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The purpose of this paper is to develop the concept and mathematical-programming methodology of matrix games with payoffs represented by Atanassov's interval-valued intuitionistic fuzzy (IVIF) sets. In this methodology, the concept of solutions of matrix games with payoffs represented by Atanassov's IVIF sets is defined, and some important properties are studied using multiobjective-programming and duality-programming theory. It is proven that each matrix game with payoffs represented by Atanassov's IVIF sets has a solution, which can be obtained through solving a pair of auxiliary linear/nonlinear-programming models derived from a pair of nonlinear biobjective interval-programming models. Validity and applicability of the proposed methodology are illustrated with a numerical example.