On existence of undominated pure strategy Nash equilibria in anonymous nonatomic games: a generalization

Int J Game Theory (2002) 31:493498DOI 10.1007/s001820300132Giulio Codognatoy and Sayantan Ghosalzy 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 2001Abstract. 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 Classication Number: C72Key words: Nash equilibrium, pure strategy, nonatomic game1. IntroductionSchmeidler (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 purication result in Schmeidler (1973). In both these papers,

players have identical, nite 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 dierent 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 rst author would like to thank Shlomo

Weber for some conversations which inspired this work.On existence of undominated pure strategy Nash

equilibria in anonymous nonatomic games:a generalization*494 G. Codognato, S. Ghosalegy sets which are compact subsets of some nite 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 nite dimensional Euclidean spaces instead of function spaces