  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.
