Common Pay-off Matrix Games and Anti-Diagonal Symmetric Coordination Games

We prove the existence of equilibrium of common pay-off matrix games as an optimal solution of a bi-linear programming problem. If in a strategy profile the randomization of the column player is completely mixed then the strategy profile is an equilibrium if and only if it is the saddle-point of the pay-off matrix. We show if the pay-off matrix is skew symmetric then the randomization of the column player that minimizes the player’s maximum expected pay-off is a symmetric saddle point of the pay-off matrix and if this randomization is completely mixed then it is a symmetric equilibrium of the common pay-off matrix game. For symmetric matrices we provide a necessary and sufficient condition for the existence of a randomization which is a symmetric saddle point as well as a symmetric equilibrium of the common pay-off matrix game. We prove the existence of equilibria for anti-diagonal symmetric coordination games and in addition provide formulas for calculating several symmetric equilibria for such games.