## Abstract

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.

## Keywords

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

### Acknowledgments

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.

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