Problem of finite аpproximation for the equilibrium set for a multicriteria bimatrix game

Recieved: 06/18/2025

Accepted: 07/12/2025

Keywords: multicriteria bimatrix games, Nash–Shapley equilibrium, linear scalarization, finite Hausdorff approximation, 2×2×2 games, degenerate bimatrix games

To cite this article

Novikova N.M., Pospelova I. I. Problem of finite аpproximation for the equilibrium set for a multicriteria bimatrix game. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2025. N 4, p.53-67 https://doi.org/10.55959/MSU/0137–0782–15–2025–49–4–53–67.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Novikova N.M. (Federal Research Center, Russian Academy of Sciences, Moscow, Russia)
Pospelova I. I. (Lomonosov Moscow State University (MSU))

Abstract

The problem of Hausdorff approximation by finite sets of the solution and the value of a multicriteria mixed strategy bimatrix game using a representation based on linear scalarization is considered. For the case of 2×2 matrices, explicit formulas are obtained for constructing nodes of a δ-net on the product of simplices of scalarization parameters. It is proved that the set of the equilibrium values obtained for the net converges in the Hausdorff metric to the solution of the initial game at δ →0. Possibile appearance of degenerate bimatrix games in scalarization is taken account. Examples are given for two-criteria 2×2×2 games.