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

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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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Bibliographic Info

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

Volume (Year): 41 (2012)
Issue (Month): 3 (August)
Pages: 553-564

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Handle: RePEc:spr:jogath:v:41:y:2012:i:3:p:553-564

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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;

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  1. John Duffy & Alexander Matros & Ted Temzelides, 2008. "Competitive Behavior in Market Games: Evidence and Theory," Working Papers 366, University of Pittsburgh, Department of Economics, revised Mar 2009.
  2. Tobias Guse & Burkhard Hehenkamp & Alex Possajennikov, 2008. "On the Equivalence of Nash and Evolutionary Equilibrium in Finite Populations," Discussion Papers 2008-06, The Centre for Decision Research and Experimental Economics, School of Economics, University of Nottingham.
  3. Duersch, Peter & Oechssler, Jörg & Schipper, Burkhard C., 2010. "Unbeatable Imitation," Working Papers 0499, University of Heidelberg, Department of Economics.
  4. Ana B. Ania, 2005. "Evolutionary stability and Nash equilibrium in finite populations, with an application to price competition," Vienna Economics Papers 0601, University of Vienna, Department of Economics.
  5. Carlos Alós-Ferrer & Ana Ania, 2005. "The evolutionary stability of perfectly competitive behavior," Economic Theory, Springer, vol. 26(3), pages 497-516, October.
  6. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
  7. Radzik, Tadeusz, 1991. "Saddle Point Theorems," International Journal of Game Theory, Springer, vol. 20(1), pages 23-32.
  8. 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.
  9. Burkhard Hehenkamp & Wolfgang Leininger & Alex Possajennikov, 2003. "Evolutionary Equilibrium in Tullock Contests: Spite and Overdissipation," Discussion Papers in Economics 03_01, University of Dortmund, Department of Economics.
  10. Brânzei, R. & Mallozzi, L. & Tijs, S.H., 2003. "Supermodular games and potential games," Open Access publications from Tilburg University urn:nbn:nl:ui:12-121822, Tilburg University.
  11. Alex Possajennikov, 2003. "Evolutionary foundations of aggregate-taking behavior," Economic Theory, Springer, vol. 21(4), pages 921-928, 06.
  12. 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.
  13. Fernando Vega Redondo, 1996. "The evolution of walrasian behavior," Working Papers. Serie AD 1996-05, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
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Cited by:
  1. Duersch, Peter & Oechssler, Jörg & Schipper, Burkhard C., 2010. "Unbeatable Imitation," Working Papers 0499, University of Heidelberg, Department of Economics.
  2. Peter Duersch & Joerg Oechssler & Burkhard Schipper, 2013. "When is Tit-For-Tat unbeatable?," Working Papers 131, University of California, Davis, Department of Economics.
  3. Burkhard Schipper, 2011. "Strategic Control of Myopic Best Reply in Repeated Games," Working Papers 115, University of California, Davis, Department of Economics.
  4. Bahel, Eric & Haller, Hans, 2013. "Cycles with undistinguished actions and extended Rock–Paper–Scissors games," Economics Letters, Elsevier, vol. 120(3), pages 588-591.
  5. Duersch, Peter & Oechssler, Jörg & Schipper, Burkhard C., 2012. "Once Beaten, Never Again: Imitation in Two-Player Potential Games," Working Papers 0529, University of Heidelberg, Department of Economics.

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