Characterizing Pure-strategy Equilibria in Large Games

Fu, Haifeng; Xu, Ying and Zhang, Luyi (2007): Characterizing Pure-strategy Equilibria in Large Games. Unpublished.

Abstract

In this paper, we divide the players of a large game into countable different groups and assume that each player’s payoff depends on her own action and the distribution of actions in each of the subgroups. Focusing on the interaction between Nash equilibria and the best response correspondence of the players, we characterize the pure-strategy equilibria in three settings of such large games, namely large games with countable actions, large games with countable homogeneous groups of players and large games with an atomless Loeb agent space. Furthermore, we also present a counterexample showing that a similar characterization result does not hold for large games under a more general setting.

• Item Type: MPRA Paper
• Language: English
• Keywords: Large games, Pure strategy equilibrium, Characterization
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

Aliprantis, C.D., Border, K.C., 1999. Infinite Dimensional Analysis: a Hitchhiker’s Guide. 2nd Ed. Berlin: Springer-Verlag.

Blonski, M., 2005. The women of Cairo: equilibria in large anonymous games. Journal of Mathematical Economics 41, 253-264.

Fu, H.F., 2007. From large games to Bayesian games: a unified approach on pure strategy equilibria. Working paper. NUS

Khan, M.A., Rath, K.P., Sun, Y.N., 1997. On the existence of pure strategy equilibria in games with a continuum of players. Journal of Economic Theory 76, 13-46.

Khan, M.A., Sun, Y.N., 1995. Pure strategies in games with private information. Journal of Mathematical Economics 24, 633-653.

Khan, M.A., Sun, Y.N., 1996. Nonatomic games on Loeb spaces. Proceedings of the National Academy of Sciences of the United States of America 93, 15518-15521.

Khan, M.A., Sun, Y.N., 1999. Non-cooperative games on hyperfinite Loeb spaces. Journal of Mathematical Economics 31, 455-492.

Khan, M.A., Sun, Y.N., 2002. Non-cooperative games with many players. In: Robert Aumann and Sergiu Hart (Eds), Handbook of Game Theory with Economic Applications Volume III , Elsevier Science, Amsterdam; p. 1761-1808.

Kim, T., Yannelis, N.C., 1997. Existence of equilibrium in Bayesian games with infinitely many players. Journal of Economic Theory 77, 330-353.

Loeb, P.A., Wolff,M., 2000. Nonstandard Analysis for the Working Mathematician. Kluwer Academic Publishers, Amsterdam.

Rath, K.P., Sun, Y.N., Yamashige, S., 1995. The nonexistence of symmetric equilibria in anonymous games with compact action spaces. Journal of Mathematical Economics 24, 331-346.

Skorokhod, A., 1956. Limit theorems for stochastic processes. Theory of Probability and its Applications 1, 261-290.

Sun, Y.N., 1996. Distributional properties of correspondences on Loeb spaces. Journal of Functional Analysis 139, 68-93.

Yannelis, N.C., Rustichini, A., 1991. Equilibrium points of non-cooperative random and Bayesian games. In: Aliprantis, C.D., Border, K.C., Luxemberg W.A.J. (eds) Positive operators, Riesz spaces, and economics, Springer, Berlin Heidelberg NewYork, pp 23-48.

Yu, H.M., Zhang, Z.X., 2007. Pure strategy equilibria in games with countable actions. Journal of Mathematical Economics 43, 192-200.