Circulant Games

This paper presents a class of finite n x n bimatrix (2-player) games we coin Circulant Games. In Circulant Games, each player's payoff matrix is a circulant matrix, i.e.\ each row vector is rotated one element relative to the preceding row vector. We show that when the payoffs in the first row of each payoff matrix are strictly ordered, a single parameter describing the rotation symmetry between the players' matrices fully determines the exact number and the structure of all Nash Equilibria in these games. The class of Circulant Games contains well-known games such as matching pennies, rock-scissors-paper, circulant coordination and common interest games.