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Maximin equilibrium

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  • Ismail, Mehmet

Abstract

We introduce a new solution concept called maximin equilibrium which extends von Neumann's maximin strategy idea to n-player non-cooperative games by incorporating common knowledge of 'rationality' of the players. Our rationality assumption is, however, stronger than the one of maximin strategy and weaker than the one of Nash equilibrium. Maximin equilibrium, just like maximin strategies, is a method for evaluating the uncertainty that players are facing by playing the game. We show that maximin equilibrium is invariant under strictly increasing transformations of the payoff functions. Notably, every finite game possesses a maximin equilibrium in pure strategies. Considering the games in von Neumann and Morgenstern mixed extension, we show that maximin equilibrium is a generalization of Nash equilibrium. In addition, we demonstrate that maximin equilibria and Nash equilibria coincide in two-player zero-sum games. We propose a refinement of maximin equilibrium called strong maximin equilibrium. Accordingly, we show that for every Nash equilibrium that is not a strong maximin equilibrium there exists a strong maximin equilibrium that Pareto dominates it. In addition, no strong maximin equilibrium is ever Pareto dominated by a Nash equilibrium. Finally, we discuss maximin equilibrium predictions in several games including the traveler's dilemma. (Submitted to MPRA for stable archival purposes. This is the first Maastricht University version.)

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  • Ismail, Mehmet, 2014. "Maximin equilibrium," MPRA Paper 97401, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:97401
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    References listed on IDEAS

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    1. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, December.
    2. Aumann, Robert J., 1974. "Subjectivity and correlation in randomized strategies," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 67-96, March.
    3. Ariel Rubinstein, 2006. "Dilemmas of an Economic Theorist," Econometrica, Econometric Society, vol. 74(4), pages 865-883, July.
    4. Bernheim, B Douglas, 1984. "Rationalizable Strategic Behavior," Econometrica, Econometric Society, vol. 52(4), pages 1007-1028, July.
    5. Jacob K. Goeree & Charles A. Holt, 2001. "Ten Little Treasures of Game Theory and Ten Intuitive Contradictions," American Economic Review, American Economic Association, vol. 91(5), pages 1402-1422, December.
    6. C. Monica Capra, 1999. "Anomalous Behavior in a Traveler's Dilemma?," American Economic Review, American Economic Association, vol. 89(3), pages 678-690, June.
    7. Ariel Rubinstein, 2007. "Instinctive and Cognitive Reasoning: A Study of Response Times," Economic Journal, Royal Economic Society, vol. 117(523), pages 1243-1259, October.
    8. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    9. Basu, Kaushik, 1994. "The Traveler's Dilemma: Paradoxes of Rationality in Game Theory," American Economic Review, American Economic Association, vol. 84(2), pages 391-395, May.
    10. R. J. Aumann & M. Maschler, 1972. "Some Thoughts on the Minimax Principle," Management Science, INFORMS, vol. 18(5-Part-2), pages 54-63, January.
    11. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    12. Rosenthal, Robert W., 1981. "Games of perfect information, predatory pricing and the chain-store paradox," Journal of Economic Theory, Elsevier, vol. 25(1), pages 92-100, August.
    13. Ariel Rubinstein, 2007. "Instinctive and Cognitive Reasoning: Response Times Study," Levine's Bibliography 321307000000001011, UCLA Department of Economics.
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    More about this item

    Keywords

    Non-cooperative games; maximin strategy; zerosum games;
    All these keywords.

    JEL classification:

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

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