On pure-strategy equilibria in games with correlated information

Abstract This paper rehabilitates a program initiated in Aumann (1974) by contributing a result to the theory of finite-player Bayesian games in environments that explicitly include correlated information. An equivalence theorem offers conditions under which the set of mixed-strategy equilibrium payoffs in a classical finite-action game of complete information coincides with the set of objective pure-strategy Nash equilibrium (PSNE) expected payoffs in an affiliated Bayesian game with type-independent payoffs. This result is motivated for a non-specialist reader by several examples. An Appendix devoted to an intuitive discussion of the so-called ‘Lebesgue extension’ is added to make the paper self-contained.

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

[2]  Thomas Kazen.  Purification , 2004, Nature Biotechnology.

[3]  R. Tourky,et al.  Savage Games: A Theory of Strategic Interaction with Purely Subjective Uncertainty , 2013 .

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

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

[6]  Wei He,et al.  On the diffuseness of incomplete information game , 2013, 1307.5271.

[7]  John Duggan,et al.  Purification of Bayes Nash equilibrium with correlated types and interdependent payoffs , 2015, Games Econ. Behav..

[8]  Konrad Podczeck,et al.  Purification and roulette wheels , 2015 .

[9]  C. Kubrusly.  Measure on a σ-Algebra , 2015 .

[10]  P. Reny,et al.  On the Existence of Monotone Pure Strategy Equilibria in Bayesian Games , 2009 .

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

[12]  Dorothiy Maharam.  Decompositions of measure algebras and spaces , 1950 .

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

[14]  J. Wolfowitz,et al.  Elimination of Randomization in Certain Statistical Decision Procedures and Zero-Sum Two-Person Games , 1951 .

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

[16]  Konrad Podczeck,et al.  On the convexity and compactness of the integral of a Banach space valued correspondence , 2008 .

[17]  F. Hayek.  Individualism and Economic Order , 1949 .

[18]  J. Wolfowitz,et al.  Relations among certain ranges of vector measures , 1951 .

[19]  Paul R. Milgrom,et al.  A theory of auctions and competitive bidding , 1982 .

[20]  William Goldbloom Bloch.  The Unimaginable Mathematics of Borges' Library of Babel , 2008 .

[21]  David McAdams,et al.  Monotone Equilibrium in Multi-Unit Auctions , 2002 .

[22]  D. Maharam,et al.  On Homogeneous Measure Algebras. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[23]  Shizuo Kakutani.  27. Construction of a Non-separable Extension of the Lebesgue Measure Space , 1944 .

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

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

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

[27]  David McAdams,et al.  Isotone Equilibrium in Games of Incomplete Information , 2002 .

[28]  Yongchao Zhang,et al.  On Sufficiently-Diffused Information in Bayesian Games: A Dialectical Formalization , 2017 .

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

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

[31]  Yongchao Zhang,et al.  Individual risk and Lebesgue extension without aggregate uncertainty , 2008, J. Econ. Theory.

[32]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[33]  Yongchao Zhang,et al.  Existence of pure-strategy equilibria in Bayesian games: a sharpened necessity result , 2017, Int. J. Game Theory.

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

[35]  Yeneng Sun,et al.  Large games with a bio-social typology , 2013, J. Econ. Theory.

[36]  Rabee Tourky,et al.  Savage games: Savage games , 2016 .

[37]  L. Castro.  Correlation of Types in Bayesian Games , 2012 .

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

[39]  M. Ali Khan,et al.  On the existence of pure-strategy equilibria in games with private information: A complete characterization , 2014 .

[40]  Roy Radner,et al.  COMPETITIVE EQUILIBRIUM UNDER UNCERTAINTY11This research was supported in part by the Office of Naval Research under Contract ONR 222(77) with the University of California. Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1968 .

[41]  Susan Athey,et al.  Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information , 1997 .

[42]  H. Jerome Keisler,et al.  Model theory of stochastic processes , 2016 .

[43]  M. Ali Khan,et al.  Set-valued functions, Lebesgue extensions and saturated probability spaces , 2012 .

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

[45]  T. Jech.  Measure Algebras , 2017, 1705.01000.

[46]  J. Nash,et al.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[47]  Yeneng Sun,et al.  Modeling infinitely many agents , 2017 .

[48]  Debraj Ray,et al.  ARTICLE IN PRESS Journal of Economic Theory 112 (2003) 365–368 , 2002 .