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Strategic uncertainty and the ex-post Nash property in large games

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  • ,

    (Department of Economics, Johns Hopkins University)

  • , P.

    (Department of Economics, University of Notre Dame)

  • ,

    (Department of Economics, National University of Singapore)

  • ,

    (Department of Economics, Ryerson University)

Abstract

This paper elucidates the conceptual role that independent randomization plays in non-cooperative game theory. In the context of large (atomless) games in normal form, we present precise formalizations of the notions of a mixed strategy equilibrium (MSE), and of a randomized strategy equilibrium in distributional form (RSED). We offer a resolution of two long-standing open problems and show: (i) any MSE {\it induces} a RSED, and any RSED can be {\it lifted} to a MSE, (ii) a mixed strategy profile is a MSE if and only if it has the ex-post Nash property. Our substantive results are a direct consequence of an {\it exact} law of large numbers (ELLN) that can be formalized in the analytic framework of a Fubini extension. We discuss how the \lq measurability' problem associated with a MSE of a large game is automatically resolved in such a framework. We also illustrate our ideas by an approximate result pertaining to a sequence of large but finite games.

Suggested Citation

  • , & , P. & , & ,, 2015. "Strategic uncertainty and the ex-post Nash property in large games," Theoretical Economics, Econometric Society, vol. 10(1), January.
    • Handle: RePEc:the:publsh:1492
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    Cited by:

    1. Chen, Enxian & Qiao, Lei & Sun, Xiang & Sun, Yeneng, 2022. "Robust perfect equilibrium in large games," Journal of Economic Theory, Elsevier, vol. 201(C).
    2. Wei He & Yeneng Sun, 2018. "Conditional expectation of correspondences and economic applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(2), pages 265-299, August.
    3. Khan, Mohammed Ali & Rath, Kali P. & Yu, Haomiao & Zhang, Yongchao, 2017. "On the equivalence of large individualized and distributionalized games," Theoretical Economics, Econometric Society, vol. 12(2), May.
    4. Kalai, Ehud & Shmaya, Eran, 2018. "Large strategic dynamic interactions," Journal of Economic Theory, Elsevier, vol. 178(C), pages 59-81.
    5. Ennio Bilancini & Leonardo Boncinelli, 2016. "Strict Nash equilibria in non-atomic games with strict single crossing in players (or types) and actions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 95-109, April.
    6. Sun, Xiang & Sun, Yeneng & Yu, Haomiao, 2020. "The individualistic foundation of equilibrium distribution," Journal of Economic Theory, Elsevier, vol. 189(C).

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    More about this item

    Keywords

    Large game; pure strategy; mixed strategy; randomized strategy in distributional form; Nash equilibrium; ex-post Nash property; saturated probability space; rich Fubini extension; exact law of large numbers (ELLN); asymptotic implementation;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D84 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Expectations; Speculations
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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