Perturbation Theory of Completely Mixed Bimatrix Games

A twoperson non-zero-sum bimatrix game (A, B) is defined to be completely mixed if every solution gives a positive probability to each pure strategy of each player. Such a game is defined to be nonsingular if both payoff matrices are nonsingular. Suppose that A is perturbed to A + aG and B is perturbed to B + aH, where C and H are matrices of the same size as A and B, and OL is a small real number, i.e., suppose that multiple elements of each payoff matrix are perturbed simultaneously. We calculate the effect of such perturbations on the solution and values of the game for each player. When a player’s payoff matrix is an M-matrix and a single diagonal element of the payoff matrix is perturbed, then that player’s value is a concave function of the perturbation. A new class of completely mixed bimatrix games is analyzed.