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

Zheng-Hai Huang; Liqun Qi

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arXiv · 1602.03280

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

Zheng-Hai Huang·Liqun Qi

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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.

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BibTeX RIS

Zheng-Hai Huang, Liqun Qi. 2016-02-10. Formulating an $n$-person noncooperative game as a tensor complementarity problem. https://arxiv.org/abs/1602.03280

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