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Continuity of the value and optimal strategies when common priors change

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  • Ezra Einy

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  • Ori Haimanko

    ()

  • Biligbaatar Tumendemberel
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    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 on beliefs. This is unlike the case of general Bayesian games where lower semi-continuity of Bayesian equilibrium (BE) payoffs rests on the “almost uniform” convergence of conditional beliefs. We also show upper semi-continuity (USC) and approximate lower semi-continuity (ALSC) of the optimal strategy correspondence, and discuss ALSC of the BE correspondence in the context of zero-sum games. In particular, the interim BE correspondence is shown to be ALSC for some classes of information structures with highly non-uniform convergence of beliefs, that would not give rise to ALSC of BE in non-zero-sum games. Copyright Springer-Verlag 2012

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    File URL: http://hdl.handle.net/10.1007/s00182-010-0248-4
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    Bibliographic Info

    Article provided by Springer in its journal International Journal of Game Theory.

    Volume (Year): 41 (2012)
    Issue (Month): 4 (November)
    Pages: 829-849

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    Handle: RePEc:spr:jogath:v:41:y:2012:i:4:p:829-849

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    Related research

    Keywords: Zero-sum Bayesian games; Common prior; Value; Optimal strategies; Interim; Ex-ante; Bayesian equilibrium; Upper semi-continuity; Lower approximate semi-continuity; C72;

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    References

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    1. Kajii, Atsushi & Morris, Stephen, 1998. "Payoff Continuity in Incomplete Information Games," Journal of Economic Theory, Elsevier, vol. 82(1), pages 267-276, September.
    2. Ehud Lehrer & Dinah Rosenberg, 2003. "What restrictions do Bayesian games impose on the value of information?," Game Theory and Information 0312005, EconWPA.
    3. Atsushi Kajii & Stephen Morris, 1997. "The Robustness of Equilibria to Incomplete Information," Econometrica, Econometric Society, vol. 65(6), pages 1283-1310, November.
    4. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
    5. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-91, June.
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