Approximation of unit-hypercubic infinite two-sided noncooperative game via dimension-dependent irregular samplings and reshaping the multidimensional payoff matrices into flat matrices for solving the corresponding bimatrix game

Vadim Romanuke

COMPUTER MODELLING & NEW TECHNOLOGIES 2015 19(3A) 7-16

Applied Mathematics and Social Informatics Department, Khmelnitskiy National University, Institutskaya str., 11, 29016, Khmelnitskiy, Ukraine

The problem of solving unit-hypercubic infinite two-sided noncooperative games is considered. The ultimate goal is to approximate the infinite game with bimatrix game, ranking the approximation accurateness. This is fulfilled in three stages. Primarily the players’ payoff functions are sampled under stated conditions of dimension-dependent irregular samplings. Then the sampled payoff functions as multidimensional payoff matrices are mapped into ordinary flat matrices under a reversible matrix map. Finally, after obtaining the solution of the corresponding bimatrix game, equilibrium finite support strategies are checked out for their consistency, being used as the approximation accurateness rank. If consistent, then the bimatrix game can be regarded as the approximation of the initial noncooperative game. For particular cases, conditions of the weakened consistency are stipulated. Different types of consistency ensure the corresponding bimatrix game solution varying reasonably by changing the sampling steps minimally. If the solution is not even weakly consistent by the most primitive consistency in ranking the approximation accurateness, then the sampling intervals should be shortened. If any shortening is impossible then the sampling points must be set otherwise. The suggested approximation tool is fully applicable to games, which are isomorphic to the unit-hypercubic infinite two-sided noncooperative game.