Generalized Multilinear Games and Vertical Tensor Complementarity Problems

This paper generalizes the multilinear game where the payoff tensor of each player is fixed to the generalized multilinear game where the payoff tensor of each player is selected from a nonempty set of tensors. We prove the existence of \(\varepsilon \) $\epsilon$ -Nash equilibria for generalized multilinear games under the assumption that all involved sets of tensors are bounded, and the existence of Nash equilibria for generalized multilinear games under the assumption that all involved sets of tensors are compact. In particular, when all involved sets of tensors are finite, we show that finding a Nash equilibrium point for the generalized multilinear game is equivalent to solving a vertical tensor complementarity problem, and establish a one-to-one correspondence between the Nash equilibrium point of the game and the solution of the vertical tensor complementarity problem.