet, the result is in accord with a common wisdom in economics: that one should expect to see efficient outcomes in a frictionless world. As Myerson [23] put it, "[s]ome economists, following Coase [ 9 ] , have argued that we should expect to observe efficient allocations in any economic situation where there is complete information and bargaining costs are small."" This raises a fundamental question about the source of inefficiency in monopoly. Namely, do we think that monopoly is inefficient only because traders overlook the possibihty of market reopening or otherwise do not act in a completely rational manner? If so, we should regard inefficiency in monopoly as a noneconomic phenomenon. This paper offers the beginnings of a response. It was shown that buyers can capture most of the efficiency gains per buyer from market reopening. The implication is that, if the seller incurs a small, upfront cost per buyer of market reopening, she will choose to open her market only once. Perhaps future research will shed light on the generality of this result. ^' Although widely accepted, this intuition does not spell out the process by which economic agents would achieve efficiency. Other examples of such processes can be found in the dynamic bargaining hterature; see [26, 29].
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APPENDIX The equilibrium quantity path is given by the recursion {^-^i-k^,)=U^^
+ 2k-3)/{2w + 2k-2mi-xi_,),
k = 2,,.„n,
where xl = 0. Finally, {i-xi)={i/2){i-xi_,). The equilibrium prices are derived from the equihbrium quantities and the equations
and Pn-k^i = [{2w + 2k-2)/{2w + 2k-l)]p„_^,
k=\,..„n-\,
REFERENCES 1. B. Allaz and J. Vila, Coumot competition, forward markets and efficiency, J. Econ. Theory 59 (1993), 1-16. 2. R. Anderson, The industrial organization of futures markets: A survey, in "The Industrial Organization of Futures Markets" (R. Anderson, Ed.). Lexington Books, Lexington, MA, 1984. 3. L. Ausubel and R. Deneckere, Reputation in bargaining and durable good monopoly, Econometrica 57 (1989), 511-531. 4. M. Bagnoli, S. Salant, and J. Swierzbinski, Durable goods monopoly with discrete demand, / . Polit. Econ. 97 (1989), 1459-1478. 5. J. Bulow, Durable goods monopolists, 7. Polit. Econ. 90 (1982), 314-332. 6. J. Bulow and P. Klemperer, Rational frenzies and crashes, J. Polit. Econ. 102 (1994), 1-23. 7. D. A. Butz, Durable-good monopoly and best-price provisions, Amer. Econ. Rev. ^{i (1990), 1062-1076. 8. D. Carlton and J. Perloff, "Modem Industrial Organization," Scott Foresman, New York, 1990. 9. R. Coase, The problem of social cost, J. Law Econ. 3 (1960), 1-44. 10. R. Coase, Durabihty and monopoly, J. Law Econ. 15 (1972), 143-149. 11. T. Cooper, "Facihtating Practices and Most-Favored-Customer Pricing," Ph.D. thesis, Princeton University, 1984. 12. A. A. Cournot, "Recherches sur les principes mathematiques de la theorie des richesses," Hachette, Paris, France, 1838. 13. A. J. Douglas and S. M. Goldman, Monopolistic behavior in a market for durable goods, J. Polit. Econ. 11 (1969), 49-62. 14. M. Dudey, On the foundations of dynamic monopoly theory, J. Polit. Econ. 103 (1995), 893-902. 15. M. Dudey, "Monopoly with Strategic Buyers, in Essays on Monopoly Pricing Strategies," Ph.D. thesis, Princeton University, 1984.
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16. F. D. Foster and S. Vishwanath, Strategic trading when agents forecast the forecasts of others, mimeo, Fuqua School of Business, Duke University, 1993. 17. D. Fudenberg and J. Tirole, "Game Theory," MIT Press, Cambridge, MA, 1991. 18. F. Gul, H. Sonnenschein, and R. Wilson, Foundations of dynamic monopoly and the Coase conjecture, / . Econ. Theory 39 (1986), 155-190. 19. J. Hadar, Optimality of imperfectly competitive resource allocation. Western Econ. J. 8 (1969), 51-56. 20. C. Kahn, The durable goods monopolist and consistency with increasing costs, Econometrica 54 (1986), 275-294. 21. F. Kydland, Equilibrium solutions in dynamic dominant player models, J. Econ. Theory 15 (1977), 307-324. 22. A. S. Kyle, Continuous auctions and insider trading, Econometrica 53 (1985), 1315-1335. 23. R. Myerson, "Game Theory," p. 506, Harvard Univ. Press, Cambridge, MA, 1991. 24. D. Newbery, Manipulation of futures markets by a dominant producer, in "The Industrial Organization of Futures Markets" (R. Anderson, Ed.), Lexington Books, Lexington, MA, 1984. 25. W. Novshek and H. Sonnenschein, Cournot and Walras equilibrium, / . Econ. Theory 19 (1978), 473-486. 26. M. J. Osborne and A. Rubinstein, "Bargaining and Markets," Academic Press, San Diego, CA, 1990. 27. I. P. L. Png, "Pricing of Capacity to a Heterogeneous Customer Population," UCLA Graduate School of Management, Working Paper 87-4, March 1987. 28. D. J. Roberts and A. Postlewaite, The incentives for price-taking behavior in large exchange economies, Econometrica 44 (1976), 115-126. 29. A. Rubinstein, Perfect information in a bargaining model, Econometrica 50 (1982), 97-109. 30. R. Schmalensee, Market structure, durabihty, and quality: A selective survey, Econ. Inquiry 42 (1979), 172-190. 31. R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, Int. J. Game Theory 4 (1975), 25-55. 32. N. Stokey, Rational expectations and durable goods pricing. Bell J. Econ. 12 (1981), 112-128. 33. J. Tirole, "The Theory of Industrial Organization," MIT Press, Cambridge, MA, 1988. 34. H. Varian, Price discrimination, in "Handbook of Industrial Organization" (R. Schmalensee and R. Willig, Eds.), North Holland, Amsterdam, The Netherlands, 1989. 35. J. Von Neumann and O. Morgenstem, "The Theory of Games and Economic Behavior," Princeton Univ. Press, Princeton, NJ, 1944.
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14 In-Koo Cho on Hugo E Sonnenschein
While struggling as a third year graduate student to begin my thesis work in the spring of 1984, Hugo advised me to look into the possibility of refining sequential equilibrium by exploiting the implicit communication between the sender and the receiver. After working for a few additional months, I was able to produce an example in early July. I used it to explain to Hugo that the implicit communication can refine sequential equilibrium ftirther than David Kreps had proposed. At the end of the discussion, Hugo said "In-Koo, you have come a long way." He asked me to poUsh the example so that he could send it to David. It took another few weeks to polish the example (which later appeared in my paper "Refinement of Sequential Equilibrium"). After dropping off the manuscript in Hugo's mailbox in the department office around midnight, I packed up to drive to Stanford the next morning to spend a year there with Hugo (along with Faruk Gul and George Mailath). It was a long and difficult trip to drive my small Volkswagen from Princeton, NJ to Palo Alto, CA. After being separated from my co-driver in Chicago, I drove the remaining 2000 miles alone. My car had a mechanical problem in Wyoming while passing through the Rockies. By the time when I was crossing the Bay Bridge, the car and the driver were about to break down. Immediately after arriving at Stanford, I went to see David to introduce myself. Watching an exhausted, somewhat awed, student standing awkwardly at the door of his office, David said "Let me guess who you are. I think you are a student of Hugo. I got it, read it and liked it." It was an end of a long journey, but also the beginning of my collaboration with David.
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May 1987
Issue 2
SIGNALING GAMES AND STABLE EQUILIBRIA* IN-KOO CHO AND DAVID M . KREPS Games in which one party conveys private information to a second through messages typically admit large numbers of sequential equilibria, as the second party may entertain a wealth of beliefs in response to out-of-equilibrium messages. By restricting those out-of-equilibrium beliefs, one can sometimes eliminate many unintuitive equilibria. We present a number of formal restrictions of this sort, investigate their behavior in specific examples, and relate these restrictions to Kohlberg and Mertens* notion of stability.
I. INTRODUCTION
Much of information economics has been concerned with situations in which the following simple signaling game is embedded: one party, hereafter called party A, possesses private information. On the basis of this information, A sends a signal to a second party B, who thereupon takes an action. Examples abound: Spence's [1974] model of job market signaling is one example, if we modify things slightly so there are two or more parties B, (We shall develop a simple case of the Spence model in this format in Section V.) Grossman [1981] examines the role of warranties and product quality using this sort of model. Models of bargaining with incomplete information (see, for example, Grossman and Perry [1986a] or *We are grateful to Anat Admati, Drew Fudenberg, Elon Kohlberg, Paul Milgrom, Richard McKelvey, Jean-Fra'^cois Mertens, Motty Perry, John Roberts, Joel Sobel, Gyu Ho Wang, and especially Hugo Sonnenschein for helpful discussion, and to three referees and an editor for helpful suggestions. The financial support of Harvard University, the Korea Foundation for Advanced Studies, the National Science Foundation (Grants SES80-06407 and SES84-05865), the Sloan Foundation, and the Institute for Advanced Studies at the Hebrew University, are all gratefully acknowledged. The material in this paper originally appeared in two separate papers, one with the above title, and a second entitled "More Signaling Games and Stable Equilibria." We hope that anachronistic references to the earlier incarnations of these ideas will not prove too troublesome to the reader.
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Rubinstein [1985]) constitute another class of examples. In the literature of industrial organization, there is the entry-deterrence limit pricing model of Milgrom and Roberts [1982a], the analyses of the chain-store game of Kreps and Wilson [1982b] and Milgrom and Roberts [1982b], and recent work on the role of advertising by Milgrom and Roberts [1986]. Theoretical accountants often employ this sort of model (see, for example, Demski and Sappington [1986]). And, on a slightly higher plane, there is the general analysis of a game of this sort due to Crawford and Sobel [1982], and related work on mechanism design by an informed principal [Myerson, 1983]. This is only a partial list (and apologies are tendered to those left off), and in most cases the models are variations on the general theme outlined above. But this theme, together with variations, has been played a lot recently. In most of these recitals, one finds a plethora of equilibria. This paper takes a noncooperative game-theoretic approach, and we mean here a plethora of Nash equilibria. One can cut back on the number of equilibria by invoking notions of perfection (or sequentiality), but this is only of minor help—in many games the wealth of off-the-equilibrium path beliefs that can be imposed gives rise to a wealth of equilibria. That is, what constitutes an equilibrium is powerfully affected by the "interpretations'* that would be given by B to messages that A might have sent, but in equilibrium does not send. In a sequential equilibrium, B is required to frame some hypothesis (probability assessment) over what is A's private information and respond accordingly. As one varies those hypotheses, one varies the optimal responses of B, and hence the incentives of A to send the various messages. At this point, in many of these contextually based analyses, the analyst(s) resorts to various intuitive criteria based on the conclusions that B "ought'' to draw from sundry out-of-equilibrium messages. If one can restrict the out-of-equilibrium beliefs (or hypotheses) of B, one can sometimes eliminate many of the equilibria. An example of this that is particularly prevalent runs as follows: suppose, for simplicity, that A's private information must be one of two things, called t and t\ Suppose that in equilibrium A sends message m with probability one. Suppose that there is a second possible message m' with the following properties: if A knows t, then A would strictly prefer (in comparison with the equilibrium outcome) not to send m\ no matter how B interprets this. And if A knows t\ then A would prefer to send m' to what A gets in the equilibrium if by sending mf A could convince B that A knew t\
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The former condition, it is argued, implies that B should not entertain the hjrpothesis that the message did come from an A who knows t, B should infer from the message that A knows t\ And, therefore, if A knows t\ he should send the message (thus upsetting the given equiUbrium). It is as if A, if he knows t\ is (by sending m') implicitly making the speech: I am sending the message m', which ought to convince you that I know t\ For I would never wish to send m' if I know ty while if I know t\ and if sending this message so convinces you, then, as you can see, it is in my interest to send it.
This particular criterion, and minor extensions to it, have appeared in several of the applications described above. It has been applied directly in Grossman [1981], Milgrom and Roberts [1982a], and Kreps and Wilson [1982b], and it is applied indirectly in Rubinstein [1985]. Its powers can be considerable: in a simple (two type) Spence signaling model, there is a single equilibrium outcome that survives this criterion (see Section V). While analyses of particular examples have been based on intuitive criteria for out-of-equihbrium beliefs such as the one just given, there have been, at the same time, further attempts to refine generally the notion of a Nash equilibrium. An important recent example of this is the Kohlberg-Mertens [1986] theory of stability and stable equilibrium outcomes. Because the Kohlberg-Mertens development takes place in a very abstract context, it is hard to see what stability entails for concrete examples. One point of this paper is to see what stability does entail for (generic) signaling games. Roughly put, we find that stability implies a number of (progressively stronger) restrictions on out-of-equilibrium beliefs in this simple class of games. Some of the restrictions we find quite intuitive; for example, stability implies the intuitive restriction given above. As the restrictions mount, however, our intuition becomes progressively weaker; until we come to implications of stability that we (at least) are unable to motivate so nicely. In the end, we have mixed feelings about stability, at least insofar as it applies to signaling games: it captures quite beautifully some restrictions that we find very satisfactory; but in other cases it seems very strong. We have two objectives in this study. Our first concern is with signaling games alone. These games, and elaborations of these games, have proved to be very important to recent work in theoretical microeconomics. By developing a sequence of (progressively stronger) general criteria for the equilibria in these games, we hope
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to provide analysts with a general language for the discussion of what level of restrictions they must impose in order to obtain a particular equilibrium outcome. Second, the theory of stability, either as it stands or as it develops, will certainly prove to be an important idea in noncooperative game theory. We hope that our examples and characterization of stability for signaling games will help in the further general development of these ideas. The paper is organized as follows. We begin, in Section II, with an example that illustrates the basic program that we are following. Then, in Section III we introduce a general framework for our analysis. In subsection III.l we define the general signaling game, and we recall in subsection IIL2 some propositions concerning equilibria of extensive games from the literature. In subsection III.3, we recall basic concepts and definitions from Kohlberg and Mertens [1986], with emphasis on a particular result that they give. Section IV is the heart of the paper. The general program that we follow for restricting beliefs in testing equilibriimi outcomes of signaling games and the connection of this program to stability are given in subsection IV. 1. The rest of Section IV develops some specifications of the general program: subsection IV.2 concerns the well-known and much used criterion of domination. Subsection IV.3 takes up what we call equilibrium domination; included here is the criterion that follows from the "speech'' given above, which we refer to as the Intuitive Criterion (the uppercase letters signifying this particular criterion). Subsection IV.4 briefly discusses (variations on) the Banks and Sobel [forthcoming] criteria of divinity and universal divinity. And subsection IV.5 discusses the "never a weak best response" criterion of Kohlberg and Mertens [1986]. In Section V we apply these various criteria to a simple version of Spence's signaling model, showing how they work to rule out all but a single equilibrium outcome in the game (namely the separating equilibrium identified by Riley [1979]). We conclude in Section VI with a discussion of the full implications of stability for signaling games, and with a summary of what (we think) we have learned. McLennan [1985] pioneers the approach of refining Nash equilibria by formal restrictions on out-of-equilibriimi beliefs. His approach dilfers from our own in one important respect, and we shall offer a few remarks on this in subsection IV.3. He should be credited, however, with initiating the general program we follow here. Contemporaneously, Banks and Sobel [forthcoming] have ana-
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lyzed the same basic problem as do we, arriving at very similar answers. We have benefited from seeing their results, and we have, for completeness, related their criteria of divinity and universal divinity to our approach. They should be given all the credit for those two criteria, and (at least) equal credit for other results t h a t appear in both papers. T h e reader will benefit from reading their treatment of these issues. Also, we are greatly indebted to Kohlberg and Mertens [1986] for many of the ideas here. In particular, they are responsible for the "never a weak best response" criterion t h a t dominates our mathematical analysis. Because our focus here is on simple signaling games, many interesting questions t h a t arise in games with a richer dynamic structure are moot. Cho [1986, forthcomilig] presents an analysis of some of our ideas, adapted to more inter^esting games. McLennan [1985] also deals with general extensive games. 11. A SIMPLE E X A M P L E
T h e basic ideas in this paper are illustrated by the following simple game. T h e reader should refer to Figure I throughout. We tell the story of a two-player game with incomplete information concerning one of the two. T h e first player is called A, and A either is a wimp or is surly. Nature has selected the disposition of A, with probability 0.9 t h a t the A selected is surly. In terms of Figure I, nature has chosen to start the game at one of the two open dots labeled t^ (for the wimp) and t^ (for the surly type of A ) . T h e prior
V^^
beer
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's
quiche ••'
quiche
, • • * -
-•••:
{•9} 3\^iP'
FIGURE I
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probability of nature's choice is indicated by the numbers in curly brackets. At the start of the game, A knows his own disposition or type, and A is faced with the choice of what breakfast to have, before setting off for the day. T h e choices available are quiche and beer. These choices are denoted by the pairs of arrows pointing out from the open dots. A^s preferences concerning breakfast depends on his type: if A is a wimp, he derives incremental payoff 1 for having quiche and 0 from beer; if A is surly, beer is worth 1, and quiche is worth 0. After breakfast, A meets with B. There are four conceivable circumstances under which the meeting could take place, corresponding to the two tjrpes of A and the two types of breakfast; these are depicted by the four filled dots. B, at this meeting, chooses whether to duel A. J5's choices are represented by the arrows emanating from the four filled dots. When B chooses whether to duel, he does so knowing what A had for breakfast, but not knowing for sure what is A's type. This is depicted in the picture by the dashed lines connecting pairs of solid dots, representing the information sets of J3—to the right is the information set of JB if he knows t h a t A had quiche for breakfast, and to the left is the information set ofB if he has observed A have beer. JB'S choice whether to duel effectively ends the game. A, whether surly or not, wishes t h a t B choose not to duel. We imagine that A gets (incremental) payoff 2 if B chooses not to duel, and 0 if B does duel. A's total payoff (the sum of the two increments) is the first number in each column vector, at the end of each sequence of choices (by nature, then A, and then B). B wishes to duel with A if and only if A is a wimp—B's payoffs, reflecting this, are t h e second number in each column vector. Note well t h a t it is more important (in terms of payoff) to A t h a t he deter B from dueling than t h a t he have his preferred breakfast. And B's prior on what type is A would, absent any further information, induce B to avoid the duel. This extensive game has two Nash equilibrium outcomes. In the first. Ay regardless of t3npe, has beer for breakfast. J5, having seen a breakfast of beer, will not duel—^this makes sense as (anticipating A's strategy) B's posterior, given a breakfast of beer, is t h a t A is surly with (prior) probability 0.9. Now to make this a Nash equilibrium, we must keep the wimpish A from having a breakfast of quiche—this will happen if, upon seeing a breakfast of quiche, A chooses to duel with probability 0.5 or greater. We can, moreover.
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"make sense'' of such a reaction by B to a quiche breakfast as follows: quiche is taken as a sign t h a t A is a wimp; B revises his probability assessment t h a t A is a wimp to 0.5 (or more). If this "posterior probabiUty'' is 0.5, then B is indifferent between dueling or not; if it exceeds 0.5, then B strictly prefers to duel. Note that, at the equilibria we have described, B's "posterior beliefs'' at the quiche information set are not computable using Bayes' rule, since there is zero prior probability t h a t B will observe the event (quiche for breakfast) t h a t he is meant to condition upon. B u t there do exist beliefs at this out-of-equilibrium information set t h a t rationalize B's dueling (with sufficiently high probability). T h a t is to say, the equilibrium outcome is sequential The second equilibrium outcome is much like the first, except t h a t the breakfast changes. A, regardless of type, has quiche for breakfast. Seeing quiche for breakfast, then, B learns nothing; his posterior on A's type is the prior, and B chooses to avoid the duel. T o keep t h e surly A from having a breakfast of beer, B, in the event of a beer breakfast, duels with probability 0.5 or more. And this response by B to the out-of-equilibrium meal of beer can be rationalized; B's out of equilibrium "posterior beliefs" are that, if A has beer, A is a wimp with probability 0.5 or more. This multiplicity of equilibria, and their source, is typical of this type of signaling game. (A general definition will be given shortly.) Even if we insist t h a t B rationalize responses to outof-equilibrium messages (such as A's choice of breakfast) with some beliefs as to A's type, the wealth of possible out-of-equilibriimi beliefs gives us a wealth of out-of-equilibrium responses by B. And, therefore, many equilibrium choices of message by A can be supported. But, in the second equilibrium, are B's out-of-equilibrium beliefs sensible? Here is an argument to say t h a t they are not. If this is the equilibrium, then a wimpish A will n e t utility 3 at the equilibrium. (He gets his preferred breakfast of quiche and no duel in the bargain.) By having beer for breakfast, the very best he could do is a payoff of 2. Sending t h e out-of-equilibrium message "beer for breakfast" makes no sense for him. B u t it might make sense for the surly A to have this breakfast, in that, in equilibrium, the surly A receives 2 in equilibrium, and he can conceivably get 3 from a breakfast of beer. Suppose then t h a t B is restricted to beliefs which p u t no weight on the wimpish A having beer for breakfast. (Think in terms of removing from the game the possibility t h a t a wimpish A could have beer.) In this case, the only beliefs B could hold are t h a t
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A must be surly, and B would not duel. If the surly A realizes all this, he knows that he can have his beer and safely anticipate no duel. This breaks the second equilibrium. The first equilibrium is unbroken by such considerations. There it is quiche that is the out-of-equilibrium meal. The surly A has no reason to defect (getting 3 in equilibrium, and getting a maximum of 2 if he has quiche), whereas the wimp could conceivably gain by a defection. So, in the spirit of the previous paragraph, we would say that B should rule out the possibility that it is the surly A that is sending the out-of-equilibrium message. This causes By in the event of receiving that message, to hold a "posterior" assessment that he faces a wimp, which in turn causes him to choose to duel. But this supports the equihbrium outcome we have described. To take the argument a level further, consider the variation in Figure IL Here we have given B three options: to duel; to walk away; to give A $1,000,000. This third option does not affect the set of equihbria, since giving away the million is always a dominated strategy for B. But this third option does ruin the specific argument we gave against the "quiche for breakfast'* equilibria. We said before that B should discount the possibility that an out-ofequilibrium breakfast of beer comes from the wimpish A, since this type of A could conceivably benefit from this defection, relative to
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Signaling Games and Stable Equilibria SIGNALING GAMES AND STABLE EQUILIBRIA
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what he gets in equilibrium. When we add the third response for B, we can no longer say that the wimpish A cannot conceivably benefit from a defection—he does benefit if this induces B to give him the million. We must modify our test to read: could the wimpish A benefit from the out-of-equilibrium breakfast, relative to his inequilibrium expected payoff, for any response by B to this outof-equilibrium breakfast that B might conceivably take? Define '^responses that B might conceivably take'' as those that are undominated by other available responses. Then with this modification, we again dispose of the quiche for breakfast outcome. It is this tjrpe of argument that we formalize here. We give answers to the following questions: what is the precise criterion being apphed in this example, and what are other, similar criteria? Can we be assured that some equilibrium outcome will always survive a given criterion? And we shall seek to connect these criteria to formal refinements of Nash equilibrium, and especially to stability. To do so requires some formal setup and a review of stability. III. FORMULATION AND PRELIMINARIES
1, Signaling Games We focus in this paper on what we call the general signaling game with two players. The first, player A, receives private information. Following standard practice, we shall say that this player learns his type t, drawn from a finite set T. The player's type is drawn according to some probability distribution % over T that is common knowledge. Player A, having learned his type, sends a message m to player B chosen out of some finite set M. We allow the set of messages available to A to depend upon A's type; we write M{t) for the set of messages available to type t, and T{m) for the set of types that have available the message m. Player JB, having heard this message, chooses a response r from a finite set of responses R. We allow the available responses to depend on the message received, writing R{m). The game ends with this response, and payoffs are made to the two players, depending on the type of player A, the message A sent, and the response B took. The utility payoff to player A is denoted u(t,m,r), and the utility to player B is denoted v{tym,r). There are very simple games in extensive form, and one can describe their (sequential) equilibria very easily. Some notation will be helpful: we shall write behavior strategies for player A as p{m;t), where, for each t, p( • \t) is a probability distribution over M{t). The 305
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interpretation is that t sends m with probability p{m;t). For behavior strategies of player B, we shall write (r;m), where, for each m, (l){^\m) is a probability distribution on R{m)\ the interpretation is that By observing m, chooses response r with probability
^(t>^>^) M(0^
tGTim)
For subsets / of T{m)y let BR(I,m) denote the set of best responses by B to probability assessments concentrated on the set L T h a t is,
BR{I,m) = [J
BR{^,m),
Write MBR{}iym) and MBR(I,m) for the mixed best responses by B to, respectively, beliefs M and any beliefs whose support is /.^ A Nash equiUbrium for a signaling game is described by the obvious conditions: given B's strategy 0, each type t evaluates the utility from sending message m as S^u(^,m,r)0(r,m), and p(-;0 puts weight on m only if it is among the maximizing ^n's in this expected utility. And given A*s strategy p, B proceeds in two steps: first, for any message m that is sent with positive probability by some t, B uses Bayes' rule t o compute the posterior assessment t h a t m comes from each t3^e t G T{m) as ii{t\m) = [7r(Op(7n;0]/ [^t'GTCm) 'J^{t')p{m\V)]. And then the Nash condition is t h a t for all m that are sent by some t j ^ e t with positive probability, every response r in the support of JB'S response must be a best response to m given beliefs M(* 1^) t h a t are computed using Bayes' rule; or, in S3nnbols, (1)
(f>{*\m) G MBRiixi'
|m),/n).
T o require t h a t the Nash equilibrium is sequential is to add the requirement that, for every message m that is sent with zero probability by A (for all m such t h a t 2^7r(t)p(m;^) = 0), there must 1. Note well that while MBRdiyin) is the set of probability distributions over BR(^Lym)j MBR(I,m) may be smaller than all probability distributions over BR{I,m)—for each 4> ^ MBR{I,m)y we must produce a single M with support / such that <> / ^ MBRi^yin), The example in subsection IV.5 demonstrates this.
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be some probability distribution over tjnpes T(m), which we shall write M(- \^)y such that (1) holds. That is, B's responses to outof-equilibrium messages must be rationalized by some beliefs on the part of JB. The general program we shall follow is to restrict the set of sequential equilibria by posing restrictions on these out-ofequilibrium beliefs. 2. Three Facts about the Equilibria of Games We next wish to record three useful facts about the sets of Nash and sequential equilibria of various classes of games. The first fact holds for all noncooperative games with finitely many players, each of whom has finitely many pure strategies. It is given by Kohlberg and Mertens [1985]. 1. The set of Nash equilibria (and also sequential equilibria) of any finite player, finite pure strategy noncooperative game, viewed as a set of probability distributions over the product space of pure strategy profiles, consists of a finite number of connected sets.
FACT
The second fact pertains to games that are generic for a given extensive form. By this is meant: fix any (finite player, finite action) extensive form. Fix as well the probabiUty distributions for any moves by nature. Let Z denote the set of terminal endpoints for the extensive form, and let / denote the set of players. Then the specification of the game is completed by an assignment of payoffs to the players, one for each player at each point in Z, That is, the space of games over the given extensive form is ^^^^ (where Ji denotes the real line). A statement is said to be true for generic extensive games if, for every fixed (finite) extensive form and probability distribution for nature's moves, the set of payoffs (for that form) for which the statement is false has closure whose Lebesgue measure in ^^""^ is zero. Put another way, if a statement is generically true in this sense, and if the payoffs for a given extensive form are chosen from 31^''^ at random, according to some probability distribution that is absolutely continuous with respect to Lebesgue measure, then there is probability 1 that the statement is true for all games in an open neighborhood of the payoffs chosen. We require one further piece of terminology. For a given extensive form with terminal endpoints Z, each strategy profile (an assignment of one strategy for each player) induces a probability distribution over which endpoint is reached. Fixing the strategy profile, call this probability distribution the outcome of the game 307
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associated with the strategy profile. If a particular outcome is the outcome for some Nash equilibrium, call it a Nash equilibrium outcome. If it is the outcome of a sequential equilibrium, call it a sequential equilibrium outcome, and so forth. With all this buildup, the following is shown by Kreps and Wilson [1982a] and Kohlberg and Mertens [1986]. 2. For generic extensive games, the set of Nash equilibrium outcomes consists of a finite number of points.
FACT
This means that, while the set of Nash equilibria for an extensive form game may be infinite, the infinite variety (generically) concerns out-of-equilibrium actions and reactions. Because the map from strategy profiles to outcomes is continuous, facts 1 and 2 combine to establish the following simple corollary: 3. For generic extensive games, a single equilibrium outcome is associated with each individual connected set of Nash equiUbria (cf. Fact 1).
FACT
The structure that is implied for generic extensive games by these three facts is illustrated by the beer-quiche example. In this example there are two connected sets of equilibria, each one resembling a line segment. The first set of equilibria corresponds to the outcome where both types have beer for breakfast, followed by no dueling. The set of equilibria associated with this outcome arises from the many possible equilibrium responses by B to an outof-equilibrium breakfast of quiche, namely any (mixed strategy) response that puts weight 0.5 or more on dueling. Similarly, the equilibrium outcome of quiche-no duel is associated with a line segment of equilibria, arising from many possible responses by B to the out-of-equilibrium breakfast of beer. Note that genericity for our signaling games is defined in the space of all payoff assignments to all endpoints. Thus, in applying Fact 2, we cannot be confident that there are a finite number of outcomes for a signaling game that is randomly selected from the subspace of signaling games in which the message sent by A has no impact on B's utility, or where A's message selection has no impact on either player's payoffs. There may be a finite number of equilibrium outcomes for such games—witness the beer-quiche game. But we cannot appeal to Fact 2 to justify an assumption that such a game, selected at random, has a finite number of equilibrium
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outcomes. In the analysis to follow, we almost always fix a signaling game and assume that it has a finite number of equilibrium outcomes, justifying this assumption by appeal to Fact 2. Thus, our justification is not valid for the games analyzed by Crawford and Sobel [1982] or Farrell [1985]. (In any event, the criteria that we subsequently develop would not have force in games in which signals are costless to the sender.) 5. Stability—A Review Kohlberg and Mertens [1986] develop a number of criteria for Nash equilibria and sets of Nash equilibria that fit under the general rubric of stability. It is not our intention to repeat their development in detail; the reader is urged to read their paper to get a complete feel for their intentions and accomplishments. (In particular, it is very important to understand why they have moved from Selten's [1965, 1975] basic program of perfecting individual equilibria to a consideration of "perfect'' (strategically stable) sets of equilibria). But to make this paper close to self-contained, salient features of their development will be summarized in this subsection. Suppose that we have a class of games F on which some metric is defined. Let 2 be the space of strategy profiles for games from F, also endowed with some metric. Let iV: F => 2 be the Nash correspondence. For a particular game 7 G F, a subset M C N{y) of the Nash equilibria of y is said to be stable if for every € > 0 there is a 5 > 0 such that every game 7' that is within 6 of 7 has some Nash equilibrium that is less than e distant from the set M, This may seem rather a strong condition, but if we allow much latitude in selecting the set M, it is rather weak. For most sensible metrics on games, the Nash correspondence is upper hemi-continuous. In such cases, one stable set of equilibria for the game 7 is N(y), the set of all equilibria of 7. ButiV(7) is "large''; one is interested in showing that a set of equilibria smaller than N{y) is stable. This is the basic mathematical structure in Kohlberg and Mertens [1986]. (The definitions that we use come from earlier versions of their paper and vary somewhat from their most recent treatment. We shall alert the reader to the single important distinction near the end of this section). For the space of games F, they fix a Normal form—a positive integer number / of players, and, for each player i = 1 , . . . , / , a positive integer number Si of pure strategies, and they consider for F all assignments of utilities to (pure) strategy profiles. That is, F = (^*»*2.-^n)/ ^ ^ ^^^ space of strategy profiles 2 is then the n-tuple of simplices of mixed strategies for the fixed game
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form (always with the usual Euclidean metric). As for the metric on r, they work with three, corresponding to their notions of hyperstability, full stability, and (plain) stabiUty. H3rperstability, for example, corresponds to the standard Euclidean metric on F. Stability (which we shall deal with in the sequel) is defined as follows. A neighborhood base for the game j is generated by taking, for each 5 > 0, the set of all games y' whose payoffs can be realized as follows: for each player i there is a (mixed) strategy ai" and a real number 5f G [0,5] such that the outcome to player j in 7' if the players chose strategies cr^, respectively, is the outcome in game 7 if they choose the strategies dia* 4- (1 — di)ai. (Kohlberg and Mertens have cr* completely mixed and 5, G (0,5); we have closed these so that we can rephrase their construction in terms of a topology on the space of games.) In what follows, we shall use stability only. But the reader should note that the basic existence result, given below as Fact 4, is true for hyperstability, and so much of our analysis can be carried over to that stronger criterion. Since the Nash correspondence is upper hemi-continuous in the metric of stability, the set N{y) is always stable. The idea, as indicated above, is to find smaller stable sets. For games that are generic in the normal form, that is rather too easy to do; it turns out that stability per se adds no restrictions to the original definition of Nash. Let us explain: genericity in the normal form is defined in the obvious way, where the space of payoff's is taken to be all payoff assignments over the normal form. Now if we think of a game in normal form as a trivial game in extensive form (where there is one information set per player, at which the player chooses among his pure strategies). Fact 2 is seen to imply that, for generic normal form games, there are a finite number of Nash equilibria. Moreover, Kohlberg and Mertens [1986] establish that, for generic normal form games, each of these Nash equilibria, taken as a singleton set, is stable (and even hyperstable). Stability acquires cutting power (for generic games) when applied to normal form games that arise from and are generic for a given extensive form. Consider, for example, the signaling game of Section III. The space of payoffs for the extensive game has dimension (TMR)^ (where we use the uppercase letters to denote both the sets and the cardinality of the sets, and we assume, for purposesof this paragraph, that M(^) = MandR{m) = J?). Viewed as a normal form game, though, A has M^ pure strategies, and B has R^y so the space of normal form playoffs for the same has dimension
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(M'^R^)^, Clearly, a generic set of payoffs for the extensive form will typically be a very small set in the set of all payoffs for the normal form, so knowing t h a t a statement is true for generic normal form games tells us nothing about the truth of statement for payoffs t h a t arise generically in the underlying extensive form. Indeed, fixing an extensive form, for games that arise from payoffs generic in the extensive form, there are sometimes infinitely many Nash equilibria (e.g., in the beer-quiche example). And it is sometimes the case t h a t no single equilibrium, taken as a singleton set, will be stable. Kohlberg and Mertens [1986], however, establish the following basic existence result: recall t h a t the set of equilibria (for any finite game) consists of a finite number of connected sets. 4. For any finite game, some one (or more) of those connected sets of equilibria, taken by itself, is stable.
FACT
Since, by Fact 3, in generic extensive games each connected set of equilibria is associated with a single equilibrium outcome, we can enlist Kohlberg and Mertens' [1986] basic existence result to say: generic extensive games possess at least one stable equilibrium outcome, where an equilibrium outcome is stable if the set of equilibria that give rises to it is a stable set. T h a t is, for generic extensive games, while we cannot guarantee the existence of an equilibrium t h a t is stable as a singleton set, we can guarantee that some single equilibrium outcome is stable. Moreover, while for games generic in their normal form every Nash equilibrium is stable, it is not t h e case that, for generic extensive games, each equilibrium outcome is stable. T h a t is, stability as a criterion for equilibrium outcomes does have cutting power in generic extensive games. It is this cutting power, in the context of signaling games, t h a t we investigate. In doing so, we make use of a result from Kohlberg and Mertens [1986]. This requires a piece of terminology: take any game and any set of equilibria for the game. A strategy for one of the players is said to be never a weak best response for the player, relative to the set of equilibria, if in no equilibrium from the set is the strategy in question as good for the player as the strategy prescribed by the equilibrium. 5. Suppose t h a t we are given a set of equilibria for a game and a particular pure strategy for a given player t h a t is never a weak best response for the player relative t o the set. Consider the game with this pure strategy removed (from the normal
FACT
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form) entirely, and consider the subset of the original stable set of equilibria that consists of all strategy profiles from the set in which the given player puts zero weight on the particular pure strategy. This subset is stable in the game that results after the particular strategy for the player is "pruned.'' The same is true if the strategy for the player that is pruned is weakly dominated by some other strategy for the player. The reader should be warned that the terminology we are using is not quite consistent with Kohlberg and Mertens [1986]. The important difference is that they reserve the term stable set for a minimal (by set inclusion) closed set of Nash equilibria having the stability property we have given. It is easier for us to to use the term stable set for any set of equilibria which has the stability property, and we shall do so.^ It makes no difference to the results, as long as one defines a stable equilibrium outcome as a single outcome that is the projection of some (minimal) stable set of equilibria. This warning is particularly apt here, as the reader searching for Fact 5 in Kohlberg and Mertens will find a slightly different formulation; namely, the subset contains a stable set of equilibria. We require a couple of preliminary results, concerning the connection between signaling games and stable equilibria. We state them in a single lemma. 1. Fix a signaling game. Suppose that an equilibrium outcome is stable, in the sense that the set of all the equilibria that give rise to the outcome is a stable set. Then the set of all sequential equilibria that give rise to the outcome is also a stable set. Also, every stable set of equilibria contains at least one sequential equilibrium.
LEMMA
The condition that the game is a signaling game is needed here: Kohlberg and Mertens [1986] present an example (which they attribute to Gul) of a game having a stable set of equilibria, each member of which gives an outcome different from the unique sequential equilibrium of. the game. (That is, not only does the stable set not contain any sequential equilibrium; every equilibriimi in the stable set gives an equilibrium outcome different from the unique sequential equilibrium outcome.) What saves us from such an unhappy situation here is that, for signaling games, equilibria that are perfect in the normal form are also sequential. From the 2. Our usage is consistent with the 1982 and 1983 versions of Kohlberg and Mertens.
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definition of stability, it is easy to see that every stable set of equilibria contains some equilibrium that is perfect in the normal form. (See Kohlberg and Mertens [1986, p. 1028].) And it is likewise easy to show that, for a given stable set of equilibria, the subset of all normal form perfect equihbria in that set will be stable. IV. SELECTION GUIDE
2. The General Program We have finally gotten through the preliminaries, and we can outline the general program of the paper. Recall the beer-quiche example. We fixed a particular equilibrium outcome, and we restricted B's out-of-equilibrium beliefs by giving and applying a criterion for saying that a particular outof-equilibrium message was "unreasonable" for a particular type. The criterion used there was that type t would not reasonably be expected by B to send out-of-equilibriiim message m if the best t could do from m was less than t got at the equilibrium outcome. This is but one criterion we might think of, and, later in this section, we shall formalize it and others. In the variation on beer-quiche, we found that this criterion might be sharpened by first discarding "unreasonable" responses by B to certain out-of-equilibrium messages. The criterion used there was that response r to message m is unreasonable if it is dominated by some other response. We shall use a criterion that is equivalent to this for ruling out responses to messages in what follows. After restricting B's out-of-equilibrium beliefs, we asked whether the originally fixed equilibrium outcome could be "supported" by out-of-equilibrium beliefs that obey the restrictions. In the case of the equilibrium outcome where both types ate quiche, the answer was no, because J3's out-of-equilibrium beliefs would cause the surly type of A to defect. We formalize all this as follows: for a given signaling game with a finite number of equilibrium outcomes, fix some one of those outcomes. Let w*(0 denote the expected utility of type t in the fixed equilibrium outcome. Construct a test of that outcome in two steps. 1. Pose some criterion for saying that a particular outof-equilibrium message cannot "reasonably" be expected to be sent by a particular type. In each of the four subsections to follow, a different criterion of this sort will be investigated.
STEP
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Also, say that message m will never "reasonably'' be expected to be met with response r if r ^ BR{T{m)ym). Now specify some order and number of times for applying these two criteria. Apply the first by deleting from Mit), for each type ty any out-of-equilibrium message m such that (t,m) is "unreasonable/' Apply the second by deleting from i?(m), for each out of equilibrium message m, any response r ^ BR (T(m),7n). Apply the two iteratively, in an order, and for a number of times that is part of the specification of the test.^ That is, the order and number of applications, together with the criterion used to strike typemessage pairs, is the specification of step 1 of the test. When step 1 is completed, we shall have, for each out-ofequilibrium message m, a set of types that have not been ruled out for that message. Call this set of types T^{m), 2. For each out-of-equilibrium message m, consider all sequential equilibrium responses of B to m in the original game. Are any of these sequentially rational for B, given that JB'S beliefs are restricted to T^{m)l If not, then the equilibrium outcome has failed the test. (If so, it has passed.)
STEP
(It may happen that T^{m) is empty. In this case, to pass the test, it is necessary that, for every probability distribution ix on T{m)y there is some 0 G MBRiiiyin) that supports the equilibrium outcome in the original game.) We shall be attempting in Step 1 (as we delete type-message and message-response pairs) to make restrictions in B's beliefs that are intuitive in the sense that no one playing the game would reasonably expect B to put positive weight, given an out-ofequilibrium message m, on a type t that has been excluded for /n. If one were trying to envisage the thought process of the players, we might imagine two stages to Step 1. First, there is the asserted fact that, given the equilibrium outcome, a given out of equilibrium message is not sensible for certain of the types of A. Second, insofar as this asserted fact is held by the players to be correct, introspection on their part will tell them that B will attach zero weight to this out-of-equilibrium message coming from those t5^es, which will cause B to consider only certain responses to the message. This restriction on B's responses, if believed by all to be valid, may lead 3. In the variation on beer-quiche, we saw that applying the second criterion could sharpen the first. Since the first will "reduce*' T(m) for a given m, it will sharpen the second. Examples are easy to concoct to show that iterating back and forth can lead to continued reductions in the set of types that are reasonable for a given out-of-equilibrium message, which is the point of this exercise.
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to more asserted facts about which types of A might conceivably send the given out-of-equilibrium messages; introspection by the players concerning this will lead to further restrictions on B*s responses, and so on. Since this cycle (and its consequences in Step 2) depends on both players coming to these conclusions through introspection, our success in this endeavor will depend upon the extent to which the asserted facts (the criterion used to delete type-message pairs) are judged to be intuitive, and on the number of iterations that are required. (Restrictions made in one or a few iterations are presumably easier to swallow than restrictions that require many iterations, since each level of introspection requires faith that the other side has made it to the previous level.) Having made those restrictions, in Step 2 we suppose that everyone expects B to respond to an out-of-equilibrium message m that is based on beliefs which give no weight to types that have been excluded for that message. With this supposition can we continue to sustain the equilibrium outcome? Again envisaging the thought process of the players, we have in mind something like Kohlberg and Merten's process of forward induction: will some t5rpe of player A, having arrived introspectively at restrictions in B's behefs (hence JB'S conceivable actions), see that deviation will lead to a higher payoff than will following the equilibrium? Note that we shall fail Step 2 if and only if there is some out-of-equilibrium message m such that, for every (f> G MBR{T'{m),m), there is some type t G T{m) with u*(t) < 2r"^(i,m,r)<^(r). We can pose a slightly weaker test, by asking whether there is some single type t (for the given m) that would send m no matter what response B picks out oiBR{T^{m),m). That is, to fail the weaker test, the restrictions from Step 1 must suffice to give us a single type who will then certainly wish to break the equilibrium; in Step 2 as formulated, we rely on the (usual) assumption that JB'S out-of-equilibrium response to m is commonly known to all types of A. The weaker test, when failed, may be slightly more convincing as evidence against the fixed equilibrium outcome, so we shall consider, in the sequel, tests that are composed of some specification of Step 1, followed by this variation in Step 2, which we refer to as Step 2A.^ 4. The reader may wonder whether a type that has been excluded for message m in Step 1 might subsequently prove to be the undoing of the equilibrium in Step 2 or 2A, in the sense that, with B*s beliefs restricted by Step 1 to exclude t3rpe t, B will respond to m in a manner that causes t to send m. If we could be sure that this would not happen, then we could pose Step 2 a bit more simply, as: is the original equilibrium outcome still a sequential equilibrium outcome in the game where only types t ^ T'{m) can send m? We shall return to this issue in subsection IV.5, at which pK)int we shall see a particular test for which the answer is no.
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Besides posing "intuitive tests'' of equilibrium outcomes in signaling games, we follow the program above in order to relate our tests to Kohlberg-Mertens stability. We seek, in general, to show t h a t any equilibrium outcome t h a t fails any of the tests we construct fails as well to be a stable equilibrium outcome. To do this, we apply Fact 5, Lemma 1, and the following lemma. LEMMA 2. For signaling games, if a response r is not sequentially rational for B in response to message m' for some beliefs over T{m')y then it is dominated by some convex combination of JB'S other available responses. This is a simple application of the separating hyperplane theorem and is left to the reader. This lemma shows t h a t deletion of message-response pairs according to the "never sequentially rational" criterion is a special case of deletion of weakly dominated strategies. Fact 5, therefore, implies t h a t any time we begin with a stable set of equilibria in a given signaling game, what remains of the set is stable in the signaling game t h a t results from the pruning of any such messageresponse pairs.^ Suppose, then, that the criterion used in Step 1 to delete t3^e-message pairs also falls under the "permitted categories" of Fact 5. T h e iterated application of Fact 5 is clearly legitimate, so if the set of equilibria is stable to begin with, what remains of it must remain so after as many of these deletions as we care to make. Now invoke Lemma 1. In a signaling game with a finite number of equilibrium outcomes, suppose that some outcome is stable. Then the set of all sequential equihbria giving rise to t h a t outcome is stable, according to the first half of Lemma 1. T h e iterated deletion from this set of type-message and messageresponse pairs according to any criteria that are permitted by Fact 5 will leave us at each stage with a stable set of equilibria that are sequential for the original game. When this is completed, apply the second half of Lemma 1 to extract an equilibrium from this set t h a t is sequential for the reduced game. Necessarily, B's response in this equilibrium must be a best response to beliefs on the types in T^{m). Hence the test in Step 2 would be passed. We summarize this discussion as 1. Insofar as the deletion of type-message pairs falls under either category permitted by Fact 5, any equilibrium outcome t h a t fails our test fails to be stable.
PROPOSITION
5. Fact 5 permits domination in mixed strategies.
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Moreover, the Kohlberg-Mertens basic existence result gives us an instant corollary. Insofar as the deletion of type-message pairs falls under either category permitted by Fact 5, some one or more of the equilibrium outcomes of the fixed signaling game must pass the test that has been posed.
COROLLARY.
2. Dominance We can now pose specific tests that follow the general scheme above. An obvious test, and one that is well-known to practitioners of signaling games, involves dominance. Pose the following criterion for eliminating type-message pairs: Elimination of type-message pairs by dominance. For out of equilibrium message m, type t may be eliminated for this message if there is some other message m' with
(We could get by with a weak inequality here, but we use the strict inequality to facilitate comparison with later tests.) From this criterion for the elimination of type-mess£^e pairs, we can construct many tests. We might allow for a single application of this criterion, which corresponds to one round of elimination of strategies dominated for A. We might allow for the deletion of message-response pairs as in IV.l, followed by one round of elimination of type-message pairs using this criterion. We might allow iterated application of the two, for as long as it is profitable. (Such an iteration must terminate eventually, as there are only finitely many message-response and type-message pairs to delete.) This corresponds to the iterated application of (sometimes weak) dominance. And we could follow any of these with either Step 2 or 2A. It is evident that the criterion above is "permitted" under Fact 5, so that Proposition 1 and the corollary apply. Are any of these tests subsumed by less stringent refinements of Nash equilibrium? The answer is no for both trembling-hand perfection [Selten, 1975] and properness [Myerson, 1978]. Consider, for example, the game in Figure III. (The pictures are as before, except that in this case, as B is given no choice of response to the message m, we do not bother to put in an information set for him.) Consider the equilibrium outcome in which both types of A send the message m, and B responds to m' by choosing ri with probability 0.5 or greater. To support this equilibrium, JB*S beliefs at 317
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m
'(^
\
°(ir
or
o)__m-J?o. {.1}
FIGURE III
the m' information set must put weight 0.5 or more on A being t5rpe ^2- But for type ^2> ^ dominates m\ So, by any of the tests constructed from the dominance criterion above, we can prune the type-message pair {t2ym') from the game. In the game that is left, B must respond to m! with r2. This causes the equiHbrium outcome to fail the test, using either Step 2 or 2A, since this response causes ti to defect. The game in normal form is given in Table I. (Note that the prior enters into the expected payoff calculations.) We leave to the reader the simple task of verifying that the equilibrium in which A chooses m regardless of type and B responds to m! with TJ is indeed proper. (Moreover, it is easily shown to be perfect in the agent normal form.) This example can be used to make another point, concerning properness for signaling games. (The material in this paragraph is a bit esoteric, and it may be skipped without loss of comprehension of most of the rest of the paper.) Consider changing the prior on A's type, from 0.9 that A is ti to 0.9 that A is tg- Since, to support the m equilibrium outcome, it is necessary that JB "assess" high posterior TABLE I GAME OF FIGURE III IN NORMAL FORM
Response
Message if m m m' m'
318
m m' m m'
0,0 -0.1,0,1 -0.9,0 -1,0.1
0,0 -0.1,0 0.9,0.9 0.8,0.9
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probability that m' comes from i^2> this change would seem to make it harder to disqualify the equilibrium outcome under scrutiny. In any event, this change should make it no easier. But if the reader constructs the associated normal form, he or she will find that this change renders the m,ri equilibrium improper. Properness now does act to disqualify this equilibrium. This seems rather counterintuitive, but it is not hard to see why this is happening. If we view this game as a two-player game {A and JB), then we must make some intertypal comparisons of payoffs. That is, we have to aggregate the payoffs of type ti and type t^oi A. The prior, in this case, serves to scale these payoffs; when the prior is high that A is typt, ^i, then it is a "worse" mistake for ti to send m' than it is for t^ Uf - is responding with Ti) simply because it is a mistake that happens with higher probability. We would see the same thing if we rescaled the utility of one type (but not the other) of ^4; if, say, we changed ti's payoffs to 0 if m, —100 if m',ri, and 100 if m',r2, then the range of priors for which m,ri is proper expands. Especially if we regard these as games of incomplete (as opposed to imperfect) information, this intertypal comparison of utility seems nearly as suspect as would be an interpersonal comparison. We shall try to avoid intert3^al comparisons of utility for the rest of the paper, which means, among other things, that we abandon properness. The most expedient means of being sure that we are not making intertypal comparisons of utility is to regard signaling games not as two-player but rather as T +1-player games, where T stands here for the number of types of A; each type of A is regarded as a separate player. We shall on occasion, use language appropriate to this interpretation in what follows. One further word on this: while the properness of the m,ri equilibrium depends on the prior, there is another equilibrium with the m outcome, namely where B randomizes evenly between the two responses, which is not proper for any prior. Moreover, the improperness of this equilibrium is unaffected by rescaling of one type's utility. The reader will be better able to see why this is happening when we move on in our development to divinity. The power of iterated dominance in signaling games has long been noted. See, for example, the development in Milgrom and Roberts [1986]. 3. Equilibrium Dominance and the Intuitive Criterion Fix a particular equilibrium outcome, and use, as before, u^(t) to denote the expected payoff at this outcome to type t of A, In subsection IV.2, the criterion used to eliminate type319
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message pairs was domination: the pair it,m) is eliminated if there is a message m' available to t that does better, no matter what was the response of B to that message, than the best that t can obtain if he sends m. Consider weakening this to read: the pair it,m) may be eliminated if (3)
a*(0 > n^ax u{t,m,r), r
Comparing with (2), we see that the difference is that the value against which the consequences of sending m are tested is the expected value that t obtains at the given equilibrium, rather than the worst that can be gotten by sending some other message. Clearly, this new criterion will allow us to eliminate more typemessage pairs in the appHcation of the first sort of substep. In the sequel, it is called equilibrium domination, or domination by the equilibrium value. Construct from equilibrium domination the following test. First, throw out all message-response pairs (myr) such that r ^ BR(T{m),m). Then use equilibrium domination to dispose of type-message pairs. Then apply Step 2A. In aggregate, this test amounts to the following. For each out of equilibrium message m, form the set S{m) consisting of all t3rpes t such that
T H E INTUITIVE CRITERION.
If for any one message m there is some type V ^T not in S(m)) such that u^if) <
min
(necessarily
u(t\m\r),
then the equilibrium outcome is said to fail the Intuitive Criterion. This is the criterion used in the beer-quiche example. (Since we begin with a round of elimination of message-response pairs, it serves both for the original example and the variation.) It has been used in a number of applications: by Grossman [1981] directly; by Kreps and Wilson [1982b] and Milgrom and Roberts [1982a] almost directly (those analyses involve a richer dynamic structure than we have here); Rubinstein's [1985] assumption B-1 is closely related (albeit again with a richer dynamic structure). Despite the name we have given it, the Intuitive Criterion is not completely intuitive. (It is certainly less intuitive than applica-
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tions of dominance.) Equilibrium dominance accords a crucial role to the particular equilibrium outcome under discussion, yet, when it works, it proceeds to discredit t h a t equilibrium outcome.^ Consider in this regard the example of Section II, beer-quiche. We argued that, at the equilibrium outcome in which both types have quiche for breakfast, the wimp would never willingly defect to a breakfast of beer, because the best he could do with this breakfast gave him a lower utility than what he got at the equilibrium. If player JB regards this as logical, then introspection would cause B t o respond to beer without a duel, based on beliefs t h a t this breakfast was a sure sign t h a t A is surly. The surly A, capable of replicating this introspection, then applies /ori^ard induction to conclude t h a t a breakfast of beer is worthwhile. But now take this a step further. If B can, through introspection, come to this conclusion, and if B believes t h a t A can come to it as well, then B will expect A, if surly, to have chosen beer. Hence quiche is a sure sign of a wimp, and should be met with a duel. And, therefore, having quiche will not net the wimpish A a utility of 3, but rather 1, which means t h a t a breakfast of beer cannot be taken as a sure sign t h a t A is surly, which breaks the chain of the previous argument just where it started. We respond to this counterargument from the following general perspective. An equilibrium is meant to be a candidate for a mode of self-enforcing behavior that is common knowledge among the players. (Most justifications for Nash equilibria come down to something like this. See, for example, Aumann [1987] or Kreps [forthcoming]). In testing a particular equilibrium (or equilibrium outcome), one holds to the hypothesis t h a t the equilibrium (outcome) is common knowledge among the players, and one looks for "contradictions.^' Thus, to argue, in our example, t h a t beer might conceivably be better for the wimp than is quiche, because quiche might engender a duel, is to accept the contention t h a t the quiche outcome equilibrium is not a good candidate for self-enforcing behavior. A comparison with McLennan's [1985] justifiable beliefs is appropriate here. Note t h a t McLennan's concept derives from an inequality t h a t is very similar to (3). The major diiference, roughly, is t h a t on the left-hand side he puts the minimum sequential equilibrium outcome to the player, for any sequential equilibrium, whereas we use the payoff from the equilibrium payoff under 6. The argument to follow was first given in our hearing by Joe Stiglitz.
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consideration. Thus, we give a much greater role to the particular equilibrium payoff under consideration. We hold very strictly, in this, to the notion that the particular equilibrium is common knowledge among the players, and we have in mind a "story'' that says that out-of-equilibrium messages should be construed as conscious defections from the equilibrium. If one thought that out-of-equilibrium messages were (probably) the manifestation of some player or other being unaware of the equilibrium, and if one further thought that this "defecting'' player (who is unaware that he is defecting) believes that some other sequential equilibrium prevails in this instance, then McLennan's weaker criterion is the more sensible. We wish to stress here that the Intuitive Criterion reUes heavily on the common knowledge of the fixed candidate equilibrium outcome and, in particular, attaches a very specific meaning (a conscious attempt to break that equilibrium) to defections from the supposed equilibrium. Whatever its intuitive merits, the Intuitive Criterion is based on a criterion for striking type-message pairs that fits into the general program we are following. PROPOSITION 2.
If message m is equilibrium dominated for type t at some equilibrium outcome, then it is never a weak best response at any equilibrium that gives the outcome.
This requires no proof; it is almost a matter of definition. Accordingly, we can post a test (stronger than the intuitive criterion), which we call the equilibrium domination test: EQUILIBRIUM DOMINATION TEST.
Fixing an equilibrium outcome,
strike type-message pairs using equilibrium domination and message-response pairs using the "never a best response" criterion, iterating between the two for as long as either has effect. (Finite termination is assured by the usual argument.) Then apply Step 2. For a generic class of signaling games, a stable equilibrium outcome will pass the equilibrium domination test. Every signaling game from this generic class (therefore) has at least one equilibriiun outcome that will pass this test.
COROLLARY.
4. Divinity Banks and Sobel [forthcoming] propose tests of equilibrium outcomes in signaling games that they call divinity and universal divinity, (In addition, they develop independently many of the
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other results here.) The reader is urged to read their paper to obtain a detailed analysis of these tests; but, for completeness, we briefly adopt their tests to the framework we are using. Consider the following two criteria for disposing of typemessage pairs. Fix the equilibrium outcome. For a given outof-equilibrium message m and for each type t, find all (mixed) responses (f> G MBR{T{m)ym) by B that would cause t to defect from the equilibrium. That is, for each t, form the set, A = {<^ G MBR(T(jn\m):
u*{t) < Y. u{t,m,r)(t>{r)i r
And define D^t-{
If for some type t there exists a second tj^e t' with A U i)? C Dfy then (t,m) may be pruned from the game.
CRITERION D 1 .
If for some type t, D^ U D? C U v^tDf, then {t,m) may be pruned from the game.
CRITERION D 2 .
The intuition that is meant to be conveyed by these criteria is that whenever type t either wishes to defect and send m or is indiflFerent, some other type t' strictly wishes to defect. Hence it should be accorded (by B) more likely that the defection came from some other t' than that it comes from t. In Dl, we require that there is a single type f that always strictly wishes to defect whenever t does. In D2, we pose the weaker requirement that for any response that causes t to defect, there is some type (which may change with the particular response) that wishes strictly to do so. Note well that D2 will permit, in general, more type-message pairs to be struck than will Dl. By striking the pair {t,m), B is assumed to believe that it is "infinitely more likely" that m has come from this other t\ One might therefore seek a milder restriction on JB'S beliefs—^for t such that DtUD^t^s nonempty, require only that S's beliefs given m do not raise the probability that A is t relative to the probability that A is some other t\ (That is, require that ixit;m)/ij,(t';m) < w{m)/7r{m'),) (When Df; U D? is empty, strike {tyin) as before.) The intuition is that t should be no more likely to send m than is t\ Divinity is, roughly, a test formed from iterated application of the milder sort of restriction just described, combined with D2. It
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does not correspond to the striking of type-message pair and thus does not fit into our general scheme, but it does have considerable intuitive appeal. Universal divinity, on the other hand, fits into our general scheme: it corresponds (again in spirit, as a precise comparison is a bit subtle) to iterated application of the strong restrictions that arise from pruning type-message pairs on the basis of D2. One could build as well weakened forms of divinity and university divinity, based on Dl instead of D2. We shall see an important difi'erence between the two in the next subsection. Note that both divinity and universal divinity subsume equilibrium domination. For if message m is equilibrium dominated for a t5^e t, then for this message, Dt and D? are both empty. As long as there is any type that would gain from defecting to m, we prune {t,m). (If no tjrpe can ever gain from sending m, then the equilibrium outcome will not fail the equilibrium dominance test or any other that we can think of on account of this message.) The connection with stability is established in the usual fashion. 3. Any type-message pair disposed of by either criterion Dl or D2 given above is never a weak best response at the given equilibrium outcome. Hence a stable equilibrium outcome will pass the Banks-Sobel test of universal divinity, and a universally divine equilibrium outcome exists, for generic signaling games.
PROPOSITION
The proof is simple. Since D2 strikes more type-message pairs, we shall work with it. If m were a weak best response for type t at some equilibrium giving this outcome, then, at that equilibrium, the response ( • ;m) would have to lie in £)?. By assumption, this (j>{ • ;m) lies in Df for some other type t\ which immediately implies that it cannot be an equilibrium response to the out-of-equilibrium message m; it would cause t' to defect. 5. Never a Weak Best Response The proof of Proposition 3 makes clear that we would not run afoul of stability if we modified the criterion to read as follows: Fix the equilibrimn outcome and an out-of-equilibrium message m, and define D^ and Dt as above. Then the pair {t,m) may be pruned if D ? C Uf^tDf^ In other words, prune (f ,m) precisely when there is no sequential equilibrium response to m at which t is indifferent between the equilibrium and sending message m; when m is never a weak best 324
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m'
-O-
m'
t2 -O-
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- • •
-^•:
FIGURE IV
response relative to the set of sequential equilibria giving this outcome* Tests built up out of this criterion will be stronger than those built up out of the criteria of the previous section, as the game depicted in Figure IV shows.^ We consider the equilibrium outcome in which both types send message m\ and JB responds to m with high weight on response r^. In this game, B's mixed best responses to m include all three pure strategies, mixtures of ri and rg, and mixtures of Tg and Tg. Simple algebra shows that D^^ consists of mixtures of TJ and r2 and mixtures of r2 and r^, where in each case rg has weight more than V2; D^^ the frontier of this set. And Dt^ consists of all mixtures of ri and rg, plus mixtures of rg and r^ that put weight greater than % on TZ; D?^ consists solely of the mixture % on rg and ^^ on Tg. By the criterion of the previous section, no pruning is possible. But by the never a best weak response criterion, we can prune type ^2 for the message m. Doing so causes B to play the pure strategy ri, which is not an equilibrium response in the original game (type ^2 would defect). With reference to footnote 5, note that in this example we prune type ^2- This restricts the beliefs of JB, who then takes a response that causes the pruned player to defect from the original equilibrium. If there is some implicit "speech'' to go with the "never a weak best response" criterion, similar to the speech that goes with the Intuitive Criterion, it would have to run something like: 7. This example is based on another, similar example, from Banks and SobeL
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We do not wish to make too much of these "speeches." But we cannot suggest an intuitive inferential process for B (of the type we have been considering) that accompanies a defection from this equilibrium outcome and that leads B to conclude that the defection cannot be from the only type that would benefit from the defection if B makes that inference. On intuitive grounds, one might wish to insist that a defection that breaks an equilibrium is accompanied by a process of inference that leads B to put weight on those types that would break the equihbrium. (This philosophy finds favor in the related work of Farrell [19851 and Grossman and Perry [1986b].) Certainly, such a restriction is obeyed by dominance and by equilibrium dominance. In other words, we could rewrite Step 2 of our tests to read: with beliefs restricted in Step 1, the equilibrium should founder or not based on a defection from a type that has not been pruned for the out-of-equilibrium message under consideration. And the tests based on criteria up through equilibrium dominance would not be affected. But the never a weak best response criterion would be changed. We do not find this intuitive. What of universal divinity in this regard? If one insisted on criterion Dl in order to strike a type-message pair, then the resulting test would be safe; an equilibrium outcome that failed the test would fail because of (at least) one uneliminated type. But if criterion D2 is used, the resulting test is not safe. One can construct examples in which, at a given stage, one t3^e is eliminated by virtue of several others, one of which is simultaneously ehminated because the first t5rpe is not yet eliminated. That is, each helps to eliminate the other. We are, in consequence, happier with tests built up out of Dl than with those built up out of D2.^ V. T H E SPENCE SIGNALING MODEL
As an example of the various tests posed above, we consider a simple case of the Spence [1974] signaling model. In so doing, we shall be a bit casual, leaving it to the reader to fill in the gaps.^ We imaging that a given worker is one of T t>pes, indexed 1, 2, 8. We are grateful to Gyu Ho Wang for this observation, and for correcting an earlier version of this paper on this point. 9. The reader, wishing to see our basic argument made both exact and more general, should consult Cho [1986].
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. . . , T. (The usual story in the Spence model is that there are many workers, divided by t3^es. We could use this formulation equally easily.) The prior probability that the worker is of type n is TT^. The worker moves first, choosing an education level e from the set [0,oo). Then two risk-neutral firms, having observed the education choice but not the type of the worker, bid for the services of the worker. The bidding is in the style of Bertrand, with each naming a wage w G [0,oo) that it is willing to pay the worker; the worker chooses whichever firm bids the most; if the firms oflFer the same wage, the worker chooses by means of a coin flip. The worker is worth ne to the firm if n is the worker's t)^e and e is the level of education he obtained. If the worker is of type n, is paid w, and obtains education level e, then the worker's utility is M; —fe„c^,for strictly positive constants k^ that satisfy ki > k2 > - - - > kx* (This particular parametric family of indifference curves is irrelevant to the analysis—any family with the property that the marginal disutility of education strictly decreases with type will do.) We should note immediately that while we have assumed only a finite number of types of players, we are allowing infinitely many actions for each type, and infinitely many responses by the firms. Thus, stability cannot properly be applied to this analysis; indeed, the definition of a sequential equilibrium must be specially adapted to this context. We shall therefore continue in the spirit of sequential equilibrium and of the restrictions on beliefs posed formally above. A sequential equilibrium is defined as follows: tj^e n selects education levels according to some probability distribution p( • ;n); we shall restrict attention to equilibria in which these distributions are discrete and have finite support, so that p{e;n) will denote the probability with which type n selects education level e}^ Firms respond to education level e according to commonly held beliefs fx{-\e) as to the type of worker that has selected e; ^,(-;c) gives the wage offered by firm i if e is selected. In equilibrium: (i) Education levels must be chosen by the worker in a way that maximizes his expected utility, taking as given the offers of the two firms. (ii) At each education level e, firms must bid optimally, given the bidding function of the other firm, and holding to beliefs about the quality of the worker given by jii. (iii) At every education level e that is selected by the worker with 10. The interested reader can verify that if any Borel distributions are allowed, all equilibria will have the character of finite support.
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positive probability, the firms' beliefs should be generated in the usual fashion by Bayes' rule. Note that in (ii) we require optimality of the firms' bids at every education level, given their beliefs. So it is (iii) that ties the firms' equilibrium strategies to the worker's. Bertrand competition among the firms ensures that in any equilibrium, they bid precisely the expected value (to them) of the worker. That is,
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Case A. Two Types When there are only two types of worker, the Intuitive Criterion suffices to make the argument. Consider first the possibility that the two types pool; they each pick some education level ep with positive probabihty. Refer to Figure V. Since each picks ep with positive probability, fJL{2;ep) < 1, and W{ep,p) must be less than 2ep. In Figure V we draw the indifference curves of the two types through the equilibrium education and wage pair {epyW(ep)), By assumption, the indifference curve of type 2 is less steeply sloped than that of t3rpe 1, so points such as those shaded along the line w = 2e exist. Since wages can never exceed twice education level in any sequential equilibrium, type 1 would be strictly worse off picking education level e* (as shown) than he is at the equilibrium. Hence by the Intuitive Criterion, we must be able to support the equilibrixmi with beliefs by the firm that e* is chosen by a worker of type 2 with probability one. But this would lead to a wage iy(e*;^) = 2e*, which causes type 2 to defect from the equilibrium. Hence only screening equilibrium can survive the Intuitive Criteria. Since in any sequential equilibrium wages at level e must be at least e, it is easy to see that, in any screening equilibrium, type 1 will select his Riley level e*. And then the Intuitive Criterion again tells us that the equilibrium must be supportable by beliefs which put weight one on type 2 for any education level e such that (4)
9o -_ k^e^ i,_^2 < ^ e* ^* -_ h(o^\^^ 2e k^(e$)
tjrpe 1 is certain to do worse (given a sequential equilibrium
wage -i w w = 2e type 2 indifference curve
w*(ep;/i)
e—education level FIGURE V
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FIGURE VI
response from the firms) at such levels e than at his equilibrium value. As in Figure VI, let e^ be the level of e where we get equality in (4). It will not be a screening equilibrium for type 2 to select an education level less than e^ and the argument just given tells us that we cannot have a screening equilibrium that survives the Intuitive Criterion if we force 2 to some education level above e^ other than his "constrained first best'' from that set. Hence the Riley outcome is the only outcome that can survive. Case B, More Than Two Types^^ The Intuitive Criterion does not suffice to get us to the Riley outcome, if there are more than two types. Consider Figure VII, for three types. Here we have drawn an equilibrium outcome at which types 1 and 2 pool at education level e^, and type 3 is screened at education level eg. To break the pool, we would want (in the spirit of the previous arguments) to have type 2 offer an education level so high that type 1 would never do so in preference to the equilibrium. But since the firms could conceivably respond as if this outof-equilibrium signal came from type 3, the education level needed to do this is at least e° (as shown). If type 2 picks a level a bit higher than this, he can be sure to get a wage of 2e or more. But this will not guarantee that he gets more from the defection than he gets from the equilibrium. Indeed, this equilibrium does survive the Intuitive Criterion (and equilibrium dominance). 11. We are especially grateful to John Roberts, who pointed out an error in our earlier analysis of this case, and who helped us to find the correct argument. 330
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w = 3e type 2 indifference curve type 1 indifference curve
FIGURE VII
But it falls when the test constructed out of criterion Dl is applied. Indeed, all pooling equilibria fail this test. Suppose in some equilibrium that we had pooling of two types or more at an education level e. Let n index the highest type in the pool. Then at any education level above e, any response (wage) that a lower index type would prefer to the equilibrium, the higher index type would strictly prefer. Hence the equilibrium outcome would have to be supportable by beliefs in which the highest index type in any pool can, by choosing a slightly higher education level than the pooling level, be assured of a wage appropriate to (at least) his type. No pooling equilibrixun can survive this. (The test built out of criterion Dl is powerful stuff indeed!) Hence only screening equilibria can survive this test. And among those, only the Riley outcome will do so. We leave to the reader the task of showing that the only way that the Riley outcome can be missed in a screening equilibrium is if, for some two successive types n and n 4- 1, the picture is as in Figure VIII. But then for education levels above the point marked e\ the "better than equilibrium" set for n (and lower types) is strictly included in the same set for type n + 1. The Dl test requires that the outcome be supported by beliefs that put no weight on types n or less, which is manifestly impossible. Does the Riley outcome survive the Dl (and even, the D2) test? We leave it to the reader to show that the answer is yes. 331
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/ ^
w = (n + 1)e
^J'
/^'''''O'^type n + 1 indifference curve ^^•'•y y / ^ t h r o u g h his equilibrium ^.^-^ / x education level
Finally, we observe that the Intuitive Criterion v^ould suffice if v^e modified the game form.^^ We have supposed that the w^orker obtains an education level, and then the firms bid for his services. Suppose that we modified things slightly, so that the worker obtains education, and then the worker proposes the wage that he wishes; the firms then (simultaneously and independently) signify whether they are willing to hire the worker at that wage. In this game, there are sequential equilibria in which the worker, in equilibrium, is paid less than his expected value to the firm. (The worker cannot ask for more because this would change beliefs.) But when the Intuitive Criterion is applied to this game, one can show that the unique equilibrium outcome that survives is the Riley outcome, no matter how many t3npes there are (as long as the number is finite). This observation is especially pertinent when one thinks of applying this sort of criterion to alternating move bargaining games, as there the party who is "on the move" is allowed to propose an entire deal, which the other party must accept or reject. This gives the intuitive criterion (and similarly based tests) more bite in those games; cf. Admati and Perry [forthcoming]. VI. CONCLUDING REMARKS—THE FULL IMPUCATIONS OF STABILITY
The arguments just given are not meant to justify restriction to the Riley outcome in the Spence signaling model. In the first place, 12. We thank Anat Admati for this observation.
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the specific game form used is cruciaL^^ But more importantly to this paper is that we do not mean to advocate all the tests we have described. The demonstration above shows the tests we have devised are very powerful in applications; perhaps too powerful. We have posed these tests in a general framework in order to provide a somewhat general typology of such tests, and to see them at work in examples. The reader must be the judge of which, if any, of them provide reasonable tests of equilibrium outcomes in particular manifestations in signaling games. We also have sought to relate these tests to stability as it applies to signaling games. Since stability entails them all, if any of them is thought to be unintuitive, then the implications of stability cannot be accepted without some further thought. We ourselves find the Dl test very strong in the context of the Spence model, and we find "never a weak best response," at least as it applies to the example in subsection IV.5, to be downright unintuitive. But the reader should be warned that stability does not end with "never a weak best response." A general characterization of stability for the outcomes of (a generic class of) signaling games runs as follows. Fix a signaling game and some equilibrium outcome. For each unsent message m, let ^ ^ denote the set of all pairs ifXyS), where jx is a probability distribution on T{m) and S is a subset of T{m) such that, at some sequential equilibrium with the given outcome, JB'S response 0(-;^) to m satisfies (i) (j>{^;m) G MBR{ix,m), and (ii) u*(t) = ^Mtymyr)(t>{nm) for all t G S. That is, at beliefs fXy B has an equilibrium response that makes m a weak best response for all the types in S simultaneously. 4. For a generic class of signaling games, an equilibrium outcome is stable if and only if: for every unsent message m and probability distribution d over T(m), there is some ifJ'yS) G "^^ such that fx is in the convex hull of 0 and the space of all probability distributions on S,
PROPOSITION
We do not attempt to prove this proposition here. It is far from trivial, and the reader should be warned that the generic class for 13. Martin Hellwig [1985] has shown that, with a different game form, C. Wilson's reactive equilibrium outcome is the only outcome to satisfy the stability-like restrictions.
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which it is true is smaller than the class of signaling games that have only a finite number of equilibrium outcomes.^^ To see this proposition in action (and to see the strength of stability), consider a signalmg game with three types, tiyt2,t^, two messages, m! and m, and three responses to m, ri,r2,r^> We shall examine the outcome in which all three types send m' with probability one, assuming that B has a unique best response which he chooses. The out-of-equilibrium data of the game are depicted in Figures IXa and IXb. Figure IXa depicts the best responses of JB to m as a function of his "posterior" assessment on the t3rpe of A. So, for example, if J5 is certain that the type is ^i, he chooses response rj. If B is certain that the type is ^3, he chooses rg. If he has an assessment that puts probability ^A on each of ^2 and ^3, he chooses r^. Note the point co; at these beliefs, B is indifferent between all three responses, and any mixed strategy is a best response. Figure IXb depicts, as a function of B's response to m, which types (if any) of A would prefer m to m! (thus breaking the equilibrium outcome we are examining). So, for each n, if B responds to m with more than weight % on r^, type T^ would prefer m to m!. At precisely weight % on r^, type t^ is indifferent. These data are consistent with the following assignment of payoflFs: let all equilibrium payoffs if m! is sent be 0. In case the message is m, payoffs to A and B, depending on the type of A and the response by B, are given in Table II, with A's payoff first. The payoffs are in "general position" as concerns the arguments we shall make—the same arguments could be made for all payoffs in some open neighborhood of the payoffs that give these data. Note that it is indeed an equilibrivim outcome for each type to send m\ This can be seen in Figure IXb, where we find a region (shaded) of responses by B at which no type strictly prefers to send m. Moreover, the outcome is sequential, since beliefs co justify any response by B. And all the tests posed above are passed: every response to m is justified by some beliefs, and sending m is a weak best response for each type, at some equilibrium. Yet the equilibrium outcome is not stable. Consider a perturbation of the game in which type t^ sends m by "trembling" much more than do types ^2 and ^3. Unless something is done to change the beliefs of B, B will respond with rj, which will break the equilibrium. 14. Banks and Sobel [forthcoming], who arrived at this proposition independently, give a sketch of the proof.
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FIGURE IXA
FIGURE IXB
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FIGURE IXC
FIGURE IXD
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TABLE II PAYOFFS TO A AND B IF MESSAGE m Is SENT
Type of A ti
h
h
1,3 -2,0 -2,0
-2,3 1,0 -2,2
-2,0 -2,3 1,2
Response of J5
^3
Now we can, by looking at equilibria where rg is played with probability %, have tg indifferent between /n and m% and so we could at such an equilibrium increase the posterior belief that m comes from ^2- But to get B to play r2 with any probability at all, B must have posterior beliefs that put substantial weight (at least V2) on m coming from ^3. (Refer to Figure IXa and the region in which r2 is a best response.) Alternatively, we can look at equilibria where r^ is played with probability %, and thus make ^3 indiiferent. But to have B respond with positive probability on r^, the posterior weight on ^2 must be at least ^4- And this would require a response that puts weight at least % on ^2. Because there is no equilibrium (at the given outcome) at which both ^2 cind t^ are simultaneously indifferent between m and m\ it is impossible to increase simultaneously B^s posterior assessment that m comes from each. And to raise the probability of either, we need B to hold beliefs that put substantial weight on the other. The equilibrium outcome is not stable. In contrast, if the data were consistent with Figure IXc instead of IXb, there would be an equilibrium response (namely, the point marked <^* in IXc) at which both ^2 and t^ are indifferent between m and m', which would allow us to move from trembles that put most of the weight on ^1 to the point o), which in turn supports the response <^*. (Payoffs for A that are consistent with IXc are easy to compute.) In terms of the proposition let 9 = (1,0,0) (where the vector refers to the probability of the types in order). With the data of IXb, the candidates for sets S in ^ ^ are {^1}, {^2}? and {^3}. Hence we can only "puir' 6 along the two faces of the simplex. Pulling in the direction of ^2 is clearly useless, as this will not change B's response at all. It is possible, moving from 6 in the direction of ^3, to reach beliefs that are equilibrium beliefs—namely those that are labeled M*. But the set of equilibrium responses that go with the beliefs M*
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includes none that make ^3 indifferent. If, on the other hand, the data were as in IXc, then the pair (cojltg^^al) is in ^^. Thus, with the equilibrium response *, we can "stabilize" any initial perturbation 9 such that co is in the convex hull of 9 and the face of types ^2 and ^3. That is, all perturbations 9 in the shaded region of Figure IXd are stabilized by co and (/>*, including (1,0,0)—the reader can verify that every other initial perturbation can be stabilized at some other equilibrium with the m' outcome. The characterization given in Proposition 4 shows that stability (for generic signaling games) entails two considerations that our earlier criteria did not. First, one must consider for which subsets of tjrpes it is possible to find an equilibrium at which all types in the subset are indifferent between the equilibrium and some outof-equilibrium message. Second (and less apparent from our example) is that perturbations that can be stabilized at a particular equilibrium depend on "direction"—one projects from the face of indifferent types, past beliefs that support the equilibrium response, to find what perturbations are stabilized at the given equilibrium. (If this second consideration is not clear, it should provide the reader with sufficient motivation to consult Banks and Sobel [forthcoming], whose example of an unstable outcome that survives "never a weak best response" trades on this second consideration.) We do not mean to say that the m' equilibrium outcome in the examples of Figure IX is not breakable by intuitive agreements. For example, the criterion proposed by Grossman and Perry [1986b] does break this equilibrium.^^ But their criterion works equally well if the data are given by IXa and IXc as if they are given by IXa and IXb, so stability makes a distinction here that we cannot motivate intuitively.^^ We conclude that, if there is an intuitive story to go with the full strength of stability, it is beyond our powers to offer it here. UNIVERSITY OF CHICAGO STANFORD UNIVERSITY
REFERENCES Admati, Anat, and Motty Perry, "Strategic Delay in Bargaining," Stanford University, Review of Economic Studies, forthcoming. 15. See also Farrell [1985], although his analysis takes place in a setting in which messages are free. 16. It is also worth noting that the example of subsection IV.5 is an unstable equilibrium that Grossman and Perry accept.
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Aumann, Robert J., "Correlated Equilibrium as an Expression of Bayesian Rationality," Eco?zometnca, LV (1987), 1-18. Banks, Jeffrey S., and Joel Sobel, "Equilibrium Selection in Signaling Games," mimeo, U.C. San Diego, Econometrica, forthcoming. Cho, In-koo, "A Refinement of Sequential Equilibrium," Princeton University, Econometrica, forthcoming. , "Refinement of Sequential Equilibrium: Theory and Application," Ph.D. thesis, Princeton University, 1986. Crawford, Vince, and Joel Sobel, "Strategic Information Transmission," Econometrica, L (1982), 1431-51. Demski, Joel, and David Sappington, "Delegated Expertise," mimeo, Yale University, 1986. Farrell, Joseph, "Credible Neologisms in Games of Communication," mimeo, MIT, 1985. Grossman, Sanford, "The Informational Role of Warranties and Private Disclosure about Product Quality," Journal of Law and Economics, (1981), 461-83. , and Motty Perry, "Sequential Bargaining under Asymmetric Information," Journal of Economic Theory, XXXIX (1986a), 120-54. , and , "Perfect Sequential Equilibrium," Journal of Economic Theory, XXXIX (1986b), 97-119. Hellwig, Martin, private communication, 1985. Kohlberg, Elon, and Jean-Francois Mertens, "On the Strategic Stability of Equilibria," Econometrica, LIV (1986), 1003-38. Kreps, David M., "Nash Equilibrium," mimeo, Stanford University, The New Palgrave, forthcoming. , and Robert Wilson, "Sequential Equilibria," Econometrica, L (1982a), 863, and , "Reputation and Imperfect Information," Journal of Economic Theory, XXVII (1982b), 253-79. McLennan, Andrew, "Justifiable Beliefs in Sequential Equilibrium," Econometrica, LIII (1985), 889-904, Milgrom, Paul, and John Roberts, "Limit Pricing and Entry under Incomplete Information: An Equilibrium Analysis," Econometrica, L (1982a), 443-59. , and , "Predation, Reputation and Entry Deterrence," Journal of Economic Theory, XXVII (1982b), 280-312. , and , "Price and Advertising Signals of Product Quality," Journal of Political Economy, XCIV (1986), 796-821. Myerson, Roger, "Refinements of the Nash Equilibrium Concept," International Journal of Game Theory, VII (1978), 73-80. , "Mechanism Design by an Informed Principal," Econometrica, LI (1983), 1767-98. Riley, John, "Informational Equilibrium," Econometrica, XLVII (1979), 331-59. Rothschild, Michael, and Joseph E. Stiglitz, "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information," this Journal LXXX (1976), 629-49. Rubinstein, Ariel, "A Bargaining Model with Incomplete Information about Preferences," Econometnco, LIII (1985), 1151-72. Selten, Reinhard, "Spieltheoretische Behandlung Eines Oligopobnodells mit Nachfragetragheit," Zeitschrift fur die Gesamte Staatswissenschaft, CXXI (1965), 301-24. , "A Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games," International Journal of Game Theory, IV (1975), 25-55. Spence, A. Michael, Market Signaling (Cambridge, MA: Harvard University Press, 1974).
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15 Faruk Gul on Hugo E Sonnenschein
During 1984-1985, Hugo Sonnenschein, Robert Wilson, and I wrote a paper entitled "Foundations of Dynamic Monopoly and the Coase Conjecture," (Journal of Economic Theory, 1986. Henceforth "DMCC") This paper offers a gametheoretic analysis of Ronald Coase's idea that a monopolist provider of a durable good would be forced to sell at the competitive price. Coase's reasoning was as follows: as soon as the consumers who are wilUng to pay the monopoly price make their purchases, the monopolist has an incentive to lower his price and sell to some of the remaining consumers. Inmiediately after the next round of sales, the monopolist would find it worthwhile to lower his price yet again. This process would continue until all buyers with reservation prices above marginal cost are served. But rational consumers would anticipate this rapid fall of prices and wait until the market price reaches its ultimate level before making any purchases. Thus, Coase's argument asserts that the durable goods monopolist incurs competition from an unexpected source: his future self. The time between subsequent market periods (i.e., offers) determines the extent of this competition. Let D(v) denote the fraction of consumers with valuation at least v. Define the competitive price as the largest p such that D(p) = D(c), where c denotes the constant unit cost of production. Hence, the competitive price is the largest price at which all consumers who would be served under perfect competition are willing to purchase the good. The case where p > c is called the gap case, while p = CIS referred to as the no gap case. DMCC shows that D can be interpreted either as the distribution of valuations in a large market or the probability distribution of the single buyer's valuation in a bilateral bargaining problem. The main results of DMCC establish that equilibrium exists for any market demand (buyer valuation distribution), that the buyers' equilibrium strategy is stationary (i.e., time-invariant) in the gap case, and that after the initial period, the sequence of observed prices is deterministic. DMCC also shows that as the time between offers becomes arbitrarily small, the entire market is served inunediately at the competitive price in the gap case, and in all stationary equilibria of the no gap case. Hence, DMCC provides a proof of the Coase Conjecture. The intuition behind this result matches Coase's argument. Consider the bargaining interpretation of the model. Suppose that buyer behavior is described by a fixed acceptance function that doesn't depend on the time between offers. Then, as the time between offers becomes arbitrarily small, in any interval of real time, the seller can make arbitrarily many offers. This means that he can achieve as much
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15 Faruk Gul on Hugo F. Sonnenschein price discrimination as he wants without incurring any delay. But this means that prices must fall to the competitive level immediately, which means that no sales will be made until they fall to that level. The main proof in DMCC shows that this argument is valid even though the equilibrium buyer acceptance function changes as the time between offers changes. "Unobservable Investment and the Hold-Up Problem," extends the analysis of DMCC to a setting where valuations are not exogenous but are determined by the buyer's unobservable investment. In the new framework, before the bargaining begins, the buyer may invest in relationship-specific capital to increase his valuation of the good. The seller observes neither the level of the buyer's investment nor the buyer's resulting valuation. Hence, whenever the buyer uses a mixed strategy, the bargaining stage is exactly like the bargaining game studied in DMCC. To see how the hold-up model relates to DMCC, consider the case where there are two possible investment levels, 0 and 1, leading to valuations 2 and 5 respectively. Since 2 - 0 < 5 — 1, the efficient decision is for the buyer to invest 1. However, if the buyer invests 1 for sure, then the seller will never charge a price below 5. Hence, the buyer will be held-up and regret making the investment. On the other hand, if the buyer invests 0 for sure, then the seller will charge 2. But this means that the buyer could have earned 5 — 2 — 1 = 2 by deviating and investing 1. Hence, in equilibrium, the buyer must use a randomized investment strategy. Part of the argument of DMCC extends to this new case with endogenously determined valuations: as the time between offers becomes arbitrarily small, the probability that the bargaining game ends (almost) immediately approaches 1. This is true, regardless of the buyer's investment strategy. But as the time between offers goes to 0, if the probability that the buyer invests 0 stays bounded away from 0, then the fact that the game ends arbitrarily quickly means that the price must converge to 2 arbitrarily quickly. But, as argued above, this would destroy the buyer's incentive to invest 0, which must happen with positive probability. So, the probability that the buyer invests 0 must stay positive but go to 0 as the time between offers goes to 0. Since a price below 2 will never be charged, and the buyer invests 0 with positive probabiUty, his payoff must be 0. We conclude that as the time between offers becomes arbitrarily small, the buyer must invest 1 almost surely and purchase the good at price 4 almost immediately so that his payoff is 5 - 4 - 1 = 0. Thus, the fact that valuations are determined endogenously enables one of the DMCC conclusions to survive without implying the other: agreement is reached almost immediately but prices do not converge to the competitive level. The main result of the hold-up paper establishes that unobservable investment and dynamic bargaining resolve the hold-up problem and restore efficiency. The key steps of the proofs resemble the corresponding arguments in DMCC. The novel argument is a formal and more general statement of the intuition of the preceding paragraph.
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Unobservable Investment and the Hold-Up Problem Econometnca, Vol. 69, No. 2 (March, 2001), 343-376
UNOBSERVABLE INVESTMENT AND THE HOLD-UP PROBLEM BY FARUK GUL^ We study a two-person bargaining problem in which the buyer may invest and increase his valuation of the object before bargaining. We show that if all offers are made by the seller and the time between offers is small, then the buyer invests efficiently and the seller extracts all of the surplus. Hence, bargaining with frequently repeated offers remedies the hold-up problem even when the agent who makes the relation-specific investment has no bargaining power and contracting is not possible. We consider alternative formulations with uncertain gains from trade or two-sided investment. KEYWORDS: The hold-up problem, unobservable investment, bargaining, the Coase conjecture.
L INTRODUCTION CONSIDER THE FOLLOWING SIMPLE MODEL of specific investment. One agent, the buyer, takes an observable action that determines his own utihty of later consumption. Afterwards, the buyer and the other agent, the seller, bargain. Such a situation is studied in a number of recent papers on the hold-up problem, contract theory, and the theory of the firm. If the bargaining process is such that the seller can extract all surplus and there is no contractual commitment, then in equilibrium the buyer will not invest at all. This is due to the fact that the investment of the buyer is sunk-cost at the bargaining stage and will not be compensated for by the seller. This result is a very extreme form of the hold-up problem and holds regardless of whether the seller makes a single take-it-or-leave-it offer or is able to make repeated offers. Next, consider the same problem, only this time assume that the buyer's investment decision cannot be observed by the seller. In Proposition 1, we show that if the bargaining consists of a single take-it-or-leave-it-offer, then in equilibrium the buyer and the seller still obtain the same payoffs as in the case of observable investment. Thus, as suggested by Gibbons (1992), in a one-shot interaction, the hold-up problem continues to be extremely costly even with unobservable investment. However, Proposition 5 establishes that if the investment decision is unobservable and the seller makes repeated offers, then as the time between offers becomes arbitrarily small the equilibrium investment decision of the buyer converges to the efficient (i.e., surplus maximizing) level. Moreover, the expected delay converges to zero. Hence, all inefficiency disappears. Regardless of the time between offers, the buyer's equilibrium payoff is zero. Therefore, the seller extracts full surplus. Neither unobservable investment I am grateful to Avinash Dixit, Gene Grossman, Aiessandro Lizzeri, Ennio Stacchetti, and Andrea Wilson for their help and comments. Financial support from the National Science Foundation is gratefully acknowledged. 343
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nor frequently repeated offers alleviates the hold-up problem; yet, the two together completely resolve it. The purpose of this paper is twofold. First, we wish to note the tendency toward efficient investment created by this interaction between unobservable investment and dynamic bargaining. In any setting where efficient trade (i.e., immediate agreement) is guaranteed, unobservable investment implies that the buyer is the residual claimant on his investment and, hence, leads to the first best outcome. In the one-sided offer bargaining setting efficient trade is guaranteed by the Coasian effect. Thus, we would expect our results to extend to other settings provided the underlying bargaining model yields immediate agreement. The second objective of this paper is to emphasize the role of allocation of information as a tool in deaUng with the hold-up problem. Audits, disclosure rules or privacy rights could be used to optimize the allocation of rents and guarantee the desired level of investment. Controlling the flow of information may prove to be a worthy alternative to controlling bargaining power in designing optimal organizations. That private information rents might substitute for bargaining power and ameliorate the hold-up problem has been noted by Rogerson (1992) and others. Our purpose is to demonstrate the importance of this effect by noting that in a Coasian setting, incomplete information may completely remedy the hold-up problem even when the investing agent has no bargaining power. There are two central assumptions in this paper. First, we assume that commitment is not possible. Undoubtedly, there are many appUcations where some commitment either implicit or through long term contracts, is feasible. Nevertheless, this extreme form of incomplete contracting may be useful both as a benchmark and for understanding the disagreement payoffs associated with any contracting game. The second main assumption is unobservable investment. Again, we recognize that there are situations where investment decisions will be observable. Nevertheless, in many settings, it will be possible for one agent to have less than perfect information about her opponent's earlier decision.^ For the argument of Proposition 5 to apply, a small amount of asymmetric information between the buyer and the seller regarding the buyer's investment level may be sufficient.^ Consequently, we would expect our analysis to be relevant in many situations, even when the seller cannot be kept totally uninformed about the decision of the buyer. With one major difference, the setting of Proposition 5 is closely related to the work on one-sided offer, one-sided incomplete information bargaining (see, for example, Fudenberg, Levine, and Tirole (1985)) and the Coase conjecture
^ This is likely to be the case if, for example, the hidden investment is investment in intangible assets or some other type of unobservable effort. ^ See the example with a random buyer valuation in Section 6.
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(see Gul, Sonnenschein, and Wilson (1986), henceforth GSW, and Ausubel and Deneckere (1989)). In the earlier papers, the distribution of the buyer's valuations is exogenously given vv^hereas in the current paper the buyer's valuation is determined by a strategic choice and uncertainty arises from the unobservabiHty of that choice and the buyer's use of a mixed strategy. Nevertheless, understanding the Coase conjecture is important for understanding Proposition 5. Assume, as is done in the literature on the Coase conjecture, that the distribution of the buyer's valuation is fixed. Then, if the lowest valuation in the support of this distribution is greater than the cost of production, the equilibrium strategy of the buyer will be described by an acceptance function that determines the v^Uingness to pay of each buyer type independent of history. The seller's problem is tofimdthe optimal level of price discrimination given her own impatience and the "demand" function defined by the buyer's strategy. As the time between offers becomes arbitrarily small, the seller will be able to move down this demand function in an arbitrarily small amount of real time. But with a fixed distribution of valuations the fact that expected time until trade is converging to zero implies that the buyer with valuation arbitrarily close to the lowest in the support of the distribution is buying almost immediately, which means that the price charged by the seller must fall to the lowest possible valuation almost immediately. This is the Coase conjecture. The conclusion that expected delay converges to zero as the time between offers converges to zero carries over to the current setting. However, since the distribution of the buyer's valuation is no longer fixed, this does not mean that the buyer with valuation arbitrarily close to the lower end of possible valuations is buying almost immediately. As the time between offers becomes small, the probability that the buyer invests at the efficient level approaches 1. Hence, only a buyer with a high valuation is buying early, in spite of the fact that investing zero is always in the support of the buyer's strategy. In equilibrium, the expected delay is small but there is positive probability that delay will occur. At each moment in time, the seller believes that there is a high probability of making a sale at a high price in the near future. This keeps the seller from decreasing prices too quickly. The fact that the distribution of the buyer's valuation is endogenously determined enables one part of the Coase conjecture to survive without the other; no delay is expected in equilibrium but the market price does not collapse immediately. This analysis explains why the Coase conjecture is compatible with Proposition 5, that is, why Proposition 5 might be true. To see why it must be true, observe that if v is the lowest buyer valuation in the support of the distribution, then a price below v will never be charged by the seller. Hence, investing zero must be in the support of the buyer's strategy and his equilibrium payoff must be zero. (If the lowest level investment in the support of buyer's strategy were strictly above zero, the buyer's equilibrium payoff would be negative.) The Coasian effect guarantees no delay in expectation. But if the buyer expects to trade immediately, it is optimal for him to invest at the efficient level. Hence, the outcome is efficient and the seller extracts all the surplus.
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We consider two extensions of our model. First, we assume that the constant marginal cost of production is random and not known at the time the buyer undertakes his investment but is commonly known before the bargaining begins. Hence, gains from trade are no longer certain. This new setting brings our framework closer to many models studied in the incomplete contracting hterature. We assume that the expected gains from trade are strictly quasi-concave in the buyer's level of investment and that the probability of positive gains from trade given zero investment is not zero. Then, we prove that if offers can be made frequently, sequential equihbrium outcomes with stationary buyer strategies are efficient. Again, the seller extracts all surplus. The restriction to stationary strategies was used by GSW for proving the Coase conjecture for the "no-gap" case (i.e., when a strictly positive lower bound on the gains from trade is not assumed). We impose the same restriction in order to extend the "no expected delay" result to the case where gains from trade are uncertain. In our second extension, we investigate the consequences of the seller having an investment decision as well. We assume that the seller makes observable, cost reducing investment prior to the bargaining stage. We continue to assume that the buyer's investment is unobservable. We show that the earlier efficiency result holds if the buyer is able to observe the seller's investment prior to making his own. However, if the two agents make their investment decisions simultaneously, then there will be a tendency for the seller to underinvest but the buyer will still invest efficiently. As noted above, a number of recent papers on incomplete contracting also deal with the hold-up problem and its impact on relation specific investment. Grout (1984) studies the relationship between shareholders and workers. In his setting, the shareholders make the relation specific investment. He compares the case where binding contracts can be made prior to investment with the case where no binding contracts can be made and the ex post distribution of surplus is determined exogenously, according to a generalized Nash bargaining solution. He notes that without binding contracts, unless the bargaining solution gives all of the power to the shareholders, there will be underinvestment. Grossman and Hart (1986) also model the ex post bargaining stage in closed form (i.e., as a cooperative game), where the ex ante specified distribution of control determines the disagreement payoffs. They observe that greater control, like greater bargaining power in Grout's model, tends to create greater incentives to invest. The optimal organizational form is determined by the relative benefits of giving control to one agent versus the other. Aghion, Dewatripont, and Rey (1994) have investigated the possibihty of designing the negotiation process optimally so as to overcome the hold-up problem. Thus, unlike the two papers above, they do not take the renegotiation or bargaining stage as exogenous. They show that first best outcomes can be obtained by the optimal renegotiation procedure (i.e., noncooperative bargaining game), in a wide range of settings. The papers discussed above and much of the contracting literature assume observable investment. Observable investment in a relation-specific asset may
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cause a hold-up problem which creates underinvestment. The focus of the literature is to investigate possible contractual resolutions and other remedies to the hold-up problem and to interpret existing institutions as expressions of these remedies. Our main result estabhshes that unobservability of investment may be an alternative remedy to the hold-up problem. The important role played by unobservability in the context of the extensive form games studied in this paper suggests that the hold-up problem may not be so severe in many contexts. Therefore, in certain applications, the relevant benchmarks for the analysis of contractual resolutions may need to be reconsidered. A common feature of many of the models studied in the hterature on moral hazard and renegotiation (see, for example, Che and Chung (1995), Che and Hausch (1996), Fudenberg and Tirole (1990), Hermahn and Katz (1991), Ma (1991,1994), and Matthews (1995)) is that a pure strategy by the agent generates too severe a response by the principal. Therefore, in equilibrium, the agent randomizes, generating asymmetric information. This phenomenon also plays an important role in our analysis. 2. THE SIMPLE MODELS
In the simple bargaining game with perfect information, the buyer chooses a level of investment x, then this choice is observed by the seller who chooses a price p. The buyer either accepts this price {a^{x,p) = l) or he rejects it \a^{x,p) = 0). In either case, the game ends. If the buyer invests x and gets the good at price p, then the seller's utility is p and the buyer's utility is v{x) —x—p. If the buyer does not end-up with the good, that is, if he rejects the seller's offer, then the payoffs for the seller and buyer are zero and —x respectively. The formal description below of the simple bargaining game with perfect information mentions only the set of pure strategies. However, we permit the use of mixed strategies throughout this paper. Hence, statements regarding the uniqueness of equilibria never entail a restriction to pure strategies. Let [0, M], M > 0, be the set of possible investment choices for the buyer. The nondecreasing function v: [0, M] -> R defines the valuation of the buyer as a function of the level of investment he has undertaken. We will assume that v(M)<M,u(Q)>0. SO is called a simple bargaining game with perfect X^ -•= [0, M] X {a^ : [0, M] X R^ -> {0,1}} and S' ==
DEFINITION: B^ = (X^,
information, where {(r^:[0,M]->R^}.
It is well-known and easily verified that in the simple bargaining game the hold-up problem rules out any possibility of investment. At the bargaining stage, the buyer's investment is sunk-cost and is not taken into account by the seller. Since the seller makes a take-it-or-leave-it offer, he extracts all surplus. Knowing this, the buyer cannot afford to undertake any investment. We record this observation as Proposition 0 below.
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PROPOSITION 0: There is a unique subgame perfect Nash equilibrium of the game B^. In this equilibrium the buyer invests zero, the seller charges p = v{0), and the buyer accepts. In equilibrium the seller and buyer obtain utility v{G) and 0 respectively.
It follows from Proposition 0 that the hold-up problem causes inefficiency whenever 0 is not an efficient investment level. Next, we show that if the investment decision is not observed by the seller, then the nature of equilibrium is changed but the equilibrium payoffs remain the same. Hence, the unobservability of the investment decision by itself does not remedy the hold-up problem. DEFINITION: B = {X^,X^) is called a simple bargaining game, where [0,M]X{C7^:[0,M]XR->{0,1}}, 5 ^ - R ^ .
X^-^
Note that the only difference between B^ and B is that in B the price charged by the seller does not depend on the investment decision of the buyer. This difference accommodates the fact that the investment choice is no longer observable to the seller. In this paper, a sequential equilibrium is a behavioral strategy profile a and an assessment ix that satisfy the following: (i) a is sequentially rational given jx, (ii) if information set h' can be reached by a given that h is reached, then, the assessment at h' is obtained from the assessment at h by Bayes' Law. This is the solution concept used throughout the bargaining hterature. PROPOSITION 1: In any sequential equilibrium of B the seller and buyer achieve utility uCO) and 0 respectively. The buyer accepts any offer p < v{x) and rejects any p> v{x). If V is strictly concave and continuous, then: Equilibrium exists and is unique. In equilibrium, the seller randomizes according to the cumulative F, where F(p) = 0 if p
PROOF: See Appendix.
Proposition 1 establishes that in equilibrium, the payoffs to the players are the same as the equilibrium payoffs when the investment choice is observable. Problem 2.23 of Gibbons (1992) makes the same point in a game with only two investment levels. For the case in which v is strictly concave and continuous, Proposition 1 also yields existence and uniqueness of the equilibrium and describes the equilibrium strategies.'* While the unobservability of the investment decision alters the In fact, existence does not require any assumptions other than continuity of v. Note that if L''(0) <^y F has a discontinuity at u(OX G has a discontinuity at j : * .
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nature of equilibrium behavior, it does not change the equilibrium payoffs (i.e., the extent of inefficiency). The source of the inefficiency changes (underinvestment is reduced but the possibility of disagreement is added) but the amount of inefficiency is not decreased by the ability of the buyer to conceal his investment decision. Next, we consider the following modifications of the games B^ and B: Instead of a single take-it-or-leave-it offer, we permit the seller to make a new offer each time her offer is rejected. Hence, after the investment stage, the seller and buyer are engaged in an infinite horizon bargaining game with one-sided offers. The payoffs of the players are computed in the standard way: If an agreement is reached at price /?, in period A: = 0,1,2... of the bargaining stage, then the payoff of the buyer is [v(x) —p]e~'^^ —x while the payoff of the seller is pe~^^, where A is the time interval between successive offers. Hence, we normalize the time units so that the interest rate describing the players^ impatience is 1. We denote the infinite horizon versions of B^ and B, B^{A) and B{A) respectively. If no agreement is reached, the seller's and buyer's payoffs are 0 and —x, respectively. The problem of bargaining with one-sided uncertainty and one-sided offers has been studied extensively. The only difference between the game B{A) and the game studied by Fudenberg, Levine, and Tirole (1985) and GSW is the initial investment stage. In the earlier models asymmetric information is assumed while in B{A) it arises endogenously, due to the unobservable investment decision of the buyer. In the game B^{A) the investment decision is observable. Therefore, there is complete information at the bargaining stage. It is easy to show and well-knovm in the bargaining literature that if one side makes all the offers in a complete information setting, then she gets all the surplus. To see this, suppose that p, the infimum of all prices ever offered in any equilibrium is strictly below v, the buyer's valuation. Then, any price strictly less than (1 — e~^)v -^pe~^ would be immediately accepted by the buyer, which means no price strictly less than (1 — e~^)u +pe~^ will be offered. So, p > (1 — e~^)u + pe'"^, which implies p>u, 3. contradiction. Since no price below u is ever charged, such a price would always be accepted if it were offered. This estabhshes that the equilibria of B^ and B^{A} are essentially identical. Hence, we have the following proposition. PROPOSITION 2: There is a unique subgame perfect equilibrium of B^(A), This equilibrium yields the same outcome as the unique equilibrium of B^; the buyer invests 0, the seller offers viO) in the initial period and the buyer accepts.
Thus, neither the unobservability of investment nor the possibility of repeated offers alleviates the hold-up problem. In either case the equihbrium payoffs for the buyer and seller remain 0 and u(0) respectively. However, in the next section we will show that the unobservabihty of the investment decision together with the possibility of repeated offers does remedy the hold-up problem.
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In this section we study the game B(A) and provide the following result: Unobservable investment together with repeated bargaining yields efficiency when offers can be made arbitrarily frequently. Hence, the ultimatum game and the one-sided repeated offers game yield identical payoffs with complete information but the corresponding games lead to very different payoffs when the investment decision is unobservable. Propositions 4 and 5 rely on the analysis of the problem of one-sided bargaining with one-sided uncertainty due to GSW. The relevant results from that paper are summarized in Proposition 3 below. Let H be any distribution over valuations such that H(0) = 0, H{v)==l for some f < 00 and let Bj^ denote the corresponding one-sided bargaining game with one-sided uncertainty. GSW characterize all stationary equilibria of the game JB^. They show that stationary equilibria can be described by two functions, a function that determines the behavior of the buyer on and off the equilibrium path and a function that determines the behavior of the seller along the equilibrium path and after some off-equilibrium path histories. GSW also provide conditions under which all equilibria are stationary. Ausubel and Deneckere (1989) use a similar construction to study stationary equilibria. The definition below is closely related to their approach. The function q describes the buyer behavior, the function r describes the seller behavior after certain histories (in particular, along the equilibrium path), and the function 77 describes the seller's expected payoff after those histories. DEFINITION: The nonincreasing, left-continuous functions g : R^ -> [0,1], r:g(R^.)->R^ and the nonincreasing, continuous function i7:g(R_^)u{0}-^ R^. are called a consistent collection if: (i) mq'')'.= i:i=^[q^^' - q^]p^e-^^,ioT ^W ^^ e ^ ( R ^ ) u {0}, where ; 7 ^ riq^Xq^^'^-^qip^) for k>{), (ii) p = r{q^) maximizes [q{p) - q^]p + e'^^niqip)) for all q^ e ^(R+). (iii) V — p = e'^^lu — riq(p))] whenever p maximizes [q(p) — q^]p + e-^'niqip)) for some q^ e q(R^) u {0} and v = sup{u'\mv') < 1 - q(p)l DEHNITION: A sequential equilibrium of Bf^ is stationary if there exists a consistent collection q, r, II such that:
(i) If p^ is in the support of prices charged in the initial period then p==p^ maximizes q(p)p + e'^lKqip)), (ii) If p^ is the lowest price charged in some (possibly off-equilibrium path) k-1 period history and p^ = r(q^) for some q^ e [0,1], then in period /:, the price riq(p^)) is charged and the probability of agreement is iqir(q(p^))) -
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It follows from the two definitions above that in a stationary sequential equilibrium, along the equilibrium path the seller does not randomize after the initial period.^ The function r describes the seller's behavior along the equilibrium path and the function q describes the buyer's behavior on and off the equilibrium path. To see how q defines the behavior of each type of the buyer, consider any p such that H is strictly increasing and continuous at v such that //(t;) = 1 — qip). If the seller charges the price p, all buyer types v' >v accept the current offer. All other types reject p,^ The seller's payoff conditional on q^
is m ^ V d - ^ ' x GSW show that given a consistent collection q, r, 77 any pricing strategy that satisfies (i) in the definition of a stationary equilibrium and subsequently chooses prices according to r is a best response to the buyer strategy implied by q. Conversely, given a consistent collection, the buyer strategy implied by ^ is a best response to any seller strategy that satisfies (i) in the definition of a stationary sequential equilibrium and subsequently chooses prices according to r. Finally, GSW establish that given any consistent collection q, r, 77, and an initial period pricing rule F satisfying (i), a stationary sequential equilibrium can be constructed by specifying off-equilibrium behavior appropriately. Hence, we sometimes refer to such F,q,r, TI ^s 3. stationary sequential equilibrium of B^. PROPOSITION 3 (GSW): Let y' > 0 denote the lower boundary of the support of H {i.e., u^ = sup{v\H(v) = 0}) in the game B^j.
(0) In any sequential equilibrium a price below u^ is never charged and the probability that agreement will never be reached is 0. After any history, if a buyer with valuation v accepts price p, then all types with higher valuation also accept p. If for every sequence Vf>v\ v^ converges to v^ and \im H(Vf)/(Vt — v^) = a implies a> 0, then: (1) For some K<^, the probability that the game ends by period K is 1. (2) There is a unique consistent collection q, r, 77. (3) A sequential equilibrium exists and every sequential equilibrium is stationary. If there exists a sequence v^> v\ converging to v^ such that Urn HiVf)/(v^ — v^) = 0, then: (4) A price p < v^ will not be charged after any history. Hence, for any K<^ the probability that the game ends after period K is strictly positive. The claims (0)-(3) above are established in the proof of Theorem 1 in GSW. The proof of (4) is in the Appendix. ^ Ausubel and Deneckere (1989) call stationary equilibria, weak-Markov equilibria. If the function q is strictly increasing, then they say that the equihbrium is a strong-Markov equilibrium. Both GSW and Ausubel and Deneckere (1989) use a slightly different description of stationary sequential equilibrium. The translation from the latter authors' definition to the one presented above is straightforward. Deriving the buyer's strategy from q is shghtly more comphcated (see the proof of Proposition 4) if H is not invertible in a neighborhood of 1 - qip).
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We would like to use Proposition 3 in our subsequent analysis. In particular, we would like to conclude that a sequential equilibrium of B(A) induces a sequential equilibrium of the game Bjj where H = G ° v~^ and G is the equilibrium investment strategy of the buyer. This is not an immediate consequence of the definition of a sequential equilibrium. A deviation by the buyer, during the bargaining stage of B(A) may cause the seller to assign positive probability to a set of valuations not in the support of H, Such a situation cannot arise in a sequential equilibrium of B^j. Nevertheless, possible deviations by the buyer play no significant role in the analysis of either game and the buyer is the only player that has a strategic choice prior to the bargaining stage. Hence, it is fairly easy to show that Proposition 3 applies to the bargaining stage of the game B(AX We refer to the collection G, F, q, r, iJ as a strategy profile for B(AX where G is a probability distribution over investment levels, i^ is a probability distribution over initial period prices and q, r, J7 is a consistent collection. Obviously, these are not the only strategy profiles one could have for the game B{A). However, in the proof of Proposition 5 it is shown that in any sequential equilibrium of BiA), Vf>v\ v^ converges to v^ and \\mH{v^)/{u^ — v^)= a implies a > 0, where H is the distribution of valuations at the bargaining stage and u^ = s\x^[v\H{v) = 0}. Hence, by Proposition 3, every sequential equilibrium of B(A) will indeed be of this form. Proposition 4 below establishes the existence of a sequential equilibrium for the game B(A). Throughout the remainder of this section and in Section 4, we assume that v is increasing, strictly concave and continuously differentiable on (0, M). Define u'iO) — lim^^ Q+ v'(x\ We also require that u'(0) < oo. PROPOSITION 4: The set of sequential equilibria of the game B{ A) is nonempty. In any sequential equilibrium, the seller's payoff is at least v{G), the buyer's payoff is 0, and 0 is in the support of the buyer's investment strategy. PROOF: The proof of existence of a sequential equilibrium is in the Appendix. Here we prove that the seller's equihbrium payoff is at least v{Q), the buyer's equilibrium payoff is 0, and 0 is in the support of the buyer's investment strategy. Let x^ := sup{x|G(x) = 0}. Clearly, x^>0. By an analogous argument to the one used in establishing Proposition 2, no price below v{x^) is charged in equilibrium and hence any such price would be accepted with probabihty 1 if it were offered. This estabhshes that the seller's payoff is at least v{x^) > z;(0). Since x^ is in the support of G, it is an optimal choice of investment (see (2) in the proof of Proposition 1). Then, since no price below v{x^) is ever charged, the equilibrium payoff to the buyer is at most -xK But 0 is an attainable payoff. Hence, jc' = 0. Q.E.D.
The proof of existence of a sequential equilibrium for the game B{A) is constructive. When v is twice continuously differentiable and v" < 0, the
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equilibrium constructed has the following features: The buyer randomizes in his investment decision according to the continuous distribution G, where G has support [0, x% is strictly increasing throughout its support, and is piecewise differentiable. The efficient investment level x* is the only point of discontinuity of G. The seller randomizes according to the distribution F, where F is continuous and strictly increasing throughout its support. Given the distribution of valuations H induced by the investment strategy G, the subsequent behavior of the agents constitutes a stationary equilibrium of the resulting bargaining game 5 ^ . The main task in the proof is to choose G and F so that the stationary equilibrium yields 0 utility for every investment decision in the support of G. When v is not twice continuously differentiable (or v"{x) = 0 for some X e (0, M)), we construct a sequence of v^ that satisfies these properties and converges to u. Then, we show that the sequential equilibria for z;„ converge to a sequential equihbrium for u. Proposition 5 below states that as the time between offers becomes arbitrarily small, the equilibrium outcome converges to efficient investment and immediate agreement at a price that compensates the buyer fully for his investment (p = v(x*) -X*). To understand the result better, consider the follov^'ng incorrect argument: As we know from Proposition 4, 0 must be in the support of the buyer's investment decision. Then, the Coase conjecture for the gap case (i.e., u^ = uiO) > 0) states that as the time between offers becomes arbitrarily small the first price charged in equilibrium must fall to v{0). But this implies that the buyer should maximize v(x) — v(0)—x, that is, choose x =x* with probability 1. Hence, 0 is not in the support of the buyer's investment decision, a contradiction. Where is the flaw in this argument? We have already proven the first assertion, that 0 must be in the support of the buyers investment decision (Proposition 4). The final step is also correct; if the price were to fall to L'(0) instantaneously, then it would indeed be uniquely optimal for the buyer to invest X*. What is incorrect is the appeal to the Coase conjecture which establishes that for a given distribution of buyers valuations, the price must fall to the lowest valuation in the support of the distribution, as the time between offers becomes arbitrarily small. But in the current setting, the distribution of buyer's valuations is not given exogenously but is determined in equilibrium. The Coase conjecture is not uniformly true over all distributions with the same lowest valuation in their support. PROPOSITION 5: For every e>0 there exists ^* > 0 such that A< A^ implies that in any sequential equilibrium of B( A) the probability of agreement by time 6 (i.e., period e/A) is at least 1 — e and the probability that the buyer's investment is in [x* — e,x*] is at least 1 — 6. That is, as the time between offers becomes arbitrarily small, the sequential equilibrium outcomes become efficient and the seller extracts all of the surplus. PROOF: Let G„ be the cumulative distribution describing the buyer's investment behavior in some sequential equilibrium a^ of BCA^). Thus, after the
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investment stage, the seller and buyer are engaged in a one-sided offer bargaining game with one-sided uncertainty as described in Proposition 3, where H^==G„° v~^. By Proposition 4, i;(0)>0 is the lowest value in the support of H„. First, we show that for all zl > 0, in any sequential equilibrium of B{ A), Uf > v(0) converges to viO) and Mm H{v^)/{uj — v{G)) = a impHes a > 0. Suppose this condition is not satisfied. Let a- be a sequential equilibrium of B{A), By (0) and (4) of Proposition 3, there exists a sequence kj converging to infinity such that vixj^y > viO), u(xf^) converges to viO), each v(Xf^) is in the support of H (i.e., x^ is in the support of G), and the buyer with valuation u(Xf^) buys in period kj. By Proposition 4, the buyer's equilibrium payoff is 0 and since a price below v(0) is never charged (Proposition 3), we have [u(x,^) — uioyje'^^j^ —x^^ > 0 for all kj. Rearranging terms in the above inequality yields hmy_^ J(t;(x^ ) — v(0))/Xf^)e~^^^> 1. But the first term on the left-hand side goes to u'(0)
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q)(p)—€viOX^—e~^). Thus, the equihbrium strategy that results in an e probability of disagreement until time e yields a payoff bounded above by /[L'(0),oo)/^^(1 —q)ip)- ^^(0)(1 —e~^) v^hen an alternative strategy that can extract all surplus and achieve a payoff arbitrarily close to j[v(oi^)Pd{l -q)(p) exists, a contradiction. So the first assertion is estabHshed. A buyer who plans to purchase at time t will choose his investment level to maximize E[(v(x) —pit))e~^]~x, where p(t) is the price charged at time t and the expectation is over the possible randomization of the seller in period 0. As t approaches 0, the maximizing level of investment x approaches x*. It follows from the first part of the proof that in the limit, the equilibrium investment behavior of the buyer must converge to the optimal investment level conditional on buying at time 0 (i.e., x*). By Proposition 4, the buyer's payoff is 0. Hence, the final assertion of the Proposition follows. Q.E.D. To understand why Proposition 5 holds, consider the simpler case in which the buyer has only two options: He can either invest 0 or x* > 0. As noted in Proposition 4, in equilibrium, the buyer must choose 0 with positive probability and his utility must be 0. As the time between offers goes to 0, the Coasian effect precludes delay and would force the initial price to ^(0) if the probability of investing 0 did not go to zero. But, this would mean that investing x* v^th probabihty 1 yields strictly positive utility to the buyer, which we argued cannot be. Consequently, the probability of investing 0 must go to 0 as the time between offers goes to 0. The key observation is that when the time between offers goes to zero, it may take a positive amount of time to ensure that trade takes place with probability 1, but the Coasian effect still guarantees that the expected time until trade approaches zero. It is of some interest to figure out what the behavior of the buyer (i.e., q) is like and in particular, how this behavior prevents the seller from succumbing to the temptation of running down the buyer's demand curve in the "blink of an eye." As the time between offers becomes arbitrarily small, the probabihty of the buyer purchasing the good at the price to be charged at time t becomes arbitrarily large compared to the probability of purchasing at the price to be charged at time t + e. This is true in spite of the fact that prior probability of purchase at or after time t is going to zero for all t greater than 0. Thus, conditional on not reaching agreement prior to time /, the probabihty of the game ending almost immediately after / is arbitrarily close to 1. This makes it not worthwhile for the seller to try to speed up the process. Proposition 5 has a peculiar observational implication: In situations satisfying the assumptions of the Proposition, in equilibrium, nearly always the buyer will invest nearly efficiently and nearly always trade will take place almost immediately at a price that covers the (sunk) cost of investment. An observer might be inclined to conclude that the compensation of the buyer's investment by the seller is due to some extra-strategic notion of fairness or a possible repeated game effect supporting such a norm. However, in our model, a purely strategic one-shot relationship is able to support this apparently paradoxical outcome.
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A crucial assumption in the preceding analysis was v(0)>c where c, the constant marginal cost of production was normalized to 0. Hence, we had assumed strictly positive gains from trade, even with 0 investment. Relaxing this assumption is important not only because the case of uncertain gains from trade is of some interest but also to facihtate comparisons between the current work and the incomplete contracts literature. To allow for uncertain gains from trade, we modify the model of Section 3 by assuming that the cost of production C, is random. We assume that the random variable C is nonnegative and has finite support. We also assume that the cost of production is incurred by the seller at the time of agreement. When the realization of C is above v(x) there are no gains from trade. The realization of C is observed by both agents prior to the bargaining stage. Let B^(A) denote the game with uncertain gains from trade. Let S denote the expected gains from trade as a function of the buyer's level of investment. That is, Six) = E^ < vix)^^(^^ ~ c)Prob{C = c} - x . Note that since u is continuously differentiable, S is continuously differentiate at all x such that Prob{C = u(x)] = 0. It is easy to verify that the left-derivative of 5 at x is u'(x)TYob{C < v(x)] - 1, Let Xc •= {x\?roh{C = vix)) > 0] and X* -= {x|L''(x)Prob{C < L'(x)} = 1}. We make the following assumption: ASSUMPTION
A: (i) X^ n ( Z * U {0}) = 0 . (ii) S is strictly quasi-concave.
Part (i) of Assumption A is a genericity requirement. For any S that fails (i) there exist e > 0 such that replacing C with C — C for any ^ e (0, e) ensures that (i) is satisfied. Moreover, if C satisfies (ii), then e can be chosen sufficiently small so that C — ^ satisfies both (i) and (ii). By (ii), there is a unique maximizer X*, of S. Since 5'(x) < 0 for all x < min X^ and 0 ^X^, (ii) of Assumption A imphes that either x* = 0 or Prob{C < v(0)] > 0. The other important consequence of Assumption A is that it ensures that the left-derivative of S is strictly positive on (0,x*). We prove below, a result analogous to Proposition 5 and show that as the time between offers becomes arbitrarily small, the buyer's investment strategy converges to x* and the probability of agreement by time e converges to Prob{C < t;(x*)}. That is, investment and trade become efficient as the time between offers converges to 0. In the current setting, it could be that at the bargaining stage, the probability that the buyer's valuation is in the interval (c, c + €) is strictly positive for all positive 6. Hence, we are in what is called the "no-gap" case of the bargaining problem.^ It is well-known that the Coase conjecture is not valid in the no-gap case without further assumptions. For the gap case. Proposition 3 guarantees
^See GSW and Ausubel and Deneckere (1989).
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that sequential equilibria are stationary. To prove the Coase conjecture in the no-gap case, GSW restrict attention to stationary sequential equilibria of B^, The existence of stationary sequential equilibria of ^ ^ , for the no-gap case is established by Ausubel and Deneckere (1989). We take the same approach as GSW and restrict attention to sequential equilibria of B^{A) that specify a stationary equilibrium for the bargaining stage. A sequential equilibrium a of B^{A) is stationary if for each c in the support of C, cr specifies a strategy profile that constitutes a stationary sequential equilibrium of the bargaining stage given cost c. In the no-gap case, the consistent collection q^,r^, U"^ associated with a given investment decision and cost c need not be unique. PROPOSITION 6: Suppose the game B^{A) satisfies Assumption A. Then, for every 6 > 0, there exists A* >Q such that A< A* implies that in any stationary sequential equilibrium of B^{A\ the probability that the buyer's investment is in [x* — 6,x*] is at least 1 — e. Moreover, conditional on C < f(jc*), the probability of agreement by time e is at least 1 — e. That is, as the time between offers becomes arbitrarily small the stationary sequential equilibrium outcomes become efficient and the seller extracts all of the surplus.
PROOF: See Appendix.
To see v^hy the restriction to stationary sequential equilibria is necessary, note that the later the buyer expects to trade, the lower will be his optimal level of investment. But when there is no-gap, it is known that many equilibria can be sustained, including ones in which there is substantial delay.^ To see why Assumption A is necessary, suppose that C has two elements in its support c^> CQ, Let x* be the unique maximizer of S and x** maximize Prob{C = Co}(z;(x) —CQ)—X. Hence, x* is the efficient investment level while X** is the efficient level of investment in the game B^{A) where Prob{C = CQ} = ProMC = CQ} and Prob{C =M} = Prob{C = Cj}. The game B^{A) is just like the game B^{A) except that instead of c^, C takes on a value that precludes gains from trade. In general, it is possible for x** to be strictly less than x*. Moreover, if Assumption A is not satisfied, then it is possible that i;(x**)
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To prove Proposition 6, we consider a convergent sequence of equilibrium outcomes as A approaches zero. The finiteness of the support of C ensures the existence of such a sequence. To see how the proof of Proposition 6 works, let G be the limiting distribution of the buyer's investment decision. If x is in the support of G, the buyer who invests x should not be able to increase his payoff with a small change in his investment without changing his buying strategy. In a stationary sequential equilibrium, the analysis of Proposition 5 suffices to show that the expected time until agreement is reached conditional on strictly positive gains from trade converges to 0. With unobservable investment the buyer invests efficiently given the probability and timing of trade. Hence, if S'(x) is well-defined and X is in the support of G, then SXx) == 0. If S'(x) is not well-defined and X is in the support of G, then the left-derivative of 5 at x must be 0. Given Assumption A, the only G consistent with this requirement is the distribution with unit mass at x*. Finally, the argument used in Proposition 4 establishes that in any stationary sequential equilibrium of B^(A) the buyer's payoff is zero and hence the seller extracts all surplus. Proposition 6 enables us to replace the requirement v(0) > 0 used in Proposition 5 with the weaker requirement that there should be some chance that C is less than v(0). Proposition 7 below shows that this condition is essential. PROPOSITION 7: If Prob{C > v(0)} = 1, then in any sequential equilibrium of B^(A) the buyer invests 0 and both players receive 0 utility.
PROOF: See Appendix. 5. TWO-SIDED INVESTMENT
In this section we consider a game in which both the buyer and the seller invest prior to the bargaining stage. For simplicity, we assume that the buyer invests either 0 or x* > 0, while the seller invests either 0 or j * > 0. As in the previous sections, the investment of the buyer x determines his valuation f. Now, the investment of the seller y determines her constant marginal cost of production c. In order to avoid trivial cases, we assume that 0 investment is inefficient for both agents. Hence v* — v^ —x* >Q and c^ — c * — j * > 0 where v^, c^ denote the valuation and cost associated with 0 investment and y*, c* denote the corresponding values for positive investment. Also, we assume u^ > c^. Hence, it is connmnon knowledge that there are strictly positive gains from trade. We consider two different extensive form games. In the first game, the buyer makes his unobservable investment after learning the investment decision of the seller. Then, the bargaining stage begins. In the second game, the seller and buyer invest simultaneously, prior to the bargaining stage. The investment of the seller is observable in both games. Hence, only the buyer has private information during the bargaining stage. The first game enables the seller to commit to a particular level of investment while the second does not.
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Let B^^iA) denote the bargaining game with two-sided investment where the seller invests first and let B^(A) denote the game in which investment decisions are made simultaneously. Proposition 8 below, estabhshes that the reasoning of Proposition 5 carries over to the case of two-sided investment if the seller can commit to her investment level. By Proposition 5, no matter what the seller invests, the buyer's investment will be efficient and the seller will extract all surplus as A approaches 1. This implies that the unique optimal action of the seller is to invest y*. Formally, Proposition 5 considers only the case in which the set of investment decisions for the buyer is an interval. However, neither the proof of Proposition 5 nor Proposition 3 from GSW require a continuum of types. Given Proposition 5, the proof of Proposition 8 below is straightforward and omitted. PROPOSITION 8: For every 6 > 0 there exists A* >0 such that A< A* implies that in any sequential equilibrium ofB^^(A) the probability of agreement by time e is at least 1 — 6, the probability that the seller invests y* is 1, and the probability that the buyer invests x* is at least 1 — e. That is, as the time between offers becomes arbitrarily small, the sequential equilibrium outcomes are efficient and the seller extracts all of the surplus.
In contrast to Proposition 8, there is a potential source of inefficiency when investment decisions are made simultaneously. To see this, note that a higher constant marginal cost of production renders the seller less impatient to run down the buyer's demand curve (i.e., ^^). Thus, for a fixed investment strategy of the buyer, a higher c results in a higher initial price and, hence, higher expected revenue in equilibrium. But this means that some of the cost of the inefficient choice of y is passed on to the buyer. Consequently, the seller has an incentive to underinvest. Therefore, when the efficiency gain from the seller's investment, c° — c* — y*, is sufficiently small, she will not invest at all. PROPOSITION 9: Fix v^, v*, x*, c^, c*. Then, there exists 6 > 0 such that for all y* > c^ - c* - 5 and e > 0, there exists Zi* > 0 such that A
PROOF: See Appendix.
In Proposition 9, the need for the efficiency gain to be small is an artifact of the discrete investment choice. Presumably, if v were a differentiable function of y, the equilibrium investment level would be bounded away from the efficient investment y*. However, the proof of Proposition 9 entails constructing the consistent collection associated with any possible investment strategy of the buyer. This task is not feasible when the investment choice is a continuous variable.
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The result that unobservable investment and the information rents it creates may provide sufficient incentives for optimal investment, even when the agent investing has no bargaining power, appears to be robust to a number of extensions or modifications of our basic model.^ We conclude by discussing a few other possible extensions and speculate on the implications of our results for the problem of organization design under incomplete contracting. If the buyer's valuation given his investment is random, then it could be possible for the buyer to enjoy strictly positive surplus in equilibrium. For example, suppose that the cost c = 0 is known. Assume that there are two possible investment levels, 0 and the efficient level x* > 0. Let t;^ > 0 be the deterministic valuation that results from 0 investment. However, assume that x* leads to the random valuation V* with support [a,b]. Suppose that only the buyer observes the realization of V* (prior to the bargaining stage) and u^
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increasing monitoring and hence forcing one agent to disclose, perhaps partially, what she knows or granting her the right to withhold such information, will influence the rents enjoyed by all agents and hence their incentives to invest. Comparing the example with a random buyer valuation above, with the model of Section 3, suggests that by altering the flow of information both efficiency and a wide range of distributions of surplus may be achieved. Dept of Economics, Princeton Universityy Princeton, NJ 08544, fgul@princeton, edu; www.princeton. edu / ~ fgul
US.A,;
Manuscript received January, 1997;finalrevision received May, 1999. APPENDIX PROOF OF PROPOSITION 1: Let p^ = s\x^{p\F{p) = 0} and p^ = \rd{p\F{p) = 1). Define x^ and x^ for G in an analogous fashion. Let z be called a point of increase of a cumulative distribution function if either z is point of discontinuity or if for every e > 0, H{z + e) > H{z). First, we will make a number of simple observations: (1) In equilibrium, the buyer will always accept any offer below his valuation and reject any offer above his valuation. The seller will not charge a price below v{x^) or above v{x^). (2) If 2 is a point of increase of F{G) then z is an optimal strategy for the seller (buyer). (3) A:^ = O a n d y = z;(0). Moreover, if f; is strictly concave and continuous, then: (4) x^ ==x* and if A: e [0, JC*), then JC is a point of increase of G. The proofs of (l)-(4) are straightforward and are omitted. Since x^ and /?' are points of increase, the first sentence of the Proposition follows from (2) and (3). (1) estabhshes the second sentence. To conclude the proof, note that since the equilibrium payoff of the buyer is 0, (2) and (4) imply jv^yivix) —p)dF(p)—x = 0 for all xe [0,A:*), establishing that F is the desired function. But this imphes, again by (2), that every p e [viOXvix*)) is optimal for the seller. Hence (1 — G{xy)v{x) = v{OX yielding the desired G. Q,E.D. PROOF OF PART (4) OF PROPOSITION 3: Let v(p) = [p-
v(0)e~^]/[l
-e~^]
and note that since a
price below v(0) is never charged, in any equilibrium, a buyer with valuation u > v(p) will accept price p immediately. Let H be the distribution of valuations after some history h. By part (0) of Proposition 3, a buyer type v
7Tj{v,p)^[\-H{v{p))]p, Straightforward calculations yield
'rTt{v,p)-vm p-vm
pHivip)) p-v{Qd
-
pH{v{p)) H-{v)[p-v{m'
Note that since v' = 1/(1 —e~^) and by assumption, lim H{Vf)/[vt — L'(0)] = 0 for some sequence y,, we can find p > viO) such that
^~
pHMp)) H-iv)[p-v(0)]^^'
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Then, 7ri(u,p) — D(0)>0. So, after any history in which uiO) has never been charged, there exists some p>0 that yields a higher payoff to the seller than u(0). Q.E.D. PROOF OF PROPOSITION 4 (Existence of a Sequential Equilibrium): Since v is concave, if v'iO) < 1, then the buyer investing 0 with probability 1 and accepting any offer less than or equal to viO) and the seller asking p = viO) after every history is clearly an equilibrium strategy profile. Henceforth, we assume v'(ff)> 1. A stationary equilibrium of the game B^ consists of G, a distribution of investment decisions, which yields a distribution of valuations, / / , an initial period pricing strategy Fy and a consistent collection (given H) q, r, TI. To ensure that G, F, ^, r, iJ is a stationary equilibrium of B^ we need to verify that q, r^ 11 is 3. consistent collection given the distribution of valuations induced by G, the sellers first period pricing strategy F is optimal given q, r, /7, and G is optimal given F, q^ r, 77. In our proof, we assume that G is piecewise differentiable and strictly increasing. Then, we construct F, q^ r, 77 v^th the desired properties, which determines an equation defining the equihbrium G. Finally, we provide a solution to this equation. Let 8-=e~^. For any v satisfying the hypothesis of the Proposition, let T=min{k>l\8'^ < (^^'(O))). Since v'(0)> 1, r > 1 is well-defined. In the equilibrium we construct, bargaining will continue for at most 7 + 1 periods. In constructing the equihbrium we consider two cases: First, we assume that v is twice continuously differentiable. This makes it possible to characterize a particular equihbrium investment strategy by T differential equations. Then, we use the fact that twice continuously differentiable functions are dense v^dthin the set of v's covered by the Theorem to construct an equihbrium for all such u. In Theorem 3 and the related definition of a consistent collection, q^, the probability of acceptance until the current period, is the state variable. In constructing an equilibrium for the entire game (i.e., including the investment stage) it is more convenient to use as the state variable the investment decision x rather than q^. Hence, instead of 77, the distribution of valuations, we utilize G, the distribution of investment decision. Similarly, instead of r, which determines the price charged in the current period given state g^, we use p, which determines the state in the next period as a function of the current state x. Finally instead of F we use F and instead of q we use P to complete the translation from the state space of <^^'s to x's. Define p, 77 :[0,M]-^[0,M] as follows: p(x) = 0 if (z;'(x)/5)> i;'(0) and pix) = v'-Hv'(x)/8) otherwise. Let x •= max{z| p(z) = 0), 17(0) =jc, rjix) = p~^(x) for all x e (0, p(x*)} and r}(x) ~x* for all xe[p(jc*),M]. Since p is continuous everywhere and strictly increasing on [Jc,M], 77 is well-defined. Moreover, rj is continuous everywhere and is strictly increasing on [0, p(jc*)]. At any state less than jc, the seller makes an offer that is accepted with probabihty 1 (i.e., p = f (0)). Let p^ix}~x for all x&[0,M] and for f > l , let p'(jc):=p(p^~Hjc)). Define V ^^ ^ similar fashion. The function P : [0, M ] -» R describes the highest price the buyer accepts if he has invested X. This function is defined as follows: P{x) = (1 — 5)E7=o ^( p'(x))5^ The function P is continuous and strictly increasing everywhere. Next, we describe the seller's pricing strategy in period 1. The distribution F has support [ pix*), x*] and 1 — Fix) is the probabihty that x will be the state at the end of period 1. F is defined as follows: Fix) = 0 if x < pix*), 1 f(x)
8
=
il-8)v'ix) 1-8 if jc e [ pix*), X*], and Fix) = 1 if x >;c*. For any G that is continuous and strictly increasing in the interval [0,x*) such that G(0) = 0, Gix*) = 1, define Tlix), the expected present value of profit conditional on x, as follows:
nix)-.= -——j^[G'ip'ix))-Gip'^Hx))]Pip'-'Hx))8'for
xGiOyX*]
and 77(0) = [;(0) where G~ denotes the left-limit of the function G. Note that G~ix) = Gix) whenever x¥=x*. The expected present value of profit at the start of the bargaining stage is denoted 77(M) := [1 - G-(jc*)]P(;c*) + 6G-(jc*)77(jc*).
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363
Next, we describe how F, q^ r, TJ can be derived from F, P, p, 77: For p
(A) G is strictly increasing and continuous on the interval [0, x*X G(0) = 0, G(x*) = 1. (B) For all xe(0,jc*), nix)
>
G(jc)-G(y) G(y) . —— Piy) + 8——n(y) Gix) G{x)
(C) [1 - G(x)]P{x) + 8G(x)nix)
for all y e [0, x ] .
is constant for all x e [ pix*X x*X
Then, G, F^ q, r, 77 is a sequential equilibrium of B{AX PROOF: Note that (B) above is necessary for the optimality of the seller behavior after the initial period while (C) is necessary for the optimahty of the initial period randomization. Clearly, if (A) is satisfied, q, r, 77 are well-defined, q, r are left-continuous, 77 is continuous except at 0 and q is nonincreasing. Since p is nondecreasing, r is nonincreasing. For y >x. G(jc)-G(p(x)) 77(x) =
Gix) Giy)-Gipix})
^ ^ ^^ Gipix)) Pi P(^)) + ^-TTTT-ni ^"^^ G(x)
.
^ • Pix))
^ ^ ^^ Gipix)) . •Pipix)) + 8—^—-nipix)) ^^ Giy)
Giy)
[vix) -Pi p'-Hz))]8'-'
+ f
dFiz)
[vix)-Pip'iz))]8^dFiz)-x
whenever y E:ip^^ ^ix*X p\x*)]. Let Wix) be the equilibrium payoff of a buyer who invests x. That is, Wix) ~ Vix, xX Both V and W are continuous. The definitions of P and p ensure that for any realization of x^ in the initial period, along the price sequence Pi p'ix^)X we have uip'ix^))-Pip'ix'))
=
8[vip'ix''))-Pip'^Hx^))].
Hence conditional on any investment level x e [0, x*] it is optimal to buy in period t — lii X e [ p'ix^X X*] and in period t if XG[ p'ix*X p'~ \x^)X That is, (1)
Vix,x)-Vix,y)>Q.
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Take any y e ( p'+Hjc*),p'U*)) for t
- V(x,y)
= [v(y) - vix)W[F(y^)
for some };^
^ipix*Xx*).
+ (1 - / ( / ) ) 5 ] - y + x .
Since y = p'(y^), v'(y) = v'(p'(y^y) = (u'(y^)/8') and v is concave with v' > 1, we have My) — uixy)/{y —x)>l. Hence, the above expression implies that Viy, y) — Vix, y)>0 for all y e (0, x*], xe[0,M]and V{y, x) - Vix, x) lim
Viy, y) - V(x, y) = lim
x-^y
y-x
=0
x-^y
y-x
whenever x e (0, ;i:*]. Then, by (1) F(y, x) - Vix, x) < Wiy) - Wix) < Viy,y) - F ( x , y ) . Dividing all terms by y — x and taking a limit as x goes to y establishes that W'iy) = 0. Since W is continuous and has zero derivative outside of a finite set, W is constant on [0,JC*]. Note that conditional on investing more than jc*, the optimal behavior in the bargaining stage is to buy in the initial period at whatever price the seller offers. Recall that given G, the bargaining behavior is optimal. Hence, the optimality of G follows from Wix) = Wix*)>Viy,x*) for all x^[0,x*X y e ix*, M] and the fact that [0,x*] is the support of G. Given Step 1, the task of establishing existence is reduced to finding a probability distribution G satisfying (A), (B), and (C) above. For the case of a twice continuously differentiable v, we will ensure that (B) and (C) are satisfied by finding G that solves the corresponding differential equations (B*) and (C*) below. Call V neoclassical if v is twice continuously differentiable, //' < 0 on (0,M), and lim^_^o+ u"ix) < 0 . Let Z^-={z^[0,M]\v'iz)=8'v'iO) for some r > 0}, Z* :={z e [0,x*]|6'i;'(z) = y'U*) for some r > 0 } , and X^ == ( 0 , M ) \ [ Z ° UZ*]. Clearly Z ^ U Z * is a finite set and x^X^ implies T)ix)GX^ and either pix) = 0 or p ( x ) e X ^ . Note that when v is neoclassical, p and P are differentiable at every x e (0, My\Z^ Z)X^ and p'ix) > 0, P'ix) > 0 for all x e (0, M)\Z^. Moreover, P' can be continuously extended to any interval [T/^CO), rj^'^^iO)] for any r = 0,..., 7 — L STEP 2: If u is neoclassical, then G satisfying the hypothesis of Step 1 exists. PROOF: Consider the following differential equations on [0, x*]: (B*)
[Gix) - Gi pix))]P'i
(C*)
[1 - Gix)]P'ix)
pix)) -gi pix))[Pi
-g{x)[Pix)
pix)) - 8Pi pHx))] = 0,
- 8Pi pix))] = 0,
where g ~ dG/dx. To prove Step 2, we first show that if G satisfies (A) of Step 1, (B*) at all xEiX^ and (C*) at all A: e (pix*), x*) nX^, then G satisfies (B), (C) of Step L Then, we conclude the proof by constructing such a G. Let G be a distribution function that satisfies (A) of Step 1, (B*) at all x eJiT^ and (C*) at all X e (p(x*), X*) nX^. For all x e (0, x*) and y e [0, x], define Gix)-Giy)
Giy)
.
Since G and U are continuous on [0,x*] so is TT. Moreover, since (B*) is satisfied and v is neoclassical, 7r(x, •) is differentiable at every y e X ^ . If (B*) is satisfied at every x e Z ° then at every such x we have JG(x)i7(x) dx = gix)Pi
364
pix)) +
8dGipix))nipix)) ^ dx
8gi pix))Pi
pHx))p'ix),
Unobservable Investment and the Hold-Up Problem THE HOLD-UP PROBLEM
365
Hence, an inductive argument establishes that dG{x)n{x} =gU)P(p(x))
for all
x&X^.
ax Since [Gir){y)) - G{y)]P'(y) -giyJPiy)
dy d7rix,y)
- 8P( p(y))] = 0 for all y e X \
-P'iy)
G(x) -Gix) Gi-niy))
whenever ye^X^. The term on the right-hand side above is strictly greater than 0 whenever X > T)iy) or equivalently, pix) >y. Similarly, this term is less than 0 whenever pix)
+ dx
8G(x)n(x)] =0
for all
x&{p{x*),x*)\Z^.
Then, (C) of Step 1 follows. Let
Recall that A is well-defined at jc e (0, M)\Z^ and that for any interval [-q^iff), rj'^ ^0)], there is a unique continuous function A^ on this interval that agrees with A at every x^irj'iOXrf^'^^iOy). Clearly, ^^ > 0 for all r = 0,..., 7 - 1. Let ^ d e n o t e the space of all continuous functions on [0, x*] endowed with the sup norm. Consider the class of first order linear differential equations of the form A{x)[fix)-Lix)]=^L'{x). Note that (B*) has this form. We say that L is a solution to this differential equation if L is a continuous function on [0,;c*] and satisfies the equation at every x e ( 0 , M ) \ Z ^ . Clearly, for any initial condition L{y) = a and / e ^ , there is a unique solution to this equation and this solution has a right-derivative at every ;ce[0,M). Let -^(/,fl) denote this unique solution. Since P has a right-derivative at every x<M so does =S!,(/,a). CLAIM 1: fix) >f(x) for all x e [0, y], L --^yif, a), and L --^-^yif, a) imply Ux) < L{x) for all :tG[0,y). PROOF: Clearly, L{x)
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PROOF: Assume I / - / I < e and \a-a\< e and let L ~^y(f, a) and L ~^yif, a). Let (f>{x) = e(^l_^B{y)-B{x)y j ^ ^ ^ ^ (f> =^^{1^,0) and - ( / > = ^ ( - 7 ^ , 0 ) where 7^ denotes the constant function €. Therefore, by Claim 1, 6e^(^> > | ^ / / - / , 0 ) ( x ) | = | ^ / / , 0 ) ( ; c ) - ^ / / , 0 ) U ) | . Also, note that ^e^(M) > |_5.(^^^ ^ _ ^^)(^)| ^ ]^^^(f^a)(x)
-^y(f,a)(x)[
Hence, |^//,a)U)-^//,a)(;c)| < 1 ^ / / , fl)(x) - ^ / / , « ) ( x ) | + \^y(f\ a)(x) - ^ / / , a)(;c)I < 26e^<^> establishing both the continuity of ^ C , -X^) and ( ^ ( ^ •)• Let y:=p'(A:*), Gy=^yo(I^,aX and for f = l , . . . , r define G^ inductively as G^ == - ^ ' « ^ r ^ ° V,G',-\y')y Note that G.^jc) = 1 - [1 - a]e^^^*>-^^^> and hence GJCJC) = 1 for alW > 0 and x e [ 0 , l ] . Since A(x)>0 for all x^X^, an inductive argument establishes that G^ is strictly increasing on [y',y^~^] whenever fle(0,1). Define the distribution G^ as follows: for x>x*, GJix) := 1, for X e [ p%x*X p ' " ^(x*)) and r = 1,...,7, GJix) ••= G'' ^{x\ for x e [0, p ^ " Hx*)), and for X < 0, G^Cx) = G J ~ HO). The construction above ensures that G^ solves (B*). To conclude the proof we need to find a <\ such that G^CO) = 0. By Claim 2, Gj~ HO) = G^CO) is a continuous function of a. Since each G^" ^ is strictly increasing on [y\y^~^\ G^ is strictly increasing on [0,x*]. Hence Go(0)<0 and Gi(0)= 1. Therefore, there exists a* <1 such that G^.(O) = 0. The function G — G^* satisfies all the desired properties. Let Xy y, Z be arbitrary compact intervals in R. STEP 3: (A) If /„ : X -> Y and g^:Y-^ Z converge uniformly to / and g respectively and /„, g„ are continuous, then g„ ° /„ converges uniformly to g ° f. (B) If each /„ : X -> Y is a continuous bijection and /„ converges uniformly to the bijection / , then f^^ converges uniformly to f~^. (C) If /„ : X -> y is nondecreasing and /„ converges to the continuous function / at every point, then /„ converges to / uniformly. PROOF OF A: Since uniform limits of continuous functions on compact sets are continuous, / , g are continuous and hence uniformly continuous. Pick € > 0. By assumption, there exists Cj > 0 such that \g(y)—g(y')\<€/2 whenever |y —y'|< Cj. Also, there exists N such that \f—fn\<€i and \g — g„\< e/2 whenever n>N. Hence for n>Ny \gif(x))
-g„(f„ix))\
< \g(f(x))
-g(f„(x))\
+ \g(f„(x))-g„(f„(x))\
< e.
PROOF OF B : Note that / is continuous; therefore f~^ is continuous and hence uniformly continuous. So, for all e > 0 there exists €^>0 such that \f~^(y) —f~^(y')\ < e whenever \y—y'\< ej. By assumption, there exists N such that n>N imphes If—fJ < ^i- For any n>N and y e y , set ^n-J^Hy) and y„ =/(JC„). By construction \y-yj<e^. So, \r^{y)-f;;Hy)\ = \r^iy)-rHyJ < €.
PROOF OF C : Suppose /„ is nondecreasing for all n (hence, / is nondecreasing). For any e > 0, pick a finite set ZczX, such that for every x^X there exists z ^ z^ ^Z satisfying z^ <x
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Unobservable Investment and the Hold-Up Problem THE HOLD-UP PROBLEM
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For any v that is not neoclassical, define p, 77, P, F as above. We construct a sequence v„ such that each v^ is neoclassical and v^^^v'^ converge uniformly to v,v' respectively. Let v'J^x) -= v'{x) for all X e [0, M] such that v'{x) = v'iOi) + {k/n)[v'{M) — f'(0)] for some integer k
p(x)
X
••=
1+x pix)~p(x)
for all 1+x for all
X G [ — 1, ;c)
and
x^[x,M].
Define p„ by replacing p with p„ and x with x„. Note that jc„ converges to x and p„ converges uniformly to p. Therefore, p„ converges uniformly to p. By (B) of Step 3, p~^ converges uniformly to p~^. Hence, rj„ converges uniformly to 77. Pick an integer K such that 8^ < u'{M)/u'{0). Then, for all x e [0,M], p^(x) = 0 and p^(x) = 0 for all n. Hence, K-l
P„(x) = (1 - a ) 53 ^n( P^(^))5' + 5''z;„(0)
and
K-l
i>U) = ( l - 5 ) X i;(p'(x))6^ + 6^i;(0). /=o Since p„, L'„ converge uniformly to p, i;, part (A) of Step 3 imphes y„( p^JCx)) converges uniformly to v( p'Cx)) and hence P„ converges uniformly to P. STEP 5: G satisfies (A) of Step L PROOF OF STEP 5: First, we prove that G is continuous at every y < jc*. Suppose not and let y > 0 be a point of discontinuity and J be the size of the jump at y. Pick x,y continuity points G sufficiently close so that , . X . M 0<\Piy)-P(x)\<-
J[Piy)-SP{p(y))] 3 + 2[J +
P(y)-8P(p(y))]
and p(y) <x
P(y) - 8P( piy}) ^""^
^^
2
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Also, choose N such that n>N imphes |P„ - F | < e, |P„ « p„ - P « p| < e (Steps 3 and 4 ensure that this can be done), \G„(x) - G(x)\ < e, and \G„(y) - G(y)\ < e. Since G„ satisfies (B) of Step 1, for ^n ~ VnWy ^C haVC
GM
'"^'^ ^ '
^M
G„K)-G„U) ^
TTT—^
'"' '"''^^ ^ '-^My''^^ '"^'^^
G„U)-G„(p„(y)) ^n(^) + 5
-——
P„( p„(3;))
Hence, 0
-2e\
<3e-[/-26][/>(y)-aP(p(y))-26]<0, a contradiction. Since G„(x) = 0 for all x < 0 and G is right-continuous, to conclude the proof of continuity we need to show that G(0) = 0. Assume not and pick /
G(0)z;(0)
G(0)\ €<
and y e (0, x) such that u(y) — v(0) < e. Again, let w„ ~ 'nj.y) and note that since G„ satisfies (B) of Step 1, we have Hniwn) ^
GAwJ-GAx) ; , ,
GJx) PnM + 8-^n„(x)
. for all
X e [0,>;].
In particular, the above inequahty holds for x = 0. Recall that p(x) = 0. Then, since y<x, for n sufficiently large
Hence, we have G„(H'„)-G„(y)
G„(y)
Since P„(y) = (1 — 8)uJ^y)-{- 6i;„(0X the above expression yields Gn(^ny[Vn(y)-vM]
^
G„(y)v„(y).
Since G is continuous at y, for /i large enough \G(y) - G„{y)\ < e. Similarly, for n large enough we have \v„{y) - v(y)\ < e, \viO) - f„(0)| < €. So, for n large enough, 36 > [!;„(>') - v(y) + v(y) - v{0) + u{0) - v„(0)] > G„{w„)[u„{y} - v^]
> G„(y}u„(y) G(0)y(0)
> [Giy) - 6]y(0) > [G(0) - €]v(0) > a contradiction.
368
,
Unobservable Investment and the Hold-Up Problem THE HOLD-UP PROBLEM
369
By construction^ G^ix) = 1 for all x>x^ and n, ;c* converges to x*. Moreover, G is right-continuous. Hence, G{x*) = 1. Since G is continuous at every x<x*, so is U. Therefore, by part (C) of Step 3, G„ and i7„ converge uniformly to G and i7 respectively on [0,fl] for any a<x*. Suppose G is not strictly increasing. Then, we can find 0 < x < j < j c * such that p\x*)^[x,y] for all / and G{x) = G{y). Hence, y < r](x). Suppose x < pix*X Since G„ satisfies (B) of Step 1 and 77„, p„ are nondecreasing, we have G„(77„(jc)) - G„(x)
(2)
n^ivnix}) =
1, , ,; G„irf„(x)) G„irj„(x))-G„{y)^^
GJx)
Pnix) + 8 ; ^
.
n^ix)
G„(7]„ix)) ^ G„{y) ^ ^ ^
G„{ri^{x)) GMrOc))-G^
G„{r)„{x)) G,U)
G„(7/„U))
G„(TJ^(X))
After some manipulation and taking hmits using Step 4, the above expression yields Gix) = Girjix)). Hence G{z) = Gix) for aU z e [x, rjiy)). If T7(X) < p(x*X repeating the above argument with rj(x) in place of X and z e (TJCX), rjiy)} in place of y yields Girj^ix)) = GCryCx)) = Gix). Continuing in this fashion, we will eventually obtain pix*) <x
G„(x)-G„(p„U)) G,(x)
G„(p„U)) P„( p„(x)) - — — - i 7 „ ( p„(x)) "^^"^ ^^+ 5 — G,(x)
< hm[[l - G„( p„(x))]i>„( p„(x)) + 5G„( Pnix))n,i
p„ix))] < Urn n„iM)
= hml[l - G„ix)]P„ix) + 5G„(x)i7„(x)] = 5i7(x). So, i7(x) > i;(0) > 0 and nix) = SlJix), a contradiction. To conclude the proof of existence we will show that G satisfies (B), (C) of Step 1 as well. Recall that G„ and 7J„ converge uniformly to G and 17 respectively on [0, a] for any « <x*. By Step 4 and (A) of Step 3, for every €>0 and x <x* there exists N such that n>N implies Gix)-Gi pix)) ^ ^ ^^ Gipix)) . ^ ^^ - — Pi pix)) + 8———ni pix)) Gix) Gix) G„ix)-G„ip„ix)) G„ip„ix)) G„ix)
-Pni Pn^x)) + 8
"^^"^
G„(x)
n„i p„ix))
- €
for all
0
But since G„ satisfies (B) of Step 1, we have G„ix)-G„ip„ix)) ^ ^ ^^
^) ^
G,ip„ix)) .
_ ,
'"' ^"^^» " '-W^''"' '^'''' ^ ( "
Pniy) + 5 - ^ i 7 „ ( j ) ,
<jr„v:c;
for aO
0
G„ixJ
Again, we can choose n large enough so that the last term above is greater than Gix)-Giy) Gix)
^ ^
Giy) . Gix)
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Hence,
Gix)-Giy) G(x)
Giy) . ^ G{x)
Since e is arbitrary, it follows that G satisfies (B). Verifying (C) requires a similar argument. Q.E.D. PROOF OF PROPOSITION 6: Since C has a finite support, if the Proposition is false, we can find an 6 > 0, a sequence ^„ > 0 converging to 0, and a sequence of stationary sequential equilibria o;,, of the games B^iA„) such that either G„(x*) - G„{x* - e) < 1 - € for all n or for some c < v{x*X the probabihty of agreement conditional on C = c < vix*) by time e is less than 1 - 6 for all n. By Helly's Selection Theorem, we can assume, without loss of generahty that G„ converges in distribution to some distribution G. Call x an active point of G if G(x + a)- Gix- a ) > 0 for all a > 0. Let jc^ denote the minimum of the support of G. STEP 1: For all e > 0 conditional on C = c
the probability of agreement by time e
PROOF OF STEP 1: If the statement above is false, then there exists e' > 0 , c
0
PROOF OF CASE 1: We estabhsh a contradiction by showing that an alternative sequence of prices r* would yield a higher expected profit for the seller than the sequence /?*, which is charged with positive probability.
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Unobservable Investment and the Hold-Up Problem THE HOLD-UP PROBLEM
371
Let p^ = max{/7^
Case 2: ( 2 ' ( e / 3 ) - ( 2 ^ ( e / 4 ) = 0.
PROOF OF CASE 2: Again, we define an alternative sequence of prices rj^ and show that for large enough «, the expected profit associated with this sequence is larger than the profit associated with the equihbrium price sequence /?*. Let /?„(«) ~ mm{p^ >P'^(e/3)), p„( j8) ~ max{/?J^
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as n goes to infinity. A price below c will never be charged. Consequently, a buyer with valuation at least v{x^) will purchase immediately at any price less than {\—e~^"')v{x)^-e~^"c. Hence, i 7 ( i 3 ) > [ l - e - ^ " ] [ t ; ( j c ^ ) - c l > 0 for n sufficiently large. Recall that Q'{e/1>)-Q'{e/^) = 0, Therefore, the net effect of switching to the alternative strategy divided by A^ is bounded below by a term converging to [\ - Q'{e/Me~^'^''^^^^^ - e' '^^lv{x^)- c]. Since Q'ie/3)
and G(x*)
= 1.
PROOF OF STEP 2: Assume x^
<x*.
To get a contradiction first assume that Prob{C = v(x^)} = 0. Since S is strictly quasi-concave. Six) > S{x^) for any x e {x^,x*]. Let c' be the maximum of the set of c's such that c < uix^) and Prob{C = c) > 0. By Assumption A, c' is well defined. Also, let c* be the lowest element in the set of c^s such that c > c' and Prob{C = c} > 0. (If this set is empty, let c* = oo.) Note that for any x such that x^<x<x* and v(x)
ProMC = c][v(x) - vix„)]E[e-''^^'n
-x+x„
c
> S{x)e-'-
(1 - e-' )x - S{x„) > 0.
But this contradicts the fact that x„ is in the support of G„ and hence, is optimal. If ?Toh{C = uix^}) > 0, let c' be the largest c such that c
372
Y, Fioh{C =
c}E{[vix,)-p,ic)]e-^'^^<^^]+\vix„)-vix^)\-x„,
Unobservable Investment and the Hold-Up Problem THE HOLD-UP PROBLEM
373
where the expectation is over the possible random choice of the seller in period 0, given c, t„(c} is the time at which the buyer reaches agreement, and p^ic) is the price at which agreement is reached conditional on x, c and the price sequence. Similarly, c
Therefore, for n sufficiently large, Step 1 yields "x-">
E ^rob{C = c]E{[vix) - vix^)]e-'"^'^] -\v(x„)
- v(x^)\-x
+x„
c
>g(x)e-' -
(l-e-nx-g{x^)-\vixj-v(x^)l
Since \x^ —xJ
where c' is the largest c such that c < vix) and PrQb{C = c) > 0. Again, the expectation is over the possible random choice of the seller in period 0, given c, /„(c) is the time at which the buyer reaches agreement, and p^ic) is the price at which agreement is reached conditional on jc, c, and the price sequence- Suppose instead the buyer invests some x' such that c'
-x\
c
Hence, u^.-u^
= [v(x')-v{x)]
Y, Prob{C = c)£-[e-'''(^>]-Jc'+Jc. c
Since v(x') — v(x) < 0 and E[e~'"^^^] < 1, the last expression yields
u^.-u^>[v(x')-vix)]
E ^rob{C = c}-x'+x
=
S(x')-S{x)>0.
c
The last inequahty follows from the strict quasi-concavity of S. Hence, investing x' yields a higher utihty than investing jc, which proves that the buyer will never invest x>x*. This concludes the proof of Step 2. By Step 2, G(jc*) = l, x^=x* and hence G(x* — e) = 0, Given Step 1, this contradicts our assumption that the Proposition is false. In Proposition 4 we showed that the buyer's equilibrium payoff in B{A) is 0. The same argument yields the same conclusion for B^(A). Hence, the outcome is efficient and the seller extracts all surplus. Q.E.D. PROOF OF PROPOSITION 7: Let CQ denote min{c|Prob{C = c} > 0) and G denote the buyer's equihbrium investment strategy. Let v{x) be the infimum of buyer valuations that purchase with positive probability, in equilibrium, ff v{x) is not well-defined, i.e., no buyer type purchases in equilibrium, we are done. Otherwise, v{x) > CQ and a price below v{x) is never charged. ff CQ > i;(OX then, the expected payoff of the buyer, conditional on investing x, is negative, a contradiction. Hence, no buyer type buys with positive probability in equihbrium and therefore the buyer invests 0 with probability L So, suppose CQ = L'(0). First, assume that there exist no y > 0 such that G{y) = G(0). Then, in any sequential equilibrium, for any A:>0, conditional on C = Co, with strictly positive probabihty the bargaining stage continues beyond period k. This follows from the fact that as long as there is some
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FARUK GUL
probability that the buyer has a valuation strictly greater than CQ, the seller can earn strictly positive profit by charging some price p> CQ that is not accepted with probabihty 1, while charging a price p
period strategy, in some sequential equihbrium oi B^iA^X conditional on the seller investing y* and 0 respectively. Let tz* := [y* - c*]/[z;* - yO], a^ - [D* - c^Mt;* - y°], K{s) •= s{s - \)/2, and define H{a, 4 S)••=Lf= 1 a'e^^^'\ Let O be the set of all sequences ( 4 , 5 * , 5 ° ) satisfying (i)-(iii) below: (i) A„,S*, S^„ > 0, and hm„ A„ = 0. (ii) T* ••= hm AJ* < log[(z;* - u^)/x*] and T^ •= lim A„S^ (T^ = oc is allowed), (iii) hmsup(/f(a*, 4 , S*)/ma^ A„, 5^) < (u^^ - c'^y/iv^ - c*\ Define 5 = [z;* - v^]M^ lim„ _^ Je'"^-^" - e"^"-^*|. To establish that 5 > 0, it is enough to show that for any sequence satisfying (i)-(iii) above, lim„^ Je"^""^" - e~^"^*] # 0. If 5 were equal to 0, we could construct a sequence satisfying (i)-(iii) such that limfe"^""^" — e~^"^"] = 0. Thus, it suffices to show that for any sequence satisfying (i) and (ii) above such that lim A^S^ = hm A„S* = 7* < oo^ Hia*, A„,S*)/Hia^, \yS^) converges to ^. To prove the latter, define 7/„ == S*/S^ and note that since a* > 1 and a^ > 1,
H(a\A„,S'J
SOf^^of'.^AKisO)
(A*/Air' where A* = (a*)'^"ei^"'^"^'^"^»~'^^ and A^ = (a^}ei'^"^^»''^\ Since rj„ converges to 1, A„ converges to 0 and A„S*, AJ^ converge to T* < oc, A*/Al converges to a*/a^ = [v*-c*]/[v*-c^]>l. Therefore, Hia*, A„,S*)/H(a^, A^, S^) converges to 00, as desired. Assume that y* >c^ — c* — 5. Next, for each investment level of the seller, we construct the unique consistent collection associated with the bargaining stage. By Proposition 3, for any A>Oy with probability 1, the game ends in finite time and PQ = v^ is the last price charged. A price below i;^ is never charged and in equilibrium the buyer accepts this v^ whenever it is offered. Moreover, the low valuation buyer will only buy in the last period. Hence, the buyer with valuation v* randomizes and is indifferent over the outcome of this randomization. Let the prices p'^ be indexed backward from the last period. The buyer's indifference implies f* —/?,+ ! = [v* —pf]e~^. This equation together with the initial condition PQ = v^ yields (1)
p', =
u*-[v*-v^]e-^\
Let m^ denote the mass of buyers that purchase in period s. That is, m^ is the unconditional probabihty that the buyer purchases in period s. For 5 > 0, the profit of the seller conditional on period 5 + 1 being reached, evaluated in period s + 1 units, is (2)
374
^s+i=f^s + i(Ps+i-c) + e
\
Unobservable Investment and the Hold-Up Problem THE HOLD-UP PROBLEM
375
where c is the cost of the seller and JTIQ denotes the probabihty that the buyer invests 0. The equihbrium levels of m^ are determined by making the seller indifferent between charging this period's price and charging next period's price. That is, (3)
'^s+i=^s+i(Ps-c)
+ 7r^-
Charging p'^ has the disadvantage that less profit is made on the mass /w^+i who would have purchased now at the higher price Pg+i, but has the advantage that it avoids the time lost on the continuation profit. The w^+i that solves (3) makes the seller indifferent between charging p^+i and p'^. Solving (1), (2), and (3) yields, for s> 1, (4)
m, = a^e^^^^>mi,
where a = [v* — c]/lu* — v^] and K(s) •= sis — l)/2. Note that TTQ = mo(po~<^)'^'^o^^^~^)• Hence, solving for mj using (2) and (3) yields m^ = brriQ where h •= {v^ — c)/f;* — y^). Let S be the smallest integer such that EQ m^ > 1. Then, 5-1
(5)
[bH{a,A,S-l}
+ l]mo=
5
Y.m^
Ylm,^[bH(a,
A,S) + l]mQ.
5=0
The mass of buyer's who do not buy at price p is T^^'^sipy^s'^ where sip) is the largest value of s for which p>p's' We define sip) to be —1 if p
p* = v*-[v*-v'^]e-^\
p^=^v*-[u*-v^]e-''\
T*
p*
Since the payoff of the buyer is 0 and the probability of the buyer investing ;c* is strictly positive, we have (7)
y*[u* -p*]^
il-y*)[v*
-p^]=x\
From (6) and (7) we obtain (8)
r * < log
By the argument used in proving Proposition 5, for any € > 0, the probabihty of agreement being reached by time e is at least 1 — e for sufficiently large n. Therefore, as n goes to infinity, the initial
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Faruk Gul 376
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price minus cost will equal the seller's profit. Since investing y* is an optimal action for the seller, we have (9)
p*-c*-y*>p^-c^.
From (4) and (5) we know that (10)
[b*Hia*, A„,S*-l)-h
l]mo,„ < 1,
where b* = {v^ - c*)/iv* - v^) and b^ = {u^ - c^VC^;* - v^X Then, (10) imphes
(11)
limsup//(fl^zl„,5:-l)///(fl^4,5„^)<^?V^7^
Thus, we have shown that the sequence iA^,S* — 1,S^) is in i7. Hence, \p^ —p*\> 8, Since pO _p* > 0 by (6), we conclude that p^ - / ? * > 5. But this contradicts (9), since y* > c^ - c* - 8. Q.E.D. REFERENCES AGHION, P., M . DEWATRIPONT^ AND P . REY (1994): "Renegotiation Design with Unverifiable Information," Econometrica, 62, 257-282. AusuBEL, L., AND R. DENECKERE (1989): "Reputation in Bargaining and Durable Goods Monopoly," Econometrica, 57, 511-531. BiLLiNGSLEY, P. (1986): Probability and Measure. New York: John Wiley and Sons. CHE, Y-K., AND T - Y . CHUNG: "Incomplete Contracts and Cooperative Investment," Mimeo, University of Wisconsin. CHE, Y.-K., AND D . B . HAUSCH (1996): "Cooperative Investments and the Value of Contracting: Coase vs. WiUiamson," Mimeo, University of Wisconsin. FUDENBERG, D . , D . LEVINE, AND J. TIROLE (1985): "Infinite-Horizon Models of Bargaining with One-Sided Incomplete Information," in Game-Theoretic Models of Bargaining, ed. by A. Roth. Cambridge: Cambridge University Press. FUDENBERG, D . , AND J. TIROLE (1990): "Moral Hazard and Renegotiation in Agency Contracts," Econometrica, 58, 1279-1320. GIBBONS, R . (1992): Game Theory for Applied Economists. Princeton, New Jersey: Princeton University Press. GROSSMAN, S., AND O . HART (1986): "The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Investment," Journal of Political Economy, 94, 691-719. GROUT, P. A. (1984): "Investment and Wages in the Absence of Binding Contracts: A Nash Bargaining Approach," Econometrica, 52, 449-460. GUL, F., AND H . SONNENSCHEIN (1988): "On Delay in Bargaining with One-Sided Uncertainty," Econometrica, 56, 601-611. GUL, F., H . SONNENSCHEIN, AND R . WILSON (1986): "Foundations of Dynamic Monopoly and the Coase Conjecture," Journal of Economic Theory, 39, 155-190. HERMALIN, B., AND M . L . KATZ (1991): "Moral Hazard and Verifiability: The Effects of Renegotiation in Agency, Econometrica, 59, 1735-1754. MA, C.-T. A . (1991): "Adverse Selection in Dynamic Moral Hazard," Quarterly Journal of Economics, 106, 255-276. (1994): "Renegotiation and Optimahty in Agency Contracts," Review of Economic Studies, 61, 109-130. MATTHEWS, S. A. (1995): "Renegotiation of Sales Contracts," Econometrica, 63, 567-590. RoGERSON, W. P! (1992): "Contractual Solutions to the Hold-Up Problem," Review of Economic Studies, 59, 777-793.
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16 Arunava Sen on Hugo R Sonnenschein
I applied to Princeton's graduate program in economics in 1982. The main reason I applied was that Vijay Krishna and Dilip Abreu were both at Princeton. They had been a couple of years ahead of me at the Delhi School of Economics and aheady had legendary status there. Both were students of Hugo and word of his quaUties as an advisor had spread on the grapevine. I was both astounded and petrified when I received word that Hugo wanted me to call him (collect). I had no idea that famous professors spoke to prospective graduate students. In fact, I was sure that I would be asked some tricky technical questions and had half a mind to keep my notes and textbooks within easy reach when I called. Instead, Hugo informed me that the department wanted to offer me a fellowship and urged me to accept. It is one of the most serendipitous moments of my life. Hugo could have let the office send me a letter but it was typical of him to take the time out to make a personal connection. At Princeton I attended Hugo's famous micro theory course in my first year. These lectures were so polished that they had an amazing clarity. Later I attended the Advanced Theory course that he offered which involved reading current research. I confess that I didn't understand ever5^hing but it was wonderfully exciting and stimulating. Soon it was time for my second year research paper. After discussions with Vijay and Faruk Gul, I decided to work on implementation theory. I made this announcement to Hugo and he agreed to supervise me. I remember the first step in this endeavor clearly to this day. On a wintry weekend morning in late '83, Hugo made me present, in the greatest possible detail, Maskin's classic result on implementation. One aspect of the result which was disappointing to him was that in the natural case where the planner's objectives are single-valued, implementability implied dictatorship. This led to the question that I tried to address in the second-year paper which later evolved into the first chapter of my thesis and which I have selected for inclusion in this volume. Suppose that the planner's objectives are single-valued but non-dictatorial. What is the way to implement it "as best as possible?" The proposal was to embed it "minimally" in an implementable social choice correspondence and then examine the distance between the two as in the limit the number of voters increases without bound. The thesis progressed infitsand starts. Hugo was always at hand to offer insight and encouragement. He had high standards and he emphasized rigor, clarity and depth. I have learnt enormously from him, and I feel both proud and privileged to have been his student.
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The Implementation of Social Choice Functions Soc Choice Welfare (1995) 12:277-292
ialt]loi= 9 Springer-Verlag1995 .
.
.
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.
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The implementation of social choice functions via social choice correspondences: A general formulation and a limit result Arunava Sen*
Indian Statistical Institute, New Delhi 110016, India Received: 2 October 1993/Accepted:2 November1993 Abstract. A corollary of Maskin's characterization theorem for Nash implementable social choice correspondences is that only trivial social choice functions can be implemented. This paper explores the consequences of implementing non-trivial social choice functions by extending them minimally to social choice correspondences which are implementable. The concept of asymptotic monotonicity is introduced. The main result states that it is not possible to find social choice rules satisfying a mild condition on its range, which is asymptotically monotonic. The implication of this result is that the multiplicity of equilibria problem which is at the heart of Nash implementation theory persists even in the limit as the number of individuals in society tends to infinity. This is true even though the opportunities for an individual to manipulate the outcome disappears in the limit.
1. Introduction
In an important contribution to the theory of incentives, Eric Maskin (Maskin (1977)) characterized the class of social choice correspondences (SCCs) which are implementable in Nash equilibrium. A SCC associates a non-empty set of social alternatives with every profile of preference orderi.ngs. A game form specifies a social alternative as a function of individual announcements. It is understood to be a representation of economic institutions which describe the way in which individuals interact. A SCC is said to be implementable if it is the Nash equilibrium correspondence of some game form. A society whose objectives are given by an implementable SCC can collectively choose to adopt the game form which
* This paper is an extensively revised version of a chapter of my Ph.D. dissertation submitted to Princeton University in June 1987. I wish to thank my advisor Hugo Sonnenschein for his valuable advice and constant encouragement. I am also grateful to Andrew Caplin, Vijay Krishna, William Thomson, Jean.Luc Vila and two anonymous referees of this journal for their numerous suggestions. All remaining errors are my own responsibility.
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implements it. If they do so they are assured that all equilibrium outcomes will be socially optimal irrespective of the preferences actually realized. Maskin's Theorem says that a SCC which is implementable must satisfy a condition called monotonicity. Conversely, if a SCC satisfies monotonicity and an additional condition called no-veto power, then it is implementable, provided that there are at least three individuals.^ The no-veto power condition is weak, so that monotonicity is almost necessary and sufficient for implementation. This paper takes the point of view that the ethic representing the social ranking of alternatives is defined by a singleton valued SCC, i.e. a social choice function (SCF). It is assumed that the domain of individual preferences is the set of all strict orderings of the social alternatives. It follows immediately from the results of Maskin, Gibbard-Satterthwaite (Gibbard 1973; Satterthwaite 1975) and Muller-Satterthwaite (1977), that the only SCFs implementable over this domain are dictatorial. This impossibility result motivates the issues raised in this paper. Suppose social objectives are specified by a non-trivial SCF. How should these objectives be implemented **as best as possible" and what if any, are the consequences of not being able to achieve the objectives "exactly"? The nature of the enquiry is "second best" in spirit and stands in contrast to the traditional approach of searching for restrictions on the domain of preferences which will permit "exact" implementation. A monotonic extension of a SCF is defined to be a SCC which includes the SCF and is monotonic. The trivial SCC which maps every preference profile to the entire set of social alternatives is a monotonic extension of all SCFs. It is shown that intersection of two monotonic extensions of a SCF is also a monotonic extension of the SCF. The intersection of all monotonic extensions is therefore also a monotonic extension and is referred to as the minimal monotonic extension of the SCF. If the SCF satisfies the no-veto power property, it is easy to show that its minimal monotonic extension is implementable. It is proposed that a SCF be implemented via its minimal monotonic extension. This procedure ensures that the socially optimal alternative is always an equilibrium outcome for all possible realization of preferences. Moreover, the number of socially non-optimal equilibria is minimized.^ The consequences of implementing the minimal monotonic extension of a non-trivial SCF is that non-optimal equilibria will arise for certain preference realizations. The focus of this paper is an attempt to provide an idea of the size of the set of profiles where such non-optimal equilibria occur. The analysis is restricted to classical "social choice" environments, i.e. environments where the set of outcomes is finite and there is no structure on preferences. Thomson (1993) extends the analysis to a variety of other "economic" environments. Consider the plurality rule which for every preference profile picks the alternative that is ranked first by the largest number of individuals. Ties are broken arbitrarily. Observe that an individual can influence the outcome of the rule only if all other individuals are (almost) exactly divided in their support of some set of alternatives. Clearly, the rule is asymptotically strategy proof in the sense that it
^ The gap between the necessary and sufficient conditions has been closed in Moore and RepuUo (1988). Moreover, the characterization has been extended to the case of two individuals - see Moore-Repullo (1988) and Dutta-Sen (1988). ^ For another approach to the question of implementing non-monotonic SCFs "approximately", see the literature on virtual implementation (Matsushima 1988; Abreu and Sen 1991).
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The Implementation of Social Choice Functions Implementation of social choice
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becomes increasingly unlikely for a profile to be realized where an mdividual can manipulate, as the number of individuals tends to infinity.^ Let a rule be called asymptotically monotonic if it is "increasingly unlikely*' that a profile will be realized where its minimal monotonic extension is multivalued, as the number of individuals tends to infinity. Since monotonicity and strategy proofness are equivalent according to the MuUer-Satterthwaite Theorem,* it would be reasonable to conjecture that the plurality rule is also asymptotically monotonic. The main result of the paper says that this is not true; indeed, it asserts that it is impossible to find rules which are asymptotically monotonic. In order to formulate the problem generally, the notion of a social choice rule (SCR) is introduced. A SCR is a collection of SCFs, one for each size of society which satisfy certain consistency properties. This notion permits a general treatment of objects like the plurality rule. Attention is restricted to SCRs which induce anonymous SCFs for all societies. This assumption implies that the outcome associated with any preference profile depends only on the proportion of various preference orderings which comprise the profile. Therefore, a profile can be represented as a point in the unit simplex of dimension equal to the total number of preference orderings minus one. A SCR is defined to be asymptotically non-monotonic if the sequence of sets of profiles where the minimal monotonic extension is multivalued, converges (in the sense of closed convergence) in the limit of a set with an area which is a strictly positive fraction of the area of the simplex. An alternative is said to belong to the non-trivial range of a SCR if the set of profiles where it is an outcome, is a set with an area which is a strictly positive fraction of the area of the simplex. The main result states that if a SCR has a non-trivial range of at least three, then it must be asymptotically nonmonotonic. A crucial element of the proof is a geometric restatement of the monotonicity condition using Hall's Marriage Lenuna, a well-known result in the theory of matching associated with bipartite graphs. This reduces the problem of finding an asymptotically monotonic SCR to a particular problem of partitioning the simplex. What is the significance of this result? It impHes that the multiplicity of equilibria problem which lies at the heart of Nash equihbrium theory persists even in large societies. Moreover, this is so even in cases where an individual's opportunities to manipulate, or to even influence the outcome, disappears in the limit. This happens because the set of profiles where multiple equilibria arise when implementing the minimal monotonic extension is distinct from and much "larger" than the set of profiles where some individual can manipulate. The paper is organized as follows: Sect. 2 presents the notation and some preliminary results. Sect. 3 considers the example of the pluraUty rule to illustrate the main result of the paper which is contained in Sect. 4. Sect. 5 concludes.
2. Notation and preliminary results The set of social alternatives is a finite K element set, A = {fli, ..• ,ajg}. A society J = {!,..., iV} is an initial segment of integers and denotes the set of individuals.
^ This result is formally proved in Pazner and Wesley (1978). ^ Provided of course, that we consider SCFs defined over an unrestricted domain.
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A preference ordering of individual ;, Pj is assumed to be a linear (i.e. complete, transitive and antisymmetric) ordering of the elements of A, Indifference is ruled out to simplify the analysis. Let ^ denote the set of all linear orderings of the elements of A. A preference profile P is a collection of individual preference orderings (Pi,..., Ps), one for every individual in J, The set ^ will be referred to as the set of admissible preference profiles. A s e c F, associates a non-empty set F(P) c A with every admissible preference profile P. A SCF is a singleton valued SCC. In Sect. 4 where variations in the size of society will be considered, SCCs and SCFs will be indexed by N. A game form G, is an AT + 1 tuple (5i,..., Ss; g) where 5 i , . . . , Sjv are arbitrary sets and ^ is a mapping, g\SiX -xSn -* A, The interpretation of G is as follows: 5; the strategy set for individual; = 1,..., AT. If j plays 5^eSj,; = 1,..., N, then the game form prescribes the outcome g(si Sj^), Note that for any preference profile P, the pair (G, P) is a game in normal form. Let N£(G, P) denote the set of Nash equilibrium strategies of the game {G,P\ Definition 1 1 . The game form G implements the SCC F if ^(iV£(G, P)) = F(P) for all admissible profiles P. A SCC is implementable if there exists a game form which implements it. The central concern of implementation theory is to characterize the class of implementable SCCs. Two concepts which are crucial to Maskin*s characterization theorem are introduced below. Definition 2.2. The profile F is an ar{ar e A) improvement over the profile P if GrPjajt -* afPjUk for all ; e / and a* e i4. Definition 2.3. The SCC F is monotonic if, for all a, 6 i4 and profile pairs P and P\ [fl,. e P(P) and F is an a, improvement over P] -* a^ e F(F), Monotonicity requires that if an alternative a^ is an outcome of F at profile P, then it must also be an outcome of F at all profiles P' where all outcomes ranked below Ur according to P are still ranked below ar according to F , for all individuals. Definition 2.4. The SCC F satisfies no-veto power (NVP), if for all alternatives a^e A and profiles P, [A, is first ranked for at least N — 1 individuals] -* a^ e P(P). The fundamental characterization theorem can now be stated. Theorem 2.1 (Maskin (1977)). / / a SCC is implementable, then it must be monotonic. Conversely, if there are at least 3 individuals, then a SCC which is monotonic and satisfies NVP, is implementable. The NVP condition is quite weak and most reasonable SCCs satisfy it. Monotonicity, on the other hand, is a strong restriction. One striking illustration of this is the fact that only trivial SCFs are monotonic. Definition 2.5. The SCF/is dictatorial if there exists an individual j eJ such that for all profiles P, / ( P ) is the P^maximal element of A, Definition 2.6. The SCF / i s manipulable at profile P if there exists an individual ; and a preference ordering Fj such that/(P) # / ( P | P } ) and f{P\P))Pjf(P).^ The SCF / i s strategyproof if it is not manipulable at any profile.
* ((P|Pj) is the profile obtained by replacing thc;th component of P by P}).
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The Implementation of Social Choice Functions Implementation of social choice
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MuUer-Satterthwaite (1977) shows that a SCF is monotonic if and only if it is strategyproof. An application of the Gibbard-Satterthwaite theorem, now yields Proposition 2.1, If a SCF has a range of at least 3 alternatives, then it is monotonic if and only if it is dictatorial This result is well-known in the literature. The reader is referred to Moulin (1983), Roberts (1979) and Saijo (1987). In this paper it is assumed that social objectives are specified by a non-trivial SCF. In particular, it is assumed that the SCFs under consideration satisfy NVP. An immediate consequence of Proposition 2.1 is that they are not implementable. A procedure to uniquely extend such SCFs to "minimal" implementable SCCs, is now described. Definition 2.7, A monotonic extension of a SCF / is a SCC F such that (i) f{P) e FiP) for all P and (ii) F is monotonic. Let M(/) denote the set of all monotonic extensions of/ Note that the trivial SCC which maps every profile onto A is a monotonic extension of all SCFs. Thus, M(/) is always non-empty. Definition 2.8, The minimal monotonic extension of a SCF/denoted by MME{fX is defined as follows:
MME{f)^f]{F\FeM(f)} The next proposition asserts that the procedure of implementing a SCF which satisfies NVP, via its minimal monotonic extension, is a valid one.^ Proposition 2.2. The minimal monotonic extension of a SCF is a monotonic extension of the SCF, Moreover, if there are at least 3 individuals and the SCF satisfies NVP, then its minimal monotonic extension is implementable. Proof It is easy to check that the intersection of two monotonic extensions of a SCF is also a monotonic extension of the SCF. Since the set of monotonic extensions is non-empty, it follows that the minimal monotonic extension is a monotonic extension. The minimal monotonic extension is therefore monotonic. If the SCF satisfies NVP, then so do all its monotonic extensions. The implementability of the minimal monotonic extension when there are at least 3 individuals can now be directly inferred from Maskin's characterization theorem. • It is assumed in the rest of the paper that the SCFs under consideration satisfy NVP. It is proposed that these SCFs ought to be implemented via their mmimal monotonic extensions. This procedure ensures that for all possible realization of preference profiles, the socially optimal alternative is always an equilibrium. Moreover, the number of non-optimal equilibria that can arise when implementing any other SCC subject to the same restriction, is minimized. Thefinalproposition in this section is easy to prove but useful in the computation of minimal monotonic extensions. Proposition 2.3. Let F denote MME{f)for an arbitrary SCFf For all a^eA and profiles P, ar e F{P) if and only if the following condition holds: either a,. =/(P) or there exists a profile F such that a, -f(P') and P is an ar improvement over P\ * Notice that the minimal monotonic extension of a SCC can also be defined. Moreover, Proposition 2.2 can easily be generalized to include this case as well.
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Proof, Sufficiency. Let f be a SCC such that for all profiles P and outcomes a^, if a, e F(P), then either a^ =/(/*) or there exists ?' such that a,. =/(P') and F is an a^ improvement over F . Suppose F # MME{f) = f. Then there exists P and a^ s.t. a, € F(P) - F(P). Since/e F, a, ¥^f{P\ Therefore, there exists F s.t. a, = / ( ? ' ) and P is an ar improvement over F, But then F is not monotonic. Necessity, Let F = MME(f) and suppose that (i) a^ 6 F(P) {«) A, « / ( P ) and (iii) there does not exist F s.t. /(P'j = a^ and P is an a^ improvement of P'. Let D denote the set of profiles F s.t. P is an a^ improvement over P'. Thejollowing property of D follows almost immediately: Let P e D and suppose P is an a,, improvement of P. Then, P ^D, Define the SCC, F as follows: jF(P)-fl,
""•'-vm
ifPeZ), o.w.
Since P e D^D ^0 and F is a strict subset of F. It is now shown that F e M ( / ) , so that a contradiction to the hypothesis^that F ^MME(f) is obtained. Since a, i^f(P) for all P 6 D, it follows that/€ F. Suppose F is not monotonic. Since F is monotonic, it follows from the construction of F that there exists P, P, with P e l ) such that ar e F{P) and P is an a, improvement of P. Using the property oiD noted previously, it follows that P e D, But then it can not be the case that a, e F{Py Therefore, F is monotonic so that F € M{/). •
3. An example In this section, the minimal monotonic extension of the plurahty rule is computed in the case where there are three alternatives. In particular, the set of preference profiles where the minimal monotonic extension is multivalued, is characterized. This example serves to illustrate the general result in the next section. Let A = {auana^}. For any profile, the plurality rule picks the alternative which is ranked first by the largest number of individuals. Ties are broken in favour of the alternative with the lower index. For the profile P, let PL(P) denote the outcome according to the plurality rule. It follows that for the purpose of determining the plurality rule outcome, it suffices to represent a preference profile P, by a point (xi, X2, X3) in the 2-dimensional unit simplex. The ith co-ordinate of this point Xf, i = 1 , 2 , 3 is the number of individuals who rank a,- first among the alternatives, according to P. The next proposition characterizes the set of preference profiles where the minimal monotonic extension of the SCF defined above, is multivalued. It should be clear from the arguments used in the proof that it is not difficult to generalize the proposition to cases where there are more than three alternatives. Proposition 3,1. Let F denote MMEiPL), Then F is multi-valued at a profile P represented by (xi,X2,X3), x^ ^ 0, i = 1,2,3 and ^Xi = 1 if and only if one of the following inequalities holds: Xi ^ X2 > X3 and Xi H- X3 < 2x2,
(1)
Xi>X3>X2
and X i + X 2 + ~ < 2 x 3 ,
(2)
X2>Xi>X3
and X2 + X 3 ^ 2 x i ,
(3)
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The Implementation of Social Choice Functions Implementation of social choice
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^2 ^ ^3 > ^1
«W^ ^1 + ^2 + — < 2X3,
(4)
X3>Xi>X2
and
X2 + X 3 < 2 x i ,
(5)
X3 > X2 > Xj
and
Xi + X3 < 2x2-
(6)
Proof, Sufficiency, Suppose (1) holds. Observe that ai e F(P). Let J2 be the set of individuals who rank ^2 first according to P, Let P' be a profile represented by (x'ljXijX'a), Xi ^ 0, I = 1,2,3 and J^x\ == 1 such that (0 all individuals in J2 rank ^2 first (u) all individuals not in J2 rank ^2 last and {Hi) x\ = xi - h- Note that such a profile can be constructed since x'2 = X2>0. Since Xi+X3<2x2, Xi -f X2 + X3 < 3x2 = 3x2. Also, x'l -f X2 + X3 = 2x^2 + xs—jj< 3x2, so that X3 £ x'2. Therefore, ^2 = PL(F), Observe now, that P is an a2-improvement of P'. It follows from Proposition 2.3 that 02 € F(P). Hence F is multivalued at P. Similar arguments can be used to show that F is multivalued at P if any of the inequalities (2)-(6) hold. Necessity, Suppose not. Therefore there exists a profile P represented by (xi, X2, X3), Xt ^ 0, j = 1,2,3 and ^Xi = 1 such that F is multivalued at P and none of the inequalities (l)-(6) hold. Suppose that in particular, Xi ^ X2 ^ X3 and Xi + X3 ^ 2x2. Two cases are considered separately. Case 1, Qi e F(P), Since ai = PL{P), Proposition 2.3 implies that there exists P' presented by (xi,X2,X3), xj > 0, i = 1,2,3 and £xj = 1 such that ^2 = PL{F) and P is anfl2improvement over P'. Clearly, xi > x\ and x^ > X3. Therefore, x'2 > i. Since P is an (i2-improvement over P', X2 > x'2. Therefore, Xi -f- X3 < f = 2x2 which contradicts the earlier supposition. Case 2. a^ e F{P), Once again. Proposition 13 implies that there exists P' represented by (x'l, x'2, x'3), Xi ^ 0,1 = 1,2,3 and ^x'l = 1 such that a^ = PL{F) and P is an 03 improvement over P'. Clearly, x'3 > x'l and x'3 > x'2. Therefore, x'3 > i. Also, X3>x'3. Since X i ^ X 2 ^ X 3 , it follows that X i + X 2 + X 3 > l , a contradiction. Similar arguments can be used to show that if P is a multivalued at P, then the following inequality pairs are mutually inconsistent; Xi > X3 > X2 and Xi-hX2+ 1^^2x3, X2 > x i >X3 and X2 + X3 > 2xi, X2^X3>Xi and Xi -I- X2 + i^ ^ 2x3, X3 > Xi ^ X2 and X2 + X3 > 2xi and X3 ^ X2 > Xi and xi + X3 > 2x2. At least, one of the following 6 cases must hold: x^ > X2 ^ X3, Xi > X3 ^ X2, X2 > Xi > X3, X2 > X3 ^ Xi, X3 ^ Xi ^ X2, X3 > X2 ^ Xi. By partitioning these possibilities into mutually independent ones, it follows that at least one of the inequalities (l)-(6) must hold. • Remark 3.1. The term ^ appears in the inequalities (2) and (6) because of the tie-breaking rule used which discriminates against a^. One way to avoid this inconvenience is to consider the minimal monotonic extension of the plurality correspondence rather than a single valued selection from it. Inequalities (l)-(6) can be conveniently represented in 2 dimensions. In Diagram 1, the shaded area denotes the set which is the limit (in the sense of closed convergence) of the sequence of sets which satisfy (l)-(6), as N tends to infinity. The most noteworthy feature of this limit set has a positive area in 2 dimensional space. The reason for this can be loosely described as follows. Consider a profile where some alternative wins without "too large" a support. There are "many*' profiles where some other alternative wins with the same or smaller
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Diagnmi 1. The shaded area represents the set of profiles where the minimal monotonic extension is multivalued, in the limit as N -•oo
^
Diagram 2. The segments AB, AC and AD represents the set of profiles where an individual can influence the outcome, in the limit as N ~»oo
support than it had in the original profile and non-supporters place it no higher than under the original profile. The original profile is thus an improvement of this other alternative from the other profile. Consequently the minimal monotonic extension is multivalued for all such profiles. It is worth contrasting this set with the set of profiles where the individual can influence the outcome of the plurality rule. This latter set is clearly a superset of the set of profiles where an individual can manipulate. It is easy to check that in the limit as N tends to infinity this set converges to the set described by the following inequalities (7) Xi,
(8)
^Xi,
(9)
Xi = X^"^ X2 = ^ 3
where, of course, X| ^ 0, j = 1,2,3 and £JC( =« 1. This set is illustrated in Diagram 2. Observe that this set is one-dimensional. The intuition for this is again quite clear
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in a large society, an individual can influence the outcome only in the *'rare" case that all individuals are exactly divided in their support for some set of alternatives. The example highlights two somewhat surprising features of minimal monotonic extensions. Firstly the set of profiles where the minimal monotonic extension is multivalued need not coincide with the set of profiles where the SCF is manipulable. Secondly, the latter set can "shrink" to a "small" set as society becomes larger without implying that the former does so as well. 4 The limit result This section presents the main result of the paper which states that the properties exhibited by the minimal monotonic extension of the plurality rule hold true quite generally. The general formulation of the problem necessitates the introduction of the concept of a social choice rule (SCR). A SCR specifies a SCF for societies of all sizes. Thus, a SCR is a sequence {/^}, N = 1,2,..., where/^ is a SCF defined for a society of size JV. However, an arbitrary sequence of SCFs is not necessarily a SCR. The notion is formally defined below. Recall that the set A has K elements. Therefore there are K\ linear orderings of the elements of ^4. Let these orderings be numbeired in some way from 1 to Kl Let A denote the K\ — 1 dimensional unit simplex. Definition 4.L A SCR {/^}, iV=l,2,..., is specified by K closed sets i4i,..., Xx cz J with the property that (Jf« j A^ = J and interior ^4,.n^l, = 0 for all r^s. Fix an integer N and let P be a profile in a society of size N. Let x e J be such that the ith component of x is the proportion of individuals who have preference ordering i, i = 1,...,X!, in the profile P. The SCF/^ is defined as follows: (i) if X € interior A, for some r e {1, ...,K}, then/^(P) = a,. (ii) if X lies on the boundary of the sets -4,.^..., A^^, then /^(P) e {fl^i,..., «rr}• The SCR is specified by the "partition", i4i,...,>4x of A, It is not exactly a partition since the boundary of two sets can belong to both sets. The set A^^ r = 1,...,K is essentially the set of profiles where the associated SCFs pick the alternative a,.. The consequence of identifying a preference profile with the distribution of the various preference orderings in the profile, is that all the SCFs specified by the SCR, are anonymous. Definition 4.2. The SCF/is anonymous if, for all permutations
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Remark 4.1. All SCRs whose associated SCFs are selections of scoring correspondences (see Moulin 1983, Chap. 1 for definitions) have the set A as their non-trivial range. One way to verify this claim is to observe that the associated SCFs are neutral (i.e. do not discriminate amongst alternatives) except over profiles where ties have to be broken. But the proportion of such profiles is negligible in the limit. Similarly, all SCRs whose associated SCFs are selections of Condorcet correspondences (once again, refer to Moulin (1983) for appropriate definitions) have a full non-trivial range. This is because there is a strictly positive proportion of profiles (m the limit) where Condorcet winners exist. On the other hand consider the SCR which picks a "status quo" alternative a^ for all profiles except those in which all individuals unanimously rank some other alternative first. In this case, ai is the only alternative in the non-trivial range of the SCR and the sets A2,".JAK are unions of particular faces of A, Definition 4.4. The SCR {/^}, N = 1,2,.., is asymptotically non monotonic if there exists B e ^ and an integer iVo with the following property: Let x e B and N > NQ. Then for all profiles P in a society of size N represented by x, MME(/^) is multivalued at P. The definition above is intuitive. A SCR is asymptotically non-monotonic if the set of profiles where the minimal monotonic extension of its associated SCFs, converges to a set whose area is a strictly positive proportion of the set A in the limit as N tends to infinity. The main result of the paper can now be stated. Theorem. If an SCR has a non-trivial range of at least 3, then it is asymptotically non-monotonic. Remark 4.2. The conclusion of the Theorem is false if the SCRs under consideration have a non-trivial range of less than three. Of course, in that case, it is possible to find SCRs which are not only asymptotically monotonic, but also monotonic in the sense that all the associated SCFs are monotonic (the constant SCR, for example). However, there exist SCRs which are asymptotically monotonic but not monotonic. In fact, all the associated SCFs may map onto the set A. Consider the following SCR. If an alternative is unanimously ranked first, it is the outcome. Otherwise the outcome is the majority winner among the two pre-spedfied alternatives, ai and ai with ties broken in favour of ai. Observe that for all societies the associated SCF is non-monotonic. Also, only ai and ai belong to the non-trivial range of the SCR. Calculations show that the minimal monotonic extension of the SCF is multivalued only at those profiles where all individuals unanimously rank some alternative among fl2> ••»%» first. Clearly, the set of such profiles shrinks to a "lower" dimensional set in A, so that the SCR is asymptotically monotonic. One difficulty with this example is that the associated SCFs do not satisfy no-veto power. Therefore, a natural question to pose is the following: does there exist an asymptotically monotonic SCR which induces SCFs which are non-monotonic and satisfy no-veto power? The answer is no. The reason is that since the sets AI,..,,AK in the definition of a SCR are defined independently of N, a SCR which induces SCFs which satisfy no-veto power must have full non-trivial range. The theorem then says that such a SCR must be asymptotically nonmonotonic. In order to gain some insight into the proof of the theorem and the crucial nature of the anonymity assumption, consider the special case where A - (ai,fl2»^3}' There are six linear orderings of the elements of A, ai > a2 > aj, fli > fl3 > az, a2>ai> a^, aa > ^3 > «i» a^>ai> 02 and as > ^2 > ^i which
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are numbered 1-6 respectively. Since the associated SCFs are anonymous, a profile can be represented as a six-tuple x s (xi,X2, ...Xe) where x,- > 0 and X,-^* = 1Now consider a profile P represented by x and consider an ai improvement of P, denoted by P' and represented by x'. An individual with preference ordering 1 or 2 must now also have either ordering 1 or 2. Likewise, an individual with ordering 3 must now have either ordering 1,2 or 3 and an individual with ordering 5 must improve to one of the orderings 1,2,5. Hall's Marriage Lemma can now be applied to infer that F is an ai-improvement of Pi if and only if xi + xi ^ Xi -I- X2 ^i + xi -f x'a > Xi -h X2 -h X3, xi -f xi + xi > Xi 4- x^ -f X5 and xi + xi + xi -Ix'5 > Xi -f X2 + X3 + X5. Thus, improvements can be given an algebraic as well as a geometric interpretation. A SCR specifies regions Ai, A^ and A^ in the sixdimensional unit simplex where outcomes au ^2 and a^ obtain, respectively. The inequalities which characterize improvements can be used to describe sets A^ A^ and ^3 where the minimal monotonic extension includes a^, ^2 and a^ respectively. The problem offindingan asymptotically monotonic SCR can be reformulated as follows: does there exist a "partition" A^^ Ai, A^ such that no two of the sets Au A2 and A^ have an intersection which has full-dimension? The argument proceeds as follows. Consider an asymptotically monotonic SCR and suppose that x is a point in the interior of the simplex which lies on the boundary of Ai and A2. It can then be shown that in a neighbourhood of x, Ai must lie entirely in the half space Xi + X2 + X5 ^ Xi + X2 + Xj and A2 in the half space Xi -1- X2 + X5 < Xi + X2 + X5. Similar conclusions hold for points on the boundaries of Ai and A^ and Az and A^ (the relevant half spaces are of course, different). These arguments can be put together to show that there must exist a point x in the interior of the simplex which lies on the boundary of Ax, Ai and A^, Moreover, it can be shown that there exists a region in a neighbourhood of X which cannot belong to any of the sets ^ 1 , ^42 and A-^, But this contradicts the initial supposition that A^.Az and A^ constitute a "partition" of the simplex. 5. Conclusion The objective of the paper was to explore the consequences of implementing social choice functions via their minimal monotonic extensions. This approach entails the possibility of multiple (and therefore socially non-optimal) equilibria for certain realizations of preference profiles. The main result of the paper established that the multiplicity problem persists even in the limit as the number of individuals tends to infinity. Appendix The proof of the Theorem proceeds in several steps. It begins with a combinatorial restatement of the definition of improvements. The set A contains K elements; therefore the set of linear orderings of the elements of A^ has cardinality K\, Let these orderings be numbered in some way from 1 to K\, The set g which denotes the initial segment of positive integers upto K\ will be identified with the set of orderings. Consider a society of size N, i.e. J = {!,...,iV}. A preference profile P can be represented by a collection of sets {T,}, isQ with the interpretation that
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individuals who belong to the set Tt have ordering i, where i e Q. Clearly, {Tj, i e 6 is a partition of J. Let a, e i4 and let P and P' be preference profiles which are represented by {Tj and {T/}, i € Q respectively. Let C,, r = 1,..., X be a mapping from the set Q to the set of non-empty subsets of Q such that for alii e Q, Cr{i) ^Qis the set of orderings which are a, improvements over ordering i. In other words, for all / e Cr(0, if a, beats alternative a^ under i, then a, also beats a, under /. For any S QQ, C,.(5) denotes the set (J ^ ^ 5 C,(0. The definition of an improvement can be restated as follows: the profile {T/}, J 6 Q is an a,(«,. e A) improvement of the profile (Tj}, i 6 g if, for alU € Q, j e T,implies that 7 eU/gQi) ^ZThe following definition is important. Definition. The profile {Tl}, ieQis an anonymous Oriar e A) improvement over the profile {T,}, i e Q if there exists a one-to-one function u:J --* J such that, for all i eQJe Ti implies that «(;) e IJ/ g c,(o ^i'Lemma 1 characterizes anonymous improvements. The proof of the lenmia uses a well-known result in graph theory, which is stated below without proof. Hall's Marriage Lemma: Let W and Z befinitesets. Let Gbea mapping from W to the set of non-empty subsets ofZ. Then^ there exists a one-to-onefunction P: W -> Z with the property that p{w) e G{w)for allweW if and only if forallV^W,
\V\<, U ^(0
Lemma A-1. {Tl}, i e Q is an anonymous a, improvement over {T,}, i e Q if and only if forallS^Q,
£ |r,| ^
£
|T/|.
Proof Let d:J-* Q be such that jeTs^j) for all j e J, For all L s J , S{L) « {i e 61 i = Sij) for some j e L}. Define the mapping G from J to the set of non-empty subsets of J such that G{j) = U / e c,(5(j)) ^ ^ ^ direct application of Hall's marriage lemma yields that {T/}, i e Q is an anonymous a^ improvement over { r j , f e 2 if and only if f o r a l l L c j , |L|:<
T{
U
(10)
It is now shown that (10) is equivalent to the condition for alls e g , X | T , U ieS
E
\Tll
(11)
i€ CriS)
Suppose (10) holds. Consider 5 £ Q. Let L = (J,-^ 5 Tt. Clearly 6{L) = S. Therefore (10) implies that
[jTi ieS
U Ti I 6 CAS)
Since { r j and {Tl}, i e Q are partitions of/, (11) follows immediately.
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Suppose (11) holds. Consider L ^ j . Let S = S(L), Therefore, (11) implies that
Z |r,|^ Z |T/|. J 6 S{L)
i 6 C,{S{L))
Since 1L| :^^^^sml^ili^^)follows immediately,
•
The next lemma underscores the importance of anonymous improvements in the characterization of the minimal monotonic extension of an anonymous social choice function. Lemma A-2. Let f be an anonymous SCF and let F denote its minmal monotonic extension. Let Pbea profile such thatf(P) = a, and suppose that F is an anonymous ar improvement of P. Then, a^ € F(P'). Proof, Let {Ti) and {T'i),ieQ denote respectively, the represenations of profiles P and P'. Since F is an anonymous a,, improvement of F, there exists a one-to-one function oiiJ -^ J such that ; e Ti implies that«(;) e |J / e c,(o ^/• Let a *" ^ denote the inverse of the function a. Define the profile P" represented by (T/'}, i e Q as follovi^s: ; € T/' if a^^(j) e T^ for all jeJ and ie Q, Clearly IT^j = \Tl% for all i e Q. Therefore, the anonymity of/implies that/(F') = a,.. It is easy to verify that F is an a,, improvement of P". The argument is now completed by invoking Proposition 2.3. • Some more notation is now introduced. For any pair of alternatives a,., a, e Ay let M{ry s)c:Q denote the set of orderings in which a, beats a,. For any profile {TJ, i e Qj let x denote the K\ dimensional vector whose ith component is \Ti\/N, Clearly x belongs to the Kl - 1 dimensional unit simplex, which will be denoted by A. The vector xe Aan N-profile if x iV is a vector of integers. For any x e J and £ > 0, let B(x, e) denote the set of {x e J | p(x, x) ^ e} where p is the Euclidean distance function. Finally, for any set X c J, d(X) denotes the boundary of X. The next lemma, loosely speaking says the following. Let {/^}, N = 1,2,..., be an asymptotically monotonic social choice rule. Recall that (/^} specifies sets ArC: A,r ^ l,,.,,K where A^ is the set of profiles on which/^ takes the value a^., for all N, Let x lie on the boundary of A^ and Ag in the interior of A, Then, in a neighbourhood of x, the sets Ar and As must lie on opposite sides of the hyperplane I^e Af(r.5)(^i - ^i) = 0Lemma A-3, Let {/^}, iV = 1,2,..., be an asymptotically monotonic SCR represented by ArCzQyr —i,..,,K. Let x e d(Ar)nd{A^) be in the interior of A where a^ and a, are in the non-trivial range of the SCR. Then there exists e>0 s,t. IxeAl Z (X|-Xi)>oinJ5(x,£)nv4/=0. I |i€M(r,s) J Proof Let F^ denote MME(f^) for all N. The following claim isfirstestablished. Claim: Let x € diA^) be in the interior of zl. Let Z be a closed subset of the set {x e J |£/g5X/ > £/e Cr{S)^iforall 5 c Q}. Then there exists an integer No such that for all integers iV > No, if x is an N-profile, then a,, e P^(P) where P is represented by {TJ, i 6 fi and Xf = Ti/N for all i e Q. Pick X in the interior of A^ suitably close to x so that for all x 6 Z, L i 6 s ^i > Z i 6 c,{S) Xi for all 5 £ fi. Assume w.Lo.g. that j3 has rational co-ordinates, in particular x, = (Xi/p for alU 6 Q where a^ and p are integers. For any integer iV,
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let w and i; be integers, v
392
d(Ai)nd{A2)nd{A^).
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Suppose not. Assume w.l.o.g. that w is in the interior of A and y^ € d{Ai)r\d{A2), It follows from Lemma A-3, that in some closed ball B^ including w, the hyperplane Sic M(i,2)(Wi — vVj) = 0 is the boundary of A^ and Ai^ Consider w e d(B^) on this hyperplane. By assumption, w^5(^43). Therefore, there exists a closed ball B^ including >v such that the hyperplane is also the boundary of Ax and A2 in B^. Applying this argument repeatedly, it follows that the hyperplane is the boundary of A^ and A2, Since a^ is in the non-trivial range there must exist another set, say A^ (of course, A4, codd be Ai or Ai) and vin the interior of J such that ve^{A^)r^^(Pi4). Applying the argument in the previous paragraph, it must be the case that the hyperplane XI€M(3,4)(^I — ^i) — ^ is the boundary of A^ and A^, The final step of the argument consists in showing that hyperplanes Zi6M(i,2)(Xi - Wi) = 0 and £,gAf(3,4)(^i — i^i) = 0 intersect in the interior of A and observing that any point in the intersection must be on the boundary of Ai, Ai and ^3. Let 5 = M(l, 2)nM(3,4). Clearly, S is non-empty and a strict subset of J. Let a and P denote respectively, X/e Af(i,2) ^i and Ii6Af(i,2)^i- By construction, a, P lie in the open interval (0,1). Assume w.Lo.g. that a < ^. Pickfisuch that a>£,i8 + 6 < l and let r = 1 — j8 — 8. Define x as follows:
1^ if i 6 Ti, where T^ = M(l, 2) - S, Xi
=
j^
if i 6 T2, where T2 = Af (3,4) - 5,
j^
if f€T3, where T3 = fi - (M(l, 2)uM(3,4)).
It is easy to check that x lies in the interior of A and on both hyperplanes. Step 2. Using step 1, it can be assumed w.l.o.g. that there exists an integer L, 3 < L < A : and xe'mXA, xeUf=i5(.4i) and x^5(i4i), j=:L + 1,...,K. From Lemma A-3, there exists a ball B(x,fi) such that inside the ball, {xlIi€M(r,s)(>^i - ^i) > 0}r\As ^ 0 for all r,s€{l, ...,L}. Let ii, 12, ••• »»2L denote the following orderings. ill ax> a2> a^> a4> ••• > ax. > ••• > an, 12: a2> ai>az>
a4> -• > ^L > ••• > QK,
i^: fl2 > ^3 > fli > «4 >
••• >ai>
••• >
ajc,
U: as > a2 > ai >fl4> •*• > a/. > ••• > a^,
J2r-i' a, >a,.+i > a i >a2 > ••• >a,_i > a^+2 > ••• >aL> ••• > ajc, i2r: «r+i > a , > a i >a2 > ••• > a , - i > a^+2 > •* > ^L > •" > «ic» i2L-i- a£.>ai >a2 >a3 > ••• >ai,> •'• > ai^, 121- ai > a^ > a2 > as > ••• > a^. > ••• > a^.
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A. Sen Define x as follows:
Xij^
Xij + ji if ; is odd and ; ^ 2L, Xij — j£ if J is even and j < 2L, Xij otherwise.
Observe that x'eBix,e). Pick an integer r from the set {1,...,L}. Observe that the ranking of a,, and ar+i{L + i = i) agrees over all pairs of orderings ij, ij+i where ; is an odd integer between 1 and 2L and j^lr1. Also, the ranking of a,, and fl,+i are opposed between J2r-i and i2f Therefore, ZfeM(r.r +1)(^! " ^i) > 0, for all r = 1,... ,L, where L + 1 = 1}. It follows that it is possible to pick a closed ball Z such that Za n{x|5;,.gjy^^^,.^i^(xi-jc<)>0, r = 1,...,L,L + 1 s 1}. Since x^d(Ai\ i^L-^ l,...,i^, it is also possible to ensure that ZnAi = 0 for all / = L + 1,...,X. Therefore, Zn{\jf^^A, = 0. However, this contradicts the assumption that Uf=i^r = ^ which is part of the definition of a SCR. •
References 1. Abreu D, Sen A (1991) Virtual implementation in Nash equilibrium Econometrica 59: 997-1021 2. BoUobas B (1979) Graph theory: an introductory course. Springer, Beriin 3. Dutta B, Sen A (1988) A necessary and sufficient condition for two person Nash implementation. Rev Econom Stud 58: 121-128 4. Gibbard A (1973) Manipulation of voting schemes: a general result Econometrica 41: 587-601 5. Maskin E (1977) Nash equilibrium and welfare optimality. mimeo 6. Matsushima H (1988) A new approach to the implementation problem. J Economic Theory 45: 128-144 7. Moore J, Repullo R (1988) Nash implementation: a full characterization. Econometrica 56: 1191-1220 8. Moulin H (1983) The strategy of social choice North-Holland, Amsterdam 9. Mullen E, Satterthwaite M (1977) The equivalence of strong positive association and strategy proofness. J Econom Theory 14:412-418 10. Pazner E, Wesley E (1978) Cheatproofness properties of the plurality rule in large societies. Rev. Econom Stud 45: 85-91 11. Roberts KWS (1979) The characterization of implementable choice rules. In J-J Laifont (ed), Aggregation and revelation of preferences. North-Holland, Amsterdam 12. Saijo T (1987) On constant Maskin monotonic functions. J Econom Theory 42:382-386 13. Satterthwaite M (1975) Strategyproofness and Arrow's conditions: existence and correspondence theorems for voting procedures and social welfare functions. J Econom Theory 10: 187-217 14. Thomson W (1993) Monotonic extensions mimeo
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17 Philip J. Reny on Hugo F. Sonnenschein
From down the hall I hear a famiUar greeting: "Helloooo Renyyyy!" The signature "hello" is as well-known to a Sonnenschein student as the theorem that bears his name. By welcoming us in this delightful manner, Hugo immediately puts us at ease, makes us feel welcome, and eliminates any notion that there might be a barrier of formality between professor and student. All of this from a simple greeting; it's vintage Hugo. Part of what makes Hugo such a great adviser, and now colleague, is his ability to listen to what one has to say, to isolate the essence of the idea, and to ask just the "right" question. "What's your best example?" is a classic Hugo question that one eventually becomes prepared to answer prior to visiting his office. Looking back on my thesis (a bit scary, I must say), I now recognize that some of the better parts of it are the examples that were the result of Hugo's challenges. Meetings to discuss my thesis with Hugo often began the same way. I would knock on the half-open door, Hugo would look up and say "Helloooo Renyyyy" and then suggest a change of venue with the phrase "Let's walk!" I thought that this was so very nice, strolling through the grounds of Princeton University, where such great minds had walked and thought before, discussing my research with a famously important microeconomic theorist. It was such a privilege. I now realize of course that, in addition to being intellectually fruitful, our walks served a more practical purpose. As a professor with students of my own, I know how important it can be to keep one's students away from one's blackboard! "Let's walk" indeed! All jokes aside, these were very special experiences. I believe that it was in 1986, my fourth andfinalyear at Princeton, when I was introduced to Hugo's paper (co-authored with Wayne Shafer) entitled "Equilibrium in Abstract Economies without Ordered Preferences" (Journal of Mathematical Economics (1975)). The question at hand was the existence of Nash equilibrium in very general settings. Like all of Hugo's work, the paper is beautifully written, the proofs are so very elegant, and the results are remarkably general, combining and extending, using novel techniques, a variety of classic and significant results from the literature. It made a lasting impression upon me. The paper I have chosen to include in this volume deals with the same question of existence of Nash equilibrium, but it explores a different generalization, namely the extent to which one can relax the assumption of continuous payoff functions. It's fair to say that my interest infixedpoint theory and the existence of equilibria in games began with Arrow-Debreu-McKenzie, took hold with Nash, and was
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permanently established with Shafer and Sonnenschein; Hugo gave our class, and others I am certain, masterful lectures on each topic. I remain actively engaged in this area of research to this day.
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On the Existence of Pure and Mixed Strategy Nash Equilibria Econometrica, Vol. 67, No. 5 (September, 1999), 1029-1056
ON THE EXISTENCE OF PURE AND MIXED STRATEGY NASH EQUILIBRIA IN DISCONTINUOUS GAMES BY PHILIP J. RENY^ A game is better-reply secure if for every nonequilibriura strategy x* and every payoff vector limit u* resulting from strategies approaching x*, some player i has a strategy yielding a payoff strictly above u* even if the others deviate slightly from x*. If strategy spaces are compact and convex, payoffs are quasiconcave in the owner's strategy, and the game is better-reply secure, then a pure strategy Nash equilibrium exists. Better-reply security holds in many economic games. It also permits new results on the existence of symmetric and mixed strategy Nash equilibria. KEYWORDS: Discontinuous games, better-reply security, payoff security, pure strategy, mixed strategy, Nash equilibrium, existence.
1. INTRODUCTION
IT IS OFTEN NATURAL TO MODEL Strategic settings in economics as games with infinite strategy spaces. For example, models of price and spatial competition (Bertrand (1883), Hotelling (1929)), auctions (Milgrom and Weber (1982)), patent races (Fudenberg et al. (1983)), etc., typically allow participants to choose any action among a continuum. However, such games frequently exhibit discontinuities in the payoffs. Consequently, standard theorems such as those found in Debreu (1952) or Glicksberg (1952) cannot be apphed to estabKsh the existence of an equilibrium (pure or mixed). Now while in many of these games equilibria can be constructed, rendering the existence question moot, there are other instances in which this is not the case. Consider for example, the pay-your-bid multi-unit auction, a game that we shall consider in detail. While its rules are simple enough to state, rather little is known at present about this auction's equilibrium strategies. Indeed, up to now, even the existence of an equilibrium in this game has been at issue since payoffs are not continuous.^ The present paper offers a pure strategy Nash equilibrium existence result for a large class of discontinuous games. Indeed, our main result on the existence of pure strategy equilibrium strictly generalizes the mbced strategy equilibrium existence results of Nash (1950), Glicksberg (1952), Mas-Colell (1984), Dasgupta ^I wish to thank Faruk Gul, Atsushi Kajii, George Mailath, Motty Perry, Arthur Robson, Al Roth, Jorgen WeibuU, and Tim van Zandt for helpful comments. Thanks also to Xin He for pointing out a misstatement in an earlier version. Three referees and the editor provided numerous suggestions that substantially improved the exposition. I also gratefully acknowledge financial support from the National Science Foundation (SBR-9709392) and the University of Pittsburgh's Faculty of Arts and Sciences. Thanks also to CERGE-EI in Prague for its warm hospitality during my visits. ^Ties in winning bids must be broken according to some fixed rule. This inevitably leads to a discontinuity in a tying bidder's payoff since bidding slightly higher would have guaranteed winning the unit. 1029
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and Maskin (1986), Mertens (1986), Simon (1987), and Robson (1994), as well as the pure strategy existence result for two-person zero-sum games due to Sion (1958). Although our main result is formally unrelated to that of Baye et al. (1993), the present hypotheses are significantly easier to check in practical applications and as we indicate below have a natural interpretation, the latter property being important for theoretical applications. Also presented here are subsidiary results on the existence of pure and mixed strategy symmetric equilibria in games possessing enough symmetry. Our results rely on a new condition called better-reply security, A game is better-reply secure if for every nonequilibrium strategy x* and every payoff vector limit w* resulting from strategies approaching jc*, some player / has a strategy yielding a payoff strictly above uf even if the others deviate slightly from jr*.^ For example, Bertrand's price-competition game is better-reply secure because given any price vector /?*, each player can obtain a payoff very close to his supremum at a price that no one else charges. Hence this payoff can be secured since it will result even if the others deviate slightly from /?*. But if /?* is not an equilibrium, then because total industry profits are continuous, for any price vector near ;?* someone's payoff must fall below that which he can secure. Our main result is that games with compact, convex strategy spaces and payoffs that are quasiconcave in the owner's strategy possess a pure strategy Nash equilibrium if in addition they are better-reply secure. Better-reply security combines and generalizes two conditions, reciprocal upper semicontinuity and payoff security. A game is reciprocally upper semicontinuous if, whenever some player's payoff jumps down, some other player's payoff jumps up. This condition was introduced by Simon (1987), and expresses a weak form of competition."^ A game is payoff secure if for every joint strategy, x, each player has a strate^ that virtually guarantees the payoff he receives at x, even if the others play slightly differently than at x? Both conditions are satisfied in many economic games and are often quite simple to check. For example, Bertrand's game is trivially reciprocally u.s.c. since the sum of payoffs is continuous. Moreover, payoff security is easily checked by noting that each ^When w* =u{x*X as results when the sequence of strategies approaching x* is X * , J C * , X * , . . . , the strategy securing some player / a superior payoff is a better reply against x*; than xf. Note that because payoffs are not necessarily continuous, there can be many payoff vector limits, u*, resulting from strategies approaching x*. '^Simon used the term complementary discontinuities. The term used here instead emphasizes the condition's relation to upper semicontinuity of a real-valued function. Indeed, the role played by reciprocal upper semicontinuity in ensuring existence of equilibrium is similar to that played by upper semicontinuity of a function in guaranteeing the existence of a maximum of a real-valued function. Dasgupta and Maskin (1986) were the first to introduce a condition of this kind in a game setting. They require the sum of the players* payoffs to be upper semicontinuous. Reciprocal upper semicontinuity is a strict generalization. ^Payoff security was introduced in an earlier version of this paper (Reny (1995)) dealing solely with the existence of mixed strategy equilibria. We have abbreviated the term used there "local payoff security" to simply "payoff security/*
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firm can guarantee a payoff that is virtually no worse than the status quo (and perhaps significantly better) by decreasing its price slightly, so long as the others' prices do not change too much. The ease with which these two conditions can be checked provides sufficient reason to highlight them, despite the fact that better-reply security is more permissive.^ The method of proof employed to obtain our main result is new. Previous techniques fail for two reasons. First, the presence of discontinuities in payoffs may preclude the existence of best replies so that best reply correspondences need not be nonempty-valued, let alone upper hemicontinuous. Consequently, both lattice-theoretical techniques as well as the standard application of Kakutahi's fixed point theorem to best reply correspondences fail. Second, to obtain the existence of a pure strategy equilibrium, one cannot approximate the original game by a sequence of finite games and expect that the sequence of mixed strategy equilibria of the finite games will converge to a pure strategy equilibrium of the limit game. Such an approximation technique has been employed by Dasgupta and Maskin (1986), Simon (1987), and Simon and Zame (1990) to obtain the existence of mixed strategy equilibria in discontinuous games. To circumvent these difficulties, we instead approximate the discontinuous payoff functions, not the strate^ sets, by a sequence of continuous payoff functions. Because strategy spaces are compact, this yields an appropriate sequence of pure strategy profiles whose limit is a pure strategy equilibrium of the original game. The remainder of the paper is organized as follows. Section 2 describes the environment and notation, and provides a number of preliminary definitions. Section 3 contains the new condition, better-reply security, as well as our main pure strategy equilibriimi existence result and its proof. Examples illustrating the theorem are also given. Section 4 considers the existence of symmetric pure strategy equilibria. Once again better-reply security plays a role. However, owing to symmetry it need hold only along the diagonal of payoffs. Perhaps more significantly, it is observed (as in Baye et al. (1993)) that the quasiconcavity condition on a player's payoff in his own action can be weakened substantially, again, by requiring it to hold merely along the diagonal of payoffs. Examples of symmetric games without quasiconcave payoffs, yet covered by this result, are provided. Section 5 considers the existence of mixed strategy equilibria and provides a number of corollaries to our main results, each of which is obtained by considering the game's mixed extension. One of the examples we consider to illustrate the corollaries is the pay-your-bid multi-unit auction. It is shown that our mixed strategy equilibrium existence result can be used to establish the existence of a pure strategy equilibrium in that game. Section 6 discusses related work. It is shown there that the mixed strategy equilibrium existence theorems of Nash (1950), Glicksberg (1952), Dasgupta and Maskin (1986), Mertens (1986), and Simon (1987) are all special cases of the corollaries pertaining to mixed In Section 3 it is shown that reciprocal upper semicontinuity and payoff security together imply better-reply security.
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Strategy equilibria given in Section 5, and so also of our main result. Section 7 considers other approaches to the existence question, and Section 8 concludes with a brief discussion of issues relating to approximating injfinite-action games by finite games.
2. PRELIMINARIES
There are N players. Each player / = 1,2,..., iV has a pure strategy set, X^, 3L nonempty compact subset of a topological vector space,^ and a bounded payoff function u^: X-^R, where X = xjl j X^,^ Under these conditions, G = (X^, Ui)fL ^ is called a compact game. Throughout, the product of any number of sets is endowed with the product topology. The symbol —i denotes "all players but /." In particular, J^_/= Xj^fXj, and jc_, denotes an element of X_|. The vector of the players' payoff functions will be denoted by u: X-^R^ and is defined by u(x) = (MJCX),..,,Up^(x)) for every x e X The graph of the vector payoff function is the subset of X X R^ given by {(jc, w) G J^T x R^|M = u(x)]. Finally, if each X, is convex and for each / and every x_, e X . , , «,-(•, x_y) is quasiconcave on X,-, we say that G == (X^, u^)f, j is quasiconcave,
3 . PURE STRATEGY EQUILIBRIA
The following definitions play a central role in all of our results. DEHNITION: Player / can secure a payoff of or G R at X G X if there exists Xi ^Xi, such that M^(JC^, jcl,) > a for all x!_, in some open neighborhood of jc.^.
Thus, a payoff can be secured by / at x if / has a strategy that guarantees at least that payoff even if the other players deviate slightly from x. The next condition employs the closure of the graph of the game's vector payofif function. While this set is perfectly well defined in our general topological space environment, it might supplement the reader's intuition to consider the metric space context. There, a pair (x*, w*) is in the closure of the graph of the vector payoff function if u* is the limit of the vector of payoffs corresponding to some sequence of strategies converging to x*. ^We follow Royden (1988) and say that a linear vector space V endowed with a topology is a topological vector space if addition is continuous from VxV into F, and multiplication by scalars is continuous from R X F into V. In particular, unlike some other treatments, this definition does not require (although it of course permits) V to be Hausdorff. ^One can in fact allow payoffs to be unbounded, but this would require introducing the extended reals, since the limiting values of the payoff functions will come into play below. A more elementary yet equivalent treatment of unbounded payoff functions (indeed, even those taking on infinite values) results by first transforming the unbounded payoffs, y^, into bounded ones, M,-, via M , = e x p y y i + exp i;,, and then following our treatment below.
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DEFINITION: A game G = (Xi,u^)fLi is better-reply secure if whenever (JC*,M*) is in the closure of the graph of its vector payoff function and x* is not an equilibrium, some player i can secure a payoff strictly above uf at x"* ^*
So, a game is better-reply secure if for every nonequilibrium strategy x* and every payoff vector u* resulting from strategies approaching x*, some player / has a strategy yielding a payoff strictly above uf even if the others deviate slightly from x*. Games with continuous payoff functions are better-reply secure, since any better reply will secure a payoff strictly above a player's inferior nonequilibrium payoff and those generated by nearby strategies. Many discontinuous economic games are better-reply secure. As already remarked in the introduction, Bertrand's price-competition game is one such example. As we shall see below, first-price auctions are also better-reply secure. Of course, auction-game payoffs need not be quasiconcave in one's own strategy, so Theorem 3.1 below may not directly apply. However, Corollary 5.2 to Theorem 3.1 guarantees the existence of a mixed strategy equilibrium in these games (see Section 5 below). We now state our main result. THEOREM 3.1: If G = (Jf^, w,)/lj is compact, quasiconcave, and better-reply secure, then it possesses a pure strategy Nash equilibrium.
A technical point of interest is that, as in Sion (1958), we do not require X to be Hausdorff or locally convex. This is in contrast to standard fixed point theorems (i.e., Glicksberg (1952)) upon which other equilibrium existence results are based.^ In Section 6 it is shown that the above result on the existence of pure strategy equilibria generalizes the mixed strategy equilibrium existence results of Nash (1950), Glicksberg (1952), Mas-Colell (1984), Dasgupta and Maskin (1986), Simon (1987), and Robson (1994). In addition (see Corollary 3.4 below), it extends to the AT-person non-zero-sum case the pure strategy equilibrium existence result due to Sion (1958).^^ While better-reply security is straightforward to verify, it is sometimes even simpler to verify other conditions leading to it. We now provide two rather useful conditions that together imply better-reply security. DEFINITION: A game G = (Z„w,)/li is payoff secure if for every x^X every 6 > 0, each player / can secure a payoff of M,(X) — e at X.
and
^ While we never require X to be locally convex, we shall later assume that X is Hausdorff when mixed strategies are considered. ^^Sion (1958) shows that compact, quasiconcave, zero-sum games possess a value in pure strategies, whenever each player's payoff is u.s.c. in his own pure strategy and Lsx. in the opponent's (i.e., whenever each player's payoff is, owing to the zero-sum property, l.s.c. in the opponent's pure strategy).
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Payoff security requires that for every strategy x, each player has a strategy that virtually guarantees the payoff he receives at x, even if the others deviate slightly from x, Sion (1958), Mertens (1986), and Simon (1987) note the importance of the robustness of one's payoff to perturbations in the others' strategies, and our notion of payoff security is a natural continuation of their ideas. DEFINITION: A game G = ( X ^ , M , ) / 1 I is reciprocally upper semicontinuous if, whenever {x, u) is in the closure of the graph of its vector payoff function and u^ix) < Ui for every player /, then u^ix) == M, for every player i.
Reciprocal u.sx. was first introduced by Simon (1987) under the name complementary discontinuities. It requires that some player's payoff jump up whenever some other player's payoff jumps down. It generalizes the condition introduced by Dasgupta and Maskin (1986) that the sum of the players' payoffs be u.s.c. Since the u.s.c.-sum condition is cardinal in nature, Simon's (1987) generalization is a distinct improvement.^^'^^ We now relate reciprocal u.s.c. and payoff security to better-reply security. The proof of the following result can be found in the Appendix. PROPOSITION 3.2: / / G = (X^, Ui)fL i is reciprocally u.sx, and payoff secure, then it is better-reply secure.
Bertrand's price competition game with zero costs provides a ready opportunity to apply Proposition 3.2. This game is reciprocally u.s.c. since the sum of the firms' payoffs is continuous. In addition it is payoff secure since each firm can secure a payoff that is at worst just below the status quo by lowering its price slightly. Consequently, as we have already argued directly, this game is better-reply secure. On the other hand, first-price auction games, while payoff secure, fail to be reciprocally u.s.c. so that Proposition 3.2 does not apply.^^ Nonetheless, as we shall show in Section 5, first-price auction games are better-reply secure. Consequently, better-reply security is strictly more permissive than the combination of reciprocal u.s.c. and payoff security. However, as mentioned before. Theorem 3.1 does not apply since first-price auction payoffs are not quasiconcave. First-price auctions and, more generally, multi-unit pay-your-bid auclk)ns are considered in Example 5.2 in Section 5 below. "None of the three conditions, better-reply security, reciprocal upper semicontinuity, or payoff security is precisely an ordinal property. However, they are all virtually so in the following sense. If fii R-> R is continuous and strictly increasing for every i = 1 , . . . , / / , then ( ^ / , M , ) / 1 I is (separately) better-reply secure, reciprocally u.s.c., and payoff secure if and only if (Xjyfi <> Ui)fL i is. ^^ I am grateful to Faruk Gul and Tim van Zandt for encouraging me to refrain from placing restrictions on the sum of the players' payoffs. Suppose there are just two bidders. Both of their (ex ante) payoffs can simultaneously jump down if each bidder suddenly ties (from below) the other bidder*s high-profit winning bids.
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Together, Theorem 3.1 and Proposition 3.2 immediately yield the following Useful result. COROLLARY 3.3: IfG = (X^, M^)/1 ^ is compact, quasiconcavey reciprocally upper semicontinuous and payoff secure, then it possesses a pure strategy Nash equilibrium.
A further imphcation is the following generalization of Sion's (1958) existence result for two-person zero-sum games. COROLLARY 3.4: Let G = {X^,u^)f^^ be a compact, quasiconcave, reciprocally u.sx, game. If each player's payoff is lower-semicontinuous in the pure strategies of the others, then G possesses a pure strate^ Nash equilibrium. PROOF OF COROLLARY 3.4: By Corollary 3.3, it suffices to show that the game is payoff secure. So, choose x^X and 6 > 0 . The lower semicontinuity of UiiXi,') on X_i implies that {JC!_, GZ_,|WJ(X^,JC'_,)>M^(X)-e} is an open neighborhood of JC_, . Consequently, the game is payoff secure. Q.E.D.
Before proceeding to the proof of Theorem 3.1, we describe the main ideas. Better-r^ly security is used to construct for each player i, a function, M,(JC), that is l.s.c. in the opponents' strategies yet capable of detecting the presence of profitable deviations in the sense that x* is an equihbrium if and only if for every player /, sup^^.^j^. W^(JK:„X*^)< w/x*). The ability to detect w^-profitable deviations via a function that is lower semicontinuous in the opponent's strategies helps provide enough continuity to reduce the general existence question to establishing that for every finite set of deviations, there is a strategy against which no w, can detect a w-profitable deviation among elements of the finite set. This is the import of Parts I and II of the proof. That such strategies do indeed exist for each finite deviation set is estabhshed in Part IIL Part III then needs only to establish the existence of a strategy with equilibrium-like properties when strategy spaces are restricted, yet must include some fixed, finite set of potential deviations. As we shall explain momentarily, the restricted strategy spaces cannot be finite so that the presence of discontinuities remains an issue. Fortunately, the lower semicontinuity of w,(x) in x_„ allows one to approximate it from below by a sequence of functions that are continuous in x_i. Consequently, standard equilibrium existence results guarantee a mixed strategy equilibrium to games with the restricted strategy spaces whose payoffs are the continuous approximations. An essential component of the approximating games is that while each player is allowed to employ mixed strategies, each views the others as employing only pure strategies. This is accomplished by assessing each player's payoff at the average (pure) strategy determined by the others' mixtures,^"* and permits the quasiconcavity of Uiix) in ^"^It is this construction that requires the restricted strategy spaces to be infinite.
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Xj^ to be exploited to construct an appropriate purification of the sequence of mixed equilibria that converges to a pure strategy "equilibrium" of the game with the restricted strategy spaces and original payoffs. A key to establishing that no Uf can detect a M-profitable deviation from the limit strategy among elements of the finite set is the property that the sequence of approximating game payoffs underestimates the value of u^ along the sequence of purified strategies, while it overestimates the w-value of a deviation at the limit. The former property follows because each u^ (and so u^) is approximated from below, while the latter property requires the approximation from below to be "as high as possible/' The following lemma provides the technical result that is needed to obtain the continuous payoff approximation described above. Its proof can be found in the Appendix. LEMMA 3.5: Suppose that Yis compact, metric, andf: y-> R is lower semicontinuous. Then there exists a sequence {/„} of real-valued continuous functions on Y such that Vy e Y: 0) fn^y)
Part (i) of Lemma 3.5 ensures that payoffs can be approximated continuously from below, while part (ii) ensures that the approximations are "high enough." PROOF OF THEOREM
Ui{x)=
3.1: For each player /, and every x^X,
sup
inf
let
Ui{x^,x'_X
where the sup is taken over all open neighborhoods, U, of jc_^. So defined, for each / and every x^^X^, W/(jc„*) is both real-valued (since u^ is bounded), and lower semicontinuous on X_^. To see lower semicontinuity, observe that for fixed x,-eX,-,M,.(x,.,^_,.) = sup^pe„f;cA'.,/yC^-A where /y(A:_,.) = inf^» .^^ w^(jc,,x'_,), if x_i^U, and fij(x_^)= —^ otherwise. Therefore, for each Xi^X^, Uiixi,-) is the pointwise supremum of a collection of Idwer semicontinuous functions on X_i, and hence lower semicontinuous itself. Note that player / can secure a payoff strictly greater than a G R at x* e X if and only if (3.1)
sup u^(x-,xti)
> a,
Xi^Xi
Let r denote the closure of the graph of the game's vector payoff function, u: X-> R^. The remainder of the proof will be broken into three parts.
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Part I (Detecting Nash Equihbria): By (3.1) and the better-reply security property, we have the following: If (jc*,M*)Gr, andforall i (3.2)
sup W/(x,,x*,-) <w*,
then jc* is a Nash equilibrium. Part II (Sufficiency of Finite Deviation Sets): For every x,y^X, let W(JC, y) = {u4^x^,y_^\,.,,Uj^(xj^yy_f^)X and for each J C G X , define £(x) = {(y,w)e r\u{x,y)
u{x,x)
for every Xei^.
To prove (3.3), let /^ = {x\...,x'"} be a finite subset of X, and for / = 1,2,,.,yN, let X^^ = {x/,...,xf}. Since each X^^ is finite, we may endow its convex hull, coX^^, with the Euclidean metric. Let the symbol "->£:" denote convergence in the Euclidean metric. Now because each X^ is a topological vector space, the topology on co X^ QX^ induced by the Euclidean metric is at least as fine as the original one.^^ Consequently, each w,(x^,-), being l.s.c. on Xj^iXj in the original product topology, is also l.s.c. on X^^, coX-^ in the Euclidean metric. Moreover, the Euclidean metric renders the sets X^^ • coX,^ compact. Hence, according to Lemma 3.5, for each i=l,2,.,,,N, and every x^^Xf, there is a sequence of functions, {w"(x,, •)}, each continuous in the Euclidean metric on X.^- coXJ^, such that for all x _ , e x^^, coXy^, (i) w"(x,,x_,)<M,(Xj,x_,), for all n, and (ii) Vx!!,. ->£ x_,,liminf„ wf(x-, xl^) > u^ix^, x_^.). For every n, construct the A^-person game, G", as follows. Player /'s set of pure strategies is AiX^^), the set of probability measures on Xj^, For /x e ^^The two topologies are identical when X^ is Hausdorff. See footnote 7.
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>f=, A(Xl^X player Vs payoff is
where Xy = JxP^j dixj e co X^^, for j = 1,2,..., AT. Consequently, each i;" is continuous on >^^j A{Xf) in the Euclidean metric and quasiconcave in ix^ for every /i._^. By standard results for continuous games (e.g., Fudenberg and Tirole (1991), Theorem 1.2), G" possesses an equilibrium /x" for every n. For every player i, we must then have for every x^ given positive weight by /x", and every x\ ^Xf u'lix'i^x'Li) < vp( fji") = u^(Xi,x"_i) < Uiix^,xli)
< u^ix^.xl^),
where the first inequality and the equality hold since /t" is an equilibrium of G", the second inequality holds by condition (i) of the constructed approximating sequences w", and the last equality holds by the definition of u^. Therefore, for each x, given positive weight by /tf, u^ix^^xl^) is at least as large as /'s equilibrium payoff i;"( jx"^). By the quasiconcavity of M,(*, X" ,) on X,., we must then have (3.4)
<(x^,x^,)
for all
x^^XP,
Since for every n, x" is an element of the compact metric space X^^ j co XP, and each u^ is bounded, we may assume without loss of generality that x" ->£ X e >^^ J CO XP, and that lim„ M,(X") = u^ exists for every f. The inequality (3.4) then yields for all x e >
where the first inequality follows by condition (ii) of the approximating sequences wf. Since F c >^^ j XP is an arbitrary finite subset of X, and x" ->£ x implies x" -> X, so that (x, u) e T, this proves (3.3). Q,E,D, REMARK 3.1: Better-reply security ensures that the set of Nash equilibria is closed and that limits of e-equilibria are equilibria. Both of these follow from Part I of the proof. To see the first, note that if x* is a Nash equilibrium, then (x*,w(x*)) satisfies (3.2). Hence, the closed set {X*|(X*,M*) e F satisfies (3.2) for some M*} coincides with the set of Nash equilibria. To see the second, note that if x^ is a sequence of e-equilibria converging to x* as e tends to zero, then for every player /, every x, G Z , , and every e, MI(^/> ^-/) ^ ^i(Xi, xl^) < MJ(X^) + e. Hence, the lower semicontinuity of each W/(x^, •) in the strategies of the others implies that (X*,M*) G F satisfies (3.2), where u* = lim^ u(x^)}^ In a metric space context, the result that limits of e-equilibria are equihbria implies that the set of Nash equilibria is closed, since one can choose e = 0 along the sequence. However, in a topological space the two results must be considered separately since the behavior of sequences and their limits is not sufficient to establish closedness of a set.
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Two examples illustrate the above results. EXAMPLE 3.1: Consider a class of two-person, non-zero-sum, noisy games of timing on the unit square.^^ Related games have been used to model behavior in duels as well as in R & D and patent races. The payoff to player i is given by
(
l^ixi),
ifx,<x_
(t>i(xi),
ifx,.=x_
m^(x_,),
if jCj>jc_
where /,(•) and m,(0 are continuous on [0,1], and l^i-) is nondecreasing on [0,1]. We have the following: If for / = 1,2 and every x e [0,1], (i) ^(:c) e CO{/^(JC), m,{x)} and (ii) sgn(/,(x) - ^,(jc)) = sgn(_,(x)-m_,(jc)), then the above game possesses a pure strategy Nash equilibrium. To see this, note first that this game is clearly compact, while condition (ii) together with the continuity of /^ and m, imply that the vector of payoffs is reciprocally u.sx. Moreover, condition (i) and /^ nondecreasing imply that the game is quasiconcave; and condition (i) combined with the continuity of /, and m^ imply that the game is payoff secure. Thus, Corollary 3.3 applies. The next example shows how Theorem 3.1 can be employed as an alternative to Debreu's (1952) social equilibrium existence result to establish the existence of a Walrasian equilibrium. Consumers' choice sets are rendered independent of the price vector chosen by the auctioneer by defining their payoffs to be (discontinuously) unattractive at unaffordable choices. EXAMPLE 3.2: Consider an iz-consumer m-commodity exchange economy in which each consumer / has a continuous, locally nonsatiated, quasiconcave utility function u^: R ^ ^ R+, and a strictly positive endowment vector e]. The following game is played by the n consumers and an auctioneer. Each consumer / chooses a nonnegative bundle x^<e +1, where e is the aggregate endowment vector. The auctioneer chooses a price vector p from the unit simplex in R^. Consequently, strategy spaces are compact and convex. Payoffs in the game are as follows. Given the joint strategy (xi,jC2,...,x„,;7), consumer /'s payoff is UfiXi) if P'Xi
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proportional reduction in his consumption of every commodity.^^ Together with the continuity of the auctioneer's payoff, this implies that the game is payoff secure. So, we may apply Corollary 3.3 to conclude that this game possesses a pure strategy Nash equilibrium, which by standard arguments constitutes a Walrasian equilibrium of the exchange economy. Theorem 3.1 can also be applied to obtain the existence of a symmetric equilibrium in nondecreasing bidding functions in symmetric mth-price auctions for m > 3. However, in order to apply the theorem, ties in bids must be broken in favor of high-value bidders. This unusual tie-breaking rule, together with the fact that ties can occur in equilibrium, suggests that equilibria in these auctions may fail to exist under the standard tie-breaking rule. See Reny (1998). 4. PURE STRATEGY SYMMETRIC EQUILIBRIA
The result of the previous section can be improved upon when the game in question possesses enough symmetry. Throughout this section, we shall assume that for all players /,;,X, =Xy. So, in contrast to the previous section, we here let Z = Z i = ••• =Xj^, If in addition, u^{x,y,.,,,y)-=^U2iy,Xyyy,.,yy)^ — = ^N^y^. •., y, x) for all jc, y e X , we say that G = (JQ, w,)/l j is a quasi-symmetric game. Note that quasi-synmietry is weaker than symmetry for games with three or more players. We shall also maintain the following convention throughout this section. For each player /, and for all jc,yGX, M,(y,...,x,...,y) denotes the function u^ evaluated at the strategy in which player / chooses x and all others choose y. Define a quasi-symmetric game's diagonalpayojf function i;: X-> R by v{x) = Wj(jc,...,jc)= -' =M;^(x,...,x) for every J c e X Recall that a Nash equilibrium, (X*,...,JC]V), is symmetric \i x^-= -- = ^ N DEHNITION: Player i can secure a payoff of aE^R along the diagonal at (x,... ,x) e Z ^ , if there exists x^X, such that w/x',..., x,..., xO > a for all x' in some open neighborhood of x e X DEHNITION: The game G = (X^, u^)f^ j is diagonally better-reply secure if whenever (x*, w*) e X X R is in the closure of the graph of its diagonal payoff function and (x*,...,x*) is not an equilibrium, some player / can secure a payoff strictly above w* along the diagonal at (x*,...,x*). DEFINITION: The game G = (Z^-, M^)/1 ^ is diagonally quasiconcave if X is convex, and for every player /, all x \ . . . , x'" e X and all x e co{x\..., x'"}
M^(x,...,x) >
min
M^(X,...,X^,...,X).
\
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Letting w-(x, y) = u^iy,..., jc,..., y) for each player /, and every x, y e X, G is diagonally quasiconcave if and only if each w^ is diagonally quasiconcave in the sense of Zhou and Chen (1988). See also Baye et al. (1993). It is worthwhile to note that diagonal quasiconcavity is strictly weaker than quasiconcavity, even for two-player games. Our main result for quasi-symmetric games is as follows. THEOREM 4.1: / / G = (JQ, i^,)/li is quasi'Symmetric, compact, diagonally quasiconcave, and diagonally better-reply secure, then it possesses a symmetric pure strategy Nash equilibrium.
As with better-reply security, diagonal better-reply security can sometimes be verified by cheeking simpler, yet more restrictive, conditions. DEHNITION: The game G = (J^,,w^)-li is diagonally payoff secure if for every x^X and every 6 > 0, each player / can secure a payoff of u^ix,..., x) - e along the diagonal at (jc,..., JC).
For games with three or more players, diagonal payoff security is strictly weaker than payoff security. For example, Hotelling's location game is not payoff secure when X is taken to be a firm's set of mixed strategies there, although it is diagonally payoff secure. The proof of the following proposition is omitted as it follows the same lines as the proof of Proposition 3.2, which can be found in the Appendix. PROPOSITION 4.2: If the quasi-symmetric game G = (X^,Ui)fL^ is diagonally payoff secure and each u^ix, ,.,,x) is upper semicontinuous as a function ofx on Xy then G is diagonally better-reply secure.
Since quasi-symmetry implies that Wi(x,...,x)= ••• =-Ujsj{x,,,.,x) for all x^X, requiring M^(X, ..., JC) to be u.s.c. in jc on X in a quasi-symmetric game is weaker than reqm^ring the vector of the player's payoffs to be reciprocally u.s.c. on X^. Consequently, for quasi-symmetric games, the hypotheses of Proposition 4.2 are strictly weaker than those of Proposition 3.2. An immediate consequence of Theorem 4.1 and Proposition 4.2 is the following result, which tends to be sufficient for most applications. COROLLARY 4.3: If G = (X/,M^)fii is quasi-symmetric, compact, diagonally quasiconcave, diagonally payoff secure, and each M^(X,...,A:) is upper semicontinuous as a function of x on X, then it possesses a symmetric pure strategy Nash equilibrium.
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We now provide the proof of Theorem 4.1. PROOF OF THEOREM
4.1: For every x,y
GX,
let
W i ( ^ , j ) = s u p inf M j ( ; c , y , . . . , / ) , where the sup is taken over all open neighborhoods UQX of y. So defined, Mi(x,y) is, for every x^X, real-valued (since Wj is bounded) and l.s.c. as a function of y on X Ix)wer semicontinuity follows as in the proof of Theorem 3.1. Consequently, player 1 can secure a payoff strictly above a e R along the diagonal at (y,..., y) e X^ if and only if (4.1)
sup Ui{x,y) > a.
Let r denote the closure of the graph of the game's diagonal payoff function. Because G is quasi-symmetric and diagonally better-reply secure, (4.1) implies that ( X * , . . . , X * ) G Z ^ is a symmetric Nash equilibrium of G if for some w* e R , ( j c * , M * ) e r and (4.2)
Mi(jc,jc*)<w*,
\fx^X.
For each x e X , let E{x) = {y G Z : Uiix.y) < Ti^iy)}, where Ui(y) = infy^ySnpy^ijU^(y\,,,,yX and where the infimum is taken over all open neighborhoods of y. Because payoffs are bounded, w/O is a well-defined real-valued function on X. Moreover, it is upper semicontinuous on X (the proof being analogous to that demonstrating the lower semicontinuity of Ml).
Now, suppose that x* e £ ( x ) for every x^X. Then, MJ(JC,JC*)<MI(X*) for every x e X However, because (X*,MJ(X*)) G T , this implies that (4.2) is satisfied with M*=Mi(x*), so that (JC*,...,JC*) is a symmetric Nash equilibrium. Thus, it suffices to show that DXGX ^ ( ^ ) is nonempty. Because Wj(0 is upper semicontinuous on X, and for every jc e X , Wj(x,y) is I.S.C. in y, E(x) is a closed (hence compact) subset of the compact set X, for each x e X Therefore, it suffices to show that the collection of compact sets {JE^(JC))^ ^ ^ possesses the finite intersection property, which by the KKM Theorem, would be the case if for every finite number of points X ^ . . . , J C ' " e X , c o U \ . . . , j c ' " } c H x O u -^ UEix""), So, suppose by way of contradiction that iceco{jc^...,jc'"}, and x^Eix^) U ••• VEix^X This means that M I ( X ^ , X , . . . , J C ) > MJ(JC^, jc)> WI(JC)>MJ(JC,..., jc) for each A: = 1,..., m, where the first and third inequalities follow from the definitions of u^ and u^, respectively. But this contradicts the diagonal quasiconcavityofG. Q,E.D.
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The following class of examples illustrates Theorem 4.1 through Corollary 4.3, In particular, it should be noted that payoffs are not required to be quasiconcave in the owner's strategy or jointly continuous in all strategies. Particular games among the class include Bertrand duopoly with continuous, but otherwise unrestricted, demand; as well as models of brand loyalty (see, e.g., Baye et al. (1993)). EXAMPLE 4.1: Consider a compact, quasi-symmetric, diagonally payoff secure game G = (X^, M,)/1 J with X^ = [0,1], and such that u^ix,..., jc) is u.s.c. in x on [0,11. If for each x e [0,1], M/Jc,..., jc,..., x) is either: (i) nondecreasing in x on [0, x], or (ii) nonincreasing in x on [ Jc, 1], then G possesses a symmetric pure strategy Nash equilibrium.^^
This follows directly from Corollary 4.3 since conditions (i) and (ii) imply that G is diagonally quasiconcave.
5. MIXED STRATEGY EQUILIBRIA
In this Section, we present a number of mixed strategy corollaries to the pure strategy existence results derived in the previous sections. Out of the need to calculate expected payoffs, we shall assume throughout this section that each u^ is both bounded and measurable. In addition, we shall assume that each X^ is a compact Hausdorff space, and weil then call G = (Z^, u^)f^ j a compact, Hausdorff game. Consequently, if M^ denotes the set of (regular, countably additive) probability measures on the Borel subsets of X^, M^ is compact in the weak* topology.^^ Extend each u^ to M = xfL^M^ by defining Ui{fx)=^ fxUiix)d/x for all /A e M , and let G = (A/^, w^X^i denote the mixed extension of G. The definitions of better-reply security, reciprocal upper semicontinuity, payoff security^etc. given in Sections 2-4 apply in the obvious ways to the mixed extension G}^ However, it should be noted that although reciprocal upper semicontinuity^ of G implies that of G, better-reply security (resp., payoff security) of G neither implies nor is implied by better-reply security (resp., payoff security) of G.^ ^^Note that the conditions allow M,(JC,...,JC,...,JC) to be nondecreasing in x on [0,x] for some values of Jc, and nonincreasing in x on [x, 1] for other values of x. ^^This follows from the Riesz representation theorem and AIaoglu*s theorem. See, for example, Dunford and Schwartz (1988). ^ Simply replace Xi by M^ everywhere in each definition. ^When one moves to mixed strategies, securing any particular payoff becomes both easier and more difficult. It becomes easier because one can now employ mixed strategies to attempt to secure a payoff and this can increase the payoff one can secure (e.g., as in matching pennies settings). On the other hand, one's payoff must be secure against perturbations from the (larger) set of mixed strategies of the others, which can reduce the payoff one can secure.
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The following result often provides a convenient means for checking reciprocal upper semicontinuity of G by reducing the task to checking that the sum of the players' payoffs is u.sx. on the set of pure strategies, X= X^^j X^. A proof can be found in the Appendix.^^ PROPOSITION 5.1: / / EfliM/(x) is upper semicontinuous in x on X, then Eflj/;^M^(x)^/x. is upper semicontinuous in fx on M, Consequently, the mixed extension of G = {Xi, M,)/1 J is reciprocally upper semicontinuous.
We now present the mixed strategy implications of Theorem 3.1 and Proposition 3.2. COROLLARY 5.2 (TO THEOREM 3.1): Suppose that G = (Z,, w,)/l j is a compact, Hausdorff game. Then G possesses a mixed strategy Nash equilibrium if its mixed extension, G, is better-reply secure. Moreover, G is better-reply secure if it is both reciprocally upper semicontinuous and payoff secure.
It is the above corollary that directly generalizes the mixed strategy equilibrium existence results of Nash (1950), Glicksberg (1952), Mas-Colell (1984), Dasgupta and Maskin (1986), Robson (1994), and Simon (1987). See subsections 6.1 and 6.2 below. Corollary 5.2 can be applied to prove the existence of mixed strategy equilibria in many standard economic games: Bertrand price competition with arbitraiy continuous cost and demand functions, Cournot quantity competition with fixed costs of production, models of adverse selection, etc. The example to follow illustrates a simple application. EXAMPLE 5.1: Consider a zero-sum concession game between two players (see Hendricks and Wilson (1983)). The players must choose a time t^,t2^ [0,1] to quit the game. The player who quits last wins, although conditional on winning, quitting earlier is preferred. If both players quit at the same time, the unit prize is divided evenly between them. Then payoffs are
iit,
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to see that the mixed extension is payoff secure: increasing slightly the time you choose to quit at worst only slightly reduces your payoff so long as the other player's mixed strategy does not change too much. Thus, Corollary 52 applies, ensuring that this game possesses a mixed strategy Nash equilibrium. It is perhaps notable that Dasgupta and Maskin's (1986) weak lower semicontinuity requirement fails in this game at the point (1,1), so that their main existence theorem cannot be applied here.^"* We now apply Corollary 5.2 on the existence of mixed strategy Nash equilibn\^ to show that the multi-unit pay-your-bid auction possesses a pure strategy equilibrium in which the bidders employ nondecreasing bidding functions. EXAMPLE 5.2: There are N risk-neutral bidders competing for K units of a homogeneous good. Bidder / receives a multi-dimensional signal x , e [ 0 , l ] ' " according to the distribution function i^, having continuous and positive density, fi on [0,1F. The bidders' signals are independent. The signal x^ determines bidder f s marginal valuation, v[{Xi) for the kxh unit of the good. Each i;[(0 is assumed to be continuous and strictly increasing on [0, I F , with u'^-) > V2{') > " > v'j^i'X and v{(0) = '•• = vj^iO) = 0.^ Knowing only their own signals, each bidder / submits K nonnegative bids b\>b[> -- >^|^. The highest K bids among tlie KN bids submitted are winning bids, and winning bidders pay the seller their winning bid for each unit won. Relevant ties are broken equi-probably.^^ We shall establish the following.
The asymmetric multi-unit pay-your-bid auction above possesses a pure strategy equilibrium in which the bidders employ nondecreasing bidding functions. We first wish to show that the auction possesses a mixed strategy Nash equilibrium in which bidders mix over nondecreasing bidding functions (although they can choose arbitrary measurable bidding functions). To begin, restrict each bidder's pure strategies to the set of ordered X-tuples of nondecreasing bidding functions (with the first majorizing the second,...,majorizing the KXh) each from [ 0 , 1 ^ into [0,D], where D is an upper bound on each bidder's marginal valuations, such that no bidder bids above his value on any unit. Consequently, the pure strategy sets are compact metric spaces in the topology of (ahnost everywhere) pointwise convergence.^^ ^"^This has been pointed out by Simon (1987). ^By strictly increasing, we mean that each marginal valuation is nondecreasing and strictly increases when all components of the signal strictly increase. We allow the possibility that marginal valuations remain constant when only some, but not all, signals strictly increase. ^^A tie is relevant if it must be broken in order to determine the allocation. In single-unit auctions all high-bid ties are relevant, but this is not always so in multi-unit auctions. ^^As in Lp spaces, two nondecreasing bidding functions here are deemed identical if they are equal almost everywhere with respect to Lebesgue measure.
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We now argue that this game's mixed extension is better-reply secure. So, suppose that m* is not an equilibrium and let (m*,M*) be an element of the closure of the graph of the mixed extension's vector (ex ante) payoff function. By definition, lim u{m") = w* for some sequence of mixed strategies, {m"}, converging to m*. Given m*^ and e>0, bidder / can achieve a payoff within e of his supremum (when restricted to nondecreasing bid functions) by employing a iC-tuple of strictly increasing bid functions, bf, say. Moreover, his payoff is continuous in the others' mixed strategies at ibf,mti)^ Now, if relevant ties occur with probability zero given m*, then payoffs are continuous at the limit, i.e., M* = w(m*). So, because m* is not an equilibrium there is a bidder / and s>0 small enough such that M,(fef,ml,) >Uiini^) = w*. Hence, the continuity of Ujibf,') at rn^i implies that bidder / can secure a payoff strictly above uf at m*. On the other hand, if relevant ties occur with positive probability given m*, then all those bidders tying with positive probability would strictly prefer to win with probability one (recall that bids are never above one's value on any unit). Moreover, along the sequence, m", one of those bidders who ties at the limit loses with probability bounded away from zero. This bidder i can then achieve a strictly higher payoff than the limiting payoff of uf = lim Ui(m") by employing bf for £r > 0 small enough. That is, M,(fcf, m",) is above and bounded away from Uiim") for large n. So once again, the continuity of M,(Z>f, •) at ml,- implies that u^ib^,m*_i) > uf and that bidder i can secure a payoff strictly above wf, at m*. Consequently, better-reply security holds. We may therefore apply Corollary 5.2 to conclude that the game possesses a mixed strategy equilibrium. So, up to this point, we have demonstrated that the pay-your-bid auction possesses a mixed strategy equilibrium when restricting attention to puresstrategies that are nondecreasing bidding functions. Assume now that all bidders' marginal valuation functions are strongly increasing. That is, they are nondecreasing, and strictly increase whenever at least one signal strictly increases and no signal decreases. As we show in the Appendix every mixed strategy equilibrium as above is then a full equilibrium in the sense that there are no profitable deviations among measurable bid functions. Moreover, it is shown there that all mixed equilibria are in fact pure. That is, in equilibrium each bidder's mixed strategy places probability one on a single nondecreasing bid function. Finally, it is shown in the Appendix that as regards existence, the restriction to strongly increasing marginal valuation functions is unnecessary. This completes, the argument. Note that because the multi-unit pay-your-bid auction fails to be reciprocally u.s.c. (see footnote 13), the results of Dasgupta and Maskin (1986) and Simon (1987) cannot be directly applied. The model of the multi-unit pay-your-bid auction presented here generalizes that of Engelbrecht-Wiggans and Khan (1995) in a number of ways. For example, their formulation does not allow the distribution of a single bidder's K marginal values to lie along a lower-dimensional surface. Thus, they rule out cases in which the niunber of objects exceeds the dimensionality of the signal
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space. In addition, they assume that the bidders are symmetric in terms of the distribution of their values. Finally, they assume, but do not prove, the existence of a symmetric nondecreasing equilibrium. We examine the symmetric case below in Example 5.3. Since the analysis above allows K=l, it applies to single-unit auctions. This then furnishes a proof of the existence of a pure strategy equilibrium in nondecreasing bid functions for asymmetric first-price auctions with risk neutral bidders and multi-dimensional signals in the independent private values case. We now provide the mixed strategy consequences of Theorem 4.1 and Proposition 4.2 for quasi-symmetric games. For the following result only, let M denote the common set of mixed strategies for each player /. COROLLARY 5.3 (TO THEOREM 4.1): Suppose that G-={Xi,u)f^^ is a quasisymmetriCy compacty Hausdorff game. Then G possesses a symmetric mixed strategy Nash equilibrium if its mixed extension, G, is better-reply secure along the diagonal. Moreover, G is better-reply secure along the diagonal if it is diagonal^ payoff secure and each M,( / I , . . . , JLI) w upper semicontinuous as a function of p. on M.
EXAMPLE 5.3: Engelbrecht-Wiggans and Khan (1995) provide a number of results on the character of symmetric equilibria in nondecreasing bidding functions in the multi-unit pay-your-bid auction when the bidders are symmetric. However, they do not establish conditions under which such an equilibrium exists. We do so here. Consider the model of this auction introduced above. In addition, assume that each bidder has the same K marginal valuation functions and the same signal distribution function. An analysis similar to that above, but now employing Corollary 5.3, establishes that this symmetric X-unit pay-your-bid auction possesses a symmetric pure strategy equilibrium in which the bidders employ a common vector of nondecreasing bid functions.
Corollary 5.3 can also be applied to provide a rather permissive symmetric mixed strategy equilibrium existence result for Hotelling's location game. However, we shall not present this result here.^* PROOFS OF COROLLARIES 5.2 AND 5.3: Both corollaries follow by simply verifying that the conditions of the associated theorem and proposition from Sections 3 and 4 are satisfied for the game's mixed extension and by noting that a pure strategy equilibrium of the mixed extension constitutes a mixed strategy equilibrium of the original game. Q.E.D. ^Dasgupta and Maskin (1986) and Simon (1987) provide conditions under which a mixed strategy equilibrium exists in Hotelling^s location game. In particular, they require that the set of locations be convex, or have smooth boundaries. By applying Corollary 5 3 , both of these conditions can be dispensed with and a symmetric mixed strategy equilibrium is nonetheless guaranteed to exist.
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We now indicate how Corollary 5.2 yields the results of Glicksberg (1952) (and hence of Nash (1950)), Dasgupta and Maskin (1986), Mertens (1986), Robson (1994), and Simon (1987).^^
6.1. Glicksberg (1952), Mertens (1986), and Robson (1994) Glicksberg (1952) shows that compact, Hausdorff games with continuous payoff functions possess mixed strategy Nash equilibria. Robson (1994), in the course of studying the informational robustness of equilibria, proves that in a compact game with metric strategy spaces, if each player's payoff is u.s.c. in all players' strategies, and continuous in the other players' strategies, then the game possesses a mixed strategy Nash equilibrium. Mertens (1986) shows that twoperson zero-sum compact games possess a value whenever player I's payoff is U.S.C. in his own strategy and Ls.c. in player II's strategy.^^ All of these results follow directly from Corollary 5.2. To see this note that by Proposition 5.1, lower-semicontinuity of one's payoff in the opponents' pure strategies implies lower-semicontinuity in the opponents' mixed strategies. But lower-semicontinuity in the opponents' mixed strategies for each of a player's pure strategies implies payoff security on the set of mixed strategies. Consequently, if every player's payoff is lower-semicontinuous in the others' pure strategies, as in each case above, then the game's mixed extension is payoff secure. Since in each case above the sum of payoffs is upper semicontinuous, the mixed extension is (by Proposition 5.1) reciprocally u.s.c. Hence, in each case CoroUary 5.2 applies.
6.2. Dasgupta and Maskin (1986) and Simon (1987X Dasgupta and Maskin (1986) impose the following conditions in order to guarantee the existence of a mixed strategy Nash equilibrium: (a) G = (X^y u^)fL i is a compact game whose sum of payoffs is u.s.c. on X; (b) each X, is a convex subset of R'"; (c) the discontinuities of each u^ lie along a finite number of "diagonal" sets; (d) each u^ is "weakly" lower-semicontinuous in /'s strategy choice. Simon (1987) strictly generalizes the result of Dasgupta and Maskin (1986) by making the following assumptions: (i) G is a compact game with metric strategy spaces and a reciprocally u.s.c. mixed extension; (ii) there exists a sequence ^^The result of Mas-Colell (1984) is a variant of Dasgupta and Maskin's (1986), and it too can be shown to follow from Corollary 5.2 Actually, this u.s.c.-l.sx. mixed strategy Nash equilibrium existence result for two-person zero-sum games follows from Sion's (1958) theorem. Mertens* (1986) main interest is in showing that in addition, both players have e-optimal strategies with finite support. Our results do not generalize this latter result of Mertens*.
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{>;• Xp}^ ^ 1 of finite subsets of the original pure strategy sets satisfying, where M/* denotes i's mixed strategies on X^: For each player /, for every (jc^,7i_,) e X , XM_,, there exists a sequence of pairs (ju,;^, V), with /nl e M^, and 7 " c Z _ , , n = 1,2,..., such that for all n: Jl_i(Y") > 1 - 1/n, and E
M?(^i) liminf M / X , , X ' _ , ) | > M^(X,,X_,) - 1/n,
for all x_i G y«. Simon's (1987), and hence also Dasgupta and Maskin's (1986), conditions imply the following. Since by (i) u^ is bounded, for every e > 0, there is an n large enough such that^^ liminf M,(/i,",/>t'_^) > f
liminf w,(/Af,xL,)|
'^X_Xx'_i-^x_i
^/
{ E
dji_i(x_i)
J
M?U)|liminfa,(x„x'_,)]U7X_,(x_,)
>Ui(Xi,Jl_i)-€,
Thus, the game's mixed extension is payoff secure. Hence, Simon's (1987) conditions (i) and (ii), and so also Dasgupta and Maskin's (1986) conditions (a)-(d), imply the hypotheses of Corollary 5.2. The failure of reciprocal u.s.c. in the better-reply secure auction game of Example 5.2 shows that Corollary 5.2 is a strict generalization of all the results discussed above. 7. OTHER APPROACHES
Like Sion (1958), Mas-Colell (1984), Dasgupta and Maskin (1986), Mertens (1986), Simon (1987), Baye et al. (1993), and Robson (1994), we have followed here what might be called the topological approach to existence of equilibrium. An alternative approach is based upon lattice-theoretical concepts, and at its heart lies Tarski's (1955) fixed point theorem. Perhaps the best examples of this method are due to Topkis (1979), Vives (1990), and Milgrom and Roberts (1990) although its flavor can also be found in Roberts and Sonnenschein (1976), and Nishimura and Friedman (1981). In each of these papers, payoffe need not be quasiconcave, and in some needn't be continuous. The key property is that ^^The first inequality follows since the function of jc_,- in square brackets is l.s.c. and less than or equal to M,( /xf, x_,) for every x_i; and the third by (ii) together with the facts that n is large enough and «,- is bounded (by (i)).
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best-replies are increasing in the opponents' strategies. This method typically yields the existence of pure strategy equilibria. Now, while lattice-theoretic methods do not require payoffs to be continuous, enough continuity must be assumed in order to guarantee the existence of best replies. So, typically payoffs are required to be u.s.c. in one's own strategy, an assumption that fails to hold in virtually all auctions, as well as in the classic games of Bertrand and Hotelling. Consequently, most practical applications of lattice-theoretical techniques tend to be confined to continuous games. A third approach, although still fundamentally topological in nature, has been introduced by Simon and Zame (1990). They show that if one is willing to modify the (vector of) payoffs at points of discontinuity so that they correspond to points in the convex hull of limits of nearby payoffs, then one can ensure a mixed strategy equilibrium of such a suitably modified game. As an example of the usefulness of their approach, Simon and Zame note that the discontinuities in Hotelling's (1929) location model arise only when two firms choose the same location, and in this case, consumers are indifferent between which firm they patronize. Consequently, rather than insist that the two firms split the market evenly, Simon and Zame suggest that one ought to be content with any division of consumers among the two firms. Simon and Zame's result ensures that there is a "sharing rule" specifying how consumers are divided among firms when indifference arises such that the resulting game possesses a mixed strategy equilibrium. While in some settings involving discontinuities this approach is remarkably helpful (i.e., Hotelling's location game), in others it is less so. For example, in a mechanism design environment where discontinuities are sometimes deliberately introduced (auction design, for example), the pai;ticipants must be presented with a game that fully describes the strategies and payoffs. One cannot leave some of the payoffs unspecified, to somehow be endogenously determined. In addition, this method is only useful in establishing the existence of a mixed, as opposed to pure, strategy equilibrium. 8. APPROXIMATION BY FINITE GAMES
For those who are more comfortable with the finite than with the infinite, a legitimate concern might be whether some infinite game under consideration is well approximated by a finite game. To be sure, the ability to approximate the infinite-action game by a finite-action game lends a degree of robustness to the analysis of the infinite-action game: the presence of literally infinitely many actions and the discontinuities that may accompany them are not essential. While space does not permit a full discussion of these issues we briefly mention some relevant approximation results that follow from Reny (1996). Call one game a finite approximation of another if the strategy spaces of the one are finite subsets of the other, and the payoff functions of the one are the other's payoff functions restricted to the finite sets. Suppose that a compact, metric game's mixed extension is better-reply secure and that for every fixed strategy of the opponents, each player's payoff is discontinuous at the resulting
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joint strategy for at most countably many of his own strategies.^^ Then there is a sequence of finite approximations with the property that the hmit of any sequence of mixed strategy equihbria of the approximating games is an equihbrium of the original game. Suppose, in addition, that every player's value function is continuous in the mixed strategies of the others.^^ Then, every equilibrium of the original game is the limit of e-equilibria of a sequence of finite approximations in which e tends to zero along the sequence.^'* In short, a reasonably large class of infinite-action games that are better-reply secure are well approximated by and robust to finite-game approximations. Consequently, the analytic convenience of the infinite-action setting is not in cdnflict with a finite-action viewpoint. Dept of Economics, University of Pittsburgh, Pittsburgh, PA 16046, U,SA.; http://www,pitt.edu/ ^ reny/preny.htm Manuscript received December 1996; final revision received September^ 1998. APPENDIX PROOF OF PROPOSITION 3.2: Suppose that (JC*,M*) is in the closure of the graph of the gamers vector payoff function, and that x* is not an equilibrium. By reciprocal u.s.c, either M,(JC*) > uf for some /, o\ M,(A:*) = u* for all /. In the latter case, because x* is not an equilibrium, some player / has a deviation, x^-, such that M,(i,-, x*,) > u^ix* ) — u*. Consequently, in either case there is a player / and a strategy Jc,- (equal to xf in the former case) such that M,(JCJ, X* ,) > wf. Fix now this player i. Choose £ > 0 so that M^(JC,,X*,)>M* + e. Because the game is payoff secure player i has a strategy jc,- such that M,(JC/,X'_,)> M,(jc,-,x*y)- €>uf for all x'_i in some open neighborhood of xt/. Consequently, player / can secure a payoff strictly above ii* at x*. Thus, the game is better-reply secure. Q.E.D. PROOF OF LEMMA 3.5: Let F = {g G CCY): giy)
(a) Ky) a , V y G £ 7 . To see that (*) holds, suppose that fiy^) > a. Choose a' ^ia^fiy^)) and note that because / is lower semicontinuous, the set {y e Y: fiy) > a'] is open. Cbnsequently, there are open balls Bj and B2 around y^ with radii rj > r2 > 0, respectively, such that fiy) > a' for all y e jBj 3 ^ 2 . This can be weakened significantly. ^^A player's value function is the supremum of his payoff as a function of the mixed strategy of the opponents. Most auctions have continuous value functions as do Bertrand's and Hotelling's price competition games. •^"^I am grateful to the editor for encouraging me to obtain this result.
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Define hiy) to be constant and equal to a' on the closure of Bj, and to be constant and equal to miny^yfW on the complement of B^. By the Tietze extension theorem, h can be extended to a continuous function on all of Y so that h{y)<,f{y) for all y^B^, Consequently, setting U = B2y both (a) and (b) above are satisfied, and this proves (*). To show now that (/„} satisfies (ii), suppose, to the contrary, that y„ -^y° and that fniy^) -^ a< f(y0)}5 -j^gj^ /(y^) > a + €, for some e > 0. By (*), there exists h^F and a neighborhood U of y^ such that My) > a + e for all y e U. Since {gug2>"') is dense in F, we may choose k so that gf^ is within €/2 of h. Consequently, for all n>k such that y„ e U, we have fniyn^ ^gk(yn) ^ hiy„) - 6/2 > « + 6/2. But this contradicts /„(y„) -^ a.
Q.E.D.
PROOF OF PROPOSFTION 5.1: Clearly, it suffices to show that E-liMiCft) is upper semicontinuous on M. Thus, it suffices to show that if / : X -* R is upper semicontinuous on X^ then F{ /JL) = fxfix)dfjL is upper semicontinuous on M. (Recall that X is compact, Hausdorff, and that M is the set of regular, countably additive probability measures on the Borel subsets of X ) Suppose that for every ft e M, there exists a sequence {/„} of elements of C^ = {^ e CiX): g>f) such that (A.1)
( fix) dp, = Km [ f„{x) dfJL, -'X n ^X
Then, / /(jc)d\L = inf \ g(x)dpu, ^x g^CfJx
for every yu^M,
implying that Fi /t) is u.s.c. in fi on Af, being the infimum of a collection of continuous functions. Thus, it suffices to estaWish that for each /x e M , there exists a sequence of continuous functions {/„} majorizing / and satisfying (A.l). Since we have not found precise^ this result in the literature, we provide a proof. Fix /JLGM. Since / is u.s.c., it is measurable. Hence, because fju is regular, Lusin's theorem (see, e.g., Cohn (1980, Theorem 7.4.3)) gives for every « > 1, a compact subset X„ of X such that fjiX„) > 1 — ( l / « ) and / is continuous on X„. Since / is u.s.c, we may assume without loss that / ( j c ) < 0 for all x^X. For each / i > l , let g„(x) =/(jc) + ( l / « ) if A:e UJ^i^jt, and g„(jc)=l otherwise. Consequently, gn'^gn + i^f ^or all n, and g„ix) -^jXx) for all x e U^^ j Xf^ and so for /x, a.e. J c e X Hence, (A.2)
lim f g„(x) dfi^ [ fix) dfi, n Jx ^X by the monotone convergence theorem. Since each g„ is lower semicontinuous, while / is upper semicontinuous and g„ > / , Dowker^s theorem (Theorem VIII.4.3 of Dugundji (1989)) implies that for each « > 1, there exists /„ e CiX) such that g„ >/„ > / . This, together with (A.2) establishes (A.1) and the result. Q.^^-D, The Pay-Your-Bid Multi-Unit Auction Below, we establish the claims made in Section 5 of the main text concerning this auction. All claims but the last employ the assumption that all bidders' marginal valuation functions are strongly increasing, i.e., they are nondecreasing, and they strictly increase whenever at least one signal strictly increases and no signal decreases. Also, all clauns but the last concern the game in which pure strategies are constrained to be nondecreasing bid functions in which no bidder bids above his value on any unit. ^^Note that a is finite since for all «, /„(y„) > gi(y„), and gj e CiY) is bounded on 7.
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CLAIM 1: In a mixed strategy equilibrium, no bidder places ex ante positive probability on any single bid. PROOF OF CLAIM 1: First, note that whenever at least one of his values is positive, a bidder must earn a positive payoff in equilibrium, since he can always do so by bidding half his value on each unit. This wins with positive probability since all others might have values near zero and would then bid below him on these units. Note that this also implies that a zero bid is submitted on a unit if and only if the unit is valued at zero, and that units with positive value are won with positive probability. Suppose, by way of contradiction, that there is a largest k>\ such that bidder one places positive probably (ex ante) on some A;th-unit bid, 5. Since values are zero with probability zero, this implies that 5 > 0. Now, fix e e (0,5) and suppose that some other bidder places competing bids between b -re and T) with positive probability.^^ If e were small enough, this could not be a best reply. The reason is that bids near T) can only come from units valued above some v>l) since every positive valuation must earn a positive surplus bounded away from zero. Consequently, the other bidder can strictly increase the probability of winning an additional unit with value at least v by raising his bid just above 5. When e is small enough, the discrete jump in the probability of winning more than compensates for the (arbitrarily) small increase in the bid. We conclude that there exists e > 0 such that other bidders place competing bids outside the interval (b- e,V) with probability one. But this means that on a set of signals having positive probability, bidder one can reduce his A;th-unit bid slightly below T? without reducing his probability of winning. (This is feasible since, by definition of k, bidder one's bid on the k + 1st unit is without loss strictly below T) on this positive probability set.) Since his probability of winning is positive (recall that positive bids win with positive probability) this strictly improves his payoff, contradicting the equilibrium hypothesis and completing the proof. CLAIM 1: If in some mixed strategy equilibrium, the pure strategy b*{x) = (6*(JC), ..., b1^{x)) is a best reply for bidder 1 for every signal x^ [O,!]"*, then each 5J(-) must be nondecreasing. PROOF OF CLAIM 2: Let x^ and x^ e [0,1]"* be two distinct signals of bidder 1 with x^ > JC°. Let b^ = b*{x^y and b^ = b*{x^\ We wish to show that b^ > b^. Let us proceed by induction on the components of the vectors of bids. Suppose we have already shown that b\>b\y...,b\> b^, (When A: = 0 this statement is empty.) To complete the induction we need only show that ^i+1 > ^?+1. So, assume, by way of contradiction, that bl+i< b^+ j. Let / > 1 be the largest ; such that bl+1 < 6?+1,..., bl^j < b^^j. Let
b'-(bl.,.,blbU,,,,.M.ry,.r.u^--X)> and let
P=(fe},...,bl,fc^„...,b,V,bLj.,„...,fcJ:). We claim that both b^ and b^ are feasible bid vectors. That is, their components are nonincreasing. Since 6° and b^ are feasible, the feasibility of b^ follows from the string of inequalities 6j > ^°+j > bl+1 and bl+j> ^ ^^+^'+ j > ^]t+y'+1» where the first string follows by asstmiption, and the second by the definition of / . Similarly, the feasibility of P follows from the strings of inequalities bl>b^> 6°+J and bl+y> bl^jf>bl+y+i, where the first string follows from the induction hypothesis, and the second from the definition of / . ^Two bidders' bids (on perhaps different units) are competing, if, with positive probability, the higher of the two bids determines who wins one of the objects. Thus, with two units and two bidders only one bidder's first unit bid and the other's second-unit bid are potentially competing, while with two units and three bidders aU pairs of bids, except two second-unit bids, are potentially competing.
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So, b' must yield an expected payoff at least as large as b' when bidder Vs signal is x\ i = 0,1. These two best reply conditions combine to yield the following inequality
X; }=k+l
[Pj{bf)-Pj(b})][ujix'^)-Vj(x')]>0 .
where Pj(b) denotes the probability that bidder 1 wins a /th unit when his bid on that unit is b.^ Now, by definition of /', bj^ > bj for every / over which the above sum is taken. Consequently, because the probability of winning a unit is nondecreasing in one's bid for that unit, we have Pjib^y — Pjit^j) ^ 0 for every such term appearing in the sum. Also, because v is strongly increasing, Vjix^) ~Vjix^) < 0 for every such term appearing in the sum. Consequently, each term in the above sum is nonpositive and so each term must be zero. We conclude that Pjibf) == Pjibj) for every 7 = A: 4-1,..., A: + / . Now, because jc* =^ 0, bidder 1 has a strictly positive value for every unit when his signal is jc^ Consequently, each probability Pjibf) == Pj(bj) > 0 since, as we argued in the proof of Claim 1 above, bidders with a positive value on a unit must, in equilibrium, win that unit with positive probability. But this implies that bidder 1, when his signal is JC^, can reduce his bids on units / = A: + 1,..., A: +f from bf to bj without decreasing the (positive) probability of winning any object, thereby strictly increasing his payoff. Thus, b^ is a profitable deviation from ^°, contradicting the best-reply definition of b^ = b*(x^) and completing the induction. CLAIM 3: An equilibrium when strategies are constrained to be nondecreasing is an equilibrium without this restriction. PROOF OF CLAIM 3: Given his signal and the equihbrium mixed strategies of the others. Claim 1 implies that each player's payoff is a continuous function of his vector of bids. Consequently, there is an optimal vector of bids, (^,*(JC,),...,6,^(JC^)) for every signal x,. By Qaim 2, each t^C*) is nondecreasing. CLAIM 4; Every mixed strategy equilibrium assigns probability one to a single pure strategy. PROOF OF CLAIM 4: To keep the notation from obscuring the point, suppose there are two units for sale, so that ^ = 2, and that the signal space is two-dimensional so that a bidder's signal takes the form (X,>')G [0,lp.^^ Consider any selection, (^i(0, ^2(')X from bidder Vs best reply correspondence. For any fixed a e (0,1], b^ix, ax) is nondecreasing in x, and must jump whenever bidder 1 possesses multiple best first-unit bids at (x, ax). (Otherwise we could construct a selection from the best-reply correspondence that fails to be nondecreasing.) Since there can be but countably many such jumps, we conclude that for every a e (0,1] and all signals of the form (x, ax), there is a unique first-unit best reply for bidder 1 for all but countably many x e [0,1]. A similar argument applies to bidder I's second-unit bid. Consequently, by Fubini's theorem, the mbced strategy equihbrium is essentially pure in that for every bidder and almost every signal, in equilibrium the bidder places probability one on a single pair of bids. Moreover, by selecting an arbitrary best reply at (the measure zero set of) signals where multiple best replies exist, we obtain a pure strategy equilibrium in which the bidding functions are nondecreasing. CLAIM 5: The assumption that the marginal valuation functions are strongly increasing can be reduced to nondecreasing. Because a bidder's /th unit bid must be his yth highest bid, this probability does not depend on the values of his other bids. ^^The argument extends naturally to the ^-unit, m-dimensional signal setting.
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PROOF OF CLAIM 5: We have established that when all bidders' marginal valuation functions are strongly increasing, the auction possesses a nondecreasing pure strategy equilibrium. Suppose now that each L>J(-) is merely strictly increasing. For 6 > 0 , let wl(x) = vl(x)+ 81 x for every signal X e [0,1]'", where 1 denotes the m-vector of I's. Consequently, each w^i') is strongly increasing, and the auction with these functions as marginal valuations then possesses a nondecreasing pure strategy equilibrium. But for any fixed e > 0, this nondecreasing equilibrium will be an ^-equilibrium of the original auction (with marginal valuations f j^CO) so long as 6 > 0 is small enough since payoffs under the wl*s are uniformly close to those under the yj^'s. Hence, by Remark 3.1 following the proof of Theorem 3.1, the limit (which exists without loss) of these nondecreasing £r-equilibria of the original game, as s tends to zero, is a nondecreasing equilibrium of the original better-reply secure game.
REFERENCES BAYE, M . R . , G . TIAN, AND J. ZHOU (1993): "Charaaerizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," Review of Economic StudieSy 60, 935-948. BERTRAND, J. (1883): *'Theorie Mathematique de la Richesse Sociale," Journal des Savants^ 499-508. BiLLiNGSLEY, P. (1968): Convergence of Probabiity Measures. New York: John Wiley and Sons. COHN, D. L. (1980): Measure Theory, Boston: Birkhauser. DASGUPTA, P . , AND E . MASKIN (1986): "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies^ 53,1-26. DEBREU, G . (1952): "A Social Equilibrium Existence Theorem," Proceedings of the National Academy of Sciences, 38, 886-893. DuGUNDJi, J. (1989): Topology, Dubuque, Iowa: Wm. C. Brown Publishers. DuNFORD, N., AND J. T. SCHWARTZ (1988): Linear Operators Part / : General Theory, New York: John Wiley and Sons. ENGELBRECHT-WIGGANS, R . , AND C . M . KHAN (1995): "Multi-Unit Pay-Your-Bid Auctions with Variable Awards," mimeo. FUDENBERG, D . , R . GILBERT, J. STIGLITZ, AND J. TIROLE (1983): "Preemption, Leapfrogging, and
Competition in Patent Races," European Economic Review^ 22, 3-31. FUDENBERG, D . , AND J. TIROLE (1991): Game Theory. Cambridge, MA: MIT Press. GLICKSBERG, I. L. (1952): "A Further Generalization of the Kakutani Fixed Point Theorem," Proceedings of the American Mathematical Society, 3, 170-174. HENDRICKS, K., AND C. WILSON (1983): "Discrete Versus Continuous Time in Games of Timing," Mimeo, New York University. HOTELLING, H . (1929): "The Stability of Competition," Economic Journal, 39, 41-57. KARLIN, S. (1959): Mathematical Models and Theory in Games, Programming and Economics. Reading, Massachusetts: Addison-Wesley. MAS-COLELL (1984): Harvard University Lecture Notes, Economics 2157, Spring. MERTENS, J. F. (1986): "The Minmax Theorem for U.S.C.-U.S.C. Payoff Functions," International Journal of Game Theory, 15, 237-250. MILGROM, P., AND R . WEBER (1982): "A Theory of Auctions and Competitive Bidding," Econometrica, 50, 1089-1122. MILGROM, P., AND J. ROBERTS (1990): "Rationalizabihty, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, 58,1255-1277. NASH, J. F. (1950): "Equilibrium Points in n-Person Games," Proceedings of the National Academy of Sciences, 36, 48-49. NisHiMURA, K., AND J. FRIEDMAN (1981): "Existence of Equilibrium in n Person Games Without Quasi-Concavity," International Economic Review, 21, 637-648. PrrcHiK, C. (1982): "Equilibria of a Two-Person Non-Zero-Sum Noisy Game of Timing," International Journal of Game Theory, 10, 207-221.
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RENY, P. J. (1995): "Local Payoff Security and the Existence of Nash Equilibria in Discontinuous Games," Mimeo, Department of Economics, University of Pittsburgh. (1996): "Local Payoff Security and the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Mimeo, Department of Economics, University of Pittsburgh. (1998): "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Mimeo, Department of Economics, University of Pittsburgh. ROBERTS, J., AND H . SoNNENScneN (1976): "On the Existence of Coumot Equilibrium Without Concave Profit Functions," Journal of Economic Theory ^ 13, 112-117. RoBSON, A. J. (1994): "An *lnformationaily Robust' Equilibrium in Two-Person Nonzero-Sum Games," Games and Economic Behaviory 2, 233-245. RoYDEN, H. L. (1988>. Real Analysis. New York: Macmillan. SIMON, L. (1987): "Games with Discontinuous Payoffs," Review of Economic Studies, 54, 569-597. SIMON, L., AND W . ZAME (1990): "Discontinuous Games and Endogenous Sharing Rules," Econometrica, 58, 861-872. SiON, M. (1958): "On General Minimax Theorems," Pacific Journal of MathematicSy 8, 171-176. TARSKI, A . (1955): "A Lattice-Theoretical Fix Point Theorem and its Applications," Pacific Journal of Mathematics, 5, 285-309. ToPKis, D. M. (1979): "Equilibrium Points in Non Zero-sum n-person Submodular Games, SIAM Journal of Control Optimization, 17,173-787. ViVES, X. (1990): "Nash Equilibrium with Strategic Complementarities," Journal of Mathematical Economics, 19, 305-321. ZHOU, J. X., AND G. CHEN (1988): "Diagonal Convexity Conditions for Problems in Convex Analysis and Quasi-Variational Inequalities," Journal of Mathematical Analysis and Applications, 132, 213-225.
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18 James Dow on Hugo F. Sonnenschein
As an undergraduate I was mad keen on mathematical economics and microeconomics. I liked the idea of doing something very pure, logical and clever. A PhD in the US was the obvious next step. I had the good fortune to be admitted to several graduate schools, and to be advised by Frank Hahn on which to choose. After pointing out the advantages of the other schools, his conmient on Princeton was: "If you're interested in theory, Hugo is your man." Of course, this was irresistible tome. Hugo Sonnenschein's course was inspirational. Apart from the emphasis on clarity, rigour, and logic there were two distinctive features, both highly characteristic of Hugo's scholarly personality. The first was his constant use of encouragement. Hugo praised us both as a cohort of students and as individuals. I believe his praise was sincere. By telling us all the time how exceptionally talented we were, Hugo unleashed a huge wave of creativity and energy. Now that I'm a teacher myself I find I have keep reminding myself to praise my students and to genuinely recognise their talents. It's too easy to fall back on the comfortable notion that the professor is better than the student. (As a parent too, I've realised that there are very similar, perhaps even more important, insights about relationships with children). The fact is that the student tends to assume that the professor's talents are far superior. This assumption can drastically limit the student's learning and achievement. The lesson to be learned from this seems very simple, but Hugo is the only professor I've known who is able (and willing) to put it into practice so consistently. The second key feature of Hugo's personality that I discovered in the course was his occasionally severe assessment of established professors on their merits. I suppose this second characteristic is just the flip side of the first. Hugo was frank and critical in his assessment of other professors, and professional or worldly success was certainly not enough to qualify them for his esteem. To be honest, I like this second characteristic of Hugo's just as much as the first. I found it was much, much harder to write papers than to pass exams. My doctoral thesis was the hardest thing I've ever done in my professional life. Although he was encouraging, Hugo set high standards. These high standards were reinforced by own high standards and my ambitions, and by the examples set by previous students of Hugo. It's interesting that the best paper in my thesis is not a project that most professors would have encouraged at all (Search Decisions with Limited Memory, Review of Economic Studies 1992). Indeed the explicit advice I
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18 James Dow on Hugo E Sonnenschein got from another faculty member was to try to start by extending existing models in minor ways (starting with the appendix to one of his own papers, of course!). Fortunately, Hugo gave me the opposite advice in no uncertain terms - the advice came in the form of polite but unmistakable hints, sometimes just in his tone of voice. I had some good friends at Princeton, especially Sergio Werlang. Our friendship developed during Hugo's course, as we were both so enthusiastic about it and the style of economics that Hugo promoted. It's actually only after our time at Princeton that Sergio and I started to write papers together and the paper reprinted here is one of that series. Like the other contributors to this volume, I must express my gratitude for the enormous amount of time Hugo spent talking to me, often while walking around the campus or getting a ridiculously large ice cream (I've forgotten the name of the ice-cream place, but I think Hugo said he owned a small share of the business). I well remember the look on the face of one of my best friends, an engineering student, when he asked how long I'd spent with my advisor that day and I casually replied "Oh, six hours."
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Nash Equilibrium under Knightian Uncertainty JOURNAL OF ECONOMIC THEORY 64, 305-324 (1994)
Nash Equilibrium under Knightian Uncertainty: Breaking Down Backward Induction* JAMES D O W London Business School, Sussex Place, Regents Park, London NWl 4SA, England AND
SERGIO RIBEIRO DA COSTA WERLANG EPGE/Fundafao Getulio Vargas, Praia de Botafogo 190, Botafogo CEP 22250, Rio de Janeiro, RJ, Brazil Received April 5, 1993; revised September 17, 1993
We define Nash equilibrium for two-person normal-form games in the presence of Knightian uncertainty. Using the formalization of Schmeidler and Gilboa, we show that Nash equilibrium exists for any degree of uncertainty aversion, that maxmin behaviour can occur even when it is not rationalizable in the usual sense, and that backward induction breaks down in the twice repeated prisoners' dilemma. We relate these results to the literature on cooperation in the finitely repeated prisoners' dilemma, and the hterature on epistemological conditions underlying Nash equilibrium. The knowledge notion implicit in this model of equihbrium does not display logical omniscience. Journal of Economic Literature Classification Numbers: € 7 2 , D81.
© 1994 Academic Press, Inc.
L INTRODUCTION
Among the well-documented phenomena that economic models do not fully explain is the fact that players of a finitely repeated game do not backward induct. The careful experiments performed by Neelin et al [39] and * The authors are especially thankful to Tommy Tan, with whom this project started. We also thank Graciela Rodriguez Marine for correcting an error in Example 3, Mario Henrique Simonsen for an improved proof for the existence theorem, and Martin Cripps, Bart Lipman, and Jose Alexandre Scheinkman for comments on the initial version of the paper. Part of this research was carried out while James Dow was visiting the Graduate School of Economics at the Getulio Vargas Foundation (EPGE/FGV) under a grant from CNPq. An initial version of the present paper [14], containing two possible definitions of equilibrium, one of which was subsequently discarded, is available to the interested reader.
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McKelvey and Palfrey [34], and the well known prisoners' dilemma tournament of Axelrod [ 4 ] show that subjects do not act according to the logic of backward induction. Several attempts to model this behaviour exist in the literature. Some of them are based on bounded rationality (Radner [42], Chou and Geanakoplos [11], Aumann [ 2 ] , Neyman [40], Meggido and Widgerson [35], and the references therein). Another type of explanation requires evolutionary behaviour with a lower bound on the number of "mutants," as in Nachbar [38]. Undeniably, however, the most successful model to justify cooperation in the finitely repeated prisoners' dilemma (which is, in fact, stronger than violation of backward induction) is due to Kreps et al. [30]. Essentially their argument is that there is always a small chance that one of the players will not act rationally. In the game they consider, there is a small chance that one of the players will cooperate. This will lead to significant levels of cooperation on the part of both players. Fudenberg and Maskin [21] and Fudenberg and Levine [20] explore the size of the entire set of equilibria obtained by means of the Kreps et al solution. In this paper, we propose to extend the notion of Nash equilibrium to incorporate agents who act in a game as if they faced uncertainty in the sense of Knight [28]. We show that this equihbrium exists for any given degree of uncertainty aversion on the part of each player (as defined in Dow and Werlang [15]). Our definition leads to the possibility of cooperation in the repeated prisoners' dilemma. We explain the relationship between this explanation and the Kreps et al explanation. At the same time, given that the phenomenon has been modelled by departing from the Bayesian view of Savage [46], it is also true that the bounded Bayesian rationahty justifications are compatible with our explanation (the evolutionary model is of a completely unrelated type, more suitable for analysis of economic institutions). Our definition also demonstrates the possibihty of prudent behaviour (maxmin) even when this is "inconsistent" with the knowledge that the other player is Bayesian rational, as in the example in Werlang [56, Chap. 3]. Finally, the definition of equilibrium allows for the possibility that the same game may be played differently by the same player when facing different opponents (even if there is a unique rationalizable outcome). The decision model on which our analysis is based is, like other nonexpected-utility models, at a fairly early stage of development. The axiomatic foundations of the representation of the preference ordering (by a utiHty function and a non-additive probabihty) are well understood. Applications to economic settings are not so well developed, and a number of outstanding issues remain, for example the question of dynamic consistency (see Machina [33], Hammond [26], and Epstein and LeBreton
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[18] or, for a discussion in the context of games, Dekel etal [13]). Nevertheless, if these decision-theoretic models are worth taking seriously for economists, they must be applied to games and other economic situations. This paper is no more than a first step in the direction of a gametheoretic application of behaviour under uncertainty (Klibanoff [27] also discusses similar issues). However, this step seems worth taking. The fact that this model formalizes the concept of Knightian uncertainty gives it a particular intuitive appeal which differentiates it from other nonexpected-utiHty models. On the other hand, it also makes the definition of Nash equilibrium more difficult since we cannot assume that mixed strategies are described by objective probability distributions, which then imply a best response function. That approach would be possible for other non-expected-utiUty models such as rank-dependent utiUty (see Weber and Camerer [55] and Epstein [17] for surveys and descriptions of these models). Here, we must use the subjective approach to mixed strategy Nash equilibrium, which imposes a consistency condition between a player's beliefs about his opponent's actions and the opponent's best response: all actions in the support of the opponent's belief must be best responses. Furthermore, with Knightian uncertainty the "support of an agent's belief" is not as simple to define as when we are dealing with standard objective probabilities (Crawford [12] defines and discusses equifibrium for games where agents have other types of non-expectedutility preferences, for which beliefs can be represented with standard probabilities). Nash equihbrium implicitly depends on a notion of knowledge: each player knows the other will play a best response. The standard definition of Nash equihbrium, like other models of knowledge used in economics, incorporates the property of logical omniscience: if an agent knows a fact, he immediately knows all the consequences of that fact. The literature on philosophy and on computer science has recognized that knowledge models with logical omniscience fail to capture some essential aspects of human knowledge, and has developed models without logical omniscience. We show that the knowledge notion implicit in our definition of Nash equihbrium under uncertainty does not display logical omniscience. The paper is organized as follows. The next section discusses the basic uncertainty model, which is due to Schmeidler [47,48] and Gilboa [23]. Section 3 gives the definition of Nash equihbrium under uncertainty and the theorem on the existence of Nash equilibrium for any given pair of values for the uncertainty aversion of the two players. Section 4 presents a sequence of examples. In Example 1, the definition is used in an example to obtain prudent behaviour which is incompatible with common knowledge of Bayesian rationahty (common knowledge of Bayesian rationahty is equivalent to rationalizability, in the sense of Bernheim [ 6 ] and Pearce
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[41], as shown in Tan and Werlang [52]). Example 2 shows the distinction between our approach and the standard (Bayes-Savage) approach with e-trembles. Example 3 is more striking, showing the emergence of cooperation in a twice repeated prisoners' dilemma, under our definition of Nash equihbrium under uncertainty. We use this example to relate the equilibrium notion to the Kreps et al model. Section 5 relates the knowledge notion implicit in the definition of Nash equilibrium under uncertainty to the one implicit in the standard (Bayes-Savage) definition. Section 6 concludes and points to directions for further research.
2. UNCERTAINTY
Schmeidler [47,48] and Gilboa [23] have developed an axiomatic model of rational decision making in which agents' behaviour distinguishes between situations where agents know the probabihty distributions of random variables and situations where they do not have this information. We refer to the former as risk and the latter as uncertainty, or Knightian uncertainty. Synonyms used in the literature include roulette lottery, for risk, and horse lottery and ambiguity, for uncertainty. The standard model of uncertainty used in economics is that of Savage [46], which reduces all problems of uncertainty to risk under a subjective probability. The Schmeidler-Gilboa axiomatization leads to different behaviour: behaviour under uncertainty is inherently different from behaviour under risk. We now give a brief exposition of the main aspects of their model (this exposition makes no claim to originality and portions are reproduced verbatim in [16]). The reader is referred to the papers by Schmeidler and Gilboa cited above for a complete description and for the underlying axioms, and to Dow and Werlang [15] for an example and an apphcation to portfolio choice (it also includes a mathematical appendix with the basic material on non-additive probabihties). Dow and Werlang [16] have an application to stock price volatihty, and Simonsen and Werlang [49] also describe the implications for portfoHo choice. Also, Wakker [54] has a model which is very similar to Gilboa [23]. Bewley [ 8 ] presents a similar model which is also designed to capture Knightian uncertainty. His model predicts that uncertainty leads to inertia, a tendency to favour the status quo, while in Schmeidler and Gilboa there is a tendency to choose acts where the agent does not end up bearing uncertainty. In a decision problem, this will lead to different predictions unless the status quo is an act where the agent bears no uncertainty. In Game Theory, it is conventional not to distinguish any particular strategy as the status quo.
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The Schmeidler-Gilboa model predicts that agents' behaviour will be represented by a utility function and a (subjective) non-additive probability distribution (also known as a capacity), A non-additive probability P reflecting aversion to uncertainty satisfies the condition P{A) + P{B)^P{AnB)-{-P{AuB),
(*)
rather than the stronger condition satisfied by (additive) probabiHties: PiA ) + P{B) = P{A nB)-\- P{A KJ B). In particular, P{A)-^P{A'') may be less than 1; the difference can be thought of as a measure of the uncertainty aversion attached by the agent to the event A. The uncertainty aversion of P at event A is c{P,A) = \-P{A)-P{A'') (Dow and Werlang [15]), We will say that P reflects strict uncertainty aversion if c(P, A)>0 for all events A (except of course where A is the empty set or the set of aU states). All the non-additive probabilities considered in this paper will reflect uncertainty aversion; i.e., they will satisfy inequality (*), Also, we will restrict attention to the case of a finite set of states of the world. The agent maximizes expected utility under a non-additive distribution where the expectation of a non-negative random variable X is defined as
E{X)=\
P{X^x)dx.
Associated with a non-additive probability P is a set of additive probabiHties called the core of P, which is defined (analogously to the core in cooperative game thoery) as the set of additive probability measures n such that n{A)^P{A) for all events A, If the non-additive probability reflects aversion to uncertainty (inequality (*)), the core is non-empty. A closely related model of behaviour under uncertainty (equivalent in many simple cases) is a maxmin formulation: the agent acts to maximize the minimum value, over the elements of the core, of expected utility (Gilboa and Schmeidler [24]). Note that this is not the same as maxmin over outcomes (prudent behaviour), except in the special case where the core contains all the distributions giving probability one to a single state. EXAMPLE. Suppose the agent has to choose between lottery tickets whose payoffs depend on whether a blue or a red ball is drawn from an urn containing balls of the two colours, in unknown proportions. He may choose between having (i) a "red" lottery ticket paying £10 if a red ball is drawn, (ii) a "blue" ticket paying £10 if a blue ball is drawn, and (iii) a
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certain payoff of £ 4 His utility function is linear (risk-neutral): U{w) = w, His beliefs are represented by the following non-additive distribution:
;^BLUE = 0 . 4 5 .
His expected utility from holding (i) the red ticket is given by the formula E{U{w))=-\
P(w^x)^jc = 0+(0.45)10 = 4.5,
sincQ P{w^x)=l i f x ^ O , P ( w ^ x ) = 0 . 4 5 i f 0 < x ^ l 0 a n d P ( w > x ) = 0if x>10. Similarly, choice (ii), the blue ticket, is worth 4.5 and choice (iii), the safe payoff, is worth 4. So the agent is indifferent between holding either ticket, and prefers holding either one of them to a safe payoff of £4. In this example, the maxmin formulation using the core of the distribution is equivalent. The core of this distribution is the set of all (additive) probabihty distributions with chances of red of between 4 5 % and 55%. So, using the maxmin model, we evaluate the payoff from choice (i), the red ticket, as min {10/7 + 0( 1 - ;?) I 0.45 ^ /? ^ 0.55 } = 4.5, as before (and similarly for (ii) and (iii)). The support of a non-additive probabihty P may be defined analogously to the additive case. One might start by supposing the appropriate analogy to be smallest event A such that P{A) = 1 (the initial version of this paper, Dow and Werlang [14], explored this notion in greater detail). However, note that if P reflects strict uncertainty aversion then the entire set of all states is the only event with probabihty one. Intuitively, the interpretation of an event of non-additive probability zero is the same as in the additive case: it is an event which will almost never happen. With a non-additive probability, however, if a set has probabihty zero, that does not mean its complement is of probabihty one. But if the complement of a zeroprobabihty set has positive probabihty (in principle it could be zero too) it has relatively infinitely more chance of happening than the original set. Hence, we are led to a definition of support based on the idea that the complement of the support has zero probabihty. The motivation for this definition is further discussed in Section 3 below when we define Nash equilibrium under uncertainty. DEFINITION. A support of a non-additive probabihty P is an event A such that P(v4^) = 0 and P(^^) > 0 for all events BaA, By^A.
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It should be clear that there may be several supports, and that a support is always contained in the smallest set with probability one. EXAMPLE.
There are three states with
Pn = Pi3 = P23 = q^i^PA-
p)
where /?, is the probabiHty of state / and Py is the probability of state / or state j . This example has constant uncertainty aversion. The (unique) support is the set of all states, {1, 2, 3}. EXAMPLE.
Again there are three states.
P3 = 0 Pi2 = qe{2p,
1-/7)
Pl3 = P23=P'
The (unique) support is the event {1, 2}. EXAMPLE. This example does not have a unique support. Again there are three states. p,=0,5
P2 = P3 = 0 Pl2 = Pl3 = ^-^ P23=0A. The supports are {1, 2} and {1, 3}. Note that in each of these three examples, the smallest set of probabiUty 1 is the set of all states,
3. NASH EQUILIBRIUM LHSTDER UNCERTAINTY
We restrict attention to two-person finite normal form games r = {Ai, A2,Ui,U2), where the A^ are pure strategy sets and the w, are utihties (payoffs). In the standard theory, a mixed strategy Nash equihbrium is defined as follows. Let (/^i,/X2) be a pair of (additive) probability measures and let supp[/xj denote the support of fi^. In Nash equilibrium, every
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aiGsupp[/Zi] is a best response to fi2 (ie., ai maximizes the expected utility of player 1 given that player 2 is playing the mixed strategy 1^,2)1 conversely, every ^2^supp[jU2] is a best response to /ii. A subjective interpretation can be given to the Nash equilibrium: the mixed strategy of player 1, /^i, may be viewed as the belief that player 2 has about the pure strategy play of player 1. Conversely, the mixed strategy of player 2, /i2, may be viewed as the belief player 1 has about the pure strategy play of player 2. This subjective interpretation is suitable for the generalization we will introduce in this paper. (The alternative interpretation, which we do not consider here, is that players act as if they actually use random-number generators to implement mixed strategies. Models of Knightian uncertainty are not suitable for describing this objective interpretation.) DEFINITION: Nash EquiUbrium under Uncertainty. A pair (Pi,P2) of non-additive probabilities P^ over Ai and P2 ^ver ^42 is a Nash equilibrium under uncertainty if there exist a support of P^ and a support of P2 such that
(i) for all ^1 in the support oi P^,a^ maximizes the expected utility of player 1, given that P2 represents player Ts beliefs about the strategies of player 2; and conversely, (ii) for all ^2 i^i the support of P2, ^2 maximizes the expected utihty of player 2 given that P^ represents player 2's beliefs about the strategies of player 1. This definition reduces to the standard definition of Nash equihbrium whenever there is no uncertainty (i.e., when the P's are additive). Clearly the definition rests upon our definition of support of a nonadditive distribution. One could speculate what would be the implications of replacing the support as we have defined it by other sets. For example, the smallest set of probability 1 is "too large," and the equilibrium notion that would result would be too strong. The reason is that such an approach would ignore the intuitive stength of the non-additive model: that an event may be infinitely more Ukely than its complement, but still have probability less than 1. Take for example the case of a strategy set with two elements, a and b. Let the other player's beliefs about the strategies be P(a) = 0.8 and P(Z>) = 0. Intuitively, b has no chance of happening, but we cannot guarantee that a will happen. If we want to be "sure" that an event will happen, this event can only be the whole strategy set {a,b}. The reader may wish to compare the discussion above with the intuitive argument used by Schmeidler as the starting point for his derivation of the decision theory of non-additive probabilities: if an agent has symmetric
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information about a number of mutually exclusive and exhaustive possible events, they should be assigned equal "probabiUties/' but the "probabiUties" need not sum to one (Schmeidler [48, pp. 571-572]). It might be enquired why the above definition is presented intuitively rather than derived axiomatically. The reason is that the standard Nash equilibrium itself has only the weakest axiomatic foundations (Tan and Werlang [52], Bernheim [7]). Axiomatic considerations lead instead to the concept of rationalizabihty (Bernheim [ 6 ] , Pearce [41], Tan and Werlang [52]). Nevertheless, Nash equilibrium rather than rationahzabihty is the standard solution concept in game-theoretic appHcations, presumably because of its intuitive appeal. Thus we cannot hope for a strong axiomatic foundation for Nash equilibrium under uncertainty. Clearly, a standard mixed strategy Nash equihbrium is also a Nash equiHbrium under uncertainty, so existence of at least one Nash equilibrium under uncertainty is no problem. But, as we discuss in Section 5 below, it is desirable to be able to view an agent's uncertainty aversion as a parameter in the description of the game. The theorem that follows allows us to do this. Note that in the statement of the theorem, the reason for the interchange of the subscripts in the notation is that P^ represents player 2's beliefs about what player 1 will do, so that the uncertainty aversion of P^ is a characteristic of player 2, and vice versa. Incidentally, note also that while we compute equihbrium where the beliefs are of the form P{A) = {l—c) Q{A) for additive Q, there are behefs with constant uncertainty aversion which are not of this form. THEOREM. Let r={Ai, A2,u^,U2) be a two-person finite game. For all (cj, €2)^ [0, 1] X [0, 1], there exists a Nash equilibrium under uncertainty (Pi, ^2)* ^^<^^ t^^i ^1 is the uncertainty aversion of P2, and C2 is the uncertainty aversion of P^.
Proof By Dow and Werlang [15] if P{A)-=^{\-c)Q{A) for some additive probabihty Q and for all events A (other than the entire set of states of the world), then P exhibits constant uncertainty aversion c. In other words, P is a "uniform squeeze" of Q. In this case, one has Ep(X)^cmmX^{\-c)EQ{X\
(**)
(Note that in case c = 1 this reduces to maxmin behaviour.) We modify the original game F to r^,^^,^^ = {Au A2, t^i, 1^2)* where i;,(a,, ay) = c,min W/(«/, «) + (l —c^Ui^a^, aj)
for z = 1, 2 andyV/.
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Let {Qi,Q2) be a standard mixed strategy Nash equilibrium of the modified game r^^ y We will show that the pair ( P j , P2X where Pi{A) = (1 — C2) Qi{A) for any event A^A^, P^iA) = {l-Ci)Q2{A)
and
for any Qwcnt A ^A2
(and naturally P^{Ai) = P2(A2)=l), is a Nash equilibrium under uncertainty for the original game T, with the specified levels of uncertainty aversion. It is immediate to verify that the uncertainty aversion of P2 is Cj and that of Pj is C2. To verify that this is a Nash equihbrium under uncertainty, note that (except in case Cj=l) the support of P, is unique, and coincides with the support of Qi ( / = 1 , 2 and j V O - Since {Qx,Q2) is a standard mixed strategy Nash equilibrium for r^^^..^^ it follows that any fl/esupp[QJ is a best response to Qj (for the modified utility t?,). In other words, a^ maximizes the following expression over aeAji f^Q.lViia, •)] = f v,{a, aj) dQj{aj) = Ci I [min Ui{a, a*)] dQj{aj) ^Aj
a*eAj
+ (l-c,)f
uM,aj)dQj{aj)
by(***)
= c, min M,(fl, a*) + (1 - c,) f^g [«,(«, •)] a*eAj
•'
= ^p.[w,(a,-)].
by(**).
Thus (2, is also a best response in the original game F. There remains the possibiHty that Cj=l. In this case, any singleton {a^} is a support of P,. Therefore any best response for player / is in a support. Thus (Pi, P2) is a Nash equilibrium under uncertainty for F, Q.E.D.
4. EXAMPLES
Example 1. Non-rationalizable Maxmin Behaviour May Occur The game shown in Fig. 1 has a unique rationalizable equilibrium (which, therefore, coincides with the (standard) Nash equihbrium), given by (w, a). Further, if player 1 knows that player 2 is rational (observe that this requires just one level of elimination of strictly dominated strategies)
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then she knows that player 2 can never play b, because it is a strictly dominated strategy. Thus she should play u. Let us imagine, however, that 10 stands for 10 milHon dollars, or, equivalently, that the parameter e is very small This game is very similar to one in Werlang [56, Chap 3 ] , who asked the following question: would you play u in this game? Note that strategy d gives a payoff very similar to the payoff obtained in the Nash equilibrium, without any "risk" that the other player does not play his part. Our definition of Nash equilibrium under uncertainty leads to the "prudent" decision d even for low uncertainty aversion of player 1. Let Pi be described by probability /?„, player 2's beHef that player 1 will play M, and probability p^, his belief that 1 will play d, with Pu-^Pd"^ 1Similarly, P2 is described by q^ and q^,, with qa + qt"^ 1- Note that from the point of view of player 2, the beliefs of Pj are irrelevant: player 2 always chooses a. Will that mean that player 1 will necessarily play ul The answer is no. We will look for Nash equilibria under uncertainty of the form ^^, = 0 and 1 > ^ ^ > 0 . Since {a} is a support (in this simple case the only one) we only have to check that a is a best response, a fact that we already know. The stategy b, since it lies outside the support, need not be a best response, again a fact that we already know. What we would like to obtain is the set of parameters which will yield d as the equilibrium action of player 1. Computing the expected values shows that she will play di( q^^l— a/20. Note that this is exactly consistent with the intuition that just a Httle bit (e/20) of uncertainty aversion is enough to make player 1 turn from the only rationalizable action u to the prudent, and intuitively sensible action d. Note also that the smaller s is, the less uncertainty averse player 1 has to be in order to play d. Finally, note that we could also justify this action by the argument of Kreps et aL; if there were a small chance S of player 2 being crazy, player 1 would behave cautiously. The following example shows, however, that the notion of Nash equilibrium under uncertainty does not Player II
a
b
10,10
-10, lO-a
10-e, 10
10-€. 10-a
Player I
a, c > 0 FIGURE 1
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coincide with the 5-craziness solution. The relationship to Kreps et al is discussed further after Example 3 below. Example 2. Nash under Uncertainty Is Not Equivalent to 5-Craziness Consider the modification of Example 1 shown in Fig. 2. Now A is a strictly dominant strategy for player 2, and we would expect player 1 to play d. Suppose we impose <5 > 0 probabihty that player 2 will play a. It is easy to see that for 0 < ^ < 1 — s/20, the standard Nash equilibrium yields d as the unique best response for player 1. On the other hand if (5 > 1 — 8/20, her unique best response is u in the standard Nash equilibrium. The "(5-craziness" approach of changing the game description by postulating that with 5 probability the other player will actually play the other strategy (perhaps because of having a "crazy" different utility function) admits w as a best response in some cases. We can compare this behaviour with the prediction of our theory. In Nash equilibrium under uncertainty, it is immediate that playing u is never a best response in any equilibrium, regardless of the level of uncertainty aversion. The reason for this difference in behaviour is that the <5-craziness solution here imposes a strategy choice for player 2 (with exogenous probabihty 6) that turns out in this example to be "optimistic" from the point of view of player 1. On the other hand, in Nash equihbrium under uncertainty preferences are always pessimistic: they give more weight to undesirable outcomes. Thus the criteria are quite distinct even in the simplest games. The difference is discussed further following Example 3 below. Example 3. Breaking Down Backward Induction We now show that cooperation may arise in the twice repeated prisoners' dilemma, thereby demonstrating that (Knightian) rational agents may not backward induct. Consider the version of the prisoners' dilemma Player II
a
b
10,lO-a
-10. 10
Player I
j
lO-e. 10
lO-G. lO--a
a, € > 0 FIGURE 2
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317
COL
F
C
0.0
a.b
b.a
1.1
ROW
FIGURE 3
of Kreps et al [30], as shown in Fig. 3, where a = 1.25 and b= —0.5, so that a-^b<2 as they require. The strategies are F (for ''fink") and C (for "cooperate"). In the twice repeated version of this game, there are eight strategies for each of the players. Four are the unconditional, or history independent strategies, F^, FC, CF, C^, which stand for, respectively, F in both rounds, F in the first round and then C in the second round, C and then F, and C in the both rounds. There are four history dependent strategies, which we name W, X, Y and Z, for convenience. W is: start with F. If the other player played C in the first round, then play C in the second round. Otherwise play F in the second round. The letter X stands for: start with F, and if the other player played C, then play F, otherwise play C. The letter Y stands for the tit-for-tat: start with C, and play C in the second round if the other player played C in the first round. Otherwise, play F. Finally, Z stands for: start with C, and play F in the second round if C was also played by the other player. Otherwise, play C. To summarize: Second round if other player's first round move was Strategy
First round
C
F
F^ FC CF C^ W X Y Z
F F C C F F C C
F C F C C F C F
F C F C F C F C
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Consider the game without discounting (the payoffs are sums of payoffs of each one-shot game). It may be verified that the following is a non-additive probability P which reflects uncertainty aversion (inequality (*) in Section 2 above): (1)
P({F^CF,C^W})=1
(2)
P ( { C ^ C F , W } ) = 0.8
(3)
P ( { F ^ CF, W}) = P ( { F ^ CF, C^}) = 0,4
(4)
P ( { F ^ C F } ) = J P ( { C F , W } ) = P ( { C F , C ' } ) = 0.4
(5)
P(W) = />(C^) = P({F2}) = P({F^C^})
=p({c^w})=i>({F^c^w})=o (6)
P(CF) = 0.4
(7)
For all events B, P{B) = P{Bn {F^ CF, C^ W}).
It is easy to see that the only support is {CF}, There is a Nash equilibrium under uncertainty in which both players have these beliefs. The expected payoffs for each of the players, given the behef P about the other player's action is, for each strategy: C/(F^) = 0,5, U{¥C) = 0, U(CF) = 0.6, U{C^) = 0.2, C/(W) = 0.3, U{X) = 02, (7(Y) = 0.3, and U{Z) = OA. Thus, the equilibrium above has CF as the prediction, with the joint probabilities of all other strategies together being zero. This means that cooperation in the first round may occur, again with (Knightian) rationahty. Comment: Relation to Kreps et al Kreps etal. [30] obtain cooperation in the finitely repeated prisoners' dilemma using an intuitively appealing argument, of the type we have described above as the (5-craziness approach. They consider a small "risk" that one of the players will always play tit-for-tat. They show that this generates cooperation in several stages of the repeated game. The irrational agent (tit-for-tat) that they "added" to the game is added exogenously, but clearly the modellers' choice of that specific type of irrationaHty Was not exogenous. It was motivated intuitively in a way which was endogenous to the particular game under analysis, A similar analysis is given in Kreps and Wilson [29] and Milgrom and Roberts [37] in their model of the chain-store paradox. They suggest that potential entrants may be afraid of encountering a "maniac" who positively enjoys playing an apparently irrational strategy; in this case, starting a ruinous price war in the chain-store entry game. In Example 3, we showed that uncertainty can lead to cooperation. There we did not use the tit-for-tat strategy to achieve cooperation, but our point was just to show that cooperation could arise. In our model the potentially "irrational" behaviour of the other agent is generated by the
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fact that, in the presence of uncertainty, the players tend to give more weight to the potential losses from taking an action. Thus in our model, the players endogenously decide which sort of agent they are "afraid" of meeting. In other words, we have a theory that explains how the "irrationality" appears in the model. However, there are two important differences between the Kreps et al approach and the approach described in this paper. The first difference was described in Example 2—where we showed that adding a "benevolent" maniac might induce agents to take less cautious decisions, whereas Nash equilibrium under ucertainty would not. The second difference is that, in Nash equilibrium under uncertainty, the type of opponent players are "afraid" of meeting may be different at different strategies. In the Kreps et al. approach, the type of opponent who has been added to the model is constant when evaluating the payoff of different strategies. In Nash under uncertainty, agents systematically evaluate payoffs relative to the worst strategy of their opponent—which will change depending on their own strategy.
5. REMARKS ON LOGICAL OMNISCIENCE
The models of knowledge common in the economics literature (see for example Aumann [ 1 , 3 ] , Bacharach [ 5 ] , Brandenburger and Dekel [9, 10], Geanakoplos and Polemarchakis [22], Milgrom and Stokey [ 3 6 ] , Rubinstein and Wolinsky [44], Samet [45], Tan and Werlang [51,52,53], and Werlang [57]) all represent behaviour of logically omniscient agents. Logical omniscience means that if an agent knows a fact, and knows that this fact implies another fact, then the agent knows that other fact. This seemingly innocuous property has powerful consequences. For example, a logically omniscient agent who knows the basic rules of propositional logic and the Peano axioms must know all mathematical results proven, and ever to be proven. Lack of logical omniscience may seem unfamiliar or even odd to economists, although the fact that human knowledge is not logically omniscient is well accepted among philosophers. Most readers of this paper know the rules of logic together with Peano's axioms, without knowing all possible mathematical results: clearly, logical omniscience is a strong property that fails to capture some essential aspects of human knowledge. For references to the extensive discussion of logical omniscience in the philosophy and artificial intelligence literature, see Fagin et al. [19]. The definition of Nash equihbrium implicitly presupposes a notion of knowledge: it assumes implicitly that an event is known when it contains a support. In other words, each player knows the opponent will play a best
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response. When probabilities are additive, this knowledge notion is very similar to the existing models in the economics literature (see Brandenburger and Dekel [9]). However, some interesting and less usual properties arise from our definition of Nash equilibrium under uncertainty. In this case there are two sources of the lack of logical omniscience in the knowledge notion. This first source was noted in a more general context by Lipman [32]. The essential idea is that when an agent learns a fact, he effectively simultaneously learns the fact and learns of another a priori unknown state of the world. Lipman [32] refers to these states as "impossible possible worlds." Gilboa and Schmeidler [25] give a related model: they show that in a decision problem under uncertainty, one can interpret non-additive probabihties as if they were additive, but defined over a suitably extended space of states of the world. This may be illustrated by Example 1 of Section 4. Since a is the rational action of player 2, and the (unique) support of P2 is [a], it follows by our definition that player 1 knows that "player 2 is rational." On the other hand, player 1 is rational and knows she is rational. Therefore, if the choice is between an action that would yield 10 and another action that would yield 10 —e, she should choose the action that yields 10. Thus, player 1 knows that "if player 2 were to choose a, she should choose w." The lack of logical omniscience comes from the fact that player 1 chooses d in equilibrium. Hence player 1 cannot know she plays w, even though this is the logical conclusion of the two facts that we argued player 1 does know. Here is an agent who knows a fact (that "player 2 chooses a"), and knows that this logically implies something else (that "if player 2 were to choose a, she—player 1—should choose w"), but does not know their imphcation (action u should be chosen). The second source of lack of logical ominiscience is the possible multiphcity of supports. Take the non-additive probability of the last example of Section 2 above. Consider the events ^ = {1, 2} and Z)= {1, 3}. An agent whose behefs are represented by P knows A and knows D, in the sense that both A and D are supports of P. However, the event {1} = ^ n D does not contain any support of P, so that it is not known by the agent. The implications of the non-closure of the knowledge operator with respect to intersections of events are immediate. It is quite possible that both the event A and the event A'^uB (which is equivalent to "A implies ^ " ) are known, but B is not known. Continuing the above example, take ^ = {1, 2} and B= {1}, Both A and A'^KJB are known, but B is not. This second source of lack of logical omniscience does not arise in Example 1 of Section 4 above, since the support of player I's belief is unique. We have shown that the knowledge notion implicit in our definition of Nash equilibrium under uncertainty does not display logical omniscience.
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Lipman [31] independently noticed this fact. In contrast to the majority of existing knowledge models without logical omniscience (see Fagin et al. [19] for references), our notion is operational: if an agent's behaviour is described by a non-additive belief, a fact is known if it contains a support of the distribution. Thus, one reason why Knightian uncertainty can be regarded as an interesting behavioural model is that it implicitly, and without additional modelHng effort, does away with logical imniscience.
6. CONCLUDING REMARKS
The definition of Nash equihbrium under uncertainty provided here explains a number of economic phenomena which, to date, have not been modelled in a satisfactory way. For example, it is possible to provide a rationale, along the lines of the above examples, for the experimental results of NeeHn et al [39] and of McKelvey and Palfrey [34]. Neehn et al [39] ran bargaining experiments with iterated offers and complete information (similar to the models of Stahl [50] and Rubinstein [43]). They found that players behaved as if they apphed only two rounds of backward induction, even when the game was repeated up to five rounds. McKelvey and Palfrey [34] document similar violations of backward induction. The definition of Nash equilibrium under uncertainty provided here may readily be used to construct examples where backward induction breaks down. An area of future research concerns rationalizability, rather then Nash equilibrium, under uncertainty. We would define the rationalizable outcomes as the set of strategies which are compatible with common knowledge (using our notion of knowledge) of Knightian rationality. There then remains to explore the relation between this rationalizabihty concept and Nash equihbrium under uncertainty. There is also the extension to n players. Finally, we end by noting that researchers have often referred to a "missing parameter" in the description of the game. Such a parameter would allow a player to play the same game differently when facing different opponents (even when the opponents have the same payoffs). We suggest that uncertainty aversion may serve as such a parameter.
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3. R, J. AuMANN, Correlated equilibrium as an expression of Bayesian rationality, Econometrica 55 (1987), 1-18. 4. R. AxELROD, Effective choice in the Prisoners' Dilemma, / . Conflict Resolution 24 (1980), 3-25. 5. M. O. L. BACHARACH, Some extensions of a claim of Aumann in an axiomatic model of knowledge, / . Econ. Theory yi (1985), 167-190. 6. D. BERNHEIM, Rationalizable strategic behaviour, Econometrica 52 (1984), 1007-1028. 7- D. BERNHEIM, Axiomatic characterizations of rational choice in strategic environments, Scand. J. Econ. 88 (1987), 473-488. 8. T. BEWLEY, "Knightian Decision Theory," Part 1, Cowles Foundation Paper No. 807, Yale University, 1986. 9. A. BRANDENBURGER AND E . DEKEL, Hierarchies of belief and common knowledge, / . Econ. Theory 59 (1993), 189-198. 10. A. BRANDENBURGER AND E . DEKEL, Common knowledge with probabiUty 1, / . Math. Econ. 16 (1987), 237-246. 11. C. CHOU AND J. GEANAKOPLOS, "On Finitely Repeated Games and Pseudo-Nash Equilibria," Cowles Foundation Paper No. 777, Yale University, 1985. 12. V. CRAWFORD, EquiUbrium without independence, / . Econ. Theory 50 (1990), 127-154. 13. E. DEKEL, Z . SAFRA, AND U . SEGAL, "Existence and Dynamic Consistency of Nash Equilibrium with Non-Expected-Utility Preferences," working paper, Israel Institute of Business Research; Tel-Aviv University, 1990. 14. J. Dow AND S. R . C . WERLANG, "Nash Equilibrium under Knightian Uncertainty: Breaking Down Backward Induction" (initial draft), working paper, London Business School, 1991. 15. J. Dow AND S. R. C. WERLANG, Uncertainty aversion, risk aversion and the optimal choice of portfolio, Econometrica 60 (1992), 197-204. 16. J. Dow AND S. R. C. WERLANG, Excess volatility of stock prices and Knightian uncertainty, Eur. Econ. Rev. 36 (1992), 631-638. 17. L. EPSTEIN, Behaviour under risk: Recent developments in theory and applications, in "Advances in Economic Theory," Vol. 2 (J.-J. Laffont, Ed.), Cambridge University Press, Cambridge, 1992. 18. L. EPSTEIN AND M . LEBRETON, Dynamically consistent beliefs must be bayesian, / . Econ. Theory 61 (1993), 1-22. 19. R. FAGIN, J. HALPERN, AND M . VARDI, A nonstandard approach to the logical omniscience problem, in "Theoretical Aspects of Reasoning about Knowledge: Proceedings of the Third Conference" (R. Parikh, Ed.), Morgan Kaufman, San Mateo, CA, 1990. 20. D. FUDENBERG AND D . LEVINE, Reputation and equiUbrium selection in games with a patient player, Econometrica 57 (1989), 759-778. 21. D. FUDENBERG AND E . MASKIN, The folk therem for repeated games with discounting or with incomplete information, Econometrica 54 (1986), 533-554. 22. J. GEANAKOPLOS AND H . POLEMARCHAKIS, We can't disagree forever, / . Econ. Theory 28 (1982), 192-200. 23. I. GILBOA, Expected utility with purely subjective non-additive priors, / . Math. Econ. 16 (1987), 279-304. 24. I. GILBOA AND D . SCHMEIDLER, Maxmin expected utility with a non-unique prior, / . Math. Econ. 18 (1989), 141-153. 25. I. GILBOA AND D . SCHMEIDLER, Additive representations of non-additive measures and the choquet integral. Annals of O.R., in press, 1994. 26. P. HAMMOND, Consistent plans, consequentialism and expected utility, Econometrica SI (1989), 1445-1450. 27. P. KLIBANOFF, "Uncertainty, Decision and Normal Form Games," working paper, MIT, 1992.
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28. F. KNIGHT, "Risk, Uncertainty and Profit," Houghton Mifflin, Boston, 192 L 29. D. M. KREPS AND R . WILSON, Reputation and imperfect information, J. Econ. Theory 27 (1982), 253-279. 30. D. M. KREPS, P. MILGROM, J. ROBERTS, AND R . WILSON, Rational cooperation in the
finitely repeated Prisoners' Dilemma, X Econ. Theory 27 (1982), 245-252. 31. B. LIPMAN, private communication, April 21, 1992. 32. B. LIPMAN, "Decision Theory with Impossible Possible Worlds," working paper. Queen's University, 1992. 33. M. MACHINA, Dynamic consistency and non-expected-utility models of choice under uncertainty, / . Econ. Lit. 21 (1989), 1622-1668. 34. R. D. MCKELVEY AND T . R . PALFREY, An experimental study of the centipede game, Econometrica 60 (1992), 803-836. 35. N. MEGIDDO AND A. WIDGERSON, On play by means of computing machines, in "Theoretical Aspects of Reasoning About Knowledge—Proceedings of the First Conference" (J. Halpern, Ed.), Morgan Kaufman, San Mateo, CA, 1986. 36. P. MILGROM AND N . STOKEY, Information, trade, and common knowledge, / . Econ. Theory 26 (1982), 17-27. 37. P. MILGROM AND J. ROBERTS, Predation, reputation and entry deterrence, / . Econ. Theory 11 (1982), 280-312. 38. J. NACHBAR, "The Ecological Approach to the Solution of Two-Player Games," working paper, 1987. 39. J. NEELIN, H . SONNENSCHEIN, AND M . SPIEGEL, A further test of noncooperative
bargaining theory: comment, Amer. Econ. Rev. 78 (1988), 824-836. 40. A. NEYMAN, "Bounded Complexity Justifies Cooperation in the Finitely Repeated Prisoners' Dilemma," working paper, 1985. 41. D. PEARCE, Rationalizable strategic behaviour and the problem of perfection, Econometrica 52 (1984), 1029-1050. 42. R. RADNER, Collusive behaviour in noncooperative epsilon-equilibria of oligopolies with long but finite lives, / . Econ. Theory 22 (1980), 136-154. 43. A. RUBINSTEIN, Perfect equihbrium in a bargaining model, Econometrica 50 (1982), 97-109. 44. A. RUBINSTEIN AND A, WOLINSKY, On the logic of "agreeing-to-disagree"-type results, / . Econ. Theory 51 (1990), 17-29. 45. D. SAMET, Ignoring ignorance and agreeing to disagree, / . Econ Theory 52 (1990), 190-207. 46. L. J. SAVAGE, "The Foundations of Statistics," Wiley, New York, 1954; second ed., Dover, New York, 1972. 47. D. SCHMEIDLER, "Subjective Probability without Additivity" (temporary title), working paper, The Foerder Institute for Economic Research, Tel Aviv University, 1982. 48. D. SCHMEIDLER, Subjective probability and expected utility without additivity, Econometrica 57 (1989), 571-587. 49. M. H, SiMONSEN AND S. R. C. WERLANG, Subadditive probabilities and portfolio inertia, Revista de Econometria 11 (1991), 1-19. 50. I. STAHL, "Bargaining Theory," Economic Research Institute, Stockholm, 1972. 51. T. TAN AND S. R . C . WERLANG, On Aumann's notion of common knowledge—An alternative approach (extended summary), in "Theoretical Aspects of Reasoning about Knowledge—Proceedings of the First Conference" (J. Halpern, Ed.), Morgan Kaufman, San Mateo, CA, 1986. 52. T. TAN AND S. R . C . WERLANG, The Bayesian foundations of solution concepts of games, / . Econ. Theory 45 (1988), 370-391.
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53. T. TAN AND S. R . C . WERLANG, On Aumann's notion of common knowledge—An alternative approach. Rev. Brasii Econ. 46 (1992), 151-166. 54. P. WAKKER, Continuous subjective expected utility with non-additive priors, / . Math. Econ. 18 (1989), 1-28. 55. M. WEBER AND C . CAMERER, Recent developments in modelling preferences under risk, OR Spektrum 9 (1977), 129-151. 56. S. R. C. WERLANG, "Common Knowledge and Game Theory." Ph.D. Thesis, Princeton University, 1986. 57. S. R. C. WERLANG, Common knowledge, in "The New Palgrave: Game Theory" (J. Eatwell, M. Milgate, and P. Newman, Eds.), Macmillan & Co., London, 1989.
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19 George J. Mailath on Hugo F. Sonnenschein
My intention, when Ifirstwent to Princeton as a graduate student, was to write a dissertation in econometrics. Thatfirstyear was a revelation. In thefirstsemester, after learning choice theory from Mark Machina, we were taught the failings of the welfare theorems by Joe Stiglitz. Then,finally,Hugo taught us the welfare theorems in the second semester, and there was no going back. Hugo is an inspiring teacher; my thoughts of becoming an econometrician disappeared that year. Hugo's track record as an advisor speaks for itself. Hugo's ability to ask just the right question was invaluable. At one point, I was discussing what I thought was a complete paper with Hugo. The model in the paper was a parametric example of simultaneous signaling in an oligopoly model, and I had explored various questions in that model concerning existence of equilibria and the relationship betweenfinitenumber and a continuum of types. Hugo asked just the right question: "How important were the parametric assumptions to the results?" Answering that question did take a little time, but I hope Hugo was not disappointed in the results. Many people have commented over the years on Hugo's ability to create an exceptional intellectual environment among the theory students at Princeton. Since many of us were fascinated by game theory at the time, an area not at the center of Hugo's own areas of research, this is even more noteworthy. His ability to always ask the "right" question was critical. Hugo then became Dean of the School of Arts and Sciences at the University of Pennsylvania, and so Hugo was again my "boss." We spent an enjoyable semester co-teaching a section of the introductory microeconomics course. Hugo also convinced me to participate in undergraduate college life (at least to the extent of having dinner one night at the undergraduate college he was living in at the time). It was at that dinner that I learned from Hugo of the overlap between the current research of a graduate student at Princeton (Jeroen Swinkels) and a project that I was working on with Larry Samuelson. I can think of no morefittingtestimony to his influence on me than the resulting series of papers.
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It is a pleasure to participate in this volume, because, unlike the other contributors, I cannot claim Hugo for my main advisor; he left for Penn at the end of my second year at Princeton. (Avinash Dixit gracefully stepped in as my advisor.) Hugo played three distinct and key roles in this paper. First, he supervised my second year paper, which is when I began exploring this topic. Hugo was, of course, also George's advisor. Finally, it is through Hugo that George and Larry on the one side, and I, on the other, learned that we had been pursuing parallel tracks of research. Thus, we learned of each other's work early enough to have a very rewarding coauthorship. The multiple ways in which Hugo is connected to this paper are, I think, symptomatic of how deeply Hugo was involved in a huge amount of what was going on in game theory at that time. There are so many papers from that period without Hugo's name on them but where his hand can be clearly seen. Hugo's role in my development became more Umited after his move to Penn. But, this paper was myfirstreal research effort, and it's when I began to learn how to write. I remember the writing critiques particularly well: at our first meeting after I'd turned in a draft, I thought maybe Hugo had been busy, because we spent 45 minutes talking mostly about thefirsttwo paragraphs. The real point, of course, was the degree of precision and effort that good writing requires. The other thing I remember most about this period is Hugo's exceptional patience and generosity. It's not just that he made the time and emotional energy for his students, but that he made it seem easy. Given the number of other things he was up to, one would suspect that this seeming effortlessness wasn't.
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Extensive Form Reasoning in Normal Form Games Econometrica, Vol. 61, No, 2 (March, 1993), 273-302
EXTENSIVE FORM REASONING IN NORMAL FORM GAMES BY GEORGE J. MAILATH, LARRY SAMUELSON, AND J E R O E N M . S W I N K E L S * ' ^
Different extensive form games with the same reduced normal form can have different information sets and subgames. This generates a tension between a belief in the strategic relevance of information sets and subgames and a belief in the sufficiency of the reduced normal form. We identify a property of extensive form information sets and subgames which we term strategic independence. Strategic independence is captured by the reduced normal form, and can be used to define normal form information sets and subgames. We prove a close relationship between these normal form structures and their extensive form namesakes. Using these structures, we are able to motivate and implement solution concepts corresponding to subgame perfection, sequential equilibrium, and forward induction entirely in the reduced normal form, and show close relations between their implications in the normal and extensive form. KEYWORDS: Extensive form games, normal form games, strategic independence, subgame perfection, sequential equilibrium, sequential rationality, information set, subgame, forward induction.
1. I N T R O D U C T I O N
with the same (reduced) normal form can have entirely different information sets and subgames. This suggests that if information sets and subgames are important features of a strategic situation, then the (reduced) normal form is an inadequate representation. This is very disturbing, because at the same time that extensive form reasoning has become pervasive in game theory, forceful arguments have been made (by, in particular, Kohlberg and Mertens (1986)) that only the reduced normal form of a game should matter or, more precisely, that all extensive form games with the same reduced normal form should be viewed as strategically equivalent by rational players.^ The transformations that relate different extensive form games with the same reduced normal form and on which Kohlberg and Mertens base their argument for the sufficiency of the reduced normal form create and destroy DIFFERENT EXTENSIVE FORM GAMES
^ Mailath thanks the National Science Foundation, Samuelson thanks the Hewlett Foundation, and Swinkels thanks the Olin Foundation and the Sloan Foundation for financial support during the writing of this paper. The authors thank Pierpaolo Battigalli, Avinash Dixit, Ron Harstad, Michihiro Kandori, Akihiko Matsui, Ariel Rubinstein, Dale Stahl, Jorgen Weibull, an editor, and three anonymous referees for helpful comments on this paper, and Hugo Sonnenschein, Vijay KLrishna, and Susan Elmes for helpful comments.on Swinkels (1989). Samuelson thanks the CentER for Economic Research at Tilburg University for their hospitality and support during the revision of this paper. The normal form subgame discussed in this paper and the basic structural theorems relating normal and extensive form subgames were first studied by Swinkels in February 1988 (see Swinkels (1989) for a report on this work). Mailath and Samuelson independently began a study of a related structure in October 1988. The authors would like to thank Hugo Sonnenschein, who brought them together in May 1989. The reduced normal form of a game is obtained by deleting, for each player, all pure strategies that are convex combinations of other pure strategies (so that no pure strategy has the same payoff implication as a mixed strategy for all plays by the other players). 273
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information sets and subgames.'* Thus, if these transformations are truly "innocuous," then information sets and subgames are apparently not strategically relevant aspects of a game! This paper suggests a resolution to the dilemma just posed: strategically relevant aspects of information sets and subgames are reflected in the reduced normal form. The choice of an action at an information set (or in a subgame) in an extensive form can only affect the outcome of the game if the remaining players' strategy profile is consistent with the information set (subgame) being reached. Thus, a player can make this choice as if such a contingency had occurred. We call this ability to restrict attention to a subset of the remaining players' possible strategy profiles when making particular strategic decisions strategic independence. We show that strategic independence has a natural description in the reduced normal form, allowing us to motivate and define analogues to extensive form structures and solution ideas entirely in the reduced normal form. The ability to work with these ideas in the normal form is exciting given the productive role they have had in the extensive form. It is also gratifying in light of a seeming tension in Kohlberg and Mertens' (1986) discussion of desirable properties of a normal form solution concept. Several of these properties, such as backward induction and forward induction, have a purely extensive form motivation.^ For example, the key step in Kohlberg and Mertens' discussion of forward induction explicitly uses the extensive form structure of the game (p. 1013, reproduced in Section 9 of this paper). This is disturbing, as Kohlberg and Mertens also explicitly endorse the strategic sufficiency of the reduced normal form, and thus the strategic irrelevance of a game's temporal structure. Our work suggests that much of the appeal of such arguments is retained when they are recast in an atemporal way in the reduced normal form. We begin by using strategic independence to define two pure strategy reduced normal form structures,^ the normal form information set and normal form subgame. Every extensive form information set and subgame generates a corresponding normal form information set and subgame. Thus, the pure strategy reduced normal form captures the strategic independence implied by any information set or subgame. Conversely, we show that every normal form information set and subgame is the image (in a sense to be made precise) of an extensive form information set and subgame. We then turn to normal form analogues of various extensive form solution concepts. Analogous to subgame perfection, we combine a requirement of optimal play on normal form subgames with a requirement of equilibrium ^ These transformations are developed in Dalkey (1953), Thompson (1952), and Elmes and Reny (1989). They are also discussed by Kohlberg and Mertens (1986, p. 1011) who "...believe that [these] elementary transformations... are irrelevant for correct decision making." ^ Note that, even though a strong form of backward induction is implied by properness, the motivation and definition of backward induction remain extensive form ones. ^ The pure strategy reduced normal form of a game is obtained by deleting, for each player, any pure strategy that is a duplicate of another pure strategy (so that no two pure strategies have the same payoff implications for all plays of the other players).
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conjectures on these subgames to define normal form subgame perfection. We show that this equiHbrium concept selects precisely those strategy profiles which are consistent with subgame perfection in every extensive form game with that pure strategy reduced normal form. Similarly, we combine beliefs on normal form information sets generated by limits of independent trembles with a condition of optimality at normal form information sets to define an analogue of sequentiality, normal form sequential equilibrium. Such an equilibrium induces a sequential equilibrium in every extensive form with that pure strategy reduced normal form. Since our solution concept is weaker than properness (Myerson (1978)), this extends the result that properness in the normal form implies sequentiality in the extensive foim? An appealing feature of the solution concept is that it is phrased in terms that have intuitive content. It may be easier to judge the relative merits of the beliefs supporting different normal form sequential equilibria than to judge whether one sequence of e-proper equilibria is more reasonable than another. This parallels a distinction between sequential equilibrium and extensive form (or agent normal form) trembling hand perfection. We also show that ideas of extensive form forward induction can be naturally motivated and formulated in the normal form. An interesting feature of our results is that the relations we find between extensive and normal form solution concepts hold only for the pure strategy reduced normal form, which ignores equivalence with mixed strategies (our results relating extensive and normal form structures hold without this qualification). The difficulties encountered in attempting to extend these results to the reduced normal form, in which equivalence to mixed strategies is also considered, supports the view that there is some fundamental difference between pure and mixed strategies. We present these concepts to demonstrate how extensive form solution ideas can be motivated in the normal form, not because of any deep belief in their reasonableness. This is an important distinction, because some of the behavioral assimiptions underlying these concepts are quite strong (and not always compelling). These assumptions are equally troubling in the extensive form^ although in some cases somewhat better hidden. In fact, it would be damaging to our claim that the normal form captures the key aspects of these extensive form ideas if it did not capture their difficulties. For example, extensive form backward induction has been criticized because it requires players to believe in the common knowledge of rationality in the face of evidence to the contrary. We shall see that this has a close normal form counterpart. Similarly, we will present an example showing that consistency of beliefs in the normal form has the same disturbing feature pointed out by Kreps and Ramey (1987) in the extensive form. The word '^motivated" in the previous paragraphs deserves considerable emphasis. Recall the dilemma generated by a belief in both the importance of ^See van Damme (1984), and Proposition 0 of Kohlberg and Mertens (1986).
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the extensive form structure and the strategic sufficiency of the reduced normal form. Without the ability to motivate extensive form ideas in the normal form, our results could be viewed as providing calculation devices for capturing the implications of extensive form solution ideas without reference to an extensive form. As such, they would provide no resolution to the dilemma, since showing it is possible to implement extensive form ideas in the normal form does not provide any justification for doing so. It is because we can also provide normal form motivations for these ideas that it is consistent to believe in both the importance of traditionally extensive form ideas and the strategic sufficiency of the normal form. We do not claim, however, that strategic independence is the only important property of an information set or subgame. In particular, extensive form information sets capture not only the circumstances under which a decision will matter, but also the last moment in the play of the game at which the decision can be changed. If the latter consideration is important (as, for example, when it influences the types of "mistake" that might be made), then strategic independence is not the only strategically relevant property of an information set. In this case, the normal form is an inappropriate representation of the situation. Similarly, when the extensive form affects the ways in which relevant nonmodeled aspects of a strategic situation (such as communication possibilities) can enter, or when differences in the extensive form affect which equilibrium is "focal", then it is unlikely that the normal form is sufficient. None the less, the relative ease with which some important extensive form intuitions and solution concepts can be interpreted in the normal form does suggest that, at least for these ideas, the reduced normal form is an adequate representation of a strategic situation. 2. PRELIMINARIES
We denote the set of players by N = {1,...,«}, and player f s (pure) strategy space by S^, i = l,,..,n. The set of strategy profiles is given by 5 = 5^ X ... X 5„. Player i's payoff function is written TT,: 5 -> 91. A set of strategy profiles S and a payoff function IT determine the normal form game (5, TT). AS usual, a subscript —/ denotes A^\{/} and a subscript — / denotes N\I. A subset of player /'s strategy space will often be written X-, Subsets of 5_, and S are similarly denoted X_^ and X. Denote the set of probability mixtures over a set X^ by AiX^X Typical strategies for player / are r,, s^, and r,. DEFINITION 1: Two strategies s^,t^ agree on X_^ if ir(5„5_,) = 7r(^^,^_,.) V^.j e X_^, The normal form game (5, TT) is a pure strategy reduced normal form game (PRNF) if V/ no strategy s^^Si agrees with any element of SiXis^} on S_i. The normal form game (S, TT) is a mixed strategy reduced normal form game (MRNF) if Vf no strategy 5, e S^ agrees with any element of id(5;\{5j) on 5_,.
The phrase "reduced normal form" is commonly used (for instance by Kohlberg and Mertens (1986)) to refer to the MRNF; we add the mixed strategy
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prefix to emphasize equivalence with mixed strategies, van Damme (1987) uses "semi-reduced normal form" to refer to the PRNF. The PRNF of a normal form game (S',7r') is that PRNF (S,TT) in which each equivalence class of strategies in 5,- that agree on 5'_^ is represented by a single strategy s^ e 5,. Thus, s'^ e s^ for 5- e S^ and s^ e S^ is a well defined (although slightly awkward) way of denoting that 5^ is one of the strategies in the equivalence class denoted by s^. We do not distinguish between PRNFs that differ only in the strategy labels. The PRNF of (S^ir') is written P{S\TT'\ For X c 5', define the image of X in P{S\ ir') by Im(J^T) = {s e S: 3 / e X such that s'i G Si VO. Analogous definitions hold for the MRNF. A typical extensive form game will be denoted F. The normal form of F is denoted (5^,77^). As a convenience, we will write (5,7r) for P(S^,7rO.^ If F has a nature player, then, following Kreps and Wilson (1982), we assume that nature moves only at the beginning of the game. For any node in F, w^ is the initial node preceding x, and piw^) is the probability with which nature chooses w^. The set of terminal nodes of F is denoted Z. For an information set h of T, denote the set of strategies in S^ consistent with reaching h by S^(h), For games without nature, h will be reached if and only if an element of S^ is played; for games with a nature player, an appropriate first move by nature will also be necessary. Define Sih) = ImiS^ih)). The functions S^iY) and S(Y) are defined analogously for arbitrary subsets Y of nodes of F. All extensive form games are assumed to have perfect recall and at least two possible actions at every information set. All games are assumed to have a finite PRNF and a player with at least two strategies in the PRNF. We largely restrict ourselves to the PRNF, rather than the mixed strategy reduced normal form. For the purpose of defining the normal form information set and subgame, and proving the theorems relating them to their extensive form namesakes, this restriction is entirely a convenience. We will outline how these structures generalize to cover these cases. As mentioned in the introduction, when studying solution concepts and proving relations to existing extensive form solution concepts, the restriction to PRNFs is substantive. 3. NORMAL FORM INFORMATION SETS
Consider an extensive form game and an information set h for some player /. While a similar discussion applies to any extensive form game, for definiteness, consider the extensive form game F of Figure 1 and the information set h for player I. The pure strategy reduced normal form of F is given by Figure 2. Note that this is also the mixed strategy reduced normal form for generic assignment of payoffs to " a " through "g". If player I chooses actions consistent with reaching h on information sets preceding h (for F, the unique such information set is her first one), further ^ It will be clear from the context whether (Syir) is to be interpreted as an arbitrary PRNF or the PRNF of a particular extensive form game.
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I
FIGURE 1
decisions will be required of her in one of two situations, depending on IFs actions. She will either find herself at h and have to decide how to continue, or she will find herself at an information set that is not reached by any path that reaches h and have to decide how to continue (for F the unique such information set is H), These decisions can be made independently. Player Fs choice when h is reached does not affect the available options nor their consequences at information sets that cannot be reached if h is. This independence is captured in the PRNF of F, The set of strategy profiles • \5'j, S2, ^3, ^ 4 / X \f | , t^ in the PRNF corresponds to reaching h in F (i.e.. X=S{h)\ (The set X is boxed in Figure 2.) Now, consider player Fs decision conditional on playing some strategy in X^ = {s^,S2yS^,s^. If player II plays a strategy from X2 = {^1, t^, then player I is indifferent between strategies s^ and ^2 and between strategies ^3 and ^4. If player II plays a strategy from S2\X2 (i.e., ^3), then player l i s indifferent between strategies s^ and ^3 and between strategies ^2 ^^^ ^4- The payoff vector in the case that player II makes a choice from X2 thus depends only on the total weight player I puts on the set of strategies {si, S2] relative to the set {^3, ^4} and is independent of the division of weight within the sets {^1,^2} ^^^ {ssyS^}, Similarly the payoff vector in the case that player II makes a choice from S2\X2 depends only on the weight player I puts on the set of strategies {s^, s^} relative to {52, ^4} and is independent of the division of weight within the sets {s^, s^} and {^2, ^4}. The key is that these two decisions, of the weight to put on {^j, ^2) relative to {^3,54} and the weight to put on {5i,.y3} relative to {52,5^}, can be made independently. The abihty to choose these weights independently is the normal form analogue of the ability in the extensive form to make a decision at h independently of the choice at information sets that cannot be reached if h is.
h h h a c ^2 a c I S3 b d
e f
S4
b d
f
H
g
Sl
g 1 S
FIGURE 2
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We will say a set of strategy profiles X is strategically independent for player / if / can make decisions over X•^ conditional on X_i independently of decisions over X^ conditional on S_\X_i_\ DEFINITION 2: The set XQS PRNF(5,7r), if (2.i)
X^XiXX_i,
(2.ii)
V r^,Si^Xi,
is strategically independent for player i in the
and 3 t^ e Z - such that
If X is strategically independent for /, there is a strategy t^ in X_^ that is equivalent, from /'s point of view, to r^ if his opponents choose X_i and to 5, if they do not. Hence, if i has beliefs cr_i over 5_, with y_/ and f_, being /'s beliefs conditioned on X_i and 5_,\J!f_,, then when r, is optimal given y_^ and 5, is optimal given f _y, f, is optimal given (r_,. While strategic independence captures an important decision theoretic aspect of information sets in the reduced normal form, the extensive form implies additional structure in the normal form. For example, in Figure 2, if player II plays a strategy from A'2 = {^i,^2)> ^^^^ player II, as well as player I, is indiiferent between strategies s^ and ^2 ^^d between strategies 5-3 and ^4. More generally, if a strategic independence is generated by an information set in an extensive form, then (2.ii) will be satisfied for all players (it turns out that for s_^ in X_i the profiles (r^, s_^) and (r,, s_^) in (2.ii) reach the same terminal node in the extensive form—see the proof of Theorem 1). In this paper, we are interested in the relationship between the extensive form and the normal form. Thus, we restrict attention to those normal form structures that could have been generated from an extensive form. In particular, we focus on those subsets of strategy profiles that are strategically independent for a player and for which the equalities in (2.ii) are satisfied for all players. We call such a subset a normal form information set:^ DEFINITION 3: The set theFRNF(5,7r), if
XQS
is a normal form information set for player i of
(3.i)
X^XiXX_i,
and
(3.ii)
V r-,5^ ^Xi, 3 t- e Z , such that t^ agrees with r^ on ^_,- and t- agrees with Si on S_i\X_i,
Note that {5,}x^_- is trivially a normal form information set for i for any Si^Xi and J f _ , c 5 _ , . ^ If the normal form was obtained from a generic extensive form, then every strategic independence is a normal form information set. We discuss issues of genericity in more detail at the end of this section.
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G, J. MAILATH, L. SAMUELSON, AND J. M. SWINKELS
REMARK 1: It will sometimes prove useful to 'index* the strategies in X^ in a way that reflects the structure of a normal form information set. For a given information set X for player /, define an equivalence relation on X^ by agreement on A"_,. Let u- denote the number of such equivalence classes, and label the equivalence classes by / = 1,..., M-. A second equivalence relation on X^ is defined by agreement on S_^\X_f. Let v- denote the number of such equivalence classes, and label the equivalence classes by A: = l,...,i;,-. Then there is a one to one correspondence between ordered pairs (;, A:) e ( 1 , . . . , u^} X {1,..., Vi}y and elements of X^. The existence of a strategy in X^ corresponding to each (j,k) pair is immediate from the definition of a normal form information set. There is only one such because we are in the PRNF: Two strategies with the same / and k indexes would agree on all of 5_,. The / index can be thought of as denoting choices for when the information set is "reached," and the k index as denoting choices for when the information set is "not reached." Given such a correspondence, we denote the (j,k) index associated with s^^Xf by (jXsjXkisf)). For example, a labelling of player Fs strategies in the information set [s^, ^2, ^3, s^} X {r^, ^2} in Figure 2 is j(si) =7(^2) "^ 1> A-^a) ^ 7(54) = 2, k(s^) = k(s^) = 1, and kis2) = Hs^) = 2.
The following theorem establishes an equivalence between normal and extensive form information sets. It is worth emphasizing the importance of the only if part of this theorem. In this sense, any normal form structure "reflecting" all extensive form information sets must be a generalization of the normal form information set, and hence characterized by a property weaker than strategic independence. THEOREM 1: The strategy subset X of the PRNF (S,7r) is a normal form information set for player i if and only if there exists an extensive form game without nature with PRNF (5,7r) with an information set h for player i such that Sih)=^X. PROOF: (<=) Let F be an extensive form game without nature, and let h be an information set for player /. Since F has perfect recafl, the actions chosen by player / which make h reachable are unique and independent of the choices of the other players. Thus, S^(h) can be written as Sfih)xS^i(h). Let r^^s^^ Sf(hX Then r^ and s^ agree on information sets for player / preceding h. Let t^ be the strategy which specifies the same actions as r^ and s^ on information sets for / preceding h, the same action choices as r^ at the information set h and those following h, and the same actions as s^ at information sets that neither precede nor follow h. Clearly t^^Sf(h), Let 5_, e5^.(/i). Then every information set for / that is reached by (//,5_^) precedes or follows h. Thus, by construction, (t^,s_^) and (r^,5_,) reach the same terminal node and so 7r(t^, s_i) = 7r(r^, s_i\ Now suppose s_^ e 5{]y\5{],(/i). Then, neither h nor any information set for / that follows h is reached by (t-,s_^X Thus (//, ^_j) and
458
Extensive Form Reasoning in Normal Form Games EXTENSIVE FORM REASONING
281
{Si,s_^) reach the same terminal node and Tr(r,, 5_,) = irC^,, 5_^). Thus, S^(h) has the required structure. This structure is clearly preserved when we pass to the PRNF and S(hX (=>) Form and label equivalence classes on X^ as described in Remark 1. If u^,Ui>2, then consider the following extensive form game F. Stage 1: Each player /' e A^\(/} chooses a strategy s^r e S^r, Player / chooses "in" or s^ e S^XX^. The choices in stage 1 are simultaneous. Stage 2: The nodes reached by a choice of "in" by player / and any choice s_i^X_^ by players N\{i} form an information set for player /, denoted /z. Player / has M, choices at this information set, labelled ! , . . . , « , . Following a choice / e { ! , . . . , « . } , the game terminates with payoff given by Tr(s^,s_f) for any s^ such that 7(^/)=; (this is well defined, since / is the label of an equivalence class under agreement on X_i), The nodes reached by a choice of "in" by player / and any choice s_f^S_f\X_i t>y players N\{i} form an information set for player /, labelled h\ Player / has u^ choices at this information set, labelled 1,..., i;,. Following a choice A: e{l,...,i;^.}, the game terminates with payoff given by Tr(s-,s_i) for any s^ such that k{si) = k. Following a choice 5^e5,\A"^ for player /, and a choice s^i^S_i for players N\[i}y the game terminates with payoff iris^, s_iX (This construction is illustrated in Figure 3 for I's information set {3^,32,5^,54} X {/j, ^2) in Figure 2.) Ignoring strategies that are obviously repetitive for player /, her strategies in this extensive form game are either a choice s^ e5,.\Z^. or a choice of "in" along with a choice of ; and a choice of k. If we associate ("in",/. A:) with the unique ^, e ^ . satisfying (;\(^/),A:,(^/)) = (;,/:), then it is obvious that (S,Tr) is the PRNF of T, and that S(h) = X, If v^ = 1 holds, then the game is the same except h' is deleted. Finally, if u^ = 1, then at h, player / has two choices, each of which results in payoffs given by 7r(^„5_^) for any 5, satisfying 7/5^) = 1 (i.e., for any 3^ e X , ) The extensive form game constructed in the proof of Theorem 1 for player I's normal form information set [3^^,32,3^,34} X {t^, (2) in Figure 2 is illustrated in Figure 3. We can extend Theorem 1 to games with moves by nature. Since a game without nature is a special case of a game with nature, the "only if" portion of Theorem 1 obviously holds for games with nature, and it remains only to consider the "if" portion. Pick a player / and an information set h for f. Let Qih) be the collection of information sets h' for player / which neither precede nor follow h and for which S(h) n SW) ^ 0 . (For sufch h\ there will exist strategies such that either h or h' will be reached, depending upon nature's choice.) Let A^^^^ be the set of action choices at these information sets, so that a^^^^ ^A^^^^ specifies an action at each information set in Q(h). Let P be the information sets for / which precede h, and let a^ be the (unique by perfect recall) actions at P which make h reachable. To simplify notation, if F is an extensive form game, denote its normal form by (7, tl/} and its PRNF by (S, TT). Let Tih,fl^^^>)be the
459
George J. Mailath, Lariy Samuelson, and Jeroen M. Swinkels 282
G. J. MAILATH, L. SAMUELSON, AND J. M. SWINKELS
II
Strategy profiles in T which reach h (given a suitable choice by nature) and take actions «^^^> on Q{h\ with 5(A,fl^(^>)= Im(r(/j,tz^(^>)X Then we have: THEOREM 2: Suppose h is an information set for player i in an extensive form game with nature. Then for each a^^^^ ^A^^^\ Sih^a^^^^) is a normal form information set for i.
PROOF: Consider T(h,a^^^^)==Ti{h,a^^^^)xT_,(h), Let s^,r,GT,(h,aQ^^^), Let t^ specify the same actions as s^ on information sets which precede or follow h; the actions a^^^^ at nodes in Q{h); and the same actions as r^ on other information sets. Let r_^ e T_i{h), Then for every choice of nature, (s^, t_^) and (tiyt_i) yield identical terminal nodes and hence have identical expected payoffs. Let t^GT_^\T_^(hX Then for every choice of nature /, and r^ yield identical terminal nodes and hence yield identical expected payoffs. This is inherited by S(h,a^^^^X which is thus a normal form information set for player /. Q,E,D. Notice that for an information set A in a game without nature, Q{h) is empty and Sihy a^^^^) = S{h). The "if" portion of Theorem 1 is then the special case II
III
L
A
D
F
c
^
11 III
R A A
D
e 1d 1
D
b 1 b 1
FIGURE 4
460
Extensive Form Reasoning in Normal Form Games EXTENSIVE FORM REASONING
283
FIGURE 5
of Theorem 2 for games without nature. This construction will be key when we consider normal form sequential equilibrium in Section 8. If X^ XX_^ is a normal form information set for player /, then X_i need not be a cross product. In the PRNF in Figure 4, for example, ^ = 53 X [{D, AX(D, D),(A,D)], the block in Figure 4, is a normal form information set for player III, and X_^ is not a cross product. This game corresponds to Selten's (1975) horse game (shown in Figure 5). We conclude this section with some remarks on genericity. While a subset of strategies can only be strategically independent (or a normal form information set) for a player if the normal form game is nongeneric, nongeneric normal form games are important from both a game theoretic and economic perspective. Nontrivial generic extensive form games, for example, have nongeneric normal forms. Moreover, nongeneric extensive form games can be generated by a generic economic setting. More precisely, a player's payoff as a function of strategies or terminal nodes in a game should be interpreted as the composition of a function mapping strategic choices into economic outcomes,^^ such as consumption, and a utility function defined on these economic outcomes. In our view, the appropriate space to require genericity is the space of utility functions. For example, the alternating offer bargaining model of Rubinstein (1982) is nongeneric as an extensive form game (players only care about the share they receive and the time of agreement, independent of the sequence of rejected offers), but allows generic preferences over the share and the time of agreement. Thus, we will take seriously ties in the normal and extensive form, since they could be generated by a player receiving the same economic outcome. It is impossible to decide which payoff ties for a player in a normal or extensive form are nongeneric without having a particular scenario or application in mind. We describe later a normal form (Figure 11 in Section 7) for which every associated extensive form is nongeneric, no one extensive form game reflects all the normal form information sets, and yet in at least one economic scenario the game is generic because all ties are generated by players receiving the same economic outcome.
This corresponds to a game form in mechanism design.
461
George J. Mailath, Lariy Samuelson, and Jeroen M. Swinkels 284
G. J. MAILATH, L. SAMUELSON, AND J. M, SWINKELS 4. NORMAL FORM SUBGAMES
The subset X can describe an information set for more than one player. For example, in a simultaneous move subgame F^ of an extensive form game F, the sets of strategies in the normal form such that each player's information set in F^ is reached are identical. The corresponding set in the pure strategy reduced normal form is thus an information set for all players. This suggests the following definition.^^ DEFINITION 4: The strategy subset X is a normal form subgame of the PRNF (S, TT) if it is a normal form information set for each player. A normal form subgame X is nontrivial if, for some player /, there exists r^^s^ eX-, 5_^ ^^-i such that 7r(r^, s_^ ^ 7r(^„ s_^).
The analogue to Theorem 1 holds for normal form subgames: THEOREM 3: The strategy subset X of the PRNF (5, TT) is a normal form subgame if and only if there exists an extensive form game without nature with PRNF (S, IT) with a subgame F' such that 5 ( r 0 = X
Given the relation between normal form information sets and subgames, it should not be surprising that the proof is a simple modification to the proof of Theorem 1 (and so is dispensed with here). The sketch of an alternate proof of the "if" direction offers some insight. A subgame in the extensive form can be replaced by another subgame with the same PRNF without changing the PRNF of the game as a whole (this observation is also key to the corollary following Theorem 4). Replace the subgame F^ by a simultaneous move subgame with the same PRNF. Any strategy profile that reaches the subgame will now reach all of the information sets in this game. Thus, the set S{F^) is a normal form information set for each player and hence a normal form subgame. Theorem 3 addresses single subgames. We now examine families of normal and extensive form subgames. If X is a normal form subgame, then X = O/eA^^/> since X = XiXX_i for all /. Neglecting the difference between TT and its restriction to X, (X,7r) defines a normal form game. DEFINITION
5: Let (5,7r) be a PRNF, and X a normal form subgame of is nested in X if, for all /,
(5,77). We say Y =TII^P/YIQS (5.i)
Y^QX^,
and
(5.ii)
if Si e Y^ and s^ agrees with t^ e Z , on X_^ then t- e y;.
**As part of their theory of equilibrium selection, Harsanyi and Selten (1988) introduce two concepts, the semicell and cell, which are somewhat related to the normal form subgame. While also an attempt to capture a form of strategic independence between agents which Harsanyi and Selten view as being characteristic of subgames, neither the semicell nor the cell is the same as our normal form subgame. In particular, these concepts are defined in what Harsanyi and Selten call the standard normal form of an extensive form: the agent normal form with the modification that agents of the same player are not treated independently.
462
Extensive Form Reasoning in Normal Form Games EXTENSIVE FORM REASONING
285
h h h k S| a I a I b I b 52 I c I c I c I c
53 d
e
d
e
FIGURE 6
The next theorem essentially states that a normal form subgame of a normal form subgame is a normal form subgame of the original game. Note that (5 Ji) is only needed to show that if the image of Y in P(XyTr) is a normal form subgame of PiXyirX then Y is a normal form subgame of (S,Tr). THEOREM 4: Suppose X is a normal form subgame of (5, v) and suppose Y is nested in X. Then the image of Yin P(X, TT) is a normal form subgame ofPiX, v) if and only if Y is a normal form subgame of (S, TTX PROOF:
The proof is a straightforward application of the definitions and so is
omitted. A simple example illustrates the need for the nestedness condition. For the PRNF in Figure 6 let X={s^,S2}xS2 and Y={s^}x[t2,t^,t^}, Then, Z is a normal form subgame of the PRNF, the image of Y is a normal form subgame of P(X, TT), and yet Y is not a normal form subgame of the original PRNF. The difficulty is that t^ is not included in player IFs strategy set. The following corollary provides a relationship between families of subgames in the extensive and normal forms. COROLLARY: A PRNF (5,7r) has a nested sequence of normal form subgames {Z"}, X"^^ QX", X""^* ^X", if and only if there exists an extensive form game F having PRNF (5,7r) with a subgame T" for each a such that 5 ( r " ) = Z " , and such that T"-"^ follows F"", PROOF:
Repeated application of Theorems 3 and 4 yields the desired result. Q,E,D.
Figure 7 (Figure 8 of Swinkels <1989)) shows that one cannot always represent II L
M
I c
a e
B
e
d d b d 1 e 1c 1
T
R
FIGURE 7
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George J. Mailath, Lariy Samuelson, and Jeroen M. Swinkels 286
G. J. MAILATH, L. SAMUELSON, AND J. M. SWINKELS
all the normal form subgames of a given PRNF in a single extensive form.^^ In particular, there is no single extensive form game that has subgames corresponding to both the normal form subgame {C, B) X {M, R) and the normal form subgame {T, C] X {L, M}. Figure 11 below illustrates the implication of this for solution concepts. Theorems 1, 2, and 3 and Figure 7 leave open the question of which collections of normal form information sets or normal form subgames can be represented in a single extensive form. An analysis of this question is provided by Mailath, Samuelson, and Swinkels (1992). 5. NATURE
The definitions of the normal form information set and subgame and the structural theorems of the last two sections are easily extended to extensive form games with moves by nature. Perhaps the simplest way of doing this is to define a normal form game with nature: DEFINITION 6: A normal form game with nature is defined by strategy sets 5o, iSi,..., 5„, payoff functions TT,: Fl^^o*^; ~^ ^ ^^^ / = 1,..., AZ, and a distribution p e A{SQ), Player 0 is the nature player. A normal form game with nature ((5Q, 5), 77, p) corresponds to a normal form game (Syir) if V 5 e 5 , and for / = 1 , . . . , « , TT^is) = I^so^SoP^^o)'^i^^O^
^^'
Note that as in the extensive form, the major difference between nature and other players in the normal form game with nature is that nature has a preassigned strategy and receives no payoffs. An n person game with nature can be considered an « + 1 person game by setting TTQCSQ,5) = 0 V C ^ Q , S ) ^ S Q X 5 . Now, consider an extensive form game with nature, and let h be an information set for some player /. Let S'^(h) be those strategies in the corresponding PRNF game with nature that are consistent with h being reached. Then one easily shows that S%h) is a normal form information set of this game considered as an « + 1 player PRNF. On the other hand, if X is a normal form information set of a PRNF with nature when that game is treated as an AI + 1 player game, then an extensive form game with an information set corresponding to h can be constructed as in the last section where the game begins with the move by nature. Theorem 2 in Section 3 shows the possibility of an alternative in which the full structure of S(h) is captured by a 3 part index similar to the 2 part index associated with S(h) for h an information set in a game without a nature player. Loosely, the first index reflects action choices in the case when the information set is reached, the second when the information set is not reached solely because of nature, and the third when the information set is made unreachable ^^ A similar game appears in Harsanyi and Selten (1988, p. 112) who make a somewhat related point. Although intended to illustrate a different point. Figure 3 of Abreu and Pearce (1984) provides another example.
464
Extensive Form Reasoning in Normal Form Games EXTENSIVE FORM REASONING
h
h
Si
2,2
5,5
h U ^5 2,2 5,5 8,8
S2
4,4
6,6
4,4 6,6 8,8
S3 1 0,0 0,0
4,4
287
4,4 8,8
FIGURE 8
by the actions of the players.^^ Entirely analogous constructions and results hold for subgames. 6. THE MIXED STRATEGY REDUCED NORMAL FORM
Extending the definitions of the normal form information set and subgame from the PRNF to the MRNF is equally straightforward. Essentially, two PRNF games with the same MRNF can diifer in their information set structure for two reasons. First, a pure strategy needed to make Definition 3 hold may be equivalent to a mixture of other strategies, and so deleted when forming the MRNF. For example, in the PRNF in Figure 8, {^j, 52} X (r^, ^2, ^3> ^J is a normal form information set for both players, and so a subgame. However, t^ is an equal mix of t^ and ^5. When t^ is removed, {^j, ^2} X '^2\{^5J ^^ not a normal form subgame, since {si,S2} X {t^, t2, t^) is not an information set for player IL The second difficulty is that the PRNF may include a strategy for a player other than the player to whom an information set belongs which is equivalent to a mixture of other strategies which put weight both on "in" and "out" strategies. As an example of this, note that adding a pure strategy equivalent to an equal mixture of ^2 and t^ destroys player Fs information set in Figure 2. However, from Theorem 1, it is easily seen that a finite set of pure or mixed strategy profiles X of the form X^ X X_i from a MRNF game (5, TT) will be the image of an information set in an extensive form game with that MRNF if and only if the PRNF formed by expanding S to include any (nondegenerate) mixed strategy in X has y\^ as a normal form information set. A similar result holds for normal form subgames. This forms the basis for an extension of the definitions of the normal form information set and subgame to the MRNF. 7. NORMAL FORM SUBGAME PERFECTION
In the previous sections, we showed that information sets and subgames imply a normal form property which we called strategic independence. In this and subsequent sections we argue that strategic independence allows us to reinterpret the intuitions traditionally associated with extensive form solution concepts, such as subgame perfection and sequential equilibrium, in the normal form. Speaking very broadly, most existing extensive form solution concepts can be thought of as having two main parts: first, a requirement of optimality of actions ' Details of the material in this and the next section can be found in Mailath, Samuelson, and Swinkels (1990).
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even at out-of-equilibrium information sets or subgames, and second, various conditions on out-of-equilibrium conjectures. The requirement of optimal actions at out-of-equilibrium information sets is often captured by "sequential rationality:" since the action chosen at an extensive form information set only matters if the information set is reached, the choice should be optimal relative to some conjecture over the set of other players' strategy profiles consistent with the information set being reached. It has been thought that such a requirement can only be motivated in the extensive form.^"^ But, the requirement of sequential rationality does not seem very different from a general requirement that if a decision only matters given some subset of the strategy profiles for the remaining players, then that decision should be optimal relative to some conjecture over those strategy profiles. These are precisely the situations characterized by strategic independence. Rational players should exploit strategic independence in their decision making, even when the strategic independence is not due to an extensive form information set or subgame. Figure 11 at the end of this section illustrates this idea nicely. Extensive form solution concepts also place two types of restrictions on the conjectures players can hold. First are requirements motivated by rationality or signaling-type arguments (such as forward induction) applied to other players. We illustrate in Section 8 how these arguments can be conducted in the normal form. Second are consistency restrictions across players' conjectures, such as requiring Nash equilibrium play on all subgames. These are easily interpretable in the normal form. We also believe that these consistency requirements are not evidently less sensible in the normal form than in the extensive form. We first make these ideas concrete in the context of subgame perfection (we consider sequentiality in the next section). Consider the game in Figure 9. The strategies (U,(\C -f {R)) are best replies in undominated strategies, and hence constitute a perfect equilibrium. However, since player II's strategy choice is irrelevant when player I plays U, i.e., her choice is strategically independent of U, she may well consider alternative possibilities. Note that if I does not play U, the result [T, M, B] X ^2 is a normal form subgame. Further, this normal form subgame has a unique Nash equilibrium (T, L). Since IFs strategy matters only if I is choosing a strategy in the normal form subgame, she should perhaps play L, the equilibrium strategy of the normal form subgame, rather than (^C + |J?). In general, since a player's choice of strategy from a normal form subgame matters only if the other players choose strategies in the normal form subgame, we might expect her to chqose a strategy that is optimal relative to some conjecture about play conditional on the other players choosing strategies in the normal form subgame. Our concept of normal form subgame perfection is obtained by requiring this conjecture to correspond to a normal form subgame perfect equilibrium of the subgame (if a game has no normal form subgames, then every Nash equilibrium is normal form subgame perfect). This is very ^^ For example, Kreps and Wilson say that "[ajnalyses that ignore the role of beliefs, such as analysis based on normal form representation, inherently ignore the role of anticipated actions off the equilibrium path in sustaining the equilibrium...'* (1982, p. 886).
466
Extensive Form Reasoning in Normal Form Games EXTENSIVE FORM REASONING II L C 1,1 1,1 u T 2,2 1,0 M 0,1 2,1 B 0,1 -2,2
289
R
1,1 1 0,0 -2,2 2,1
FIGURE 9
Strong, but not evidently less plausible than requiring that strategy profiles in an extensive form game prescribe optimal actions relative to conjectures on unreached subgames which are subgame perfect equilibria of those subgames, i.e., not evidently less plausible than extensive form subgame perfection. In order to define normal form subgame perfection, v^e need some notation. Let cr be a (possibly) mixed strategy profile for (S, TT), i.e., a = (o-j,..., cr^), with cr^ e A{S^. If a> e A{X) and X^ c S^^ then we can treat cr^ in the obvious manner as an element of AiS^), For X^ QS^ and A^= Fl^^/, define 0-1;!^ to be the vector (o'i\xi,'",orn\x„X where (r^\x^ is the projection of a-, onto X^. Notice that a^lx^ will not be a probability distribution on the set X^ if o} has assigned positive probability to 5 , \ ^ , . If (X, TT) is a normal form subgame, then for s^ e P^{X, ir), define o'i\\p(x,'jT)iSi) = Z[,'^^s,}^i\x,(s'i) and c7'||/>(;^,^) = (o'il|p(Ar,^),...,cr„||/>(A:,^)). Since s^ is the equivalence class of strategies in P(X,TrX o-^Wpix^tryiSi) is the probability that any strategy in the equivalence class s^ is played according to a^. DEFINITION 7: The vector a is proportional to a' if, for all /, either there exists dsts ia^eSl^^, such that a^^a^al ^^ ^^ \^2iSi one of cr^ and tr/ is the zero vector.
Thus a\x is always proportional to some mixed strategy profile on X, We now define normal form subgame perfection. DEFINITION 8: If (5,7r) does not have any proper nontrivial normal form subgames, then cr is a normal form subgame perfect equilibrium if it is Nash. In general, cr is a normal form subgame perfect equilibrium if it is a Nash equilibrium and if, for all normal form subgames (Z,7r), cr\\p(^x,7ry is proportional to a normal form subgame perfect equilibrium of P{X, TT).
This is the coarsest refinement of Nash that has the property that the projections onto normal form subgames of equilibria satisfying the refinement also satisfy the refinement. The corollary to Theorem 5 below establishes the existence of normal form subgame perfect equilibria. The game in Figure 10 illustrates the recursive nature of normal form subgame perfection. The profile {T,L) is a Nash equilibrium, but it is not normal form subgame perfect. It projects onto a Nash equilibrium, (7, C), of the subgame S^ X {C, R) (and onto any Nash equilibrium of {M, B) X {C, R}). However, the game S^ X {C, R) has (M, B) X {C, R) as a subgame, which has a unique Nash equilibrium given by (fM + \B, | C + \R), Flayer II must thus play 467
George J. Mailath, Larry Samuelson, and Jeroen M. Swinkels 290
G. J. MAILATH, L. SAMUELSON, AND J. M. SWINKELS
L T' 2,2 I M 2,2 B 2,2
II C
R
" 3,0 3,0 2,1 1,5
5,3 6,1
FIGURE 10
| C + |i? in any normal form subgame perfect equilibrium of S^ X (C, 7?}, which thus has (fAf + jB, \C + \R) as its unique normal form subgame perfect equilibrium. The profile (7, L) is not proportional to this equilibrium and thus is not a normal form subgame perfect equilibrium. The unique normal form subgame perfect equilibrium of the game is ( | M + \B, \C + \R\ yielding payoffs ( 3 | , 2}). A useful characterization of normal form subgame perfection is provided by the following (the proof is a straightforward implication of Theorem 4 and so is omitted): LEMMA 1: A strategy profile a of the PRNF (5, ir) is a normal form subgame perfect equilibrium if and only if for every sequence of normal form subgames {X'')^^Q such that X"^ has no nested proper normal form subgames, X"^^^ is nested in X"^, and X^ = Sy there is a sequence {O-^I^^Q, a-^ = or,cr" a Nash equilibrium of X" for a = 0,...,m, such that a^lx^^^ is proportional to a"'^^, for a = 0,.. ,,m — 1.
We now show that normal form subgame perfection is equivalent to extensive form subgame perfection in every extensive form game with the given PRNF. We begin with a definition: DEFINITION 9: A strategy profile o-^ of an extensive form game F is induced by the strategy profile a of its PRNF if C7-^IIP(5'',^^)(^/) = <^/(^/) V5, e S^, V/. THEOREM 5: The strategy profile cr is a normal form subgame perfect equilibrium of a PRNF (5, TT) if and only if a induces an extensive form subgame perfect equilibrium in every extensive form game without nature with PRNF (5, TT). PROOF: (=>) We proceed by induction on the number of proper normal form subgames, denoted n^, in G = (5,7r). Let F be an extensive form game with PRNF (5,77). Let o- be a normal form subgame perfect equilibrium of {S, irX If n^j = 0, there are no proper normal form subgames and hence any subgame F^ of any extensive form with PRNF (Syir) has S(FO=^S (by Theorem 3). The strategy profile a induces a Nash equilibrium on each of these subgames, and hence induces an extensive form subgame perfect equilibrium on F. Suppose «^ > 0 and that the result holds for n = 0 , . . . , w^ — L Let F^ be a maximal proper subgame of F satisfying S(F^)¥^S. Now, S(FO is a proper normal form subgame of (S,Tr), so that a\\p(Sir')) is proportional to a normal
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form subgame perfect equilibrium of S(r^). By the induction hypothesis, any normal form subgame perfect equilibrium of FiSiF^X TT) induces an extensive form subgame perfect equilibrium on F\ We thus induce subgame perfect equilibria on every proper subgame of F, Since a is Nash, cr then induces an extensive form subgame perfect equilibrium on F, (<=) Suppose (7 is a strategy profile of the PRNF (5,7r) that induces a subgame perfect equilibrium on every extensive form F with PRNF (5, TT). By Lemma 1 it is sufficient to show that for every sequence of normal form subgames {y^^'lJLo, such that X"^ has no proper normal form subgames, X"'*'^ is nested in X", and X^=^S, there is a sequence {or"}^^Q, a^ = a,o'" a Nash equilibrium of X" for a = 0 , . . . , m , such that o-'^lx"^^ is proportional to a"^^, for a = 0,..., m — 1. By the Corollary to Theorem 4, there is a single extensive form representing all of the normal form subgames in this sequence, with the extensive form subgame representing X"'^^ succeeding the extensive form subgame representing X''. By hypothesis, cr induces a subgame perfect equilibrium on this game. But this yields a sequence of Nash equilibria on the subgames with the property that the first term is a, and each term is the projection onto the subgame of the previous term, yielding the result. Q,E.D, Normal form subgame perfection can thus be read a "subgame perfect in every equivalent tree." It captures any restriction on equilibrium play implied by subgame perfection on some equivalent extensive form. However, this equilibrium concept has a surprising feature. Consider the game in Figure 11. The equihbrium ( 7 , L) is normal form subgame perfect. It projects onto the equilibrium ( r , R) of the subgame S^ X {C, R), which in turn projects onto the equilibrium {B^R) of the subgame [M,B)x[C,R], The profile ( r , L ) also projects onto the equilibrium (M, L) of the subgame {M, B) X 52, which in turn projects onto the equilibrium (M, C) of the subgame {M, B) X {C, R). By the corollary to Theorem 4, there is an extensive form game with subgames corresponding to each of these sequences (though no extensive form capturing both sequences). The interesting aspect is that ( r , L) is sustained by conflicting prescriptions for the game {M, B) X {C, R), Player I thinks that the equilibrium that will appear in the {M, B) X {C, R) subgame is the one that is most favorable to I (and least favorable to II, so that II will not play into it if I plays to it), while player II has the opposite expectation. Because the only remaining Nash equilibrium of {M, B) X {C, R) is the mixed strategy profile ((fM + \BX (fC-h \R)\ which yields payoffs,(T>TX there is no single equilibrium of the normal form subgame {M, B] X {C, R} that can justify both players' decisions to II L c R T 4,4 4,4 4,4 I M 4,4 8,3 0,0 B 4,4 6,6 3,8 FIGURE 11
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avoid it. The equilibrium (7, L) is subgame perfect in every extensive form associated with this game, but different extensive forms require different equilibria for the subgame corresponding to {M, B) X {C, R}, It is common in the refinements h'terature to require players to hold identical expectations about out-of-equilibrium strategy profiles (though private-beliefs equilibria (Kalai and Lehrer (1991)) and self-confirming equilibria (Fudenberg and Levine (1991)) do not). Anyone who finds this requirement compelHng might be troubled by the inconsistent beliefs which support (T, L) as a normal form subgame perfect equilibrium. The appropriate response to this inconsistency is unclear, and a full debate of the merits of various responses would take us too far afield. Defining a version of normal form subgame perfection which imposes consistent conjectures amongst players about unreached subgames, and showing that such a solution always exists is a straightforward exercise. Our concept of normal form sequential equilibrium will imply this consistency, so we omit the definition here. Suppose Figure 11 represents a scenario in which two players have the option of entering a relationship, with the relationship modelled by the 2 x 2 subgame (Af, B) X {C, R), The subgame is reached if and only if both players choose to enter the relationship. If either player vetoes the relationship, then the same economic outcome results. There are two extensive forms of interest with the PRNF of Figure 11, reflecting differing orders of vetoing the relationship, depicted in Figures 12 and 13. Note that the extensive form in Figure 12 does not capture the normal form information set S^ X {C, i?}, while the extensive form in Figure 13 does not capture {M,B)xS2^ The outcome (no, no) (corresponding to (T, L) in Figure 11) is subgame perfect in both trees. However, reflecting the inconsistency discussed above, the outcome is supported by the specification of (M, C) on the 2 x 2 subgame in Figure 12, while it is supported by the specification of {B, R) on the 2 X 2 subgame in Figure 13. That is, while (no, no) is subgame perfect in both trees, differing trees require differing specifications of play on the 2 x 2 subgame. Consider now the argument that if the "true'" underlying extensive form is as depicted in Figure 12, then only player II should use his strategic independence.
1 8,3 0,0 6,6 FIGURE 12
470
3,8
Extensive Form Reasoning in Normal Form Games EXTENSIVE FORM REASONING
M
8,3
0,0
B
6,6
3,8
293
I
FIGURE 13
However, if one is convinced that players will exploit some strategic independences, then it appears very difficult to preclude the possibility that players will exploit all strategic independences. For example, it seems to us at least possible that player I, in making his decision between no and yes, will recognize that only if II says yes will the decision be relevant. Thus, even if I is pretty sure II will say no, I may still base her decision between yes and no on her expectation of play in the 2 x 2 subgame. The observation that the subgame perfect equilibrium ((no, M),(no,C)) does not respect Fs strategic independence is reflected by the fact that it is not extensive form trembling hand perfect.^^ Drawing a sharp distinction between the situation of I and II in this scenario becomes even more tenuous when one considers that many strategic situations one might want to model do not generate a unique "right" extensive form. Thus, we may be forced to analyze interactions of the type given in Figure 11 with no sharp information as to the nature of the extensive form, and perhaps with no extensive form at all. In particular, it would be quite surprising if employment relationships, which one might know yield the (generic) assignment of payoifs to economic outcomes given in Figure 11, were actually negotiated according to a fixed extensive form. We think that players may still exploit strategic independences in such cases, and find it very hard to argue that they will exploit some but not all instances of strategic independence. 8. NORMAL FORM SEQUENTIAL EQUILIBRIUM
Kreps and Wilson's (1982) definition of sequential equilibrium required first that actions at each information set be a best response to some behefs about how that information set was reached (sequential rationality) and second that these beliefs be "reasonable" (consistency). While the definition of "reasonable" beliefs is somewhat problematic, Kreps and Wilson use as their definition that beliefs at information sets must be the limit of the Bayesian beliefs generated by completely mixed behavior strategy profiles converging to the equilibrium strategy profile. We thank an editor for drawing our attention to this fact.
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In this section, we define a PRNF solution concept, normal form sequential equilibrium, similar in spirit to sequential equilibrium. We begin by using limits of completely mixed strategies to generate beliefs over normal form information sets. DEFINFTION 10: For a completely mixed strategy profile a** and XQS, define (T^{'\X) by aKs\X)^(T\s)/{Z,^x<^\t))ior s^X, and 0 elsewhere. For a sequence [a^} of completely mixed strategy profiles, define ai'\X)^ \\mj^_^^&^{' \X) when this is defined. If [cr^} has the property that cr(- \X) is defined for all XQSy then it is termed conditionally convergent, \i X^ X^ X j ^ _ . , then C7i('|v^) and o-_,.(-|Z) are defined by ai{5,-|Z) = E , ^ ^ a-((5,.,^_,)|Z), 2inda_,{sjX)=^Z,^^xa{{t,,s_-)\X\
It is clear from the independence of mixed strategies and the definitions that a,{s,\X)=^\imj,^^cTt{s,)/{Z,^^x,(^t^s,)) for 5, e j^. and a_,{s_,\X)^ lim^_^^ a^iis_i)/{Y.t_^^x_,^t^^-i)^ ^^^ ^_, e Z _ „ so that in particular, a^iSilX) is independent of X_^, cr_^(^_jZ) is independent of X^y and a{s\X) = (T,{S,\X)CT_,{SJX). DEFINITION 11: The strategy profile a- is a normal form sequential equilibrium if there exists a conditionally convergent sequence cr* of completely mked strategy profiles with a^ -^ a such that for any player / and any normal form information set X for player /, a;(*|^) is a best response on X^ to a_^{'\X), i.e.. Si e supp(a-,(• \X)) => ir.is,, cr.,( • | ^ ) ) > ^,(f„ a_,{ • \X)) V/, e X,,
There is no inclusion relationship between normal form sequential equilibria and normal form trembling hand perfection. Recall that normal form trembling hand perfection does not imply subgame perfection in the extensive form, while normal form sequential equilibria does (see below). Furthermore, weakly dominated strategies can be played in normal form sequential equilibria, but not in trembling hand perfect equilibria. Kreps and Wilson (1982) and Kreps and Ramey (1987) note that the notion of a sequential equilibrium might be reformulated with a number of alternative consistency requirements. Our concept of normal form sequential equilibrium, in imposing consistency of beliefs, inherits the difficulties which arise in connection with the latter in the extensive form. Consider the PRNF in Figure 14 (this is the PRNF of the first game in Kreps and Ramey (1987) with the payoff to (L, L, L) changed). This game has a unique Nash equilibrium outcome, given by {R, R, aL + {l—a)R), where a e [ | , | ] . Player III has a normal form information set [LL, LR, RL] x[L,R} (where the first set gives strategy pairs for players I and II). Now, any conditionally convergent sequence in which the play of I and II tends to RR must in its projection onto Ill's normal form information set place
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III
L L
L
295
R
-2,-2,-2] 3,0,0 1
R 0,0,1 1 2,2,0 II III
R
L
L
R
-1,-1,-11 0,0,1 1
R 0,3,0 1 2,2,0 FIGURE 14
no weight on LL, Further, the only such projection supporting Ill's choice of a G [}, | ] must put equal probability on LR and RL but no probability on LL. No pair of strategies for I and II can produce such an outcome. This is the normal form analogue of the failure of structural consistency noted by Kreps and Ramey. Normal form sequentiality thus captures the disadvantages as well as advantages of its extensive form counterpart. THEOREM 6: A proper equilibrium of a PRNF game (S,7r) is a normal form sequential equilibrium of (5, TT). PROOF: Take a sequence justifying the proper equilibrium. By repeatedly taking convergent subsequences, we obtain a sequence cr^ that is conditionally convergent. Let X he a normal form information set for player /. We need to show that a^('\X) is a best response to cr_,(-1^^"). So, suppose not. Then, there exists Sf e supp (cr,(- IX)) and t^ e X^ such that 7r,(^„ cr_,(- \X)) > 7ri(s^,(T_.('\X)X and therefore such that Tr,(/,-,cr^.(-|X))>'jrX5,.,c7-^,(-l^)) for all k sufficiently large. But, as ^ is a normal form information set, there is a strategy r^^X^ that agrees with t^ on X_^, and with s^ on S_i\X_i, and therefore, 7ri(r-yO-/^)> Tr^is^^a/') for all k sufficiently large. But, by the definition of a proper equilibrium, o'/^(s^)/a/'(ri) -> 0, which contradicts a^is^lX) > 0. Q,E,D.
It is clear that normal form sequential equilibria are normal form subgame perfect (and, in fact, satisfy the stronger consistency criterion discussed near the end of the previous section). Since every finite normal form game has a proper equilibrium, we thus have the following corollary. COROLLARY: Every PRNF has a normal form sequential equilibrium and so a normal form subgame perfect equilibrium.
It is straightforward to see that Theorem 6 would remain valid if the requirement of Definition 11 were strengthened to require optimal play on every strategic independence.
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The major theorem of this section shows that a normal form sequential equilibrium induces a sequential equilibrium in every game with that PRNF, even if that game includes moves by nature. Note that the PRNF is not a normal form game with nature (as defined in Section 5). For an information set /?, let Q now be the collection of nodes which neither precede nor follow h. Let a^ specify actions at these nodes. Then a special case of the proof of Theorem 2 gives that S{h, a^) is an information set. We are now in a position to prove Theorem 7. THEOREM 7: A normal form sequential equilibrium of a PRNF induces a sequential equilibrium in every extensive form game with that PRNF, PROOF: Let F have normal form (T, if/X and PRNF (5, ir). Let c be a normal form sequential equilibrium of (5, TT), with associated sequence a^. We extend cr^ to T by dividing o'l'is) over those strategies in 7} which agree with s^. For each element s^ e S,, let Nis^) = {t^ e 7}: t^ e 5,}. Then, for all t^ e Nis^), define Vi(h^ = o'/^(s^)/#NiSi). Because or* is completely mixed and ratio convergent, so is 7]^, Similarly, define rJ^(t^}=^o•^(s•)/#N(Si), We first generate consistent beliefs. For y^ any normal form mixed strategy, h an information set belonging to / such that yilT^ih)] =5^ 0, and a an action at /z, let
y,[[t,^T,ih)\t,(h)=a}]
'^^'^^—mm • Setting g^ equal to an arbitrary completely mixed distribution over actions at other information sets yields a behavior strategy for /. By Kuhn's Theorem (1953), gi is realization equivalent to %. For each 77*, let b^ be the behavior strategy generated in this way. Because 77* is completely mixed, so is b^. For h an information set, and x a node of h, let /x*(x) be the conditional probability under b^ of reaching x given h is reached. Set 6 = limj^._^^^* and /z = lim^^-^^jLt^ Because {77*} is conditionally convergent, these are both well defined. Since /i* is derived by Bayes' rule from Z?*, {b,ix) is an assessment. Pick an arbitrary player / and an information set h for /. For d any behavior strategy profile, define P^''^{z\h) to be the probability that z is reached given that play starts ^i x^h with probability fiix). We now show that b is sequentially rational given ft, i.e., for all h^ the problem (*)
max Y.
P'"'^^''iz\h)Ui{z)
has b^ as a solution, where u^(z) is player /*s payoff at terminal node z and ^ \ c ^ = (^i,...,6,_j,c„^,^i,...,^„). Let bi be a behavior strategy realization equivalent to 7/^(- iTih)), For any c,, let c, be the behavior strategy which agrees
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with c^ at and beyond h, and with b^ at all other information sets. As /*s strategy affects (*) only in what it specifies at or beyond h, restricting our search in (*) to c, does not affect the maximization. Now, the argument of (*) is the limit of
(**)
E^'^''^'^''(^l^)"/(^)-
For each k, b^\Cf reaches h with positive probability. By perfect recall, JJ!^ can be taken as generated by Bayes' rule from b^Xc^, (For this calculation, only what jjJ^ specifies on h matters. By perfect recall, this is independent of fs strategy.) Thus (* *) is equal to
z>-h
By perfect recall, the denominator is independent of c-, so henceforth we ignore it. Now, for each c„ it is easily verified that there exists a mixed strategy % realization equivalent to c, and such that yi(T(hya^y) = rif(T(h,a^)\T(h)) Va^ ^A^. As b^\c^ is realization equivalent to (y,, 77^,), (* * *) is proportional to (where w^ is the initial node preceding z and nature chooses w^ with probability p(w^)) Ep(H'jy,(7;(z))T,i,(r_,(z)K(^). Dividing through by r]^_^(T(h)) and taking limits, an equivalent problem to (* ) is
max E ^'
p{w^)ym^))v-i{nz)\T(h))u,iz).
Z^h
As c, varies only after h, we change nothing by letting the sum range over all z. But, then, an equivalent problem is max^,(r,,77-.(-in^)))As the weight y, puts on 7}(/i, a^) for each a^ is independent of c,, it is enough to show that for each a^ such that this weight is positive, y.(') = rj^i- iTih)) is maximal on T^ih^a^) given T ? . / * \T{h)), But, when this weight is positive, the restriction of 77/• \T{h)) to T^(h,a^) is just a scaling of Q7^(- |7(A,a^)). The proof is completed once we show that rf^i- \T(h)) is optimal. It is enough to show that the optimahty of cr.(-|5(/i,fl^)) against Gr_i(^\S(h)) implies the optimality of T]i('\T(h,a^)) against rf_^('\T(h)X To prove this, let r,,^,^ Ti(h,a^X ri,Si^Si(h,a^X n^NirX and f-cM^,). For subsets of 5„ define /(;^.) = [ 5 . e 5 _ , : 7r(',r_-) constant on X^}, Also denote by I the similar
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function defined on subsets of 7]. Then,
L
V->{L,\T_,ih))[^,{fJ_,)-4>,{Sj_,)]
'-,Gr_,V(7-,{A,aO))
E
E
E
t,_,(f_,ir_,(/«))
^-,('-,l5-,(/'))[l^,('-„^-i)-TT,(5,.,(_,)]
(_,e5_,('i)V(5,(/i,a2))
= ^,('-,>'^-,(-15(/«)))-TT,(^,,cr_,( • |5(/i))) and we are done.
Q.E.D.
We have the following version of a converse to this theorem. We will say that a conditionally convergent sequence a^ on the PRNF (5, TT), converging to a, induces a sequential equilibrium on an extensive form game F with PRNF (5,7r) if the assessment (b,fi) constructed from cr^ by the method used in the proof of Theorem 7 yields a sequential equilibrium of F, Then Theorem 1 can be used to show the following. THEOREM 8: Let a be a normal form strategy profile and cr^ a conditionally convergent sequence converging to a in the PRNF (5, TT). Let cr^ and a induce a sequential equilibrium on every extensive form F with PRNF (5, IT). Then a is a normal form sequential equilibrium supported by the sequence a^.
Figure 11 provides a counterexample to the conjecture that if a corresponds to a sequential equilibrium outcome in every extensive form with a given PRNF, then o- is a normal form sequential equilibrium in that PRNF. It is difficult to extend the equivalences we have found between solution concepts in the PRNF and the extensive form to the MRNF. To see why, consider the extensive form game in Figure 15, along with its PRNF in Figure 16. The unique sequential equilibrium of this game has I choosing B, and II choosing A, then R. This corresponds to (B,A) in the PRNF, which is the
c r2,i 0,0 3,3 0,1 2,0
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II
T B
A 6,6 6,6
L
C
2,1 0,1
0,0 2,0
FIGURE
R 3,3 4,3
16
unique normal form sequential equilibrium. Now, the strategy R for II is just a 50/50 mix of A and C. If we prune R from both the extensive and normal forms, then the unique sequential equilibrium of the new game has I choosing T, and II choosing A, then L. This corresponds to {T,A) in the new PRNF, which is the unique normal form sequential equilibrium of that game. As the new PRNF is also the MRNF of the original game, this implies that there cannot exist a single strategy profile in the MRNF consistent with a sequential equilibrium in every extensive form game with that MRNF. However, the normal form subgames in the two FRNFs do generate the same payoffs and outcome in the extensive form. We conjecture that every MRNF has a normal form sequential equilibrium with payojfs consistent with a sequential equilibrium in every extensive form with that MRNF, but have been unable to prove this.
9. FORWARD INDUCTION
We now show that extensive-form forward induction notions can be captured in the normal form. Several formulations of forward induction have appeared. One of the most common, especially in signaling games, is the requirement that an equilibrium survive the elimination of never-a-weak-best-response strategies. This is a normal-form property, while our interest is in showing that extensive form forward induction requirements can be captured directly in the normal form. Kohlberg and Mertens (1986, p. 1008) first motivate forward induction in the extensive form game given by Figure 17 below, where forward induction is equivalent to the combination of requiring invariance to coalescence of exten-
FlGURE 17
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George J. Mailath, Lariy Samuelson, and Jeroen M. Swinkels 300
G. J. MAILATH, L. SAMUELSON, AND J. M. SWINKELS II L
R
TT 2,2 2,2
I M 3,3 0,0 j
B[ 0,0
1,1
FIGURE 18
sive-form moves and invariance to the elimination of dominated strategies.^^ We will confine our remarks to showing that the forward induction argument in this game is readily captured in the normal form. Kohlberg and Mertens (1986, p. 1013) argue that the equilibrium in which player I plays T and player II plays R fails forward induction because " . . . it is common knowledge that, when player II has to play in the subgame, implicit preplay communication (for the subgame) has effectively ended with the following message from player I to player II: 'Look, I had the opportunity to get 2 for sure, and nevertheless I decided to play in this subgame, and my move is already made. And we both know that you can no longer talk to me, because we are in the game, and my move is made. So think now well, and make your decision.'" Player II is then to realize that player I would have forsaken the payoff of 2 only in the expectation of a payoff of 3, indicating that player I must have played M and prompting player II to play L. In light of this, T is an inferior choice for player I, disrupting the "equilibrium" ( r , R), This reasoning is explicitly extensive form and seems to rely on the fact that player I can present player II with a fait accompli. However, we can use the notion of a normal form subgame to conduct this type of argument in the normal form. The PRNF of the game is given in Figure 18. Under the profile ( r , /?), player IFs choice is irrelevant. In the spirit of our earlier discussion, in evaluating the choice between L and i?, player II can observe that this choice matters only if player I, rather than choosing T, chooses a strategy in [M, B), But surely the only reason player I would choose in this set is if he expects to receive more than 2, which only occurs in the strict Nash equilibrium (M, L) of the normal form subgame {M, ^} X ^2. Thus player II should choose L, since // player I chooses to play in {M, B], he will choose M. It is important to observe that this reasoning was ex ante. There is no necessity to present player II with a fait accompli. Thus, while forward induction was originally motivated ir^ the extensive form, it can be motivated naturally in the PRNF. Further, this reasoning does not rely on dominance. It can be shown that replacing the outside option T with a constant sum game with unique equilibrium payoffs (2,2) does not alter in any way the logic of the above argument.
van Damme (1989) provides a more elaborate extensive-form notion of forward induction based upon similar intuition.
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10. CONCLUSION
In closing, we suggest that while much of the power of extensive form reasoning survives in the normal form, its problems do as well. It has been argued (by, among others, Rosenthal (1981), Binmore (1987, 1988), and Reny (1985)) that backwards induction is flawed because it requires players to believe in the rationality of an opponent in the presence of evidence to the contrary. The type of normal form reasoning discussed in this paper has similar problems. Players are not asked to make decisions in the face of evidence that their opponents are irrational, but they are asked to make decisions as if such evidence existed. For example, suppose X is a normal form subgame of the two player game (S, ir), cr is a Nash equilibrium of (S, TT), and player IFs equilibrium strategy projects onto X while player Vs equilibrium strategy does not (i.e., a^iX^) = 0 and a-^X'^ ^ 0). Suppose further that all of player Fs strategies in X^ are strictly dominated. Our concept of normal form subgame perfection requires that player IPs equilibrium strategy project onto a Nash equilibrium of (Jf,7r), because player II reasons that if player I were to choose from X^, then I would choose an equilibrium strategy in X^ But, given that any action from ^1 is irrational, why should player II be confident that player I, if choosing from ^ 1 , would play according to any rationality standard, let alone play his part of a Nash equilibrium of (^,7r)? Once again, the vices as well as the virtues of extensive form reasoning appear in the normal form. DepL of Economics, 3718 Locust Walk, University of Pennsylvania, Philadelphia, PA 19104-6297, USA,, Dept, of Economics, 1180 Observatory Dn, University of Wisconsin, Madison, WI53706, USA,, and Dept, of Economics, Stanford University, Stanford, CA 94305, USA, Manuscript received January, 1990;finalrevision received July, 1992.
REFERENCES ABREU, D., AND D . PEARCE (1984): "On the Inconsistency of Certain Axioms on Solution Concepts for Non-cooperative Games," Journal of Economic Theory, 34, 169-174. BINMORE, K. (1987): "Modeling Rational Players: Part I," Economics and Philosophy, 3, 179-214. (1988): "Modeling Rational Players: Part II," Economics and Philosophy, 4, 9-55. DALKEY, N . (1953): "Equivalence of Information Patterns and Essentially Determinate Games," in Contributions to the Theory of Games, Volume 11. Annals of Mathematical Studies 28, ed. by H. W. Kuhn and A. W. Tucker. Princeton NJ: Princeton University Press. ELMES, S., AND P . RENY (1989): "On the Equivalence of Extensive Form Games," mimeo, Columbia University and the University of Western Ontario. FuDENBERG, D., AND D. LEVINE (1991): "Self-Confirming Equilibrium," Department of Economics Working Paper 581, MIT. HARSANYI, J., AND R. SELTEN (1988): A General Theory of Equilibrium Selection in Games. Cambridge, MA: MIT Press. KALAI, E., AND E . LEHRER (1991): "Private-Beliefs Equilibrium," Center for Mathematical Studies in Economics and Management Science Discussion Paper 926, Northwestern University.
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George J. Mailath, Lariy Samuelson, and Jeroen M. Swinkels 302
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KoHLBERG, E., AND J.-F. MERTENS (1986): "On the Strategic Stability of Equilibria," Econometrica^ 54, 1003-1037. KREPS, D . , AND R . WILSON (1982): "Sequential Equilibria/' Econometrica, 50, 863-894. KREPS, D., AND G . RAMEY (1987): "Structural Consistency, Consistency, and Sequential Rationality," Econometrica y 55, 1331-1348. KuHN, H. W. (1953): "Extensive Games and the Problem of Information," in Contributions to the Theory of Games, Volume II. Annals of Mathematical Studies 28, ed. by H. W. Kuhn and A. W. Tucker. Princeton NJ: Princeton University Press. MAILATH, G . J., L. SAMUELSON, AND J. M. SWINKELS (1990): "Extensive Form Reasoning in Normal Form Games," CARESS Working Paper #90-01, University of Pennsylvania. (1992): "Normal Form Structures in Extensive Form Games," CARESS Working Paper #92-01, University of Pennsylvania. MYERSON, R . B . (1978): "Refinements of the Nash Equilibrium Concept," International Journal of Game Theory, 7, 73-80. RENY, P . (1985): "Rationality, Common Knowledge and the Theory of Games," mimeo. University of Western Ontario. ROSENTHAL, R . (1981): "Games of Perfect Information, Predatory Pricing and the Chain-Store Paradox," Journal of Economic Theory^ 25, 92-100. RUBINSTEIN, A. (1982): "Perfect Equilibrium in a Bargaining Model," Econometrica, 50, 97-109. SELTEN, R . (1975): "Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games," International Journal of Game Theory, 4, 22-55. SWINKELS, J. M. (1989): "Subgames and the Reduced Normal Form," ERP Research Memorandum No. 344, Princeton University. THOMPSON, F . B. (1952): "Equivalence of Games in Extensive Form," Research Memorandum 759, The Rand Corporation. VAN DAMME, E . (1984): "A Relation between Perfect Equilibria in Extensive Form Games and Proper Equilibria in Normal Form Games," International Journal of Game Theory, 13, 1-13. (1987): Stability and Perfection of Nash Equilibria. Berlin: Springer Verlag. (1989): "Stable Equilibria and Forward Induction," Journal of Economic Theory, 48, 476-496.
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20 James Bergin on Hugo E Sonnenschein
I ended up in academe, working in the field of economic theory by accident. I came to Princeton from LSE where I had obtained an MSc, with primary focus on Econometrics. While at LSE I took one course in theory -with Douglas Gale who was then starting his academic career. I had not intended continuing study further than that, but a friend in London recommended pursuing a PhD: I asked Steve Nickell at LSE for some advice. He had just returned from a visit to Princeton and suggested applying there. I applied and was delighted to be accepted. At Princeton, continuing in econometrics was a possibility, but, as this volume testifies, there was great energy and enthusiasm surrounding theory at the time to the extent that students organized an independent theory seminar at one point. I was caught up in that enthusiasm, but after completing the comprehensive exams, was short of a thesis topic. Around that time I encountered the ACDA (Arms Control and Disarmament Agency) papers on games of strategy with as3aiimetric information and decided to pursue research in that area. I'm not sure of the practical relevance of that literature, at least in terms of clear or direct application. But, there is no doubt as to the elegance and beauty of those papers - hardly a surprising observation given that the group of authors included Aumann, Nash, Selten, and many other legendary names. With a project under way, I "signed up" Hugo and Joe (Stiglitz) as joint supervisors, and worked for Joe as a research assistant. Because the work was on a tightly articulated model, there was little room for reflection on issues of modeling: the focus was on developing results within the existing framework. This has advantages and disadvantages: the direction is relatively clear, which is good, but modeUngflexibilityis limited and this restricts the scope for maneuver in obtaining results. For me, the presence of Hugo and Joe electrified the academic life of the would-be theorist on campus, attracting enthusiastic students from all over, and creating an environment of learning and academic excellence second to none. It was my good fortune to arrive there at the time. It was a wonderful experience - thanks to Princeton for making it possible, and to Hugo, Joe and many others, such as Ed. Mills, for emiching the experience during my time there. The paper selected, a chapter from my dissertation, addresses two questions: the structure of behavior in dynamic models, and the strategic use and revelation of information over time. The paper considers these two issues in the context of infinitely repeated games of complete information. These are briefly discussed in
481
20 James Bergin on Hugo E Sonnenschein turn in the next two paragraphs. In multi-period models, the study of behavior is greatly simplified when the actions can be conditioned on a suitable summary statistic of the past, rather than on the entire history. In dynamic programming, this is both natural and taken for granted: a history of investment can be summarized by current capital stock or asset portfolio, optimal depletion of a nomenewable resource depends not on the history of depletion, but on the remaining stock. And so on. However, in problems with more than one decision maker where other strategic issues arise, the availability and sufficiency of a summary variable is less clear. In stochastic games withfiniteaction and state spaces an old result confirms that there are equilibria with actions at any point in time depending only on the current state, previous actions and states don't affect current behavior. However, this result doesn't extend in general where the state space is continuous. In the incomplete information game, the natural state variable is the player type distribution, which belongs in a continuous state space. With this in mind, the paper shows that for zero sum games (where players interests are opposed), if the value function is differentiable, then the posterior distribution is a sufficient statistic for the history of behavior, and equilibrium actions may be conditioned solely on the posterior distribution. The second part of the paper considered the nature of information revelation over time in this class of game. The earUer Uterature had focused on games where the payoff flow was averaged over time, and in that context, a conmion pattern of play for an informed player is to strategically reveal some information initially, dispersing the posterior distribution relative to the prior, and then subsequently make no further use of information. With discounting, this is generally not an optimal strategy: for many games optimal behavior requires the strategic use (and hence revelation) of information over time, so that it takes infinitely many periods for information to be fully revealed. In the discounted case, it turns out that on the space of distributions over characteristics, computation of the gain for use of information is a larger order of magnitude that the resulting loss from facing a now better informed opponent. This leads to slow revelation of information over time in many games, with the informed player exploiting information period by period. I will end with a poem, "Pangur Ban", due to an 8th or 9th century anonymous Irish academic which describes the research process by analogy with the task of a cat, Pangur Ban, (for the record, the stroke called a "fada" in Irish Gaelic, lengthens the sound and Ban is pronounced "Bawn"). I think it captures many of the important features of the research endeavor. Tlie translation is by Robin Flower and is taken from 1000 Years of Irish Poetry, ed. by Kathleen Hoagland.
482
20 James Bergin on Hugo F. Sonnenschein PANGUR BAN
I and Pangur Ban my cat, Tis a like task we are at: Hunting mice is his delight, Hunting words I sit all night.
'Gainst the wall he sets his eye Full and fierce and sharp and sly; 'Gainst the wall of knowledge I All my Uttle wisdom try.
Better far than praise of men 'Tis to sit with book and pen; Pangur bears me no ill-will. He too plies his simple skill.
When a mouse darts from its den, O how glad is Pangur then! 0 what gladness do I prove When I solve the doubts I love!
'Tis a merry task to see At our tasks how glad are we, When at home we sit and find Entertainment to our mind.
So in peace our task we ply, Pangur Ban, my cat, and I; In our arts wefindour bliss, 1 have mine and he has his.
Oftentimes a mouse will stray In the hero Pangur's way; Oftentimes my keen thought set Takes a meaning in its net.
Practice every day has made Pangur perfect in his trade; I get wisdom day and night Turning darkness into light.
483
Player Type Distributions as State Variables and Information Revelation MATHEMATICS OF OPERATIONS RESEARCH Vol. 17. No. 3. August 1992 Printed in U.S.A.
PLAYER TYPE DISTRIBUTIONS AS STATE VARIABLES A N D INFORMATION REVELATION IN Z E R O SUM R E P E A T E D GAMES WITH DISCOUNTING * JAMES BERGIN This paper examines the role of the player tyi>e distributions in repeated zero sum games of incomplete information with discounting of payoffs. In particular the strategic "sufficiency" of the posterior distributions for histories and the limiting properties of the posterior sequence are discussed. It is shown that differentiability of the value function is sufficient to allow the posteriors to serve as "state" variables for histories. The limiting properties of the posterior distributions are considered and a characterization given of the set of possible limit points of the posterior distribution. This characterization is given in terms of the "value" of information in the one-stage game.
L Introduction. This paper focuses on two aspects of repeated incomplete information zero sum games: the role of the posterior distribution as a state variable and the extent to which information is revealed in equilibrium. The potential role of the posterior distribution as a state variable can be motivated by the fact that, in this class of game, the value function satisfies a recursive equation. For discounted games the recursion is, vj,p,q)
= min max{(l - 6)Ep*(?'';c*^^'-y^ + 6 E ^ / y ^ U p ( ^ . 0 . ^ ( y ' » ) } y
Jf
and for finitely repeated games with summation of payoffs, y
X
(The notation is explained in §2, where these functions are discussed.) One might conjecture from these equations that how players play in successive periods depends only on the past as it aifects the posterior distributions—or that one could find equilibria where this is true. For finitely repeated games at least, this is not the case: there are examples of games where a player cannot guarantee the value of the game with a strategy that depends only on histories through the posterior distributions.* It is of interest to investigate the circumstances under which the posterior distributions are "sufficient'* for histories. This question is developed in §3, where it is shown that, 'The following example (due to J. F. Mertens) illustrates the i[)oint. The game is a game of one-sided information repeated twice. The stage games are:
^• = |o -\\^'-Vl
2I
and the prior distribution is /? = | . This game has a unique optimal strategy for player I, the informed player and maximizer. Play type independent (5, | ) in the first period and play the type dominant strategy
•Received June 15, 1988; revised December 5, 1990. AMS 1980 subject classification. Primary: 90D05. lAOR 1973 subject classification. Main: Games. OR/MS Index 1978 subject classification. Primary: 239 Games/Stochastic. Key words. Zero-sum games, incomplete information, discounted payoffs. 640
485
James Bergin PLAYER TYPE DISTRIBUTIDNS IN ZERO SUM REPEATED GAMES
641
for the infinitely repeated game, posteriors are "sufficient" for histories when v„ is differentiable (see Theorem 2 below for a precise statement of the result). This resuh may be used as follows. If one solves the recursive equation for v^ the problem of describing the optimal strategies still remains. At this point if v^ is differentiable it is sufficient to find the first-period component of the strategies at each (/?,
1 0
0] Oj'
2^[0 [O
0 1
In this game, with the limit of means payoff criterion, for any prior distribution /?, p^= p almost surely in any equilibrium. With discounting of payoffs the posterior p^ is in the set {0,1) with probability 1 in any equilibrium. The explanation is as follows. At any prior distribution /?, the informed player can guarantee an expected payoff of u{p) = p{\ - p) by playing a type independent, nonrevealing strategy.^ However to guarantee a higher payoff at any stage requires that he use a type dependent strategy. But, given any strategy a for player I which reveals information, player II can find a strategy which exhausts almost all the information in a within some fixed length of time N (say). Then, given any posterior, p^^ (depending on the history to A^), II can in each subsequent period play the stage game strategies which are optimal in the zero sum game with payoff matrix Aip^^) = p^^jA^ + (1 — p^)A^: with the limit of means criterion this achieves over the entire game, a payoff roughly equal to E^ (uipi^)}.
in the second and last period—i.e. top if type 1, bottom if type 2. For player 11 any optimal strategy requires playing (|, 5) in the first period. In the second period there are four possible histories /i, = (1,1), /12 = (2,1), k^ = (1,2), h^ = (2,2), where (/,/) is a history with 1 = 1 denoting top for I and / = 1 denoting left for 11. Let y, = yihj) denote the probability that II plays left given history hj. Then any optimal strategy requires that ^i + ^3 = 5, and yj + y* — 3/4. Thus, although the posterior equals 5 at all four histories, the uninformed player cannot play the same following each history. ^Throughout the paper no attempt is made to distinguish between type independent and nonrevealing strategies. It serves no purpose here. With a nonrevealing strategy, the prior and posterior coincide with probability 1. A type independent strategy requires that each player type plays the same strategy. A nonrevealing strategy may not be type independent at extreme points of the set of player type distributions (see Kohlberg 1975b for a discussion of this.matter). The function u(p) is defined below—in §2.
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Player Type Distributions as State Variables and Information Revelation 642
JAMES BERGIN
Since u is strictly concave, for n sufficiently large E^^pMp^)} < u(p). This means that a strategy which reveals no information guarantees a higher payoff to the informed player. The key point is that, with the limit of means criterion, gains from information usage must be sustainable indefinitely. With discounting, since the distant future is unimportant, current gains from information usage may more than offset losses arising from the fact that the opponent is "better informed" in future periods. 2. Description of the game. A matrix is selected from the set [A^''\k e iC, r G /?) according to the independent prior distributions p = ( p \ . . . , p ^ ) e A'^ and q = ( ^ ^ . . . , (y'^) e A^ (A'" denotes the simplex of dimension m - 1, and without confusion, K will sometimes denote an integer and sometimes the set {1,2,...,/C}; a similar interpretation applies to R): The matrix A^'' is chosen with probability p'^q''. Player I is informed of the choice of k ^ K and II is informed of the choice of r ^ R. The game is played repeatedly, each player observes the history h^ = ( / j , ; , , . . . , /,_J, y,_ j) before moving at time t. Given that the pair (/c, r) is chosen and history /i = (/,, j j , . . . , i^J,..,) occurs player I receives a*''(/2) from player II, where «*'(/.) = (1 - 5) E 5'-«f;, (where A"^ = (fl^'U,.;,,). Histories are denoted H^ = (lx jy-\ H^ = U X JT and / / ' = n7=;(/ X J) with the convention, H^ = 0 . Thus h,^H^,h ^H and h* = (f,,;,>'/ +i>^"/+i>--) ^ ^'^ Strategies in the game are sequences of functions cr = (jCj, jCj,..., x , , . . . ) and 7" = (y^ y2y'"^yty"-^ ^^^ players I and II respectively, where: x^: H^X K -^ dJ and y^: Hi X R -^ A"'. The sequence of functions (;cf, ^ 2 , . . . Jcf . •.) = o"^, with x^: H^-^ A^, denotes a strategy for player I type k. The strategy r'' for player II type r is defined similarly. Let 3^, be the finite field determined on H^ by //, and set 5^ = V7=i«^. ^t represents the information available to players at time /. Let ^ (g> 2^^^ be the sigma field on H^ X K X R, A pair of strategies (o-, T) and a prior distribution (/?, q) on Kx R determine a probability measure fi^^p,j on 5^ <8> 2^^^, with corresponding expectation operator E^^^^, For fixed /i (= p-^^rpq^ ^^^ "^^V compute the sequence of posterior distributions {Pt^qt)'- P^ = fiik\^f) and q^ = jLi(rl^). Sometimes these will be written p*(/i) or pj^ih^) (with h = (/z,, h)) to denote fxik\S^^Xh\ and similarly with q[. With this notation, the probability of a history hf G //^ is given by
keK,reR
\s=i
The posteriors p/^, ^;^, ^ e A", r e /? are bounded martingales which converge almost surely. These almost sure limits are denoted p^, q^. A strategy pair (&, f) is an equilibrium if for a\\ k ^ K and r e / ? ,
E,iJa''^{h)\k)>supE,,Ja'''(h)\k}
and
The term E^fp^[a'"(.h)\k) will be called the payoff to player I type k and denoted f *.
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James Bergin PLAYER TYPE DISTRIBUTIONS IN ZERO SUM REPEATED GAMES
643
Let a'^'ih) = (1 - 8)ir,^,8'- 'a^^ where h = (/j, j \ , . . . , / , , / „ . . . ) and h, = (/j,/,,...,/,_„/,_i). Then E^j^p^{aY{h)\h^yk} is called the continuation payoff to player I type k given history h^. The discussion in the remainder of the paper is based on three functions, v^, v„ and M, defined A^ X A^ and denoting respectively the value of infinitely repeated game, the value of the n-stage repeated game and u, the value of the one-stage game where both players are restricted to type independent strategies. Let x = (x\x^,...x^X y = (y^y^,...y'^) where x\.,.,x^ G /^^ and y \ . . . , y ^ e A''. Denote by g(x, y) the function Lk^rp'^^'^^^^'^'^y'^ ^^^ J^^ /(^> y) = (1 - 8)g(x^ y). The value of the one-stage game is defined: v^ipyq) = max^ min^ gix, yX For the game repeated n times with payoff in the rth period weighted by (1 - 8)8'~^ the value function t;„, n > 2, satisfies the recursive formula (using an argument given in Sorin 1980): v,(p,q)
= max nnn{{l-8)j:p'Q'^'A"'y
+
8j:x,yjV„_,(p{xJ),q{yJ))).
The posterior distributions are p{x,i) = [p''(x,i)}j^^^ and g(y,y) = {^''(y,y)}/Li, where pKx,i)-= [p^x^/x^ and q'(yyj) = {q^y-Zy) with ^, = L p \ ^ and y^ = Y.q'^yj^ Thus, for example, p^(x, i) is the posterior probability that player I is type k, given the strategy x and the outcome /. The function w, defined by ^{pyQ) = max mini!
^p^q''A^''\y
where i G A^ and y G A^, gives the value of the one-stage game when neither player is allowed to play type dependent. For infinitely repeated games the value function satisfies the recursion (see Theorem 1 below): VXP^Q)
= max minlil y
^
- 8)j:p'q'x'A''y^
+
k,r
8j:x,yMpix,i),q{yJ))\, iJ
f
When R = \, these functions become vj^p), u{p) and vSp)—given by the expressions above with summation only over k and with r superscripts and q variables deleted. The following result will be used extensively throughout the paper. THEOREM 1. The value function, VSPJ QX of ^he infinitely repeated game exists and is continuous in (p,qX It satisfies vSp,q) = \iT[i^^^vJ^Pyq) and is the unique function satisfying the recursion: LUp,q) = max mm{(l
- 8)ZpVx'A''y'
+
8Y.x,yjV^{pixJ),q(y,j))).
PROOF. Consider the recursion determined by the n-stage game: Vn{p,q) = max min{/(j:,y) "t-
8Y,XiyjV„_^{p{xJ),q{yJ))\.
Let p{x) denote the random variable with prob(/7(jc) = p{xj)) = jc, (with a similar definition for q{yX) and let E^y denote the expectation determined by (x, y). With
488
Player Type Distributions as State Variables and Information Revelation 644
JAMES BERGIN
this notation: v„{p,q) = max min{/(x,y) + X
8E,^[u„_^{p{x),q(y))]}.
y
Let ( i , y) and (x, y) be solutions for the n -V \ and n period games, respectively. Then Vn^iiP^Q) - v„iP,q) = { / ( i , y ) +
8E,,[v„(p(x),q{y))]}
-{f{xJ)+dE,,[v„_,(p{x)^g{y))]} <{f{x,y)+8E,,[v„(p(x)^q{y))]} -{f{xj)
^
8E,,[u„_,{p(x)^g{y))]},
(The inequality follows since y may not be optimal in the « + 1 period game and x may not be optimal in the n period game.) Thus, ^n^iiP^Q) - v„{p,q)
-
[v,-^ip(x),q{y))]}.
For each (/?, q)y 3 some ( i , y) such that this inequality holds and so: V(p,^),
v„^^{p,q) -u„{p,q)
^8sup\v„{p,q)
-u„_^{p,q)\.
Similarly, ^(PyQ)^
^n+iiPy^) - Vn(PyQ) > -8sup\v„(p,q)
-
v„_^(p,q)\.
Using the sup norm—if / , g: A'^ X A^ -> R, then p(/, g) = sup|/(/7, q) - g(p, q)\ yields p(y„+i, v„) < 8p{v„,u„_i)y so the minmax operator is a contraction. Now v„ is continuous in (pyq) since it is a finite zero sum game with parameters varying continuously on A^ X A^. Noting that IvJ is a cauchy sequence, it converges uniformly to a continuous function—say vSPy qX Finally, observe that given e > 0,3n such that if ain) is an optimal strategy for player I in the n-period game (n > n) then in the infinitely repeated game the strategy a = iain), x,+j, Jc,^.2> - • •) guarantees v„(py q) — e, where Xf+^ is arbitrary for T ^ 1. (Since payoffs are discounted, when n is large the payoff over the remainder of the game—from n on—is small (less than e).) Thus player I can guarantee a payoff in the infinitely repeated game of at least vjipy q) — €. Similarly, player II can guarantee a payoff no larger than v„(py q) + e for n sufficiently large. Since [vj converges to i;«„ in the limit both players can guarantee uSpy q) and so ^ J p , ^) is the value of the infiriitely repeated game. QED It should be pointed out that when the limit of means payoff criterion is used, vjip,q) still converges-^even when the infinitely repeated game has no value. This is discussed in Mertens and Zamir (1971). 3. Posteriors as state variables. This section shows that when the value function is differentiable everywhere in the simplex of player types, then the posterior distribution is a sufficient statistic for histories: there exist equilibria such that if at two different histories the posterior distributions coincide then the strategies of all player types are equal at both histories. In the following discussion iPtih^X QM^))
489
James Bergin PLAYER TYPE DISTRIBUTIONS IN ZERO SUM REPEATED GAMES
645
denote posterior distributions following the history /i, (the determining strategies are not made explicit). THEOREM 2. Let VSP,Q) be the value function of an infinitely repeated incomplete information zero sum game with discounted payoffs and player type distributions given by ipy q\ If VSP,Q) is everywhere differentiable on A^ X A^ (the simplexes of player typesX then the posterior distributions (/?,, q^) are sufficient state variables to determine the strategies of both players. That is, there exists an equilibrium such that if at h, and h^, (p,(h,),q,(h,)) = ipMrXQM,)X then x^(h,) - xi(h,X "ik^K and y;(h,) = y;(/i,), Vr G R, PROOF. Denote by ^ e IR^ and ^ e R^ respectively, vector payoffs to players I and II. Let a(p, q)bc the equilibrium correspondence from player type distributions to vector payoffs (i.e. (f, O G aip, q) implies that there exists an equilibrium strategy pair (a, T), such that in this equilibrium the expected payoff to player I type k (player II type r) is f * (C^))- The important point in the proof is to show that on (A^ X A^T (the notation X° means the interior of XX oc(p,q) is a function—i.e. f and ^ are uniquely determined. Thus, if two different histories lead to the same posteriors in (A^ X A^)° then the expected payoff to each player type following each of these two histories (the continuation payoff vectors) must be the same. This essentially allows the players to "play the same" at histories h^yh^ where the posteriors are the same: ip,{hX qf^hj) = ip^ih^ q^{h^)X The only problem arises on the boundary of (A^ X A'^) where differentiability of y„ does not imply uniqueness of the vector payoffs To see that a is a function on (A'^ X A^)°, take ^ e A^, /? e (A^)° and (f,^) e (x(py qX The vector f satisfies:^ (i) p • ^ = VSP, q) and (ii) p ' i> VSP, q) Vp e A^. Condition (i) follows directly from (^, C) ^ «(p, qX To check that (ii) holds, suppose not. Then there is some p with p • ^ < vSp, q) and since vS'yq) is continuous there is some p* e (A^)° with p* • f < vjp*, q). In the game with player type distributions (/?, q) pick T, an optimal strategy for II, with (f,^) e « ( p , ^ ) ,
vj,p,q)
= Ei^'supi&,*^-,{a^^(/i)|/c} =
ZP'^'
(where E^k-- is the expectation operator on histories determined by or*, r and q). Thus II has a strategy bounding above (coordinate-wise) by ^, the vector payoff to I. Therefore if II uses the strategy r in the game with priors (p*,^), he can hold the expected payoff to I to p* • f. This contradicts the assumption that vSp*y q) > P* ' ^ and so (ii) is verified. Therefore, with q fixed, f defines a hyperplane tangent to vS'yq) at p. Since vS',q) is differentiable, f is uniquely determined. A similar argument shows that C is also uniquely determined for all (p, ^) e (A^ X A^)°. For any (p, q) on the boundary of (A^ X A^) there is a unique pair (^, f) satisfying lim a{p„, q„) = (f, () for all (p„, q„) G (A^ X A^)°, with (p„, q„) -^ (p, q) (since the one-sided derivatives are assumed to exist). Thus, a can be extended to the boundary of (A'^ X A^) as follows: define the function a by a{p,q) = a(p, q) on (A^ X A^)° and extend a to (A^ X A^) uniquely by setting a(p, ^) = lim a(p„,^„) where ^Pn^qn^ ~^ ^PyQ^y ^^^ ^^^Y (p, ^) lu (A^ X A^). Siucc the correspondence from priors to equilibrium strategies is upper hemicontinuous, the function a associates to any pair of priors (p, q) an equilibrium vector payoff (f, 0 (i.e. (f, O ^ ctipy q)X In one-sided infonnation games with the h'mit of means payoff criterion, this would follow directly from Blackwell approachability. See Hart (1985). With discounting, approachability theory does not apply. Also, the game may not have a value in two-sided information case with the limit of means criterion.
490
Player Type Distributions as State Variables and Information Revelation 646
JAMES BERGIN
The function a may be used to generate an equilibrium of the infinitely repeated game with priors (/?, e„ and yij > ^/i» ^^jfjyj' (There are no restrictions on strategies for the second or later periods.) This game has an equilibrium. Given some equilibrium of this game, denote the first-period vector payoffs (f |„, f,„), and second-period continuation vector payoffs following history (/,/): (^2n(^/X^2«(^/))- To make explicit the dependence of (fin^fin) on (p„,q„ye„) write (fi(/7„,
lim ( f i( P „ , q„y€„),Cl(
Pn^ ^n^^n)) = « ( P> Q)-
Therefore, take a subsequence e^(„) of e„ such that j^^^^{il(Pn^Qny^m(n))^Cl{PnyQny^m(n)))
= ^{ p, q) = ( f i , ^ i ) »
Say.
Take convergent (sub)sequences in second-period variables so that {P2n{hJ).Q2n{^J)^^2n{iJ)^C2n{hj))
With this construction (f2(^^X^2('>i)) ^ ^^^2^^ A^2('>/)) ^ ' » ^ Finally, define Xu^p, q) and yfy(/?, ^) as the limits of the first-period component of the equilibrium strategies: A:f,(n) and y[y(n) (where jcf,(w) and y\f.n) are greater than or equal to e^(„)). Note that with this construction, if for some (/,;), iP2(U)XQ2^hjy) = (/?>A^2(^/)) = (fi>fi)Using this procedure functions xf,(/7, q) and yIy(/7, q) can be defined on (A^ X A^), V/, 7, k, r. These functions may in turn be used to define an equilibrium satisfying the conditions of the theorem. If the priors are (p,q), let first period strategies and continuation payoffs be determined exactly as above. If history (/i,7i) occurs leading to posteriors (/?2^^i»/i)>/i)X the continuation payoffs are uniquely determined as a(i72^/i,7iX^2(''i>/i))Define Jc|,(/,,7,) = A:f^(/72('i>yiX ^z^'j^/i)) ^"^ y2/'i>/i) analogously. Treating (Pi^^y JX ^2^'»/)^ ^^^ ^s priors and applying the above procedure again gives posteri-
491
James Bergin PLAYER TYPE DISTRIBUTIONS IN ZERO SUM REPEATED GAMES
647
ors and continuation payoffs: {P3{hJuh^J2)rQ3{hJvhJ2))
and
a{p^{i^J^,i2J2)r^3(h^h^hJ2))-
Proceeding inductively with this procedure, at each history /z, = (/,,;,,/2,y2,--• i,_i,/;_iX the strategies defined there are best responses, given the continuation payoffs. Thus the strategies constructed define an equilibrium of the game. QED 4. Information revelation. Define the set B = {(p,q)\vSpy(]) = u{p,q)). This is the set of player type distributions where the value of the infinitely repeated game is the same as the value of the one-stage game where players are not allowed to use their information. Observe that B contains the extreme points of A^ X A'^. Theorem 3 below asserts that the limits of the posterior distribution must lie in this set almost surely (relative to the equilibrium measure ^i). Since v^ is concave in p and convex in q this immediately excludes from B (almost surely) any point (/?, g) if given q, u is not concave at p and given p, u is not convex at q. However since v^ is in general very difficult to compute, an additional set A defined in terms of one-shot games is introduced (in §4.1 and §4.2) and it is shown that if the prior is in this set, then it cannot be in B, and so the posterior must be in the complement of the set A. Thus the set B is used indirectly to identify the set of possible limit points of the posterior distribution. THEOREM 3. Let (cr, f ) be equilibrium strategies in the game with prior distributions (Py q) (irid let fl = M^fp^ ^^ ^^^ equilibrium measure on J^ ® 2^^^. Then (/7^, q„) e B a.s. jl. PROOF.
The proof is given in four lemmas.
LEMMA 1. Denote the equilibrium correspondence from priors to strategies by (p{p,q). Then ((T,T) ^ ipip,q) and vSp^q) ^ u{p,q) imply E^ {\\p2- p\\-^ 11^2 - ^11} > 0." PROOF. TO see this, note that E^^pJ(\\p2 - p\\ + ll - " > y / > - - ) and y, = (y/, yf^...,y/^)). Take vSp,q) > uip,q) and note Lp^x^A^y, = Lp'^q^x^A'^yi Thus miny{Lp^q'x^A^')y < uip.q), so let y satisfy iLp''q'^x^A'"^)y < uip, q). If player II plays y in period 1, then, on any first-stage history with positive probability the posterior distributions equal the prior distributions and so from then on II can guarantee vSp, ^ ) following that history. Thus, given cr^ an optimal (minimizing) strategy of II guarantees a payoff no larger than (1 d)uip,q} + SuSp,q) < vj,p,q). This contradicts the assumption (cr,T) e cpip^qX Taking vSp, q) < u(p, q) yields a similar contradiction. QED LEMMA 2. Let m{p, q) = inf,,{£^,^^{||/?2 - p\\ + 11^2 " 'jt&Ko', r ) e cp{p, q)). If vSp,q) =5^ uip^qX then for any closed neighbourhood jK(p,q) of {p,q) on which vSPy q) ^ uipy qX for all (p, q) in J^ip, qX inf{w(p, q)\{p, q) e J^{p, q)} -= rn > 0. PROOF. Since vSp^q) ¥^ u(p,q) for all ip.q) ^ ^(p^qX we can partition J^{p,q) into two disjoint sets, ^lip^q) = ^(p,q) n {(p,q)\u„ip,q) > u(p,q)) and ^ ( / 7 , q) = ^(py q) O {(p, q)\vSPy q) < u{p, {p, q) and (p„, q„) e ^{p, q) n {(p, q)\vSp, q) > u{p,q)) then ip,q) is in {{pyq)\vSPyq) > u{p,q)], using continuity of v^ and u. Since ^(p, q) is closed, {p, q) is also in ^(p, qX This then implies that vSp-, q) > u(p,q) and so {p,q) G J^lp.q). Suppose J^^ipyq) ^ 0 . Since E^^pj^\\p2 - p\\ + ^WPi - p\\ = ^ke K^P2 ~ P*f» where Ix| denotes the absolute value of x.
492
Player Type Distributions as State Variables and Information Revelation 648
JAMES BERGIN
11^2 ~ ^IB i^ ^ continuous function of (a-, r) and (p is a closed valued correspondence, the function m(/7, q) is lower-semicontinuous. Therefore mip, q) has a minimum on ^\^P,Q)—say at ip.q) with m(pyq) = mj. If m, = 0 then there exist ((7,f) G (pip^q) and £^^fp^{llp2 "/^H + 11^2 "~ ^Itt "= 0' contradicting Lemma 1, so m, > 0. Similarly, if e/^(/?, 0, on e//^(/7, 0 otherwise one set, ^ip, q) is nonempty and then m = m-> 0. QED LEMMA 3. Ler n be a measure on 5^. TTiere exists a set //(<») c //„, //(c») G 5^ 5Mc/i that ii(H{oo)) = 1 flnt/ for any h e //(oo), where h = (/i„ /lO e H, X //', /x({/zJ
XH')>Oforallt. PROOF. Let //(/) = {(/z,, /lO e //JM({/I,} X / / ' ) > 0}. Clearly, ^iiH(t)) = 1. Note that Hit + 1) c H(t). Define H{^} = 07=1^(0. Since the measure fx is continuous from above, lim,_<« iiiHU)) = ju,(/f(oo)) = L Pick any / and /i = (/i„ /lO e //(oo) and note that since //(oo) c //(/), Tz e //(/) and so niih) X / / ' ) > 0. QED LEMMA 4. Ler {CT.T) e (pip^q} and let jx be the measure on //„ ® 2^^^ determined by ((7, r)flATt/(/?, <7X Then {p^, q^) e 5 a.s. fi. PROOF. The posteriors converge almost surely to {p^,q^ so pick a set H with measure 1 on which the posteriors converge pointwise—i.e., H c //^, yiiH) = 1 and iPtih), q^ih)) -> ipJhX qSh)X Vh e //. Let //(oo) be determined by JLI, as in Lemma 3, set / / * = / / n / / ( o o ) and observe that ^(H*) = \. If for some h^H*, i^'SpSh\ qSh)) > uipjh)yqj,h)) (the following argument can also be applied to the case vSpSh\qSh))
0 0
A^ 1 2 0 0
0 -1
^Eissentially, the continuation game determined by the posterior distributions at h and the branches of the extensive form, from there on. Note that, by restricting attention to //*, we focus only on equilibrium paths. This argument makes use of the special structure of the game: there is full monitoring, so at time / the history up to this lime is fully observed and the player type distributions are independent.
493
James Bergin PLAYER TYPE DISTRIBUTIONS IN ZERO SUM REPEATED GAMES
649
Thus u{p) = - p ( l - p) and Vi(p) = vSp) = 0, Vp e [0,1], Define strategies a and T for players 1 and II respectively as follows. Given a history /i^ = , ; c , , . . . ) b e given by O'l, iv hy Jjy • • •. it-\yj't-lX let (T = (jf,, X2 x,ih,, k) = (1,0) for A: = 1,2 if ;; = 2 for all T ^t - 1. (Both types play L) x,ih,, 1) = (0,1) if ;; = 1 for some T < r - 1. (Type 1 plays 2.) x,{h,y2) = (1,0) if ;V = 1 for some r < r - 1. (Type 2 plays 1.) For T = (yi, y 2 , . . . , y „ . . . ) , y(;i,) = ( 0 , l ) if /^ = 1 for all T < r - 1. Otherwise y(/i^) = ij, | ) . Both strategies guarantee 0—the value of the game. However, for any p e (0,1), E^^p{\p„ - p\) = 0—the posterior limit coincides almost surely with the prior. The following two sections deal with the one- and two-sided information games separately. 4.1.
One-sided information.
In this case take R = 1 and define the set A as:
A = [p\ (i) For every e > 0 and sufficiently small, 3x = {x\ x ^ , . . . , jc'^) such that min^ Lp^x'^A'^y = u(p) + O^ie) and Z^^jllpixJ) - p\\ < O^ie), (ii) M is differentiable relative to the face in which /? lies}. (Here O^ie) means a positive term of the same order of magnitude as e, 0{e) and o(e) mean term of the same and smaller orders, respectively. If p is not an interior point of A'^ then u is diiferentiable when viewed as a function only of those elements of p that are strictly positive. Note that condition (i) implies that u^(p) > u{p). In the context of one-sided information games, the set B becomes: B = {p\vSp) = u{p)}. Denote by X, the complement of a set X. THEOREM 4.
B
QA.
PROOF. It is enough to show that: p ^A =^ vSp) > nip) (so that B QAX Therefore, take p &A and consider the following strategy for / (the informed player). In period 1 play a strategy x which satisfies (i) in the definition of A. If in the second period, the posterior is /?(jc, i) (i.e. / chooses / in the first period) play a type independent strategy guaranteeing uip(x,i)) in that and subsequent periods. This strategy guarantees an expected payoff to player I of (1 - 8)iu(p) + O^ie)] + 8T,i^/X^uip(x, /)). Since u is diiferentiable relative to the face in which p lies and all posteriors derived from p must lie in this face also, expanding u around p for p(x, i) close to p gives u{p{xj))
=u{p)
+ Vu(p)'
[p(x,i)
-p]
+o[\\p{xJ)-p\\],
Thus, since L,e/-^i * ipix, i) - p] = 0, E ^ / w ( p ( x , 0 ) = "(/?) + o\ ll^i\\p{x,i) iei
-p\\ =
u{p)+o{€).
\
Therefore, this strategy guarantees:
u{p) ^ {{I - 8) + Since [(?(€)/0+(6)] -^ 0 as e -> 0 and small the last expression strictly exceeds higher payoff than «(/?), so that ujp) > In view of Theorem 3, Theorem 4 has
494
a[o{e)/0^e)])0^e). O^ie) > 0 for all € > 0, for e sufficiently u(p). Thus I has a strategy guaranteeing a u{p) and therefore B QA. QED the following corollary.
Player Type Distributions as State Variables and Information Revelation 650
JAMES BERG IN
COROLLARY. Let {a, r) be equilibrium strategies in the game with prior distribution p and let jl = Mafp be the equilibrium measure on ^® l'^. Then p„ ^A a.s. p,. Referring back to Example 1, it may be confirmed that A = (0,1) and so p^ ^A = {0,1} almost surely. In that example, fix a discount rate 8 and prior p e (0,1). Then, if in stage 1 the informed player plays f^^^"!^/'] if type 1, [^'"+^'1 *^ ^P^ ^ (e sufficiently small), and then plays an optimal nonrevealing strategy thereafter, this strategy guarantees a payoff of p{l -p)
{2p - i r - 1
+ 1(1 - 8)
[{1 -p)
+ e(2p - l)][p - e{2p - 1)]
For € sufficiently small this is greater than p(l - p). One might conjecture that the set A could more simply be defined as y4 = {p\v^(p) > u(p)}y however this is not the case—as Example 3 below illustrates. The key property of this example is that if the informed player is to achieve a higher payoff than u(p) (the payoff achievable with no information usage by a type independent strategy) he must reveal a "lot" of information: any strategy guaranteeing the informed player a stage game payoff higher than u{p) leads to a posterior distribution close to either 0 or 1. However, in either case the posterior distribution is strategically very unfavourable to the informed player in the sense that vSp) is relatively small for p close to 0 or 1. This leads to the informed player having an optimal nonrevealing strategy in the infinitely repeated game. In this example at some calculations show that y,(|) = 4 and w(|) = 0. However, in the repeated game any strategy for the informed player guaranteeing more than 0 in the first period leads to posteriors (in the second period) which are either greater than 51/55 or less than 4/55. On the set E = [0,4/55] U [51/55,1], v^ip) ^ - 1 and so ujp) < u^(p) < - 1 on this set also. To push the payoff above 0 in the first period, the informed player must "spread" the posteriors into the set E and so obtains a payoff no greater than - 1 in the remainder of the game. When 6 > | (so that the remainder of the game is important) the unique optimal strategy in the first period is a type independent strategy. This implies that vj,^) = 0. Thus, condition (1) in the definition of y4 is a necessary condition. The details are as follows: EXAMPLE 3. In this example there are two player types for player I, with prior probability p that player I is type 1. For the infinitely repeated game, take 8 e (|, 1). A'
A' 10 2 100
-2" -2 -2
-2 -2 -2
-100' 2 10
A{p)=-pA'
+ ( 1 -/7)yi2
' Up - 2
- 1 0 0 + 98p
2(2/7 - 1)
2(1 -
-2-98/7
2p)
10-12/7
Thus, v,{p) = - 2 + 12/7, i;j(/7) = 1 0 - 1 2 p ,
/?< I, p>l
In particular, VjiiO) = y / l ) = - 2 , vj[\) = 4 and L'J(-^) = Vji^) = - 1 . Some calculation yields: «(p) =2(1 - 2p){[Sp - 102{1 - p ) l / [ l 0 2 ( l -p}+Sp]}, u(p) = 2(1 - 2p){ll02p
- 8(1 - p ) l / [ 8 ( l -p)
+ Wlp]}
P < i p>l
495
James Bergin PLAYER TYPE DISTRIBUTIONS IN ZERO SUM REPEATED G A M E S
651
Let p =^ { and recall that vji^) satisfies: u^{^) = max(min(l - d)(\xWy
+ \x^A^y) +
^Y.^,vJ^p{i))\,
where x--=\(x]-\- xf). Therefore, I 1 •
[ ; 4 | ) < max 1(1 -
8)Ux^A
2 1 L 2 .
+ xM^
' 1 1 2 1
+ 5Ev«>(p(0)
. ^ J
Since the centre row of A(\) is (0,0), uS{) > 0. We show that vS\) = 0, and this is achievable only by a type independent strategy: only a type independent strategy ensures that the last expression is nonnegative. Let
2
1 1 1
I
2 I
and note that r
4]
1 • 2 1
L2 .
_
0 -51
r
and
A^
'-511 0 4J
1 • 2 1
L 2
.
in the present example. Expanding the right-hand side of the expression gives: |max{(l - 8)[4x\ - 51JCJ + 4 x | - 51x2] -^8[(x\+x',)v^{p{l)) + ((1 -x\-xi)
+
(xl+xl)v^{p{3))
+ (1 - ^ ]
-X|))^4P(2))]}.
The coefficient on (1 - 5) may also be written: [(4JC{ - 51ji:f) + (4JC| - 51JI:3)]. Consider the following two cases separately: x\-\- x\ = 2 and x\+ x\ < 2. In the first case {x\ + x | = 2), the coefficient on (1 - 5) is 0 and p{i) = | , for all /. In this case then, the strategy of player I with x\-^ xl^^l ensures a payoff of 8vS\)(Note the " | " outside the "max" expression.) Since vS\) ^ 0 , if x\-\- x\ = 2 is optimal then vS\) = 0. In the second case (x\ -^ xj < 2), one of the terms x\, xj, x^ or xj must be strictly positive. The coefficient on d is no greater than vj{) since vj\) ^ vjp), V/?. (This follows from two important facts—v^ is always concave and for a certain class of games, such an Example 3, vSp) = vS^ - p)- See the appendix for some discussion.) Thus in this case, the payoff cannot be higher than in the first case unless the coefficient on (1 - 8) is strictly positive. This implies that at least one of the terms {4x\ - Slx^Xi^xj - 51x3) is strictly positive. First consider the case where one term is positive and one is negative, taking (4x{ - 51x?) > 0 and (4x| - 51xp < 0. Since (4x} - 51xf) > 0 we may write 51
496
xj + 17, with 77, > 0.
Player Type Distributions as State Variables and Information Revelation 652
JAMES BERGIN
Note that
P(l) = [Arl/(^i+^?)] = [l/(l + (^?A!))] > [1/(1 + (4/51))] = (51/55), using the fact that (Arf/jt}) < (4/51). The associated payoff is: ^ { ( 1 - 3)[47,, + {4x1 - 51^1)] + s[(l + ^)xi
+
+
v,]uj,p{l))
S{xl+xl)vJip(3))
+ 5[l - (i + ^ ) ^ 2 _ ^_ + 1 _ ^. _ ^2]„^p(2))}. This may be rewritten as: ^ [ ( 1 - 5)4 + avJip{l))]7j,
+ i ( 4 x | - 51jc^){l - 8)
^ l [ l - (^ + T")^? - ^1 + 1 -
xl-xiyip(2))y
Consider the third term (5{}) in this expression: the coefficient on each vJpiO) (the probability of /) is nonnegative and since ujj) > ujp) and vS\) > 0, the third term is no greater than SuJ^j). Thus the whole expression is no greater than H ( l - 5)4 + 8V4P{1))]7J,
+ | ( 4 x | - 5U])(1 - 5) + 8vJi\),
Recall that pil) > 51/55 and observe that - 1 = f;,(^) > vS^) > vSpil)). Consequently, since (1 - 8)4 < 8 this expression is negative, as we have assumed that (4A:f - Sljc]) < 0. In the case where both of the terms {4x\ - 51jcf) and (4A:| - 51x]) are positive then with 5^
2^
-j-xf + 77,
^
2
and xi=
51
-^
so that p(l)> § and p(3) < ^ . A calculation similar to that above gives the payoff: ^ { ( 1 - 5)[477, + 4773] + ^[(l + ^^xf
+ 77,]MM1))
+ 5[(l + ^ ) x l + 773]^„(p(3»
497
James Bergin PLAYER TYPE DISTRIBUTIONS IN ZERO SUM REPEATED GAMES
653
This may be rewritten as: ^[(1 - 5)4 + 8v^(p{l))]rj,
+ 5{i[(l + ^^xjjv^pil))
+ ^[(1 - 5)4 +
8uJip(3))]v,
+ ^[(l + ^yj^v^{pi3))
Again the third expression is no greater than 8vS^) and with (1 - 8)4 < 8, each of the first two expressions is negative (since p(l) and /?(3) are both less than — 1). Thus, when f < 5, z;J|) < 8uJ,j). Since vjj) > 0, this imphes that vjj) = 0 and this value is achieved only by a nonrevealing strategy. This completes the example. 4.2. Two-sided information. The previous discussion relied on the fact that an informed player could control the variation of the posterior distribution on player types. In the two-sided information case, actions of one player affect the information revealed by an opponent indirectly through the history. Difficulties of this sort are avoided by perturbing the game. Let: X^ = {x = ( x \ JC^,. .. x^)U* e A', xf > e, V/ e /} and Y^ = {y = ( y \ y ^ . . . y^)|y'' e A-', y[ ^ e, V; e /}, where e is a small positive number. With this restriction, define functions u^(p,
PROOF. The proof uses essentially the same ideas as the proof of Theorem 2 and is omitted. Appendu. Let a be a ( / X 1) vector, b a (1 X / ) vector and y4 an / X / matrbc. Thus
,
b-[h,..,bjl
a,i,fli2,...,fl^y
A =
«21'«22>--->«2y «/i,«/2>-'.«/y
498
Player Type Distributions as State Variables and Information Revelation 654
JAMES BERGIN
where a, is the ith row of A. Let
P(a) =
^/-i
, P(b) = [bj,bj.„...,b,]
and
i3(a/) /S(a,_,) /3(^) =
Note that /3 is symmetric in the sense that p(piX)) = X, X 2i vector or matrix. This is essential for the following result—the proposition is false when j8 is an arbitrary permutation. PROPOSITION 1. Let A^,A^ be two I Xj matrices and (/?, 1 - p) a probability distribution on the indices 1 and 2. Let vj^p) be the value of the infinitely repeated game defined by A^y A^ andp. Then p(A^) =A^ implies that vSp) = vS^ ~ pX PROOF. Denote by vj^p), the n-period repeated game with weight (1 — 5)5'~ * on the tih period payoff. Then v„{p) = v„(l -p) implies that v„^^{p) = f^+iCl -p\ To see this, let ^^ f ^ be optimal first-period strategies for 1 in the « + 1 period game with prior p. Thus, with |y = p^} + (1 - /?)f/, i^nAP) = nun j ( l - 8)\peA'
Now, we show that y„^.,(l -p)>
+ (1
-p)eA^]y
f„+,(p). Put ij' = /3(^^), 77^ = /3(f'). Then
''n+iCl -P) > nun{(l - 5 ) [ ( 1 -PWA'
+Pv^A^]y
Observe that -qW = {r,'a\,.... ly'a)) with A' = (a\,..., a]). Also, 7;' = /3(^^), a] = Thus,
so that 17I4' = piiM^). Similarly, tiM^ = /3(f l4>) and therefore (I - 5 ) [ ( l - p ) r , ' ^ ' + ; ; 7 , l 4 2 ] = (1 - 5)^[pf •^' + (1 - p)f ^^2]
499
James Bergin PLAYER TYPE DISTRIBUTIONS IN ZERO SUM REPEATED GAMES
Since min^ ^ ^ or - y = min^ ^^pia)
655
y,
min(l - 5)[(1 -/7)T7l4^ + p7)W\y y
= mm(l - b)[p^^A^ + (1
-p)^^A^]y,
y
Also, given any /, 3 ; such that (1 — P)T]] + pr)^ = (1 - p)^f H- p^] and
(l-p)7,!+p,,,?
(\-p)^f+p^}-
This last fact implies that:
{\-p)v] {i-p)vl+Pvf
Pi} pij +
{i-pHf
Symmetry of v„ implies then
"i{i-p)vl+Pvfj
""{pij + (1 -p)if
Thus
Therefore v^^Jil - p) > t;„ + i(/?). Reversing the argument given above yields v„^^ip) > v„ + i(l — p) and so i;„ + ,(l - p) = v^^^ip). The discussion above implies directly that t;,(/?) = i;j(l - p). Consequently, for any n,v^^^{\ - p) = v„^yip). By Theorem 1, Hm„_^„t;„^,(p) = Iim„^„z;„^,(l - p) = vjp) = uS^ - pX QED PROPOSITION 2.
v^ is concave in p.
PROOF. This result is given in Aumann and Maschler (1966) (see Zamir 1974 for a short proof) in the case where the limiting average payoff criterion is used. The same proof can be applied with discounting—a short sketch is as follows. Consider a finitely repeated game of length n. Take two prior distributions /?, and P2 and some a e [0,1] and let p = ap^ + (1 - a)p2^ Consider two games. In the first a prior, either /?, or P2> is chosen with probability a and (1 - a) respectively. Player II is informed which prior is chosen, I is fully informed and they play the n stage game. The second game differs from the first in that player 2 is not informed as to which prior (pi or p^) is chosen—so player IPs information is represented by the distribution p = a/7 J + (1 - Qf)/>2- The first game has value avj^p^) + (1 - cc)v„ip2) ^^^ the second has value f„(/?). Since player II, the minimizer is better informed in the first case avj^p^) + (1 — a)v„ip2) < v„(p). Hence, V«, t;„() is a concave function and so the uniform limit vS') is also concave. QED Acknowledgements. Thanks are due to Vijay Krishna, Jean-Francois Mertens, Hugo Sonnenschein and Joe Stiglitz for helpful conservations. The constructive
500
Player Type Distributions as State Variables and Information Revelation 656
JAMES BERGIN
suggestions of two anonymous referees and an associate editor are gratefully acknowledged. All errors are my own. References [1] Aumann, R. J. and M. Maschler (1966). Game Theoretic Aspects of Gradual Disarmament. Report to the US. Arms Control and Disarmament Agency, Final Report on Contract ACDA/ST-80, prepared by MATHEMATICA, Princeton, NJ, June 1966, Chapter 5. [2] Hart, S. (1985). Nonzero-Sum Two-Person Repeated Games With Incomplete Information. Math. Oper. Res.m, 1, 117-153. [3] Kohlberg, E. (1975a). The Information Revealed in Infinitely Repeated Games of Incomplete Information. Internal. J. Game Theory 4. [4] (1975b). Optimal Strategies in Repeated Games with Incomplete Information. Internal. J. Game Theory 4. [5] Mayberry, J. P. (1967). Discounted Repeated Games with Incomplete Information. Report to the ACDA, Contract ACDA/ST-116, prepared by Mathematica, Inc., Princeton, NJ. [6] Mertens, J. F. and S. Zamir (1971), The Value of TVo-Person Zero-Sum Repeated Games with Lack of Information on Both Sides. Internal. J. Game Theory 4. [7] Sorin, S. (1980). An Introduction to Two-Person Zero Sum Repeated Games with Incomplete Information. IMSSS Technical Report No. 312, Stanford University. [8) Zamir, S. (1974). Convexity and Repeated Games, in J. P. Aubin (Ed.), Analyse Coniexe el ses Applications. Lecture Notes in Economics and Math. Systems. 102, Springer Verlag, Berlin and New York. DEPARTMENT OF ECONOMICS, QUEEN'S UNIVERSITY, KINGSTON, ONTARIO, CANADA K7L 3N6
501
21 Daniel R. Vincent on Hugo E Sonnenschein
"Repeated Signalling Games and Dynamic Trading Relationships" was the first paper I wrote after leaving Princeton and becoming an academic economist. For me, it was a natural successor to the three papers in my thesis, papers which marked me clearly as a student of Hugo Sonnenschein. My thesis treated dynamic trading games where two or three agents attempted to determine a price and time of trade for a single indivisible, durable good. The thesis papers owed a great deal to Hugo's well-known paper with Robert Wilson and Faruk Gul [2], insofar as they employed a similar model and even used much the same strategy of proof. Indeed, the first paper of my thesis [4] owed its primary significance to the fact that it was able to reverse a famous prediction about durable goods monopoly offered in Gul, Sonnenschein and Wilson [2]. One interpretation of the model in their paper implies that when a bargainer is uncertain solely about the private valuation of her rival, then bargaining will only take a significant time to complete if the technology of making offers requires it. In contrast, I demonstrated that when the uncertainty is over the quality of the good to be traded, often bargaining games had to last a signficant time to ensure trade. In "Repeated Signalling Games" I wanted to extend the class of games in this literature to enable an examination of the effects of allowing players a far richer signalling space. This goal was accomplished by introducing trade in nondurable goods (so buyers could return after consummating a trade) and divisibility (so buyers could signal by consuming different quanitities). A consequence of this extension, as many contributors to this volume would have known, was to introduce a large multiplicity of equilibria. The equilibrium I focused on in the paper derived its justification indirectly from Hugo's contribution as well. Solving the game by a type of backward induction generated an iterated two stage signalling game. In the final stage of a given period, an informed buyer selected a quantity that served as a 'signal' to the uninformed seller about his type, a signal that (because of iteration) determined a subsequent continuation utility for the game. To address the multiplicity, I adopted the spirit of the refinement suggested by Hugo's student, In-Koo Cho (along with David Kreps) [1]. The striking feature of the equilibrium that I identify is that even in multi-period games, players typically reveal their types immediately and yet continue to incur separation costs throughout the game. This behavior can only be supported by a belief system of the seller that allows her to move from a conviction that no high demand buyer is present to one that such a buyer may well be present. Ordinarily such a reversion of beliefs may have been
503
21 Daniel R. Vincent on Hugo F. Sonnenschein unacx:eptable in the literature. However, previous work by other students of Hugo, (Madrigal, Tan and Werlang) [3] showed that not only were such belief systems reasonable, they were often necesssary in order for perfect Bayesian equilibria to exist. This paper thus reflects the enormous contribution Hugo has made to my professional life. It exploits on the training I gained as a student of Hugo at Princeton, it builds on a body of Uterature in which Hugo was a major player, and it draws on the results of many of Hugo's students as well. [1] Cho, In Koo and David Kreps, "Signaling Games and Stable Equilibria," QJE 102 (1987): 179-221. [2] Gul, E, H. Sonnenschein, and R. Wilson. "Foundations of Dynamic Monopoly and the Coase Con]&ctwc&/' Journal of Economic Theory 39,1 (1986): 155-190. [3] Madrigal, V, T. Tan, and S. R. de Costa Werlang, "Support Restrictions and Sequential Eqailibna,''Journal of Economic Theory 43 (1987): 329-334. [4] Vincent, Daniel R., "Bargaining With Common YahiQS,''Journal of Economic Theory 48,1 (1989): 47-62.
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Repeated Signaling Games and Dynamic Trading Relationships INTERNATIONAL ECONOMIC REVIEW Vol. 39, No. 2, May 1998
REPEATED SIGNALLING GAMES AND DYNAMIC TRADING RELATIONSHIPS* BY
DANIEL R . VINCENT^^
University of Western Ontario, Canada
A seller of a nondurable good repeatedly faces a buyer who is privately informed about the position of his demand curve. The seller offers a price in each period. The buyer chooses a quantity given the price. The quantity demanded reveals information about the buyer. An equilibrium is characterized with the feature that buyer types separate completely in the first period. This equilibrium uniquely satisfies a modified refinement of the Cho-Kreps criterion. Despite the immediate separation, the buyer distorts his behavior throughout the game. The requirement to signal types can raise the utility of all types of informed players.
1.
INTRODUCTION
Playing hard to get is as time-honored in markets as it is in love. The coy and clever buyer knows that betraying too much eagerness to a seller can often place him at a disadvantage as their relationship develops. However, feigned indifference comes at a cost. Delayed consumption destroys irrevocably some opportunities for satisfaction- A careful buyer must always balance his wish for immediate gratification with a caution against betraying his true desires; an interested seller must balance her desire to benefit from the current transaction with the need to extract information about the future of the relationship. In dynamic games, this phenomenon gives rise to the so-called ratchet effect. When an uninformed agent learns information early in a game, she can be expected to exploit it subsequently to her opponent's disadvantage. One result is that the cost of inducing information revelation grows the longer the trading relationship is expected to persist. The consequence of this behavior has been found to suppress the revelation of information in dynamic games (see Freixas et al., 1985, Hart and Tlrole 1988, or Laffont and Tirole 1987). These results have generally been derived in models in which the uninformed agent is in an unusually strong strategic position either because of her contracting power, or because of very simple preferences of the informed agent that offer agents of one type very little scope to separate from agents of another type. If the consequences of revealing information to an unin-
* Manuscript received August 1994; revised May 1996. ^E-mail: [email protected] This paper has benefitted from conversations with Morton Kamien and Alejandro Manelli, and from the detailed comments of two referees.
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formed agent are severe, then we can expect little information revelation. And, in dynamic trading games, if the only way agents can separate is through the stark choice of zero purchases or single unit purchases, again we can expect little information revelation. What happens in a different contracting environment and when preferences are such that the possibilities for screening are richer? In Section 2, a dynamic contracting game is described in which two agents desire to trade a divisible, nondurable good or a service, period by period. The buyer has private information about the position of his demand curve. Since the relationship is dynamic, the problem of an uninformed seller is twofold: to extract surplus from the current trade, and to extract information that may be exploited in future trades. Allowing an agent in a bilateral monopoly the sole right to offer nonlinear contracts yields that player a substantial amount of strategic power. This power and the temptation to extract surplus in later periods reduces the ability to extract information in the current period. One way to relax this stark formulation is to restrict the seller to another commonly observed type of offer: hnear contracts, in which the seller offers a price and the buyer chooses a quantity. This game possesses a perfect Bayesian equihbrium with a very stark feature. For a large subset of the parameter space, the equilibrium path is characterized by immediate information revelation by both types. Despite the revelation, economic behavior continues to be distorted along the equilibrium path as the low-type buyer is forced to continue to convince the seDer that he is indeed a low type. The low-type buyer separates from the high type by selecting a quantity determined by a demand curve lower than his true demand in every period but the last, even though at intermediate stages of the game the seller has acquired enough information to know with probability one whether the buyer is a high or low type. In this sense, the equilibrium illustrates that how an agent knows information as well as what the agent knows can play an economic role. Dynamic games in which the informed player has a large strategy space are typically plagued by a large set of equilibria. This model is no exception and, indeed, there also exist other separating equilibria as well. It is natural, therefore, to investigate the plausibility of this particular equilibriimi. In Section 4,1 show that if the model is extended to allow for any small but positive probability that a buyer's type might change in every period, then the equilibrium characterized in Section 3 is the imique one to survive the iterative appUcation of a well-known refinement of perfect Bayesian equilibrium. Section 5 examines other features of this equilibrium. As a result of the persistent concern that a seller may revise her beliefs, the equilibrium behavior confers a surprising benefit on both buyer types. In signalling games, one type of informed agent often incurs a cost due to the asymmetry of information as she attempts to separate from the other type. In this environment, it is shown that both types of buyers can benefit from the presence of asynmietric information. In a repeated context, the fear that the uninformed agent may update in an unfavorable way following some actions of the informed agent can serve as a valuable commitment device in earlier stages of the game, that will allow the agent to commit to strategies that would otherwise not be credible.
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THE MODEL
In a r-period game,^ an uninformed supplier faces a buyer who is privately informed about his preferences for a nondurable good. The seller can provide the good at constant (zero) marginal cost and seeks to maximize total discounted expected profits. A buyer of type a, in period t obtains a per-period utility from a quantity, q^, of the good purchased at a per-unit price, p^, given by (1)
2{a,-p,)q,-qf,
«,e{a^,fl^},
q>0.
Observe that the marginal rates of substitution between q^ and the money good differs depending on the buyer type. The buyer wishes to maximize the expected value of the T-period discounted sum of (1) (the discount factor, 5, is the same for buyer and seller) and has private information about the true value of «„ which may be high or low. The prior probability that the buyer is a high type at the beginning of the game (Prob[af='afj]) is ftj^e (0,1).^ An example generating these payoffs is a market where the seller sells to a downstream retailer or importer who incurs a quadratic cost of distributing the good. In addition, the true price the retailer receives upon reselling the object, a^, remains private to the retailer, perhaps because of unobserved taxes or rebates or other linear costs. In a myopic or static framework, the preferences of the buyer yield a demand curve of the form ^, = max {0, a,-/?,}, and the monopoHst's optimal price for fl£ > fl///2 is just a weighted combination of her monopoly price against each of the possible demand curves where the weight is given by her prior belief about the buyer type. It is never optimal for the seller to charge a price above a ^ / 2 , so there is no loss of generaUty in restricting attention to prices below this bound. However, if ^L < ^ / / / 2 , even in the static game, the seller's profit function is nonconcave in prices for some beliefs. In this case, her optimal price is either the weighted combination or just fl///2, depending on her beliefs. Since these issues are not the focus of this paper, I rule out this possibility by maintaining the restriction. The game consists of the following moves. At the beginning of each period, the seller offers a per-unit price and commits herself to providing any quantity the buyer chooses at that price. The buyer then chooses a nonnegative quantity. In the dynamic game, the strategic power of the seller is determined by the space of contracts available to her. The extreme case in which the seller may choose only linear pricing contracts is restrictive, but it is of interest for two reasons."^ A truly bilateral monopoly should have the characteristic that in the case of complete information, there is a nontrivial sharing of surplus. Restrictions on the strategy ^ The denotation, period r, refers to the period in which t periods remain to the end of the game. Thus period 1 is the last period and period T is the first. With the exception of the treatment in Section 4, for most of the paper I assume that once a buyer's type is chosen by Nature, it remains fixed for the remainder of the game. *The case in which a seller offers nonlinear contracts is examined in the two-period case in a slightly different model by Laffont and Tirole (1987).
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Space are often exploited to induce this division. One way to model bilateral bargaining power would be to incorporate explicitly offer-counter-offer negotiations in each period, as in Rubinstein (1982). Solving this game would be a daunting task. The restriction to linear pricing schemes allows the construction of a game that forces the sharing of some surplus and, as it turns out, is tractable. Furthermore, one often observes such restrictive pricing schemes. For example, negotiations between unions and firms often have the characteristic that a wage is determined (in this model, set by the union) and employment is then selected by the firm. In the next section, a perfect Bayesian equilibrium^ of the linear pricing game is illustrated with the feature that for a large class of parameter values the buyer types separate in every period.
3.
EQUILIBRIUM BEHAVIOR IN THE LINEAR PRICING GAME: THE CASE OF COMPLETE SEPARATION
Dynamic signalling games typically possess many perfect Bayesian equilibria (pBe). The focus of this paper is on equihbria that are fully revealing in every period. It is well-known that even in 'well-behaved* two-stage signalling games, fully jseparatmg equilibria may not exist if, given preferences, the signalling space is not large enough to allow one type to profitably distinguish himself from another type (Cho and Sobel 1990). Since I am interested in examining the nature of separation in many-period signalling games, that issue is side-stepped by placing restrictions on the parameter space in order to allow the possibility of full and immediate revelation. The equilibrium characterized in this section is extremely simple. In each period, buyer types separate for every price by choosing a quantity determined by a linear demand curve independent of seller beliefs and of the history of the game. The high type chooses the demand corresponding to his static demand curve, the low type, in every period, chooses a quantity off a linear demand curve that is generally lower than (but parallel to) his static demand curve. The actual position of the demand curve is such that in any period if the buyer type is high, he is just indifferent between mimicking the low type in this period and for the rest of the game and choosing his high demand. As a result, the more periods remaining to the end of the game, the lower this low-type equilibrium demand curve must be (see Figure 1). The proof that such behavior is the outcome of a perfect Bayesian equilibrium is best seen by construction. Define a monotonic sequence, {aj^f} as follows. Let X* = ajf(4 - 35)/(4 - 8). If a^ <x*, set Uj^^ = fl^. Otherwise define fl^, iteratively ^y «Li = «L and
(2) 5i
^Lr = «// - V^C^// - «L/-i )0^H - «L/-i) / 2 ,
for/>2.
For a definition of perfect Bayesian equilibrium, see Fudenberg and Tirole (1991). This definition, which more closely corresponds to that of sequential equilibrium, is slightly more restrictive than that used in earlier applications such as Freixas et ah, (1985).
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REPEATED SIGNALLING GAMES
FIGURE 1
The intercept of the low-type demand curves is given by Uj^^ in every period. It can be shown that aj^^ converges to x* as / becomes large and that for a fixed t, Oj^^ falls with increases in 8. In order to ensure that the signalling space is large enough to allow complete separation, I maintain the following restriction on parameters: (Al)
TySyaj^^aff 2Lit such that a^j>
fl///2.
Notice that for 8 < 4 / 5 , A l holds for all values of T as long as aj^ > a^/2 holds. In a perfect Bayesian equilibrium, we must specify seller beliefs following any history of the game. In all of what follows, I consider equilibria in which the seller's beliefs may change only following a move by the buyer. Given Bayes' rule, then, it is sufficient to characterize seller beliefs solely by the sequence, {ju.,},^i, which is the probability the seller places on the buyer being a high type in period / just before the seller posts a price, p^. THEOREM 1. Assume Al. The following behavior can be supported as a perfect Bayesian equilibrium outcome of the game. For any pricey p^, a buyer who is a high type in period t demands qfj ^a^—p^ and a low-type buyer demands qj^^ = a^^^ —p^. For any history resulting in seller ^s beliefs in period t of /i,, the seller offers a price p^ = PROOF.
For some period r onwards define strategies as:
Hl^. In every period, / < r, for every history, for every seller belief, and for every price offer, /?,, a high-type buyer in period / demands a^-pf and a low-type buyer demands (z^, —/?,•. H2^. In every period, / < r, for every history of the game, the seller offers a price, Pi( Mi) == (M/^H + (1 - Mi)«L/)/2-
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Clearly, H I and H2 are perfect Bayesian equilibrium strategies for r = 1.^ Suppose that they can be supported as pBe strategies following some period T = t — \.\ show that they can also be supported as pBe strategies for T = r and the theorem follows by induction. Observe that by assumption A l , /?,(/i,) < aj^- in every period, so positive quantities will be demanded in every period. HI and H2 imply that in period ^ — 1, whatever his behavior in the past, a high type expects to separate by demanding aff—pf_i in period t—1 and that the subsequent equilibrium path is stationary. Thus, his payoff following period t-1 can be represented simply by some constant. Now let /?, be a seller price offer in period t and define ^^, so that
2(fl^ -p,)qu
- qh + ^{{^H -Pt-i{^)f
= («// -Ptf + ^((«// -Pt-Mf
+ %/) + ^Ht)
That is, qj^f is the highest quantity choice such that the high type is just indifferent between revealing himself now by demanding cifj—p^ or demanding ^^, in this period, persuading the seller he is a low type and receiving the lowest possible price, Pi^fiff) in period t—1. By HI, in either case, he will demand ciH~Pt-\ ^^ the following period and reveal his type. Note that ^^, = a^^ —p^ where aj^^ is defined by (2). In order to show that demanding a^j -p, is optimal for the high type we need to describe the consequences of other choices. Observe that given H l , _ j , the seller's strategy defined by H2^„i is sequentially rational and determined solely by her t — I'st period beliefs. The consequences of a deviant quantity in period t, then, are determined by the effects of this quantity on the t — Tst period beliefs of the seller, fJ^t-i' P^^ ^^y history and any quantity choices, q^ <^z.,, the seller believes that the type is a low type in the current period with probability one and therefore fXf_^ = 0. If q' > qi^fy define p^^iq') implicitly by
2{a^,-p,)cf-q'^
+
3[{aH-p,_,{p-)f^-v^)
For any history with qt> quy ^be believes that the buyer is a high type with probability at least iLt*(gO and therefore, /i,_i > ^t*. (Note that even for p^ = 0, so the seller initially believes she is facing a high-type buyer with probability zero, whenever q'>qu ^s an out-of-equilibrium event, seller updating from ^t, = 0 to Pt-i > 0 is consistent with pBe) With these beliefs, the definition of ^t* ensures that the high type at least weakly prefers to demand afj—Pt to any lower quantity. Lemma 1 in the Appendix uses the induced preferences of the two types of buyers to show via a single-crossing property that the low type strictly prefers ^^^, which ^ Where it is clear, the T subscript is dropped.
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results in the lowest plausible price in the next period to any higher quantity and the higher next-period-price, Pf_^( fi*). Thus the behavior of both buyer types satisfy H I for period t, and since buyer types separate for any price, there is no dynamic role of pricing for the seller. For any seller belief in [0,1], given the separating behavior of the buyer for any /?,, the monopolist's optimization problem is exactly the same as the static problem confronting a monopolist with a linear demand curve with intercept a^j or a^^^. Given A l , this problem is concave, / ? / /JL^) < a^/l for all [L^ and the behavior described in H2 is optimal for period t as well. Since H I and H2 is satisfied for T = 1, induction imphes that the behavior satisfies the conditions of a perfect Bayesian equihbrium for all periods T such that a^^ > «///2. D Notice that in the extreme case of 5 = 0, buyers discount the future completely. In this case, x* = a^ and by definition, a^ = a^, for all t. Separation implies no distortion since there is no incentive for the high-type to under-demand. Similarly with low 6's, separation will always occur and the equilibrium corresponds to the repeated static solution. More interesting, however, is the case where the future matters because of a higher discount factor. In this case, where x* < a^, the succession of demand curves, a^,, is monotonically decreasing in t. As the length of the game increases, the low-type buyer must under-demand more in order to dissuade imitation by the high type. Observe that, lim «Lr = «///3 S->1
If b is high and T is large, it is possible that assumption Al is violated. This possibility is also a potential source of nonconcavity in the seller's optimization problem and her pure strategy best-response correspondence may not be convex-valued. In such cases, complete separation cannot be supported by the perfect Bayesian equilibrium described in Theorem 1. Some partial pooling will typically occur in early stages of the game, such that a^j- < a^/l. Initially, then, there may be some gradual learning. In these cases, the equilibrium behavior is not as simple as that of Theorem 1 since it will typically involve mixed strategies in equilibrium for the high-type buyer and mixed strategies (out of equilibrium) for the seller. These complications are accounted for in an earlier version of this paper (Vincent 1994), which provides a characterization of equilibrium strategies over the full parameter space. For values of a^, a^, 5, and r, such that aJ^^ > a^/2, the equilibrium paths coincide. Otherwise, if the seller believes relatively strongly that the buyer is a high type, it may be in her best interest to offer prices which induce only gradual revelation by the high-type buyer. In these periods, behavior very similar to the original ratchet effect of Laffont and Tirole emerges. The high-type buyer reveals himself with some probability, j8^. Only as the game approaches the later periods does complete separation emerge. Theorem 1 characterizes only one of many possible pBe, Even in two-stage signalling games, pooling equilibria often can coexist with separating equilibria. This is true here for 7 = 2 and, therefore, for T>2 as well. In two-period games, many
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such pooling equilibria fail to survive common belief-based refinements of perfect Bayesian equilibria. I show next that a similar approach may be applied in the multi-period game, A recursive application of the Cho-Kreps (1987) intuitive criterion is defined. The original game is modified to allow for stochastically changing types (but vnth arbitrarily small probabilities of changes). Theorem 2 illustrates that, in this game, the equilibrium path described in Theorem 1 is the only one to survive this restriction. 4.
A REHNEMENT OF PERFECT BAYESIAN EQUILIBRIUM
The equilibrium characterized in Theorem 1 has the feature that the private information of the buyer is revealed in the first period of the game. This stark learning behavior points to an intriguing and controversial feature of the equilibrium. If the seller observes an equilibrium low quantity in an early period of the game, she must believe she is facing a low type mth probability one. Furthermore, she must continue to beheve she is facing a low type throughout the rest of the game. Even so, it is not sequentially rational for her to revert to the optimal monopoly price, a^^/l against a low type. Instead, since the low type is demanding off a lower demand curve, «^, —/7^, the seller^s sequentially rational price falls to a^y2. Despite the equilibrium generated knowledge that the low type is in the game, behavior continues to be distorted along the equilibrium path. Is this apparently counterintuitive feature simply a curiosum of the large size of the set of perfect Bayesian equilibria of repeated signalling games? One way to approach this question is to determine whether the equilibrium would survive the application of a common refinement of sequential equilibrium. A difficulty arises, however, in the attempt to extend the definitions of these refinements to multistage games with full separation. After separation occurs, the seller will believe with probability zero that the buyer is of a particular type. But many belief-based refinements require the comparison of the value of a candidate outcome to various values derived from potential continuation paths following a deviation. In this environment, such a comparison would be vacuous if we did not allow the seller to consider the possibility of changing her belief from probability zero to a positive probability. The equilibrium characterized in Theorem 1 exhibits this phenomenon of increasing supports off the equilibrium path. Seller updating includes the feature that for high-quantity deviations the seller wall change her belief that the buyer is a high type with probability zero to a belief that the buyer is a high type with some positive probability. Note that perfect Bayesian equilibria (and sequential equilibria) of signalling games not only allows for this type of updating, but in some games it is required to ensure the existence of sequential equilibrium. (See Madrigal et al., 1987, and Noldeke and van Damme, 1990.) Beaudry and Poitevin (1993) also use a dynamic extension of a standard refinement to analyze a one-shot signalling game with the possibility of later renegotiation. Their model also yields equDibria where the uninformed agent's beliefs feature increasing supports off the equilibrium path. In this section, I skirt the issue by extending the model to ensure that the uninformed agent can never believe she is facing any given type with probability
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one. The extended model introduces the possibility that informed agents' types may change exogenously in every period. Bayesian updating by the seller must take this possibility into account when she formulates her new beliefs. Specifically, let a^^^ be the buyer's type in period r 4-1 and assume that in any period, buyer types follow a stationary Markovian process of the following form: (3)
?iob[a, = ajj\a,+i
=fl,] =^,,
i = L,H,0
< e^< Sf,
for/ = l , 2 , . . . , r - l . Thus, conditional on being of type i in period t + 1, the buyer is relatively more likely to be of type / in period 1.1 focus on the limiting case where s^ approaches one and £i^ approaches zero, but the model could, of course, be interpreted literally as a description of a game where there is a significant probabDity that types change over the course of the game. Consistent with full rationality, both the buyer and the seller factor in the possibility of future type changes in the determination of current optimal strategies. The operative role of this modification is that for any Sj^ > 0, in no period does the seller believe with probability zero she is facing a high-type buyer, and therefore the issue of nonincreasing supports does not arise. This feature is used to describe the extension of the refinement. Before defining the refinement formally, it may be helpful to walk through a short example to illustrate its application in a three-period game with « / / = ! , 6 = 1 , a^ = 3 / 4 and ^r^ = 1.^ Consider the subform of the game beginning at the second stage of the second last period with the seller's price, P2, already set and with some current seller belief, )Lt2> strictly above 0. Sequential rationality imposed on the buyer in the last period implies that one can represent buyer types's preferences in (^2> Pi^ space by the following induced utility function:
Figure 2 illustrates these preferences in (q2,Pi) space. It can be shown (Lemma 1) that these indifference curves are concave and that the slope of the high-type indifference curves are steeper than those of the low type. Seller behavior in the last period is determined for any belief fx^y and since buyer behavior in the last period is a function only of seller price, p^, a two-period signalling game emerges from the overlap of the second stage of period two and the first stage of the last period. The buyer's second-period quantity demand acts as a signal that the seller observes and generates a response that is the final period price. In a two-period signalling game, the Cho~Kreps (1987) Intuitive Criterion (what will be refinement R2 in the multi-period game) rules out candidate pBe outcomes such as point A in Figure 2 as follows.^ Trace the indifference curve of the high I also set €1^ very small so some of the actual numbers are not precisely correct but are the limits as s^ goes to 0. Point A represents a pooling outcome since any seller offer strictly between 3/8 and 1/2 in the last period can only be generated by a seller belief strictly between 0 and L
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High type Indifference / Curves
Low type Indifference / Curve
FIGURE 2
type downward and to the left until it crosses the Pi = 3/8 line at q'. Suppose a deviant quantity, q' — A. is demanded at this stage. There is no seUer belief and subsequent best response that could yield an outcome for the high-type buyer that the buyer prefers to the candidate outcome, A. On the other hand, if the seller updates her beliefs following the deviation by putting zero weight on the high type and responds with the price p^ = 3/8, for small enough A, the low type gains a strictly higher payoff than from A. In a simple two-stage signalling game, the Cho-Kreps criterion would imply that the seller then should believe that only a low type would demand such a low quantity, and therefore update with a belief fjLi = Si^ and respond with a price in the last period of p^ = 3/8. This argument can be used to eliminate all pooling outcomes. The only outcome that survives the restriction to this type of seller updating rule is the separating outcome B for the high type and C for the low type shown also in Figure 2. The high type receives his full information equilibrium payoff while the low type underdemands just enough so as to dissuade imitation from the high type. This last condition yields the quantity ^^2 "= ^LI ~P2^ ^/^ ''Pi ^^^ ^W price offer /?2- Th^s a lower 'effective' demand curve determines the low type's behavior in the secondto-last period. Given the necessary buyer behavior in period 2, the seller's best response in period 2 is again a simple function of her beliefs: p( 1x2) = (/i,2 + (1 - iXj)*^/^)/'^The fact that the buyer types separate for any sequentially rational price offer of the
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Repeated Signaling Games and Dynamic Trading Relationships REPEATED SIGNALLING GAMES
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seller implies that there is no informational (and therefore no dynamic) role of prices. Instead, the seller is again in a situation analogous to that of the static monopolist facing one of two possible demand curves. This time, though, the low demand curve is lower than before because of the low type's desire to separate from the high type. Observe that, except for the determination of the seller's price, />2> this argument is made independent of the actual value of the seller's current prior, 1x2. What is required is that, fixing any candidate equilibrium, for 'low enough' deviant quantities, the seller will believe that a low type made the demand. This requirement, in turn, requires that to support the only eqm'librium outcome satisfying this feature, we be able to place a high-enough probability that a high type makes a deviant quantity above the low type's quantity, ^^2- ^^ r = 2, then these requirements are satisfied as long as «^ > a / / / 2 and iJij=^ ^2^^-^ ^ three-period game, the issue becomes more delicate if ^j^ = 0, because then it is possible that along the equilibrium path, At.2 = 0. Given that buyers expect to separate in this manner for all prices in the second stage of the middle period, the utility that each buyer type expects from any continuation path at the beginning of period 2 will be a function only of the seller price offer, p2y independent of the history of the game. Furthermore, if we move forward in the game to the second stage of the initial period, with a seller price, p^y outstanding, it can be shown that the induced buyer preferences over the quantity they demand given this price and the subsequent /?2 that this generates from the seller are qualitatively similar to those in Figure 2. The equilibrium path isolated in Theorem 1 is obtained by applying the intuitive criterion {R-^ after replacing the continuation paths of the game with the expected payoffs (which depend only on the seller beliefs and through them on subsequent seller price offers). An argument similar to that for the overlap of period 2 and 1 applies here as well and yields complete separation again in the first period, this time with the low-type buyer demanding a quantity, 9^3 = 1 - 0.5((1 - 5/8X3 - 5/8)>^ = 0.528 - / ? , . The equilibrium price path follows one of two patterns. Independent of the buyer type, the initial price is given by (/i3 4- (1 — ii^ai^^/2. If the buyer is a high type, the quantity demanded is relatively high, and subsequent prices move immediately to 1/2 in each of the remaining two periods. If the buyer is a low type, the initial demand is lower than the low type's static demand, the seller's next period price falls to a price below her static monopoly price against a low type and then rises as the game continues. The characterization of the formal refinement requires some more notation and definitions. Let h^ denote a history of a game up to the end of period t and (htyPf_i) denote the history to the middle of period t — l^ Since a strategy determines the continuation play of the game for any given history, we can compute expected payoffs from t — 1 onwards for buyer and seller given a history and a strategy, or. Denote the buyer's expected payoff by ^/^(/j/, A - i , « ; ) when the history is ih^,Pt_^X the buyer type in period / is fly, and the strategies from t onwards are ^ For buyers, a history includes the realized price offers, demand choices and the reah'zation of buyer types to that period. For the seller, a history only includes the first two sequences.
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determined by a. The seller's expected payoff from the same stage in the game onward, conditional on the buyer type in period t being a^ is uj^(/i,,/7^_i,fly). Given a history, (/if,/?,_i) and a profile of pure strategies,^^ o-, let q!Li(ht,Pf_^,aj) be the strategy choice of a buyer who is of type aj in period t-L Similarly, p^-iih^) is the corresponding seller price choice fixed by a. A pBe also characterizes seller beliefs after any history. Let ^f^-iihf) denote the seller's interim probability that the buyer is a high type at the beginning of period t — 1 (following the random move by nature at the beginning of period / — 1). Recall that I consider only pBe such that beliefs may change only after buyer deviations.^ ^ DEFINITION 1.
A subset of perfect Bayesian equilibrium strategies, 2 , satisfies
condition C, if for all strategy profiles, a- G 2 , for all / < /, for all /i,+i, /i/+i, for all
Definition 1 characterizes a class of strategies that exhibit a strong type of stationarity from some period t to the end of the game. Since for any pBe, ^f (/i2»Pi>^i) = ^1 ~Pv ^^^ ^^^ histories, the set of all pBe strategies satisfies Q . DEFINITION 2.
For any set of pBe strategies, 2 , satisfying C,, define
^^2(M) = {PrlA^argmax(/x<(/i,+i,/7„«;^) + ( l - A t X ( / i , ^ i , p , , a ^ ) ) for some cr^'%}.
BR^^ is not necessarily the set of seller equilibrium strategies since any given seller belief p, may never arise in a pBe, However, in order to apply the refinement, I want the ability to conduct thought experiments that range over all possible seller beliefs following a deviation. This device allows that flexibility. Notice that w*^ is defined as a function of the strategy profile alone, not seller beliefs. The variation of beliefs, p, in the current period is not assumed to affect future play of the game.^^ This definition does not require pure strategies but (i) as long as Al is satisfied, the equilibrium will be in pure strategies, and (ii) restriction to pure strategies requires less notation so I will refer only to the pure strategy case here. There is another somewhat technical restriction. The original definition of perfect Bayesian equilibrium (for example, Freixas et aL, 1985) placed relatively few restrictions following out-ofequilibrium histories. Suppose that an out-of-equilibrium history occurs and the pBe assigns subsequent beliefs {iLt,}/=,-i following it. Even if the continuation path follows the prescription of the pBe following the out-of-equilibrium history, the original definition did not force ^t, and /jt,_| to be consistent with Bayes* rule and the equilibrium strategies. However, in finite games this additional restriction would be implied by sequential equilibrium via the condition of consistency and it seems natural to require it here as well. This restriction requires that beliefs following an out-of-equilibrium move be what Fudenberg and Tirole (1991) term 'reasonable* beliefs. ^ More generally, one might prefer to consider how the continuation path for the rest of the game changes also with changes in the period / belief. However, since pBe fixes strategies and then appends beliefs, the machinery of pBe does not allow us to specify variations of continuation strategies following changes in period t beliefs. An alternative approach that could achieve this would be to use the concept of meta-strategy introduced in Grossman and Perry (1986).
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I now define a refinement of a subset of pBe, DEFINITION 3.
If E is a subset of pBe of the T period game satisfying C^_i,
J^,(S) = {c7-eS|V/i,^i,V/7,
{^-(/i,^i,/7,.fl^),^,^(Vi,/7,,a^)}
such that if
and <(ht+uP,.q.P.af,)>v,''(h,^^,Pt,a^), then,
iorsomc p ^BR'^^Sj,) J,k^ {L,H},
k^j\
f^^-i{h,,p,,q)='£f,}.
If 2 does not satisfy Q _ i , then i?,(2) = 2.'The refinement is generated by applying R^ iteratively. Let 2 | be the full set of pBe. The T-fold application of the refinement yields the subset of pBe, Xj- = Rj'iRj'_^(... R^i^i)•..)) = i^^(2|). In words, the r'th refinement states the following. Suppose that all the pBe under consideration, 2^_i, have the feature that buyer behavior is stationary in all periods following period t, and so continuation payoffs will depend only on the seller's subsequent price offers, which in turn depend only on her subsequent beliefs. Then, if a deviant demand occurs in period / with, the characteristic that, given the hypothesized equilibrium continuation, one type does worse for any sequentially rational seller price offer while there is a sequentially rational seller price offer for which the other type does strictly better, then the seller must believe that it was the latter type who deviated in period t. Theorem 2 shows that for any Sj^ > 0, the equilibrium path described in Theorem 1 is the only one to survive the J-fold application of this refinement. THEOREM 2. Let Sj be the set ofpBe and define l,j = R^(^^), Assume Al and suppose f£ > 0. If (T^'^jy then a generates the equilibrium path described in Theorem 1 with aj^f defined as (4) a^t = min{a^,a„ - ylh{e„ - ei^){3{£„an + (1 - .e//)«Lr-i) - (^L«// + ( 1 " ^LWI-I))
PROOF.
/^}
The proof of Theorem 2 is found in the Appendix.
The direct application of refinement, R^ puts restrictions on how the seller can update when she observes lower than expected quantities demanded. In a way, it allows the low type the opportunity to destroy any pooling equilibrium by signalling his type with a low quantity demand. The richness of the preferences implies that the high type is not willing to sacrifice high quantity consumption now for lower
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prices in the future. However, this restriction eliminates many pBe and as a consequence it also forces restrictions on how she may update when higher than expected quantities are observed. Even though information is fully revealed, screening costs are incurred throughout the game. After the first period, both the buyer and the seller know all the relevant information for the rest of the game. Nevertheless, the equilibrium strategy of the low type is to under-demand for the remainder of the game (except the final period). He acts as if he had a demand curve with a strictly lower intercept. Given this behavior, the seller can do no better than to post a lower price. The buyer who is informed that he is of a low type signals this to the seller in the first period but is 'forced' to continue to convince the seller throughout the game. There is a sense in which although all the information is revealed immediately, complete separation has really not occurred until the game is fully over. Observe that a condition of the theorem is that s^^ > 0. If Sj^ = 0, there are other pBe satisfying R^, For example, in the three-period game,^^ a pBe exists with the following characteristics. In the first period, for any price, p^y the low type demands a low enough quantity, ^3, that even if the high type demanded q^, was offered PL ^ ^ L / 2 for the rest of the game and the buyer was able to demand Qff —pi^ in the last two periods, the high type still prefers to demand a^j —p^ and reveal himself. In this equilibrium, the seller believes At2 = 0 (/t2 = 1) if she sees the low (high) quantity in the first period and never changes it for the rest of the game. In the subsequent periods, she offers the full information static price, aj^/2 or fl///2. The equilibrium is not eliminated by R^, It may seem odd that if the seller sees first a low quantity and then the quantity afj—Pi she never wavers from her belief, ^t = 0. However, the demand «// — P2 ^ ^i^^s^Piy ^H^ ^^^ therefore, according to Definition 3, there is no restriction implied for how she should update. This type of equilibrium does not result in the peculiar phenomenon of reaching a point in the game where it is common knowledge that the buyer is a low type and yet under-demanding persists and perhaps, is attractive for that very reason. On the other hand, it implies a sort of dogmatic belief formation by the seller. She forms her belief in the first period of the game and never changes her mind thereafter. Not surprisingly, for very long g^mes, it is much harder to support complete separation with this type of equDibria since the temptation for the high type to deviate in the first period and get a low price for many periods in spite of high demand is very strong. To support these dogmatic equilibria for games of arbitrarily many periods and for any a^ > a^/^ requires a discount factor below 4 / 9 that is much lower than the 4 / 5 bound in the equilibrium characterized in Theorem 1 (see the comment following assumption Al.) Of course, there may also be other separating equilibria with less dogmatic behavior by the seller. For example she may change her mind only after observing some fixed number of deviations, but then some assessment would have to be made concerning what is a reasonable number of deviations before the seller should switch her beliefs.
^^In a two-period game, the equilibrium path described in this paragraph and that of Theorem 1 coincide.
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No equilibrium of this type survives the application of the refinement with £^£ > 0 because A^2 ^ ^ ^^^^^ ^^Y history leading to period 2 and, by Bayes* rule, if the seller observes a high demand in period 2 she must believe it came from a high type and respond with a high price in period 1. The definition of q^ would then fail to satisfy the incentive compatibility constraint on the high type. Of course, the pBe described in Theorem 1 also survives in the limit as Sj^ goes to zero.
5.
IMPLICATIONS OF EQUILIBRIUM
The equilibrium exhibits some intuitive comparative statics. As long as we continue to assume that Al holds, the low type's demand falls as 8 rises. He must distort his demand even further the more important the future becomes. Similarly, as T becomes large, the more the low type must under-demand in early periods, $mce longer games offer greater rewards to a high type who successfully mimics a low type. Finally, holding ^^ + ^^ fked, demand rises as Sff - s^^ falls. The closer the ^'s, the less valuable is current information, and therefore the less costly it is for the low type to separate from the high type. The equilibrium characterized in Section 3 exhibits some additional noteworthy characteristics. A sort of ratchet effect is still present although in a different sense than in the nonlinear model. There, the principal is forced to offer a more generous scheme in order to induce information revelation. In the case of the linear contracting game, the informed agents will often reveal following any price offer of the seller. However, it is this behavior that forces the seller's price offer to be lower than in a static price-setting problem. For any given belief of the seller, her optimal price is higher with the same belief as the game nears the final period. Since, in equilibrium, information is completely revealed in each period, it is interesting to compare the results here with those of a similar model where the buyer has the same preferences but acts nonstrategically. Both types can benefit from the strategic behavior. To see this, note that the price offer of the seller is typically lower in the strategic game. If the true state is high, the lower price is a straight gain to the buyer. When the true state is low, the lower price is a benefit even though the buyer is also forced to under-demand relative to his true demand curve. For a^, > aj^/3, a condition implied by Al, the low-type buyer is made strictly better off by the lower price. Consider the simple T = 2 game. In a game where there is no possibility of a high type, subgame perfection forces the buyer to choose q=aj^—p in every period and, therefore, the equilibrium price path is just p = aj^/2 in each period. In the game with a small initial probability of a high type, the initial price is (close to) ^L2/2 < « L / 2 ^^^ then reverts to a^/2 in the last period if the type is in fact low. Even with the lower quantity demanded in the first period, the low type does better than in the game with no possibility of a high type.^'* This result is noteworthy since it represents a situation in which informed types are in a position in which they are = 2, the *dogmatic* equilibrium discussed after Theorem 2 and the equilibrium described in Theorem 1 coincide so this result does not necessarily rely on the refinement.
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forced to separate but the separation can benefit both types. In a typical screening model, some types of the informed players are usually forced to incur screening costs to separate themselves from other types. Here the screening environment can bestow an advantage. In a static or nonstrategic monopoly pricing game, a low-type buyer would prefer it if, by committing to demand a lower quantity, he could convince the seller to offer a lower price. In general, the technology for such a commitment is lacking. Here, though, the low type has a credible concern that the seller will mistake him for a high type in the remainder of the game. The concern serves as a commitment device and allows him to induce a lower price from the seller to shift some of the surplus from the trade in his direction. This differs from results in standard two-stage signalling games because of the addition of at least one earlier stage where the actions of the uninformed agent (the seller's initial pricing stage) is affected by the later signalling concerns of the informed agent. In addition, the strategy space of the uninformed agent is rich enough to allow for strategy choices that both types strictly prefer to the strategy choices in the complete information game. For example, in Kreps and Wilson's (1982) multi-period signalling game, a version of the chainstore paradox, the uninformed entrant can only decide whether to enter or not. In that environment, never enter is the outcome in a complete information game with the strong incumbent and there are no other strategies that can benefit both types, as in this game where both types prefer the lower price. As a result, the possibility for this beneficial commitment feature did not arise in their model.
6.
CONCLUSION
The ability of a seller to extract information depends on her strategic power. The more powerful the seller, the more dangerous it is for an informed buyer to reveal his private information, and in long games the ratchet effect shows us that the result is very little information transfer. If the strategic power is more equally shared, however, as in linear contracting games, the likelihood of information revelation rises dramatically. In this class of bilateral monopoly games, the signalling space is rich enough for informed buyers to separate in each period. However, even though this separation conveys a great deal of information, it does not relieve the players of the burden of separation in subsequent play of the game. Some transactions are not consummated even if the players are virtually certain that they should be. The disinterested partner shows his true colors in the first period and proves it over and over for the rest of the relationship by demanding less of his partner than he truly desires. Playing hard to get results in the persistence of an under-requited love.
APPENDIX PROOF OF THEOREM 2.
If aj^^ = a^^y then the proof is similar but simpler so I
focus on the harder case with a^^ <«/,. Let 2,_i be the subset of pBe, such that buyer behavior satisfies Hl,_j from Theorem 1. Note that this stationary and separating buyer behavior implies that for o-e X,_i, after any history /i-^j resulting
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in a seller belief, /it^, / < / - 1, the seller's unique sequentially rational response must be /?,(A>t,) = (/i^a^4-(l-^t.)a^.)/2. Since 2,_i satisfies C,_j, BRy^ fx) =-p^i fx). Also note that the set of all pBe equals 2i and satisfies Sj = J?i(2i) since any seUer posterior p,Q is irrelevant because the game has ended. By Theorem 1, '^j = i^T-C/^r-iC.. • i^i(Xi)...)) is nonempty. If we can show that for X^.j = Rt-,liRf-i^•' • i^i(2i)))...), c7 G Rtdf_I) implies that buyer behavior according to cr satisfies H l „ then Theorem 2 follows by induction. Therefore, suppose that 2,_j = i^,_i(i^f_2("^i(2i))). Let o-ei^/X^.j) and let /?, be any price offered after any history. Since Hl,_i fixes buyer behavior in the following periods, for any quantity, qy chosen in this period, and price ^p^.j offered in the next period, the continuation utility of a buyer of type j in period r, given a is
= {2{a^ -p,) -^ {a^-pf
- q)q + 8(e,{a^ -pf - {a^~p,-
+ (1 - e.){{a^ -pf
qf ^- d[s^{a^-pf
+ {I-
- {a^- a^-.f))
s^){aj^-pf)
+ K],
-¥K^,,
for constants i^y,, j = L,H, By Hl^.j and sequential rationality on the seller, Pf_^ must lie between Pt-i(^L)^PLt-i ^^^ A-i(^//)=/^///-i- The worst that can happen to the buyers in the next period is the highest of these prices, therefore a buyer who is a high type in period t has continuation utihty that is bounded by what he would get if he chose the current period utility-maximizing quantity, a^j-p^ and received the highest price in the next period, p^t-i' (5)
Viq,P;cifjyPt)
I use the following result concerning the continuation function in the proof. LEMMA 1. For any s^, 6-^, with 1 > % > ^^^ > 0, // (^, p) satisfy (5), the indifference curves generated by V are concave and the slope of the H curve is greater than that of the L curve. PROOF. Let dfj = SfjCifj + (1 - ^//)fl^ and a^^ = Sj^ttfj + (1 - €i}aj^. In what follows I focus on the case, q
^aj-p,-q S{aj-p)'
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Daniel R. Vincent 292
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If (q,p) satisfy p = dj-a^ +Pt + qy the indifference curve has a slope of 1 / 6 > 1. Since pjft-i < f l ^ / 2 , the point (icifj-pj),Pfj,_^) lies strictly below the line /?== dH~^H'^Pt'^^> which has a slope of one. An indifference curve which passes thro.ugh i(aff—p,Xpfft_i) cannot cross that line, since at the point of crossing, it would have a slope that exceeds the slope of the line. Therefore the set defined by (5) lies below the line p = aff — ajj+Pf-\- q. The following results rely on this fact. Taking differences gives. dp
dp dqlff
dqk
llaL-Pi-q 8\ aj^-p
^//-A-g\ a^-p /'
Rearranging, gives 1 l^L-P+P-i^L-^L-^Pi-^q)
^H-P+P-i^H-^H+Pi-^q)
^L-P
^H-P
Cancelling the one in each term and noting that a^ - A^ > 0 > a^ — a ^ we get dp dqlt
dp dqln
}_(p-{aH-aH-^Pi-^q) S\ i^L-p)
P-i^H-^H+Pt ^H'P
1
/ I
1
^L-P which is less than zero for p
^
+ q)
^H-P
SH^^L-
= llT7.-Tf|-l|/(«>-A-.).
which is less than zero for p
aj 4-/7,4- ^. This yields the concavity of the D
Let qjf be a quantity prescribed by a for the high type in period t and let w be the next period seller price. Since £j^ > 0, q^^ must occur with positive probability and if H alone demands ^ ^ , we must have '7T=PHt-v ^^ general, we must have '^>Pu-i' If '^'^Pnt-iy then the low type must also choose ^^ with positive probability. Define qiqffy TT, p^) by H^^PLt-i\(^HyPt)
=
{^H-Ptf-{^H-Pt-(iHf 4- B[efj{afj - irf 4- (1 - s„){aj^ -
7Tf]+K„,.
(In Figure 2, q corresponds to q'.) Substituting in the definitions also yields that if ^H^^H ~Pt ^^^ '^="PHt-iy Q^^^Lt"^ ^Lt ~Pty whcrc fl^, is defined in equation (4) (if aj^,
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Repeated Signaling Games and Dynamic Trading Relationships REPEATED SIGNALLING GAMES
293
Suppose that a deviant offer q = q- ^ is demanded. By Lemma 1 and assumption Al, there is an A small enough that ^ > 0 and the high type strictly prefers qjj to q and its consequent continuation to the deviation and the best possible contmuation that he can hope for, while the low type strictly prefers demanding q, receiving the subsequent price Pu-i ^^^'^^(^0 ^ ^^^ ^^^ period and abiding by the behavior described by Hl,_i subsequently. By the refinement i?,, then the seller must believe that a low type demanded q and sequential rationality along with the buyer behavior, Hl,_j requires her to respond with the price, Pu-i ^ the next period. This breaks any pooling behavior in period t. A similar argument follows to show that any separating outcome must satisfy condition HI,. Therefore, 2, = i?/2,_j) and yields buyer behavior HI, and induction yields the result.
REFERENCES BEAUDRY, P. AND M. POITEVIN, "Signalling and Renegotiation in Contractual Relationships," Econometnca 61 (1993), 745-782. CHO, I.K. AND D . KREPS, "Signalling Games and Stable Equilibria," Quarterly Journal of Economics 102 (1987), 179-221. AND J. SoBEL, "Strategic Stability and Uniqueness in Signalling Games," Journal of Economic Theory 50 (1990), 381-413. FREIXAS, X-, GUESNERIE, R., AND J. TiROLE, "Planning Under Incomplete Information and the Ratchet Effect;'Review of Economic Studies LII (1985), 173-191. FuDENBERG, D. AND J. TiROLE, "Perfect Bayesian Equilibrium and Sequential Equilibrium" Journal of Economic Theory 53 (1991), 236-260. GROSSMAN, S. AND M. PERRY, "Sequential Bargaining Under Asymmetric Information," Journal of Economic Theory 39 (1986), 120-154. HART, O. AND J. TIROLE, "Contract Renegotiation and Coasian Dynamics," Review of Economic Studies LV(4) (1988), 509-540. KREPS, D.M. AND R. WILSON, "Reputation and Imperfect Information," Journal of Economic Theory 27 (1982), 253-279. LAFFONT, J.J. AND J. TiROLE, "Comparative Statics of the Optimal Dynamic Incentive Contract," European Economic Review 31 (1987), 901-926. MADRIGAL, V., T. TAN, AND S.R. DE C WERLANG, "Support Restrictions and Sequential Equilibria," Journal of Economic Theory 43 (1987), 329-334. NOLDEKE, G. AND E . VAN DAMME, "Switching Away From Probability One Beliefs," Discussion Paper A-304, University of Bonn, 1990. RUBINSTEIN, A., "Perfect Equilibria in Bargaining Models," Econometrica 50 (1982), 97-109. VINCENT, D.R., "Repeated Signalling Games and Dynamic Trading Relationships," mimeo, Northwestern University, 1994.
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22 Lin Zhou on Hugo F. Sonnenschein
In 19851 was brought to Princeton by Gregory Chow, hugely famous in China for two tests named after him. The first was a test used to select young people to study economics in US, and the second was the popular Chow test in econometrics. Although I passed the first test withflyingcolor, I never really wanted to become an expert of the second test - an econometrician, that is. At the reception for the new graduate students, Gregory introduced me to Hugo, who just came from Stanford with his troops. I naturally thought that I might be able to work under Hugo's guidance. Little did I know then how lucky I would be. As it turned out, I was the last student Hugo officially supervised during his tenure at Princeton. Hugo did not take many new students after he returned from Stanford. This allowed me to have easy access to him. Looking back, I probably have taxed much more valuable time of his than I should, but I have certainly taken full advantages of such opportunities. During my third year, Hugo presented his work with Salvador Barber a on voting by quotas, which grabbed my attention. After working on the problem for weeks, I realized that their main result could be extended considerably. As I reported the news to Hugo, he was very pleased and graciously invited me to be co-author of the paper. I was honored and excited at the same time. While I was dreaming of more work with Hugo or under Hugo, I was shaken as he announced that he would leave Princeton to become the dean at Penn. Sensing my uneasiness with the situation, he sent me to Salvador, first to finish our joint paper, and second, to learn more from Salvador. Again, Uttle did I know how much I would learn on this trip. A few days after I arrived in Barcelona Salvador broke his leg. Although it was by no means a direct fault of mine, I felt awful. Like most Chinese, I would believe everything was connected. As a result, I paid frequent visits to Salvador's house to work with him. But it was really a blessing in disguise. While he was in the office, Salvador was always busy with either meetings or people, but now he was at home, and he was with me alone. As we took time to complete our paper on voting by committees, Salvador also told me about his then current work, including his work with Peleg on the Gibbard-Satterthewaite theorem with continuous preferences. Prior to that, I had thought about the problem of extending the Gibbard-Satterthwaite theorem to economic models, where preferences are both continuous and convex. When the commodity space is a straight line, the median voter scheme is both strategy-proof and non- dictatorial. But if the commodity space is of multidimensional, there was no general result at the time. (The Grove
525
22 Lin Zhou on Hugo F. Sonnenschein mechanisms work only when preferences are quasi-additive.) I had not had any success up to that point. I immediately revisited the problem in light of Barbera and Peleg's work. By refining their technique to deal with the additional structure of convexity, I managed to prove the Gibbard-Satterthwaite theorem in the economic environment: any strategy-proof mechanism whose range is not contained in a straight line must be dictatorial even when preferences are both continuous and convex. For this result I am mostly indebted to Salvador. But I also owed it to Hugo since it was Hugo who arranged my trip in the first place! I am happy to include it in this volume to honor Hugo, as well as Salvador.
526
Impossibility of Strategy-Proof Mechanisms Review of Economic Studies (1991) 58, 107-119
Impossibility of Strategy-Proof Mechanisms in Economies with Pure PubHc Goods LIN ZHOU Cowles Foundation^ Yale University First version received May 1989; final version accepted May 1990 {Eds.) This paper investigates the structures of strategy-proof mechanisms in general models of economies with pure public goods. Under the assumptions that the set of allocations is a subset of some finite-dimensional Euclidean space and that the admissible preferencees are continuous and convex, I establish that any strategy-proof mechanism is dictatorial whenever the decision problem is of more than one dimension. Furthermore, I establish a similar result when preference relations also satisfy the additional assumption of monotonicity. These results properly extend the Gibbard-Satterthwaite theorem to economies with pure public goods.
1. INTRODUCTION When a society consisting of several individuals has to select from a set of alternatives, it often relies on certain rules to make a choice. Such rules are often called mechanisms (or voting schemes, or social choice functions). These mechanisms may be inherited from earlier generations, or they may be adopted via democratic processes. In order that a mechanism represent an optimal compromise (in any well-defined sense) of the conflicting interests of the individuals in society, it must take into account their preferences over the alternatives. These preferences, however, are usually privately known and they have to be solicited for public use. Thus utility-maximizing individuals can manipulate the final outcome by misrepresenting their preferences. As a result of such manipulation actual outcomes may be far from satisfactory from the social point of view. Hence in order to have a better understanding of social decision-making processes it is important to know how severe the problem of manipulation is, and whether mechanisms immune to manipulation can be devised. In the framework of social choice theory, Gibbard (1973) and Satterthwaite (1975) independently proved that, subject to a minor qualification, a mechanism is manipulable if it is non-dictatorial. However, the original theorem is stated only for the case in which all possible preferences are admissible, and it leaves unanswered the question of whether similar results are true under various restrictions of the domain of admissible preferences. For example, let us consider a canonical problem from public finance.' There are three public goods to be provided: education, telecommunication, and transportation. It is assumed that individuals have continuous, and convex preferences over these goods. Since they are real "goods", it is also assumed that individuals' preferences are monotonically increasing. The feasible set is given by A = {(x,, Xj, Xy) \ x, ^ 0, X ^i = !}• Society 1. This example was suggested by H. Moulin. 107
527
Lin Zhou 108
REVIEW OF ECONOMIC STUDIES
has to choose an allocation from A In this problem it is natural to consider mechanisms that satisfy the property of unanimity, i.e. mechanisms that choose x if all individuals regard x as the best alternative in A. The question is whether there exists a nonmanipulable, and non-dictatorial mechanism. Surprisingly, this seemingly simple question is not answered by any existing result. The results derived in this paper will provide an answer. I consider in the paper two basic models of economies with pure public goods: in the first one admissible preferences are continuous and convex, and in the second one they are increasing as well. Although the existing literature on this subject contains some results that suggest that the Gibbard-Satterthwaite theorem could be extended to such economies, it does not have a general and fully satisfactory answer. The difficulty on this subject stems from the fact that most work on strategy-proofness relies heavily on pure logical induction, thus leaving very little room to deal with the properties of continuity and convexity. For example, in Schmeidler and Sonnenschein's (1978) proof of the Gibbard-Satterthwaite theorem, it is assumed that if one changes an individuaPs preference relation by moving any pair of alternatives to the top of the original one, then the new preference relation is still admissible. This procedure, however, certainly destroys the continuity of the preferences. A recent paper by Barbera and Peleg (1990) presented a new proof of the GibbardSatterthwaite theorem that is based on the pivotal-voter technique developed earlier by Barbera (1983). It is direct and simple, invoking neither the Arrow theorem nor any monotonicity argument. Yet it is so powerful that under its framework many interesting issues can be addressed. The authors used it to prove that if the set of allocations is a metric space (not necessarily a subset of some finite-dimensional Euclidean space) and the space of admissible preferences contains all continuous functions, then any strategyproof mechanism is dictatorial. The shortcoming of their work is that some double-peaked utility functions are used in an essential way. Thus it failed to deal with the convexity of preferences, which is regarded as an important property for most economic models. In this paper I am able to use the pivotal-voter technique to establish some impossibility results for economies with pure public goods. It is shown that one simple dimension condition, analogous to the cardinality condition in the Gibbard-Satterthwaite theorem, plays an important role in our results. It is also shown that although the space of all continuous, convex preferences is usually associated with economic environments, some smaller subspace of it (for example, the subspace of quadratic utility functions) is sufficient for a negative result to emerge,^ The paper is organized as follows. In Section 2, the main results are stated and they are compared to some existing works on this subject. A formal proof is presented in Section 3. Section 4 contains an application of the result to public goods economies in which admissible preferences are further assumed to be strictly increasing. Finally, we have a brief discussion on private goods economies and mixed economies in Section 5. 2. In the work by Maskin (1976), Kalai and Muller (1977), and Ritz (1985), it is established that a restricted domain admits a non-dictatonal Arrovian welfare function if and only if it admits a non-dictatorial, strategy-proof social choice procedure. However, it is important to recognize that the above relationship does not hold for Arrovian welfare functions and strategy-proof mechanisms (as they are usually defined). Otherwise our work would be vacuous since it is known that there exist no non-dictatorial Arrovian social welfare functions in our models. The definition of a sodal choice procedure is different from that of a mechanism as the former requires substantially stronger consistency conditions. Unfortunately, the lack of a common terminology has lead to misconceptions about these results—even among many economists. One can find in Barbera, Sonnenschein, and Zhou (1990) an example of a domain that admits a class of non-trivial strategy-proof mechanisms but no non-dictatorial Arrovian social welfare functions; and in Kalai, Muller, and Satterthwaite (1979) a domain with the opposite set of characteristics (Example C in Section 1).
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Impossibility of Strategy-Proof Mechanisms ZHOU
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2. THE MODEL AND MAIN THEOREMS There are n agents in society. Society has to choose an allocation from a set of feasible allocations. A is the set of all conceivable allocations. It is a convex set in some finite-dimensional Euclidean space. For any set BczA^Co{B) denotes the convex hull of B; dim {B) the dimension of B, which is the dimension of the smallest affine superset of B; and #{B) the cardinality of B. Each agent has a complete and transitive preference relation over A H denotes the space of all admissible preferences of the agents over A Particularly, He denotes the space of all continuous and convex preferences over A; and £IQ denotes the space of the preferences that can be represented by a quadratic function M(X) = - ( X —a)^H(x —a), in which H is a positive definite matrix, and a is a point in A Given a preference relation /? in n and a set B e A, Argmax (R; B) denotes the set of maximal points of R on B, and argmax {R\B) denotes the same set if it contains a unique maximal point, fl" denotes the product space: n" = n x n x - - x n . A generic point R = {Rx, JRJ, • -, ^n) in H" is called a preference profile, in which R, is the /-th agent's preference relation over A. Sometimes R = (i?,, R2y...» Rn) is simply written as (i?,, i?_,). A direct mechanism, or simply a mechanism, is a function f:€i"-^A, which maps each preference profile to an allocation.^ Since A may contain allocations that are not feasible, / is usually not onto A The range of / is denoted by Af. If each individual i announces a preference relation -R/eH, then f{Ri, R2,,,, y R„) is society's chosen allocation. Definition 1. A mechanism/is strategy-proof IHOT any profile /? = (/?,, i?2, •••, ^n), any agent f, and any Q, in H, f(R,,R.,)R.J(Q,,R^dIt follows directly from the definition that if a mechanism / is strategy-proof, then it is always a dominant strategy for each agent to report the truth. For more detailed discussion of the concept of strategy-proofness and related issues, readers are referred to existing literature, for example, Muller and Satterthwaite (1986). Our aim here is to determine the structures of strategy-proof mechanisms. We first look at a special class of mechanisms. Definition 2. A mechanism / is strongly dictatorial if there is an agent i (the strong dictator) such that for all preference profiles 1? = (R,, K 2 , . . . , /^«), /(i?) = argmax (/?.; A/). A strongly dictatorial mechanism, if it exists, is strategy-proof. But it does not always exist. The unique maximal point on the right-hand side is not well-defined unless additional assumptions on either Q. or Af are made. To avoid this non-existence problem in our general discussion, we consider the following class of mechanisms. Definition 3. A mechanism / is weakly dictatorial, or dictatorial, if there is an agent i (the dictator) such that for all profiles R = (R^, /?2, • • •, Rn), f(R)eATgm2Lx{RryAf). 3. I will only discuss direct mechanisms in the paper. Nevertheless, the results can easily be generalized via the "revelation principle" to arbitrary mechanisms that allow agents to use a variety of strategies.
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A dictatorial mechanism is hardly a satisfactory solution for social decision-making since it mainly represents a single agent*s interest. A natural question then is: does there exist any non-dictatorial strategy-proof mechanism? The Gibbard-Satterthwaite theorem gives a negative answer to this question when admissible preferences are unrestricted. It states that any strategy-proof mechanism / on an unrestricted domain of admissible preferences is dictatorial whenever #^4/ ^ 3 . This result reveals the essential difficulty in social decision-making when the relevant information is private. However, the assumption of unrestricted domain severely limits its application to any interesting economic model in which individuals* preferences have structures prescribed by economic theory. Since then many researchers have been investigating the validity of the Gibbard-Satterthwaite result in various economic models. Here we consider a general model of economies with pure public goods. We assume that the allocation set >4 is a convex set in some finite-dimensional Euclidean space E'' and that the preference space is He, the space of all continuous and convex preferences over A. It turns out that a negative result prevails. Theorem 1. Any strategy-proof mechanism f on 0.c satisfying dim (Ay) ^ 2 is dictatorial The dimension condition dim (Af)^2 in Theorem 1 is exactly the counterpart of the cardinality condition #Af^ 3 in the Gibbard-Satterthwaite theorem. These two conditions are related in two ways. First, they are generally equivalent. Obviously the former implies the latter. Although the converse is not always true, if # A / ^ 3 and these points do not lie on the same straight line, then dim {Af) ^ 2. Secondly, they play the same role in different contexts. When #A/ = 2, simple majority voting provides a counter-example to the Gibbard-Satterthwaite theorem. In our model, if dim(A/) = l, then a convex preference relation restricted on Af becomes single-peaked. It is easy to verify that the mechanism that always chooses the median voter's most preferred outcome is strategyproof and non-dictatorial (see Moulin (1980)). We now compare Theorem 1 to two of the most relevant previous results on this subject. Border and Jordan (1983) established a negative result similar to the GibbardSatterthwaite theorem for a specific case in which the allocation space is some finitedimensional Euclidean space E'^ (or at least a direct product set in E^) and the mechanism is onto the allocation space. Their work, however, depends strongly on these assumptions; therefore it does not generalize. An earlier result by Satterthwaite and Sonnenschein (1981) linked the property of strategy-proofness to that of local dictatorship. An agent i is a local dictator at some preference profile if a small perturbation of other agents' preferences does not change the local structure of the set of allocations he can achieve. They considered the following model: the set of allocations A is a convex subset of E'^ and F is a differentiable allocation mechanism on an admissible utility function space n that is an open convex subset of C~{A). It was proved (under some other technical conditions): if F is strategy-proof, then at each regular point of F there is a local dictator. They concluded that this type of degeneracy is almost like that of global dictatorship; and therefore their result is parallel to the Gibbard-Satterthwaite theorem. The median voter mechanism, nevertheless, reveals a serious deficiency of their work. In this case, at any regular point the median agent is a local dictator. Hence it bears the degeneracy asserted by Satterthwaite and Sonnenschein. Still one is quite satisfied with this mechanism since it is the median voter's choice that best represents the compromise that the society is seeking. Thus there is no evidence against local dictatorship. In fact, if one could find
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for cases in which dim {Aj) ^ 2 any strategy-proof mechanism with characteristics similar to the median-voter mechanism, it would be considered a positive result, regardless of whether it is locally dictatorial or not. Consequently, a real negative result is called for to demonstrate the impossibiHty of such mechanisms in higher-dimensional cases. This point was not observed in Satterthwaite and Sonnenschein's work. As we have already argued that Theorem 1 gives us a result corresponding to the original Gibbard-Satterthwaite theorem in a model usually associated with pure public goods economies. It applies directly to the location problem and many other similar problems- However l i e , the space of admissible preferences considered in Theorem 1, is much larger than is needed. One important subspace of He, the space of all quadratic preferences is large enough for a negative result to emerge. Definition 4. An admissible preference space ft is abundant on some set B c A if it contains all quadratic preferences on B, i.e. for any u € ilc there exists some veCt, such that U\B= visTheorem 2. Assume Q, is abundant on some convex set Be: A. Any strategy-proof mechanism f on ft" satisfying dim (Af) ^ 2 and Af^B is dictatorial. Theorem 2 demonstrates that ftp has enough variety of preferences to make strategic manipulation inevitable. Of course, this is again by no means necessary. One will see from the proof that many other spaces can serve the same purpose. Roughly speaking, an important point is that such a preference space should be rich enough so that it is closed under any nonsingular transformation. This is also supported by the work of Border and Jordan (1983).^ It is easy to see that Theorem 1 is just a si>ecial case of Theorem 2. We single Theorem 1 out because it has a very clean form that matches the original GibbardSatterthwaite theorem. On the other hand. Theorem 2 is much sharper and its flexibility allows us to solve problems with different restricted domains of preferences. Such an application is shown in Section 4. 3. PROOF OF THEOREM 2 Since we mainly deal with preference relations that have utility function representations, we v^ll use utility functions instead of preferences in our discussion. We begin with two basic properties of any strategy-proof mechanism / on any domain. If / is strategy-proof on some ft", then for any pair of utility function profiles (M, , M2, • • •» "n) and (t?,, t;2, • • •, ^n)> the following inequalities hold: « . ( / ( « ! , U-,))^M,(/(t?,,P_,)),
UnifiUy,
M2, ' . . , W n ) ) ^ Wn(/(Wl, W2, . • • , t)„)).
From these inequalities, we can derive two lemmas. 4. In their paper, they derived a class of non-dictatorial strategy-proof mechanisms for the space of all separable quadratic preferences, which is not closed under linear transformations with non-zero off-diagonal elements. Then they were able to establish a negative result for preference spaces which allow arbitrarily small off-diagonal perturbations. However, this strong result was established under the strong assumption that mechanisms are onto. It seems unlikely to be true for general mechanisms.
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Lemma L For any wefl, Argmax{u;A,) Argmax (M; Ay).
is nonempty, and
f{u,u,..,,u)e
Proof. For any a e Aj, find some (u,, i?2, • • -, Un) such that /(t;,, i?2,..., t?„) = a Successive applications of the above inequalities lead to «(/(«, M, . . . , u)) ^ w(a). || Lemma 2. For any preference profile ( M| , "2, • •, "n), '/ ^''^''^ '-^ 5ome a € y4/ 5MC/I r/ia/ a = argmax (M,; Af) for allj, then / ( M , , M 2 , . . . , M „ ) = a-
Proof Take any profile (u,, t;2,. • . , v„) such that / ( v , , t;2r • • -, ^n) = «- Apply the above inequalities successively. Since a is the unique maximal point for each Uj, not only the utility values but also the outcomes on both sides of each inequality must be equal. || The properties stated in Lemma 1 and Lemma 2 are two different expressions of conditional unanimity, i.e. unanimity on the range of the mechanism. The first requires that when all agents have the same utility function the mechanism choose an allocation that is at least as good as any other allocation in its range. The second requires that when all agents consider a specific allocation as the unique best allocation in its range the mechanism choose this allocation. To prove Theorem 2, we first look at a subspace H/ of O consisting of preferences that are more nicely-behaved and thus more easily dealt with: D,f = {ueD.\u is continuous and strictly quasi-concave, and argmax (M; A,) exists.} (Note that a utility representation for a strictly convex preference relation is strictly quasi-concave.) In most part of the proof we will consider/*, the restriction o f / on ft", and try to demonstrate that/* is dictatorial. Then a simple argument leads to that/ is dictatorial. It is clear that/* is still strategy-proof. Furthermore, the range o f / * remains the same as that o f / Therefore, the dimension condition in Theorem 2 still holds for/*. To show Aj* = Af, we take any aeAf. Since ft is abundant on some B 3 A,, there is a M6ft such that u\B = -\\x-a\\^[». Obviously a = argmax (M; A,), hence by definition, u e ft/, and a =f(u, « , . . . , u) by Lemma 2. Thus aeAf*. We now work with/* on ft". For convenience of notation, however, we keep using / instead of / * • We first introduce the basic idea of the pivotal-voter technique. It is captured in the following concept. For each agent r, given his utility function M„ define the option set for agents other than i by: 0_,(«j) = {ae A [there exists M_J eft""*such that a =f{Ui, «_,)}. This is the set of allocations which agents other than 1 can achieve collectively when agent Vs utility function is fixed at M,. Or in other words, (?_,(«,) is the range of the restriction of / on ft/"* 'when w, is fixtd. It is direct to observe that if an agent i is a dictator, then for any u„ 0_,(Wi) = argmax (w,; A,). More importantly, the converse of the above observation is also true, i.e. if there is an agent i such that 0_,(ti,) = argmax (M,; A/) for all ti^, then he must be the dictator. Therefore, we can prove the theorem by showing the existence of such an agent r. We do this in several steps.
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Step 1. 0_ j(Mj) IS closed. Proof, Wefirstshow that Af is closed. For any point a e Bd{Af)^ the boundary of Afy M(X) = —||x--a||^ belongs to O, where ||-1| is the Euclidean distance.^ By Lemma 1, Argmax (w; Af) is non-empty. However, it can be nothing but {a} because a e Bd{Af), Hence a G Af. This means that Aj is closed. Now take any ae Bd(0-i{ui)). Since Af is closed, aeAf. Thus u(x) = ~\x — a^^ belongs to H/. Notice that when w, isfixed,the restriction of/onft"~*is still strategy-proof and its range is nothing but 0_,{Wj). It then follows from what we have proved that 0-i{Ui) is closed. || Step 2.
For any «,, argmax (u,\ Af) e 0_,(M,).
Proof, By the definition of Hf, argmax {Ui\ Af) is always well-defined. Now take any (u,, t?_,) such that /(t?„ t;_,) = argmax («,; Aj), f is strategy-proof implies "/(/("i,t?-J)^M,(/(t;,-,i;_,)).
This means that argmax ( MJ , A^ ) = / ( M, , t;_i) G 0_ j( M, ). || In order to continue our discussion we need some notation. Let a, b be two points in some £ ' \ denote (a, 6), (a, 6], [a,fe),and [a, 6] the segments, open or closed, connecting them. Given two sets S and T, we say that S is star-shaped (relative to T) with respect to a base point 6 e 5, if for any ceS^^^c^bl^r^T^ S. Step 3. 0-.;(M,) is star-shaped {relative to Af) with respect to argmax (M,; Af), Proof Suppose the statement is false. Then we can find a and b such that a € 0-i{Ui)ybe Aj\0-i{Ui), be {a, argmax (Ui;Af)), Since 0_,(w,) is closed, we can further assume, without loss of generality, that there exists /? = A(6 — a), A > 0 , such that (a, 6 + 2p]n O_i(M,) = 0 . (See Figure 1.) Let 11 denote the straight line passing a and b. Choose c = j{a-hb)+p and construct a sequence of utility functions u^""^ in ilfi M<'">(x) = - ( x - cyH^"'\x - c), in which the positive definite matrix f/*'"Ms chosen so that any indifference curve of M^*"* is an elliptical ball obtained by shrinking a standard ball by a factor of 1/m to 11 in all directions orthogonal to II. Consider the sequence of profiles {(«,, wL^O}, in which 0_.(u,>
argmax («,; Ay)
FIGURE 1
5. The precise statement should be that there exists weft such that M(X)|^^ = -j|x-fl||"|/»,. However, our use of language makes the argument a little simpler, without any effect on its validity. This same remark applies on several subsequent occasions.
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for all j V i, and the sequence of allocations {/(«,, M-7 )}- Since when M, is fixed, / is a strategy-proof mechanism on [1}'^ for agents other than r, u^'"\f{Ui, uL70) = u^'"\a) for all m by Lemma L This means {/(«,, uL70} is a bounded sequence; therefore it (or a subsequence of it) converges to some point d. By the construction of the sequence, d is on [a, b + 2p]. And since 0_,(M,) is closed, d is also in 0_i(M,). Thus d = a because (a,6 + 2p]nO.,(ti,) = 0 . On the other hand, let Vi{x)^ - \\x- b\\^ and consider the sequence {/(u,, MLT')}For the same reason as above, {/(t^„ w-7^)} converges to some e€ [a+2p, b]. When {/(M„ tiL7^)} converges to a and {/(t?„ uL70} converges to e, the corresponding sequences {w,(/(w„ tii.70)} and {Ui{f(vi, ML7^))} will converge to M,(a) and w,(e) because Wf is continuous. Since/is strategy-proof, uAfiUi, Mi.7^)) = M,(/(i?f, wL70) for all m. Taking the limit, we have M,(a)^ M,(e). But the strict quasi-concavity of u, implies M,(argmax (ti,-; Af))> u,{e)> Ui(a). We have a contradiction. This proves our claim. Step 4.
||
For any pair u, and v-, in Q.f^ if argmax (ii;; Aj) = argmax (t;,; Af)^ then 0_,(M,)=0_,(t?,).
/Voo/ Suppose it is not true. Since both 0_,(M,) and 0_f(t?,) are star-shaped (relative to Af) with respect to the same point J = argmax (Mj;i4/) = argmax (t;,;i4^), they must differ on some ray starting from d. We can assume, without loss of generality, that there exist a and h such that a 6 0_,(u,), 6G 0_,(Mi), &€(a,argmax (M,; A/)], and [a^h)r\ O^,{Ui) = 0. (See Figure 2.) Denote II the straight line passing a and h As in step 3, we construct a sequence of utility functions u^^"^ in Clfi M<->(x) = -(x -ay&'"\x
- a),
in which G^""^ is also similarly defined. Consider the sequence of allocations {/(M„ ui70} in which Uj'"^ = u^""^ for all j9^ i. Using a similar argument as in Step 3, we can show that {/(W„ MI.7^)} converges to b. Since/is strategy-proof, t?,(/(t?„ ML7^))^ r,(/(w,., ML7^)) forall m. Taking the limit, we have t?,(a)^ t?,(6). But the strict quasi-concavity of u, implies Vi{d)^V:(b)>vM^ This is a contradiction. Step 5. all u, G H/.
||
Either (i) O.i(Ui) = argmax («,; Ay), for all u, G£1/, or (ii) 0_,(M,) = A/, for
0_.(u.) f/ = argmax (u^\ A,) = argmax (z;,;v4/)
FIGURE 2
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Proof. We first show that for any given w? 6 ft/, either 0_i(M,) = argmax (M,; Af)^ or 0_,{M,) = Af. If this is not true, then there exist a and h such that a 9L 0.i{Ui), b e 0_;(w,), and 57^ argmax (M,.; Af). (See Figure 3.) The condition that dim (Af)^2 guarantees that we can find a pair like this also satisfying that a, h, and argmax (M;; AJ) are in general position, i.e. they do not lie on a single straight line. Let 11, denote the straight line passing a and argmax (u,; Af)^ and 112 the straight line passing a and h. Take M(X) = - ||x - a |p and shrink the indifference curve of it to 112 in all directions orthogonal to 112. We can get a utility function veQ.f such that a = argmax (t;; A,), and no point on [a, argmax (ii,; A/)] belongs to Argmax (t?; 0_^(M,)). Since Argmax (t?;0_;(Mf)) is compact, we can also apply the above procedure to - | | x - argmax (M,-; Af)f to find a t?,- e ft/ such that argmax (t?,; Af) = argmax {ur, Aj), and Vi(a)>Vi{c) for any c in Argmax (i;; 0_,(t/j)). Consider the profile (u„U-,) in which Vj = v for all y 7^ IRVi, V-i) e Argmax (t?; 0_j(t?,)) by Lemma 2. Since 0_,(Ui) = 0_i(U|) by Step 4, /(t?f, t;_,) 6 Argmax (i?; 0_,(Mi)). But when agent i with utility function v, falsely announces that he has utility function - [ | x - a f , the allocation would be a by Lemma 2, which is better to him than/(t>„ t>_,). This contradicts the fact that/ is strategy-proof. Now we show that if O.^w,) = argmax(Uf;A/) for some M;eft/, then 0_i(t?,) = argmax (t?,; Af) for all v, e ft/. Suppose that it is not true. By what we have just shown, there must exist some M,-€ft/ such that 0_,(Mi) = A/. Notice that argmax (t?,; A,^) 5^ argmax (M,-; Af) by Step 4. Add to them some ceAf such that these three points are in general position. Then an argument similar to that in the above paragraph will lead to a contradiction. t| Step 6.
There is an agent i such that for all Ufenj, 0-,{Ui) = argmax (M,; A,).
Proof We proceed by induction on n, the number of the agents. The case /i = 2 is simple. If the statement is not true, then for any (wi, Uj), 0_,(M,) = Af, and 0_2(«2) = Af, f is strategy-proof implies /{M, , M2) = argmax (MJ ; A,), and /(M, , U2) = argmax (M2; Af), But it is impossible for profiles (w,, U2) in which argmax (u,; Af) ^ argmax (MJ; A,). Now we assume the statement is true for n = k. Let us consider the case n = k+l. If the statement is false, then by step 5, for any profile (wi, M2, - •»w^t+i), ^-i(w») = ^ / for all agents. Therefore, when we fix any agent's utility function,/is still a mechanism for the other k agents that satisfies the conditions of the theorem. If we first fix some t>, Gft/, then by the induction hypothesis, we can find some i?^ 1 such that f(vi,U2,...,
Ufc+i) = argmax (ti,-; Af),
for any «_, e ftp'.
FIGURE 3
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But if we fix some Vi such that argmax (u,; Af) 7^ argmax (uj; Af), again by the induction hypothesis, we can find another agent j 1^ i such that / ( " i , "2, • • -, t^„ • • -, Wfc+i) = argmax (uj; Af)
for any M_, G H;"*.
If we choose some M, such that argmax (M,; Ay) 7^ argmax (i?,; A/), and some M_, in which M, = Vi (these two conditions are consistent even when/ =1), then the above two equations lead to a contradiction. Thus we complete the induction. || Up to now, we have found a dictator i for the restriction o f / on H". To conclude the proof of Theorem 2, we have to demonstrate that he is indeed a dictator f o r / o n H". To see this, consider profiles in which the dictator's preference relation belongs to ft/ while the other agents might have preference relations outside it. If there were such a profile that could lead to an allocation different from the dictator's best, then it is easy to find a manipulation by some agent other than the dictator (using an argument similar to that in Step 5 and applying induction on the number of agents who have preferences outside D.f), Thus the dictator can enforce any allocation aeAf by claiming that he has the utility function — ||a - x||^. Therefore,/is strategy-proof implies/(ti) € Argmax (ui; Af) for any preference profile u e ft". This completes our proof.
4. AN APPLICATION In the problem introduced at the beginning of the paper, admissible preferences are also assumed to be monotonically increasing. This property is very common in many problems associated with public goods. Theorem 2 cannot deal with this type of problem directly because ft^ contains preference relations that have bliss points. Nevertheless, we can apply it in a roundabout way. Let us consider A* = JB+, the non-negative orthant of A:-dimensional Euclidean space JEfc, k^3. ft* is the space of continuous, strictly concave, and strictly increasing utility functions on E+. This model is standard and has been considered by many authors.^ Since here we are considering allocation mechanisms that are related to decision-making instead of just ranking alternatives, some feasibility constraint should be imposed. It not only makes the model more realistic, it also keeps the problem well-defined (given that the utility functions are strictly increasing the range of any strategy-proof mechanism must be properly bounded by Lemma 2 in Section 3). For simplicity, we assume that it is given by A** = { x e £ i | Z p ^ ; ^ / } , where p.'s and / are all positive numbers with Pi representing the price of public good i and / the total budget. Let a(A**) denote the set {xe E+|Xp,^. = /}• A mechanism is again a function/:(ft*)"-> A**, which chooses a feasible allocation for every preference profile in (ft*)". Finally, we impose another condition that is intuitively appealing. Definition 5. A mechanism / is unanimous if for any weft*, /(M, M,..., M) = argmax (M; A**). 6. For example, Kalat, Muller, and Satterthwaite (1979) considered this model and proved the nonexistence of Arrovian social welfare functions.
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Unanimity, as defined above, is a very natural and mild requirement. In fact, if we require that / be onto A**, then it is a consequence of strategy-proofness by Lemma 1. Our next theorem demonstrates that a unanimous mechanism violates incentive compatibility unless it is strongly dictatorial. Theorem 3. Any unanimous mechanism f: (ft*)" -^ A** is strategy-proof if and only if it is strongly dictatorial Proof It is trivial that / is strongly dictatorial implies that / is strategy-proof. Now assume that / is a strategy-proof and unanimous mechanism. If we can show that ft* is abundant on the range of f then / is dictatorial according to Theorem 2. Thus / is strongly dictatorial since Argmax (M; A**) is always a singleton when u is strictly concave. We first show that the range o f / is 5(A**). Given a e d(A**), it is easy to find some weft* such that a = argmax (M; A**). Thus, / is unanimous implies that a belongs to the range off On the other hand, suppose a belongs to the range o f / Rnd a preference profile (u^, "2, • • •, "n) such that a —f(ux, t i 2 , . . . , M„). We normalize M,, 112, • • •, "« so that M,(a) = ti,(a) for all 1*5^ 1. Then we construct a utility function u as follows: M{X) = Min,<,-^„ {ii;(x)} + Min,^,^, {(x, -f l)/(a,- -f 1)}. Since ti^eft* for all j , Min, {M^(x)}€ft*. It is obvious that Min, {(x,-f l)/(fl| + l)} is continuous, weakly concave, and weakly increasing. Hence, when adding it to some utility function in ft*, the sum still belongs to ft*. Thus weft*. And we further claim that if M(X)^ w(a) for some x?^ a, then M,(X)> Mj(a) for all i. This is because there are only two possible cases: (i) x, ^ a, for all i; or (ii) x, < a, for some i. In the first case, since M, is strictly increasing, M,(X) > M,(a) for all i. In the second case, Min, {(x, + l)/(af + 1)}< 1 = Min, {(ai + l)/(ai +1)}. Because M ( X ) ^ 11(a), it must be true that Min^ {Uj{x)}> Miny {M^(a)}. This also leads to iij(x)> w,(a) for all i. We now replace M, in (M, ,112,..., u„) by M. / is strategy-proof implies "(/(w, "2, •. • , "n))= "(/(MI , W2,..., u„)) = u(a),
and M , ( a ) = W I ( / ( M , , M 2 , . . - , t i „ ) ) ^ W , ( / ( M , W2,. - . , M«».
These two inequalities lead to /(w, ti2, .••,«„) = a. Doing this repeatedly for all i leads to /(M, M, . . . , W) = fl. Since / is unanimous, a = argmax (M; A**). This is possible only when a Gd(A**). Therefore, the range o f / is diA**). Second we show that ft* is abundant on a(A**). Given any quadratic function w(x) = - ( x —fl)^//(x-a), we define Vs(x) (s is a scalar) by:
vAx)=--(x-a~sH-'pyH{x-a-sH-'p), We can write t),(x) as
v„{x) =
-(x-a-sH~'pyH(x-a-sH-'p)
= M(X)-2S(X-CI)V + 5 V W ~ V
Since the second and the third terms are constants on a(A**) = {x e E+ |Z Pi^i = ^}> "(^) and u,(x) represent the same preference relation on 5(A**). The gradient of v^ix) is proportional to s p - H ( x - a ) . If we choose an 5 large enough, then all the first-order derivatives of Vs{x) are positive on any bounded set in E+, especially on A**. Therefore, t), is also increasing on any bounded set in E+. Then it is not difficult to construct some
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weft* such that w coincides with v^y thus u on d(A**). Hence, H* is abundant ond{A**). II 5. DISCUSSION In this paper we use the pivotal-voter technique to establish some Gibbard-Satterthwaite type results for pure public goods economies in which agents' preference relations satisfy classical assumptions of continuity, convexity, and monotonicity. For mechanisms in such environments, the requirement of strategy-proofness alone essentially leads to dictatorship. This strong negative result does not hold for private goods economies or mixed economies. The best known example is the Groves scheme (see Groves and Loeb (1975), and Holmstrom (1979)). But this does not mean that the problem of manipulation is absent in private goods economies or mixed economies. There are other properties, say Pareto optimality, that arc important when we consider a mechanism. In case of the Groves scheme, it has been shown that it is generically non-optimal (Hurwicz and Walker (1990)). In general if one considers mechanisms that are both strategy-proof and Paretooptimal (together with some other necessary qualifications), then negative results very often will arise. Contributions by Dasgupta, Hammond aand Maskin (1979), Hurwicz (1972), Moreno and Walker (1989), and Satterthwaite and Sonnenschein (1981), have provided many such results in private goods economies or mixed economies. However, there is still not a result that is as clean and general as the results we have for pure public goods economies. It remains to be seen whether the technique or the results in this paper can lead to some substantial progress in this direction. Acknowledgement. I started this research while I was visiting Universitat Autonoma de Barcelona in 1988. It was motivated by a paper of S. Barbera and B. Peleg. I thank Barbera for his insightful suggestions. I also thank H. Moulin, H. Sonnenschein, and the referees for helpful comments. My visit was partially supported by Research Grant PB 86-0613 from the Direccion General de la Investgacion Cientifica y Tecnica, Spanish Ministry of Education. I am grateful to the Alfred P. Sloan Foundation for the financial support during the academic year 1988-1989. REFERENCES ARROW, K. J. (1963) Social Choice and Individual Values (New York: Wiley). BARBERA, S. (1983), "Strategy-Proofness and Pivotal Voters: A Direct Proof of the Gibbard-Satterthwaite Theorem", International Economic Review^ 24, 413-417. BARBERA, S. and PELEG, B. (1990), "Strategy-Proof Voting Schemes with Continuous Preferences", 5acia/ Choice and Welfare, 7, 31-38. BARBERA, S., SONNENSCHEIN, H. and ZHOU, L. (1990), "Voting by Committees", Econometrica (forthcoming). BORDER, K. C. and JORDAN, J. S. (1983), "Straightforward Elections, Unanimity and Phantom Agents", Review of Economic Studies, 50, 153-170. DASGUPTA, P., HAMMOND, P. and MASKIN, E. (1979), "The Implementation of Social Choice Rules", Review of Economic Studies, 44, 153-170. GIBBARD, A. (1973), "Manipulation of Voting Schemes: A General Result", Econometrica, 41, 587-602. GROVES, T. and LOEB, M. (1975), "Incentives and Public Inputs", Journal of Public Economics, 4, 311-326. HOLMSTROM, B. (1979), "Groves' Scheme on Restricted Domains", Econometrica, 47, 1137-1144. HURWICZ, L. and WALKER, M. (1990), "On the Generic Non-Optimality of Dominant-Strategy Allocation Mechanisms: A General Theorem that Includes Pure Exchange Economies", Econometrica, 58, 683-704. KALAI, E. and MULLER, E. (1977), "Characterization of Domains Admitting Nondictatorial Social Welfare Functions and Nonmanipulable Voting Procedures," Journal of Economic Theory, 16, 457-469. KALAI, E., MULLER, E. and SATTERTHWAITE, M. A. (1979), "Social Welfare Functions When Preferences Are Convex, Strictly Monotonic, and Continuous", Public Choice, 34, 87-97. MASKIN, E. (1976), "Social Welfare Functions on Restricted Domains" (Mimeograph, Harvard University). MOULIN, H. (1980), "On Strategy-Proofness and Single Peakedness", Public Choice, 35, 437-455. MORENO, D. and WALKER, M. (1989), "Nonmanipulable Voting Schemes When Participants' Interests Are Partially Decomposable", Social Choice and Welfare (forthcoming).
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MULLER, E. and SATTERTHWAITE, M. A. (1985), "Strategy-Proofness: the Existence of Dominant-Strategy Mechanisms", in Hurwicz, L., Schmeidler, D. and Sonnenschein, H. (eds.) Social Goals and Social Organization, (Cambridge: Cambridge University Press). RITZ, Z. (1985), "Restricted Domains, Arrow Social Welfare Functions and Noncorruptible and Nonmanipulable Social Choice Correspondences: The Case of Private and Public Alternatives", Journal of Economic Theory, 35, 1-18. SATTERTHWAITE, M. A. (1975), "Strategy-Proofness and Arrow's Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions", Journal of Economic Theory, 10,187-217. SATTERTHWAITE, M. A. and SONNENSCHEIN, H. (1981), "Strategy-Proof Allocation Mechanisms at Differentiable Points", Review of Economic Studies^ 48, 587-597. SCHMEIDLER, D. and SONNENSCHEIN, H. (1978), "Two Proofs of the Gibbard-Satterthwaite Theorem on the Possibility of a Strategy-Proof Social Choice Function", in Gottinger, H. and Ensler, W. (eds.) Decision Theory and Social Ethics, (Dordrecht, Holland: Reidel).
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23 Zachary Cohn on Hugo F. Sonnenschein
I met Hugo as a mathematics undergraduate at the University of Chicago, when I took a game theory course he was teaching the fall of my junior year. I greatly enjoyed the class, though my background in economics at the time was limited to whatever he had done the previous day. After the midterm exam, Hugo invited me to his to his office to chat, which started a wonderful conversation that continued each week for the next two years. Though simple, I am still struck by how welcoming Hugo was; I cannot recall another time when a professor was as inviting, or as genuinely interested and excited in talking with a student. I can no longer recall what grand plans I had for our meetings, but Hugo mentioned his work with Matt Jackson and soon I was trying to work out some details about the efficiency when appUed to linked bargaining. After a few diversions, that work eventually evolved into the paper here. My meetings with Hugo would always start with a trip to the Divinity School Coffee Shop, where he would get a cup of coffee and I some tea (evidence, he would say, that I was a mathematician), and often wind up some time later when his office phone would ring, wondering when he would be coming home for dinner. The meetings were always engaging, and were a defining point of my undergraduate career. I can only hope to one day repay him by trying to be as patient and capable with my own students. To a great friend and educator, happy birthday! Editors' note: The paper "On Linked Bargaining" is in process for publication, but is not yet published. Thus, due to copyright issues we do not include it here.
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