International Journal of Game Theory

, Volume 41, Issue 3, pp 553–564 | Cite as

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

  • Peter Duersch
  • Jörg Oechssler
  • Burkhard C. SchipperEmail author


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.


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

JEL Classification

C72 C73 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alós-Ferrer C, Ania AB (2005) The evolutionary stability of perfectly competitive behavior. Econ Theory 26: 497–516CrossRefGoogle Scholar
  2. Ania A (2008) Evolutionary stability and Nash equilibrium in finite populations, with an application to price competition. J Econ Behav Organ 65: 472–488CrossRefGoogle Scholar
  3. Brânzei R, Mallozzi L, Tijs S (2003) Supermodular games and potential games. J Math Econ 39: 39–49CrossRefGoogle Scholar
  4. Brown GW, von Neumann J (1950) Solutions of games by differential equations. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games (Annals of Mathematics Studies no. 24). Princeton University Press, Princeton, pp 81–87Google Scholar
  5. Duersch P, Oechssler J, Schipper BC (2011) Unbeatable imitation mimeo. University of Heidelberg and the University of California, DavisGoogle Scholar
  6. Gale D, Kuhn HW, Tucker AW (1950) On symmetric games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games. Annals of Mathematics Studies, Princeton University Press, Princeton, pp 81–87Google Scholar
  7. Hehenkamp B, Leininger W, Possajennikov A (2004) Evolutionary equilibrium in Tullock contests: spite and overdissipation. Eur J Polit Econ 20: 1045–1057CrossRefGoogle Scholar
  8. Hehenkamp B, Possajennikov A, Guse T (2010) On the equivalence of Nash and evolutionary equilibrium in finite populations. J Econ Behav Organ 73: 254–258CrossRefGoogle Scholar
  9. Kaplansky I (1945) A contribution to von Neumann’s theory of games. Ann Math 46: 474–479CrossRefGoogle Scholar
  10. Leininger W (2006) Fending off one means fending off all: evolutionary stability in quasi-submodular games. Econ Theory 29: 713–719CrossRefGoogle Scholar
  11. Matros A, Temzelides T, Duffy J (2009) Competitive behavior in market games: evidence and theory, mimeo. University of Pittsburgh, PittsburghGoogle Scholar
  12. Monderer D, Shapley LS (1996) Potential games. Games Econ Behav 14: 124–143CrossRefGoogle Scholar
  13. Nash J (1951) Non-cooperative games. Ann Math 54: 286–295CrossRefGoogle Scholar
  14. Nydegger RV, Owen G (1974) Two-person bargaining: an experimental test of the Nash axioms. Int J Game Theory 3: 239–249CrossRefGoogle Scholar
  15. Possajennikov A (2003) Evolutionary foundation of aggregative-taking behavior. Econ Theory 21: 921–928CrossRefGoogle Scholar
  16. Radzik T (1991) Saddle point theorems. Int J Game Theory 20: 23–32CrossRefGoogle Scholar
  17. Roth AE, Malouf MWK (1979) Game-theoretic models and the role of information in bargaining. Psychol Rev 86: 574–594CrossRefGoogle Scholar
  18. Schaffer ME (1988) Evolutionary stable strategies for a finite population and a variable contest size. J Theor Biol 132: 469–478CrossRefGoogle Scholar
  19. Schaffer ME (1989) Are profit-maximizers the best survivors?. J Econ Behav Organ 12: 29–45CrossRefGoogle Scholar
  20. Schipper BC (2003) Submodularity and the evolution of Walrasian behavior. Int J Game Theory 32: 471–477Google Scholar
  21. Shapley LS (1964) Some topics in two-person games. In: Dresher M, Shapley LS, Tucker AW (eds) Advances in game theory (Annals of Mathematical Studies no. 52). Princeton University, Princeton, pp 1–28Google Scholar
  22. Tanaka Y (2000) A finite population ESS and a long run equilibrium in an n-players coordination game. Math Soc Sci 39: 195–206CrossRefGoogle Scholar
  23. Topkis DM (1998) Supermodularity and complementarity. Princeton University Press, PrincetonGoogle Scholar
  24. Vega-Redondo F (1997) The evolution of Walrasian behavior. Econometrica 65: 375–384CrossRefGoogle Scholar
  25. von Neumann J (1928) Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100: 295–320CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Peter Duersch
    • 1
  • Jörg Oechssler
    • 1
  • Burkhard C. Schipper
    • 2
    Email author
  1. 1.Department of EconomicsUniversity of HeidelbergHeidelbergGermany
  2. 2.Department of EconomicsUniversity of California, DavisDavisUSA

Personalised recommendations