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Formulating an n-person noncooperative game as a tensor complementarity problem

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  • Zheng-Hai Huang

    (Tianjin University
    Tianjin University)

  • Liqun Qi

    (The Hong Kong Polytechnic University)

Abstract

In this paper, we consider a class of n-person noncooperative games, where the utility function of every player is given by a homogeneous polynomial defined by the payoff tensor of that player, which is a natural extension of the bimatrix game where the utility function of every player is given by a quadratic form defined by the payoff matrix of that player. We will call such a problem the multilinear game. We reformulate the multilinear game as a tensor complementarity problem, a generalization of the linear complementarity problem; and show that finding a Nash equilibrium point of the multilinear game is equivalent to finding a solution of the resulted tensor complementarity problem. Especially, we present an explicit relationship between the solutions of the multilinear game and the tensor complementarity problem, which builds a bridge between these two classes of problems. We also apply a smoothing-type algorithm to solve the resulted tensor complementarity problem and give some preliminary numerical results for solving the multilinear games.

Suggested Citation

  • Zheng-Hai Huang & Liqun Qi, 2017. "Formulating an n-person noncooperative game as a tensor complementarity problem," Computational Optimization and Applications, Springer, vol. 66(3), pages 557-576, April.
    • Handle: RePEc:spr:coopap:v:66:y:2017:i:3:d:10.1007_s10589-016-9872-7
    DOI: 10.1007/s10589-016-9872-7
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