On existence of undominated pure strategy Nash ... - Springer Link

Int J Game Theory (2002) 31:493–498 DOI 10.1007/s001820300132

On existence of undominated pure strategy Nash equilibria in anonymous nonatomic games: a generalization* Giulio Codognatoy and Sayantan Ghosalz y

Dipartimento di Scienze Economiche, Universita` degli Studi di Udine, Via Tomadini 30, 33100 Udine, Italy, and SET, Universita` degli Studi di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy. Financial support from CNR Research Contribution 99.01549.CT10 is gratefully acknowledged. z Department of Economics, University of Warwick, Coventry CV4 7AL, United Kingdom. Received August 2001

Abstract. In this paper, we generalize the exitence result for pure strategy Nash equilibria in anonymous nonatomic games. By working directly on integrals of pure strategies, we also generalize, for the same class of games, the existence result for undominated pure strategy Nash equilibria even though, in general, the set of pure strategy Nash equilibria may fail to be weakly compact. Journal of Economic Literature Classification Number: C72 Key words: Nash equilibrium, pure strategy, nonatomic game

1. Introduction Schmeidler (1973) shows that a game with an atomless continuum of players has a pure strategy Nash equilibrium. Subsequently, Le Breton and Weber (1997) show, in the same framework as Schmeidler (1973), that an undominated pure strategy Nash equilibrium exists. Their proof relies on showing that the set of mixed strategy Nash equilibria is weakly compact which allows them to infer the existence of an undominated pure strategy Nash equilibrium, by using the purification result in Schmeidler (1973). In both these papers, players have identical, finite sets of pure strategies and linear payo¤ functions. In this paper, in a framework borrowed from Khan (1985) and Rath (1992), we generalize both these results to the case where players have di¤erent strat* We are indebted to Michel Le Breton and two anonymous referees for their comments and suggestions. We are also indebted to Marcellino Gaudenzi and Fabio Zanolin for their valuable contribution to the proof contained in the Appendix. The first author would like to thank Shlomo Weber for some conversations which inspired this work.

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egy sets which are compact subsets of some finite dimensional Euclidean space and payo¤ functions which are required to be continuous and measurable but not necessarily concave or integrable. Our results rely on the fact that we are able to work in finite dimensional Euclidean spaces instead of function spaces endowed with the weak topology. In particular, our result on the existence of pure strategy Nash equilibria generalizes Rath (1992) as it allows for heterogeneous strategy spaces. Under the same assumptions that guarantee the existence of a pure strategy Nash equilibrium, we construct an example which shows that the set of pure strategy Nash equilibria may fail to be weakly compact. Instead, our main result, the existence of an undominated Nash equilibrium in pure strategies, relies on the fact that we are able to work on the compact set of integrals of pure Nash equilibrium strategy selections, thereby avoiding the complication of mixed strategies. In Section 2, we introduce the model. The statement and the proof of our two results are contained in Section 3. Section 4 concludes. 2. The model The space of players is denoted by the atomless measure space ðT; T; mÞ, where T is the set of players, T is a s-algebra of subsets of T and m is a real valued, nonatomic, countably additive measure defined on T. We assume that ðT; T; mÞ is complete and finite, i.e., T is the s-algebra of all m-measurable subsets of T and mðTÞ < y (see Khan (1985)). A null set of players is a set of measure 0. Null sets of players are systematically ignored throughout the paper. Thus a statement asserted for ‘‘all’’ players, or ‘‘each’’ player in a certain set, is to be understood to hold for all such players except possibly for a null set of players. The word ‘‘integrable’’ is to be understood in the sense of ! Rn Lebesgue. Let R n denote the n-dimensional Euclidean space. Let K : T ! be the strategy set correspondence, i.e., KðtÞ is the set of strategies available to player t. We make the following assumption (see Schmeidler (1973), Khan (1985) and Rath (1992)). Assumption 1. (i) K is nonempty and compact valued; (ii) K is integrably bounded, i.e., there exists an integrable function h : T ! Rþn such that hðtÞ a KðtÞ a hðtÞ, for each t A T; (iii) K is measurable, i.e., its graph belongs to the product s-algebra T BðR n Þ, where BðR n Þ denotes the Borel s-algebra of R n . A strategy selection is an integrable function f : T ! R n such that, for all t A T, f ðtÞ A KðtÞ. ÐLet FK denote the set of all strategy selections and, for any f A FK , let sð f Þ ¼ T f ðtÞ dm and SK ¼ fsð f Þ : f A FK g. The utility function of player t is given by uðt; ; Þ : KðtÞ SK ! R. We make the following assumption. Assumption 2. (i) uðt; ; Þ is continuous on KðtÞ SK , for all t A T; (ii) uð ; ; qÞ is a measurable function on the graph of K, for all q A SK . We conclude this section with the following two definitions. Definition 1. A strategy selection f^ is a Nash equilibrium if, for all t A T, uðt; f^ðtÞ; sð f^ÞÞ b uðt; x; sð f^ÞÞ, for all x A KðtÞ.

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Definition 2. A strategy selection f is Pareto dominated by a strategy selection f 0 if uðt; f 0 ðtÞ; sð f 0 ÞÞ b uðt; f ðtÞ; sð f ÞÞ, for all t A T, with the strict inequality for a set of players with positive measure.

3. Two theorems First, we prove the existence of a Nash equilibrium. Theorem 1. Under Assumptions 1 and 2, there exists a Nash equilibrium f^. Proof: We follow and generalize a proof sketched by Rath (1992). For ! KðtÞ by Bðt; qÞ ¼ each t A T, define a correspondence Bðt; Þ : SK ! argmaxfuðt; x; qÞ : x A KðtÞg. By Assumptions 1 and 2 and by the maximum theorem (see Berge (1997) p. 116), Bðt; Þ is uppersemicontinuous, for each Ð ! SK by GðqÞ ¼ T Bðt; qÞ dm. SK is t A T. Define a correspondence G : SK ! nonempty, compact and convex, by Assumption 1 and, respectively, by Theorem 2 p. 62, Proposition 7 p. 73, and Theorem 3 p. 62, in Hildenbrand (1974) (these results generalize, respectively, Theorems 2, 1, and 4, in Aumann (1965)). For each q A SK , GðqÞ is nonempty, by Theorem 2 in Hildenbrand (1974) p. 62, because Bð; qÞ has a measurable graph, by Assumptions 1 and 2 and by the Application of Lemma III.39 in Castaing and Valadier (1977) p. 86. Moreover, by Theorem 3 in Hildenbrand (1974) p. 62, GðqÞ is convex, for each q A Sk . Finally, since Bðt; Þ is uppersemicontinuous, for each t A T, G is uppersemicontinuos, by Assumption 1 and by Aumann (1976). By Kakutani’s fixed point theorem, G has a fixed point q^. From the definition of G, there is a strategy selection f^ A FK such that sð f^Þ ¼ q^ and f^ A Bðt; q^Þ, for all t A T. But then, the strategy selection f^ is a Nash equilibrium. 9 In order to prove the existence of an undominated Nash equilibrium, we also need the following assumption (see Le Breton and Weber (1997)). Assumption 3. uð ; ; Þ is bounded, i.e., there exists a number M > 0 such that juðt; x; qÞj a M, for all t A T, for all x A KðtÞ and for all q A SK . Let F^K denote the set of the pure strategy Nash equilibria and let S^K ¼ fsð f Þ : f A F^K g. Khan (1985) shows that, under Assumption 1, the set of strategy selections Fk is compact in the sðL1 ðTÞ; Ly ðTÞÞ topology. The following example (see also Bewley (1972) and Le Breton and Weber (1997)) shows that, under Assumptions 1, 2, and 3, F^K may fail to be closed and, hence, compact in the sðL1 ðTÞ; Ly ðTÞÞ topology. This motivates why we need to show the compacteness of S^K in order to prove the existence of an undominated pure strategy Nash equilibrium. Example. Consider the following specification of a game satisfying Assumptions 1, 2, and 3, where T ¼ ½0; 1, T is the s-algebra of Lebesgue measurable subsets of T, m is the Lebesgue measure on T, KðtÞ ¼ fðx1 ; x2 Þ A Rþ2 : x1 þ x2 ¼ 1g, for all t A T, uðt; x; qÞ ¼ q1 x12 þ q2 x22 , for all t A T. For this game, the set of the pure strategy Nash equilibria is not compact in the sðL1 ðTÞ; Ly ðTÞÞ topology.

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Proof: Let f f^n g be a sequence of FK such that ( 2j ð1; 0Þ if there exists j ¼ 1; . . . ; n, with 2j1 2n a t < 2n ; ^ fn ðtÞ ¼ 2j a t < 2jþ1 ð0; 1Þ if there exists j ¼ 0; 1; . . . ; n 1, with 2n 2n : 1 1 For each n ¼ 1; 2; . . . ; as q^n ¼ sð f^n Þ ¼ 2 ; 2 , Bðt; q^n Þ ¼ fð1; 0Þ; ð0; 1Þg, and, then, f^n A F^K . The sequence f f^n g converges in the sðL1 ðTÞ; Ly ðTÞÞ topology to a point f^ A FK such that f^ðtÞ ¼ 12 ; 12 , for all t A T (for a detailed proof, see ^ B F^K since q^ ¼ sð f^Þ ¼ 1 ; 1 and, hence, f^ðtÞ ¼ the Appendix). Nevertheless, f 2 2 1 1 ^ 2 ; 2 B Bðt; qÞ ¼ fð1; 0Þ; ð0; 1Þg, for all t A T. As the sðL1 ðTÞ; Ly ðTÞÞ topol9 ogy is Hausdor¤, this implies that F^K is not compact in this topology. We are now ready to prove the existence of an undominated Nash equilibrium. Theorem 2. Under Assumptions 1, 2 and 3, there exists a Nash equilibrium f which is dominated by no other Nash equilibrium. Proof: First, we show that S^K is compact. Let f^ qn g be a sequence of S^K and let f f^n g be a sequence of F^K such that sð f^n Þ ¼ q^n , for each n ¼ 1; 2; . . . : By Assumption 1, the sequence f f^n g is uniformly integrable and, since SK is compact, the sequence fqn g has a subsequence (which we denote in the same way to save in notation) which converges to a point q^ A SK . But then, by Theorem A in Artstein (1979), there exists an integrable function f^ such that sð f^Þ ¼ q^ and such that f^ðtÞ is a limit point of f f^n ðtÞg, for all t A T. First, this implies that f^ A FK because f^ðtÞ is the limit of a subsequence of f f^n ðtÞg, for all t A T. Now, consider a player t A T and let f f^n ðtÞg denote a subsequence (which we denote in the same way as the original sequence to save in notation) which converges to f^ðtÞ. The fact that uðt; f^n ðtÞ; q^n Þ b uðt; x; q^n Þ, for each n ¼ 1; 2; . . . ; and for all x A KðtÞ, implies, by Assumption 2, that uðt; f^ðtÞ; q^Þ b uðt; x; q^Þ, for all x A KðtÞ. But then, f^ A F^K and q^ A S^K , thereby implying that S^K is compact. For each t A T, define a function mðt; qÞ : SK ! R by mðt; qÞ ¼ maxfuðt; x; qÞ : x A KðtÞg. By Assumptions 1 and 2, the maximum theorem (see Berge (1997) p. 116) implies that mðt; Þ is continuous, for each t A T, and Lemma III.39 in Castaing and Valadier (1977) p. 86 implies that mð; qÞ is measurable, for each q A Sk . As the space of players ðT; T; mÞ is finite, Assumption 3 implies that mð; qÞ is integrable, for each q A SK (see, for instance, Theorem E in Halmos (1974) p. 113). Define the function m : SK ! R Ð by mðqÞ ¼ T mðt; qÞ dm. We show that m is continuous on SK . Let fqn g be a sequence of SK converging to q. Since, by the continuity of mðt; Þ, the sequence fmðt; qn Þg converges to mðt; qÞ, for each t A T, Assumption 3 and the Lebesgue dominated convergence theorem imply that the sequence fmðqn Þg converges to mðqÞ, thereby implying that the function m is continuous on SK . The compactness of S^K and the continuity of m imply that there exists a point q A S^K such that mðq Þ b mðqÞ, for all q A S^K . Let f A F^K be a Nash equilibrium such that sð f Þ ¼ q and suppose that it is Pareto dominated by another Nash equilibrium f 0 A S^K . This implies that mðt; q 0 Þ b mðt; q Þ, where q 0 ¼ sð f 0 Þ, for all t A T, with the strict inequality for a set of players with positive measure. But then, mðq 0 Þ > mðq Þ, generating a contradiction. This

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implies that f is a Nash equilibrium which is dominated by no other Nash equilibrium. 9 4. Conclusion We generalize two existence results for pure strategy Nash equilibria in anonymous nonatomic games, provided by Schmeidler (1973) and Le Breton and Weber (1997), by exploiting, as suggested by Rath (1992), all the implications of anonimity formulated in terms of averages.

5. Appendix In this Appendix, we show that the sequence f f^n g, defined in the Example ^ of Section 3, converges, 1 1 in the sðL1 ðTÞ; Ly ðTÞÞ topology, to the function f ^ such that f ðtÞ ¼ 2 ; 2 , for all t A T ¼ ½0; 1. fgn g such Consider the sequence 1 1 ^2 1 1 2 ^ that, for each n ¼ 1; 2; . . . ; ðgn ðtÞ; gn ðtÞÞ ¼ fn ðtÞ 2 ; fn ðtÞ 2 , for all t A T. Clearly, the sequence f f^n g converges, in the sðL1 ðTÞ; Ly ðTÞÞ topology, to the function f^ such that f^ðtÞ ¼ 12 ; 12 , for all t A T, if and only if, the sequence fgn g converges, in the same topology, to the function g such that gðtÞ ¼ ð0; 0Þ, in the for all t A T. As Ly ðTÞ H L1 ðTÞ, the sequence fgn g converges, Ð1 sðL1 ðTÞ; Ly ðTÞÞ topology, to the function g if the sequence f 0 gn1 ðtÞfðtÞ dm; Ð1 2 0 gn ðtÞfðtÞ dmg converges to ð0; 0Þ, for each function f A L1 ðTÞ. As we can proceed componentwise, we denote by fgn g one of the components of the sequence fgn g itself. Let Ccy ðTÞ denote the vector space of all functions on T with continuous derivatives of all orders and with compact support in ð0; 1Þ. Ð1 We first show that the sequence fÐ 0 gn ðtÞcðtÞ dmg converges to 0, for each t function c A Ccy ðTÞ. Let Gn ðtÞ ¼ 0 gn ðtÞ dm, for each n ¼ 1;Ð 2; . . . : Consider 1 now c A CÐcy ðTÞ. As, for each n ¼ 1; 2; . . . ; 0 gn ðtÞcðtÞ dm ¼ Ð 1 a function 1 0 0 M 0 Gn ðtÞc 0 ðtÞ dm Ð 1 and j 0 Gn ðtÞc ðtÞ dmj a 2n , where M ¼ maxfc ðtÞ : t A Tg, the sequence f 0 gn ðtÞcðtÞ dmg converges to 0. Consider now a function f A L1 ðTÞ. Let e > 0. As Ccy ðTÞ is norm dense in L1 ðTÞ (see, for instance, Theorem 12.10 in Aliprantis and Border (1999) p. 433), there is a function Ð1 c A Ccy , depending on e, such that 0 jfðtÞ cðtÞj dm < e. Moreover, by the resultÐ shown above, there is a natural number n, depending on e and c, such 1 that 0 jgn ðtÞcðtÞj dm < 2e , for all n > n. But then, ð 1 ð 1 gn ðtÞfðtÞ dm ¼ gn ðtÞðfðtÞ cðtÞ þ cðtÞÞ dm 0

0

a
n. This completes the proof.

ð1 0

jgn ðtÞcðtÞj dm

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