On the existence of pure-strategy equilibria in games with private information: A complete characterization

This paper reports a definitive resolution to the question of the existence of a pure-strategy Bayesian–Nash equilibrium in games with a finite number of players, each with a compact metric action set and private information. The resolution hinges on saturated spaces. If the individual spaces of information are saturated, there exists a pure-strategy equilibrium in such a game; and if there exists a pure-strategy equilibrium for the class of games under consideration and with uncountable action sets, the spaces of private information must be saturated. As such, the paper offers a complete characterization of a longstanding question, and offers another game-theoretic characterization of the saturation property, one that complements a recent result of Keisler–Sun (2009) on large non-anonymous games with complete information.

[1]  Roy Radner,et al.  Private Information and Pure-Strategy Equilibria , 1982, Math. Oper. Res..

[2]  Roy Radner,et al.  Approximate Purification of Mixed Strategies , 1983, Math. Oper. Res..

[3]  H. Jerome Keisler,et al.  Adapted probability distributions , 1984 .

[4]  Yeneng Sun,et al.  Purification of measure-valued maps , 2006 .

[5]  M. A. Khan,et al.  Nonatomic games on Loeb spaces. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Erik J. Balder,et al.  Generalized Equilibrium Results for Games with Incomplete Information , 1988, Math. Oper. Res..

[7]  M. A. Khan,et al.  Non-Cooperative Games with Many Players , 2002 .

[8]  S. Rashid.  Equilibrium points of non-atomic games : Asymptotic results , 1982 .

[9]  Konrad Podczeck,et al.  On purification of measure-valued maps , 2009 .

[10]  R. Aumann.  Subjectivity and Correlation in Randomized Strategies , 1974 .

[11]  Yeneng Sun,et al.  A general Fatou Lemma , 2007 .

[12]  Peter A. Loeb,et al.  Purification and saturation , 2009 .

[13]  H. Jerome Keisler,et al.  Why saturated probability spaces are necessary , 2009 .

[14]  M. Ali Khan,et al.  Pure strategies in games with private information , 1995 .

[15]  Yeneng Sun,et al.  Pure strategy equilibria in games with private and public information , 2007 .

[16]  Robert J. Weber,et al.  Distributional Strategies for Games with Incomplete Information , 1985, Math. Oper. Res..

[17]  M. Ali Khan,et al.  On a private information game without pure strategy equilibria1 , 1999 .

[18]  D. Schmeidler.  Equilibrium points of nonatomic games , 1973 .

[19]  M. Ali Khan,et al.  Non-cooperative games on hyperfinite Loeb spaces , 1999 .

[20]  J. Harsanyi.  Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points , 1973 .

[21]  M. Ali Khan,et al.  On games with incomplete information and the Dvoretsky–Wald–Wolfowitz theorem with countable partitions☆ , 2009 .

[22]  Haifeng Fu.  Mixed-strategy equilibria and strong purification for games with private and public information , 2008 .

[23]  Jianwei Wang,et al.  Purification, saturation and the exact law of large numbers , 2012 .

[24]  Yeneng Sun,et al.  The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games , 2006, Int. J. Game Theory.