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Reinterpreting Mixed Strategy Equilibria: A Unification of the Classical and Bayesian Views

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  • Reny, Phil
  • Robson, Arthur

Abstract

We provide a new interpretation of mixed strategy equilibria that incorporates both von Neumann and Morgenstern's classical concealment role of mixing as well as the more recent Bayesian view originating with Harsanyi. For any two-person game, G, we consider an incomplete information game, IG, in which each player's type is the probability he assigns to the event that his mixed strategy in G is 'found out' by his opponent. We show that, generically, any regular equilibrium of G can be approximated by an equilibrium of IG in which almost every type of each player is strictly optimizing. This leads us to interpret i's equilibrium mixed strategy in G as a combination of deliberate randomization by i together with uncertainty on j's part about which randomization i will employ. We also show that such randomization is not unusual: For example, i's randomization is nondegenerate whenever the support of an equilibrium contains cyclic best replies.

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  • Reny, Phil & Robson, Arthur, 2004. "Reinterpreting Mixed Strategy Equilibria: A Unification of the Classical and Bayesian Views," Microeconomics.ca working papers robson-04-02-12-12-44-46, Vancouver School of Economics, revised 12 Feb 2004.
    • Handle: RePEc:ubc:pmicro:robson-04-02-12-12-44-46
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    File URL: http://microeconomics.ca/arthur_robson/sc-09-26-03.pdf
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    1. Reny, Philip J. & Robson, Arthur J., 2004. "Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views," Games and Economic Behavior, Elsevier, vol. 48(2), pages 355-384, August.
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    4. Stephen Morris, 2006. "Purification," Levine's Bibliography 321307000000000470, UCLA Department of Economics.
    5. Stefanos Leonardos & Costis Melolidakis, 2018. "On the Commitment Value and Commitment Optimal Strategies in Bimatrix Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-28, September.
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    8. Lee, Natalie, 2023. "Feigning ignorance for long-term gains," Games and Economic Behavior, Elsevier, vol. 138(C), pages 42-71.
    9. von Stengel, Bernhard & Zamir, Shmuel, 2010. "Leadership games with convex strategy sets," Games and Economic Behavior, Elsevier, vol. 69(2), pages 446-457, July.
    10. Zhongmin Wang, 2009. "(Mixed) Strategy in Oligopoly Pricing: Evidence from Gasoline Price Cycles Before and Under a Timing Regulation," Journal of Political Economy, University of Chicago Press, vol. 117(6), pages 987-1030, December.
    11. Xiao Luo & Yi-Chun Chen, 2004. "A Unified Approach to Information, Knowledge, and Stability," Econometric Society 2004 Far Eastern Meetings 472, Econometric Society.
    12. Gallice, Andrea, 2007. "Best Responding to What? A Behavioral Approach to One Shot Play in 2x2 Games," Discussion Papers in Economics 1365, University of Munich, Department of Economics.
    13. Mohtadi, Mohammad Mahdi & Nogondarian, Kazem, 2015. "Presenting an algorithm to find Nash equilibrium in two-person static games with many strategies," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 442-452.
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