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On the closed-graph property of the Nash equilibrium correspondence in a large game: A complete characterization

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  • Qiao, Lei
  • Yu, Haomiao
  • Zhang, Zhixiang

Abstract

We show that if every large game with a given player space and any given uncountable trait space (or action set) is a proper idealized limit, then the player space must be saturated. When the player space is allowed to be an arbitrary atomless probability space, even a non-saturated one such as the classical Lebesgue unit interval, we establish the following: (i) If a large game has a countable action set and a countable trait space, then the game has a closed Nash equilibrium correspondence, and is thus proper as an idealized limit; (ii) If every large game having a given action set and a given trait space is proper as an idealized limit, then both the action set and the trait space must be countable.

Suggested Citation

  • Qiao, Lei & Yu, Haomiao & Zhang, Zhixiang, 2016. "On the closed-graph property of the Nash equilibrium correspondence in a large game: A complete characterization," Games and Economic Behavior, Elsevier, vol. 99(C), pages 89-98.
  • Handle: RePEc:eee:gamebe:v:99:y:2016:i:c:p:89-98
    DOI: 10.1016/j.geb.2016.07.007
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    References listed on IDEAS

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    More about this item

    Keywords

    Closed-graph property; Large game with traits (LGT); Lebesgue unit interval; Nash equilibrium; Nash equilibrium distribution; Saturated probability space;

    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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