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Abstract <p>The problem of finding Hausdorff approximation by finite sets of the solution to and the value of a multicriteria bimatrix game in mixed strategies is considered using a representation based on linear scalarization. For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2\times 2\)</EquationSource> <!--CMatCMGU2570023Novikova-m1--> </InlineEquation> matrices, explicit formulas are found for constructing nodes of a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\delta\)</EquationSource> <!--CMatCMGU2570023Novikova-m2--> </InlineEquation>-net on the product of simplexes of scalarization parameters. The convergence in the Hausdorff metric of the set uniting the equilibrium values obtained for this net to the solution of the original game as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\delta\to 0\)</EquationSource> <!--CMatCMGU2570023Novikova-m3--> </InlineEquation> is proven. The possibility of appearance of degenerate bimatrix games under scalarization is taken into account. Examples are given for two-criteria <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2\times 2\times 2\)</EquationSource> <!--CMatCMGU2570023Novikova-m4--> </InlineEquation> games.</p>

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Problem of the Finite Approximation of an Equilibrium Set for Multicriteria Bimatrix Games

  • 期刊论文

  • 发表日期：

    2025年12月15日

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  • N. M. Novikova,
  • I. I. Pospelova

摘要

Abstract

The problem of finding Hausdorff approximation by finite sets of the solution to and the value of a multicriteria bimatrix game in mixed strategies is considered using a representation based on linear scalarization. For \(2\times 2\) matrices, explicit formulas are found for constructing nodes of a \(\delta\) -net on the product of simplexes of scalarization parameters. The convergence in the Hausdorff metric of the set uniting the equilibrium values obtained for this net to the solution of the original game as \(\delta\to 0\) is proven. The possibility of appearance of degenerate bimatrix games under scalarization is taken into account. Examples are given for two-criteria \(2\times 2\times 2\) games.