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Pure strategy equilibria in symmetric two-player zero-sum games

Author

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  • Peter Duersch

    ()

  • Jörg Oechssler

    ()

  • Burkhard Schipper

    ()

Abstract

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.
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Suggested Citation

  • Peter Duersch & Jörg Oechssler & Burkhard Schipper, 2012. "Pure strategy equilibria in symmetric two-player zero-sum games," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 553-564, August.
  • Handle: RePEc:spr:jogath:v:41:y:2012:i:3:p:553-564
    DOI: 10.1007/s00182-011-0302-x
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    References listed on IDEAS

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    1. Alex Possajennikov, 2003. "Evolutionary foundations of aggregate-taking behavior," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 21(4), pages 921-928, June.
    2. Duersch, Peter & Oechssler, Jörg & Schipper, Burkhard C., 2012. "Unbeatable imitation," Games and Economic Behavior, Elsevier, vol. 76(1), pages 88-96.
    3. Branzei, Rodica & Mallozzi, Lina & Tijs, Stef, 2003. "Supermodular games and potential games," Journal of Mathematical Economics, Elsevier, vol. 39(1-2), pages 39-49, February.
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    5. Hehenkamp, Burkhard & Possajennikov, Alex & Guse, Tobias, 2010. "On the equivalence of Nash and evolutionary equilibrium in finite populations," Journal of Economic Behavior & Organization, Elsevier, vol. 73(2), pages 254-258, February.
    6. Tanaka, Yasuhito, 2000. "A finite population ESS and a long run equilibrium in an n players coordination game," Mathematical Social Sciences, Elsevier, vol. 39(2), pages 195-206, March.
    7. Ania, Ana B., 2008. "Evolutionary stability and Nash equilibrium in finite populations, with an application to price competition," Journal of Economic Behavior & Organization, Elsevier, vol. 65(3-4), pages 472-488, March.
    8. Brânzei, R. & Mallozzi, L. & Tijs, S.H., 2003. "Supermodular games and potential games," Other publications TiSEM 87c16860-0596-4448-808d-c, Tilburg University, School of Economics and Management.
    9. Fernando Vega-Redondo, 1997. "The Evolution of Walrasian Behavior," Econometrica, Econometric Society, vol. 65(2), pages 375-384, March.
    10. Hehenkamp, B. & Leininger, W. & Possajennikov, A., 2004. "Evolutionary equilibrium in Tullock contests: spite and overdissipation," European Journal of Political Economy, Elsevier, vol. 20(4), pages 1045-1057, November.
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    Citations

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    Cited by:

    1. Ɖura-Georg Granić & Johannes Kern, 2016. "Circulant games," Theory and Decision, Springer, vol. 80(1), pages 43-69, January.
    2. Peter Duersch & Jörg Oechssler & Burkhard Schipper, 2014. "When is tit-for-tat unbeatable?," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(1), pages 25-36, February.
    3. repec:eee:joepsy:v:63:y:2017:i:c:p:86-92 is not listed on IDEAS
    4. Duersch, Peter & Oechssler, Jörg & Schipper, Burkhard C., 2012. "Unbeatable imitation," Games and Economic Behavior, Elsevier, vol. 76(1), pages 88-96.
    5. Michel Grabisch & Antoine Mandel & Agnieszka Rusinowska & Emily Tanimura, 2015. "Strategic influence in social networks," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01158168, HAL.
    6. Ismail M.S., 2014. "The equivalence between two-person symmetric games and decision problems," Research Memorandum 023, Maastricht University, Graduate School of Business and Economics (GSBE).
    7. Peter Duersch & Joerg Oechssler & Burkhard Schipper, 2011. "Once Beaten, Never Again: Imitation in Two-Player Potential Games," Working Papers 1112, University of California, Davis, Department of Economics.
    8. Bahel, Eric & Haller, Hans, 2013. "Cycles with undistinguished actions and extended Rock–Paper–Scissors games," Economics Letters, Elsevier, vol. 120(3), pages 588-591.
    9. Schipper, Burkhard C, 2011. "Strategic control of myopic best reply in repeated games," MPRA Paper 30219, University Library of Munich, Germany.
    10. Wang, Hua & Meng, Qiang & Zhang, Xiaoning, 2014. "Game-theoretical models for competition analysis in a new emerging liner container shipping market," Transportation Research Part B: Methodological, Elsevier, vol. 70(C), pages 201-227.
    11. repec:spr:jogath:v:46:y:2017:i:3:d:10.1007_s00182-016-0555-5 is not listed on IDEAS
    12. Ismail M.S., 2014. "A sufficient condition on the existence of pure equilibrium in two-person symmetric zerosum games," Research Memorandum 035, Maastricht University, Graduate School of Business and Economics (GSBE).
    13. Takuya Iimura & Takahiro Watanabe, 2016. "Pure strategy equilibrium in finite weakly unilaterally competitive games," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(3), pages 719-729, August.

    More about this item

    Keywords

    Symmetric two-player games; Zero-sum games; Rock-paper-scissors; Single-peakedness; Quasiconcavity; Finite population evolutionary stable strategy; Saddle point; Exact potential games; C72; C73;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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