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Continuity of the value and optimal strategies when common priors change

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  • Einy, Ezra
  • Haimanko, Ori
  • Tumendemberelz, Biligbaatar

Abstract

We show that the value of a zero-sum Bayesian game is a Lipschitz continuous function of the players?common prior belief, with respect to the total variation metric (that induces the topology of setwise convergence on beliefs). This is unlike the case of general Bayesian games, where lower semi-continuity of Bayesian equilibrium payo¤s rests on the convergence of conditional beliefs (Engl (1995), Kajii and Morris (1998)). We also show upper, and approximate lower, semi- continuity of the optimal strategy correspondence with respect to the total variation norm, and discuss approximate lower semi-continuity of the Bayesian equilibrium correspondence in the context of zero-sum games.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Einy, Ezra & Haimanko, Ori & Tumendemberelz, Biligbaatar, 2009. "Continuity of the value and optimal strategies when common priors change," Discussion Papers 2009-09, Graduate School of Economics, Hitotsubashi University.
    • Handle: RePEc:hit:econdp:2009-09
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    File URL: https://hermes-ir.lib.hit-u.ac.jp/hermes/ir/re/17505/070econDP09-09.pdf
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    References listed on IDEAS

    as
    1. Atsushi Kajii & Stephen Morris, 1997. "The Robustness of Equilibria to Incomplete Information," Econometrica, Econometric Society, vol. 65(6), pages 1283-1310, November.
    2. Lehrer, Ehud & Rosenberg, Dinah, 2006. "What restrictions do Bayesian games impose on the value of information?," Journal of Mathematical Economics, Elsevier, vol. 42(3), pages 343-357, June.
    3. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
    4. Kajii, Atsushi & Morris, Stephen, 1998. "Payoff Continuity in Incomplete Information Games," Journal of Economic Theory, Elsevier, vol. 82(1), pages 267-276, September.
    5. Dov Monderer & Dov Samet, 1996. "Proximity of Information in Games with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 707-725, August.
    6. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-391, June.
    7. Ezra Einy & Ori Haimanko & Diego Moreno & Benyamin Shitovitz, 2008. "Uniform Continuity of the Value of Zero-Sum Games with Differential Information," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 552-560, August.
    8. Paul R. Milgrom & Robert J. Weber, 1985. "Distributional Strategies for Games with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 10(4), pages 619-632, November.
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    More about this item

    Keywords

    Zero-Sum Bayesian Games; Common Prior; Value; Optimal Strategies; Upper Semi-Continuity; Lower Approximate Semi-Continuity;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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