Approximating Multicriteria Matrix Game Solution with Finite Sets

The problem of representing the solution and the value of a multicriteria mixed strategy matrix game by finite sets is considered. According to Shapley’s ideas, linear scalarization is applied to the vector payoffs in the game. By this, the game is reduced to a parametric set of scalar bimatrix games. On the sets of the parameters, we construct a special finite δ $$\delta $$ -net for approximating the solution and the value sets of the initial game. For multicriteria games with 2 × 2 $$2\times 2$$ matrices, explicit formulas for the nodes of the δ $$\delta $$ -net are given by solving the degenerate bimatrix games obtained in the scalarization process. We prove the convergence of the finite set formed of game equilibria corresponding to the δ $$\delta $$ -net nodes to the solution of the initial game in Hausdorff metric. Examples of two-criteria 2 × 2 $$2\times 2$$ games are given; they visualize finite approximation of non-convex equilibrium sets.