Perturbation theory of completely mixed bimatrix games (Q1122510)

The perturbation theory of nonsingular, completely mixed games is developed. The perturbations are given by the addition of real bimatrices multiplied by a real number. The game solution and the corresponding values for each player are presented in dependence on the multiplier. The formulas of first and second derivatives with respect to the multiplying parameter are obtained. In the general case it is difficult to find general conditions on the payoff and perturbation matrices which determine the sign of the second derivative in the zero point. The following special cases are considered. Both matrices are symmetric and the payoff matrix is positive definite (in particularly the diagonal- cyclic case), zero-sum and M-matrix cases with the perturbation of a single element.