Theory and Decision

, Volume 80, Issue 1, pp 43–69 | Cite as

Circulant games

  • Ɖura-Georg Granić
  • Johannes KernEmail author


We study a class of two-player normal-form games with cyclical payoff structures. A game is called circulant if both players’ payoff matrices fulfill a rotational symmetry condition. The class of circulant games contains well-known examples such as Matching Pennies, Rock-Paper-Scissors, as well as subclasses of coordination and common interest games. The best response correspondences in circulant games induce a partition on each player’s set of pure strategies into equivalence classes. In any Nash Equilibrium, all strategies within one class are either played with strictly positive or with zero probability. We further show that, strikingly, a single parameter fully determines the exact number and the structure of all Nash equilibria (pure and mixed) in these games. The parameter itself only depends on the position of the largest payoff in the first row of one of the player’s payoff matrix.


Bimatrix games Circulant games Circulant matrix   Number of Nash equilibria Rock-Paper-Scissors 



We would like to thank Carlos Alós-Ferrer, Tanja Artiga Gonzalez, Wolfgang Leininger, participants at the SAET13 conference in Paris and seminar participants in Cologne and Innsbruck for helpful comments and discussions. We also thank two anonymous referees and the coordinating editor for helpful comments and suggestions. Johannes Kern gratefully acknowledges financial support from the German Research Foundation through research projects AL-1169/1-1 and AL-1169/1-2. Ɖura-Georg Granić also gratefully acknowledges financial support from the German Research Foundation (DFG) through research project AL-1169/2-1.


  1. Alós-Ferrer, C., & Kuzmics, C. (2013). Hidden symmetries and focal points. Journal of Economic Theory, 148(1), 226–258.CrossRefGoogle Scholar
  2. Bahel, E. (2012). Rock-paper-scissors and cycle-based games. Economics Letters, 115(3), 401–403.CrossRefGoogle Scholar
  3. Bahel, E., & Haller, H. (2013). Cycles with undistinguished actions and extended rock-paper-scissors games. Economics Letters, 120(3), 588–591.CrossRefGoogle Scholar
  4. Casajus, A. (2000). Focal points in framed strategic forms. Games and Economic Behavior, 32(3), 263–291.CrossRefGoogle Scholar
  5. Davis, P. J. (1979). Circulant matrices. New York: Wiley.Google Scholar
  6. Diekmann, O., & van Gils, S. A. (2009). On the cyclic replicator equation and the dynamics of semelparous populations. SIAM Journal on Applied Dynamical Systems, 8(3), 1160–1189.CrossRefGoogle Scholar
  7. Duersch, P., Oechssler, J., & Schipper, B. C. (2012). Pure strategy equilibria in symmetric two-player zero-sum games. International Journal of Game Theory, 41(3), 553–564.CrossRefGoogle Scholar
  8. Durieu, J., Haller, H., Querou, N., & Solal, P. (2008). Ordinal games. International Game Theory Review, 10, 177–194.CrossRefGoogle Scholar
  9. Hofbauer, J., & Sigmund, K. (1998). Evolutionary games and population dynamics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  10. Hofbauer, J., Schuster, P., Sigmund, K., & Wolff, R. (1980). Dynamical systems under constant organization ii: Homogeneous growth functions of degree p = 2. SIAM Journal on Applied Mathematics, 38(2), 282–304.CrossRefGoogle Scholar
  11. Janssen, M. C. (2001). Rationalizing focal points. Theory and Decision, 50(3), 119–148.CrossRefGoogle Scholar
  12. Keiding, H. (1997). On the maximal number of nash equilibria in a bimatrix game. Games and Economic Behavior, 21(1–2), 148–160.CrossRefGoogle Scholar
  13. Keiding, H. (1998). On the structure of the set of Nash equilibria of weakly nondegenerate bimatrix games. Annals of Operations Research, 84, 231–238.CrossRefGoogle Scholar
  14. Loertscher, S. (2013). Rock-Scissors-Paper and evolutionarily stable strategies. Economic Letters, 118, 473–474.CrossRefGoogle Scholar
  15. Maynard Smith, J. (1974). The theory of games and the evolution of animal conflicts. Journal of Theoretical Biology, 47(1), 209–221.CrossRefGoogle Scholar
  16. Maynard Smith, J., & Price, G. R. (1973). The logic of animal conflict. Nature, 246, 15.CrossRefGoogle Scholar
  17. McKelvey, R., & McLennan, A. (1997). The maximal number of regular totally mixed Nash equilibria. Journal of Economic Theory, 72(2), 411–425.CrossRefGoogle Scholar
  18. McLennan, A. (1997). The maximal generic number of pure Nash equilibria. Journal of Economic Theory, 72(2), 408–410.CrossRefGoogle Scholar
  19. McLennan, A., & Park, I. (1999). Generic \(4\times 4\) two person games have at most 15 nash equilibria. Games and Economic Behavior, 26(1), 111–130.CrossRefGoogle Scholar
  20. Nash, J. F. (1951). Non-cooperative games. Annals of Mathematics, 54(2), 286–295.CrossRefGoogle Scholar
  21. Quint, T., & Shubik, M. (1997). A theorem on the number of Nash equilibria in a bimatrix game. International Journal of Game Theory, 26(3), 353–359.CrossRefGoogle Scholar
  22. Quint, T., & Shubik, M. (2002). A bound on the number of Nash equilibria in a coordination game. Economics Letters, 77(3), 323–327.CrossRefGoogle Scholar
  23. Sandholm, W. (2010). Population games and evolutionary dynamics. Economic learning and social evolution. Cambridge, MA: MIT Press.Google Scholar
  24. Shapley, L. (1974). A note on the Lemke–Howson algorithm. Mathematical Programming Study, 1, 175–189.CrossRefGoogle Scholar
  25. Sinervo, B., & Lively, C. M. (1996). The rock-paper-scissors game and the evolution of alternative male strategies. Nature, 380(6571), 240–243.CrossRefGoogle Scholar
  26. von Stengel, B. (1997). New lower bounds for the number of equilibria in bimatrix games. Technical Report 264, Department of Computer Science, ETH Zürich.Google Scholar
  27. von Stengel, B. (1999). New maximal numbers of equilibria in bimatrix games. Discrete & Computational Geometry, 21(4), 557–568.Google Scholar
  28. Weibull, J. (1995). Evolutionary game theory. Cambridge, MA: MIT Press.Google Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of CologneCologneGermany

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