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In this paper, we consider a class of $n$-person noncooperative games, wherethe utility function of every player is given by a homogeneous polynomialdefined by the payoff tensor of that player, which is a natural extension ofthe bimatrix game where the utility function of every player is given by aquadratic form defined by the payoff matrix of that player. We will call such aproblem the multilinear game. We reformulate the multilinear game as a tensorcomplementarity problem, a generalization of the linear complementarityproblem; and show that finding a Nash equilibrium point of the multilinear gameis equivalent to finding a solution of the resulted tensor complementarityproblem. Especially, we present an explicit relationship between the solutionsof the multilinear game and the tensor complementarity problem, which builds abridge between these two classes of problems. We also apply a smoothing-typealgorithm to solve the resulted tensor complementarity problem and give somepreliminary numerical results for solving the multilinear games.

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