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

• Published in: Computational optimization and applications Vol. 66; no. 3; pp. 557 - 576
• Main Authors: Huang, Zheng-Hai, Qi, Liqun
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2017; Springer Nature B.V
• Subjects: Applied mathematics; Convex and Discrete Geometry; Equilibrium; Game theory; Games; Management Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimization; Players; Polynomials; Statistics; Studies; Tensors; Utilities; Utility functions; Tensor complementarity problem; Nash equilibrium; person noncooperative game; Game theory; Bimatrix game
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-016-9872-7

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.