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 \(\varepsilon \)-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.

本文将每个玩家的收益张量固定的多线性博弈推广到每个玩家的收益张量从非空张量集中选择的广义多线性博弈。我们在所有涉及的张量集都有界的假设下证明了广义多线性博弈的\(\varepsilon \) -纳什均衡的存在性，并且在所有涉及的张量集有界的假设下证明了广义多线性博弈的纳什均衡的存在性袖珍的。特别是，当所有涉及的张量集都是有限的时，我们证明找到广义多线性博弈的纳什均衡点等价于解决垂直张量互补问题，并在纳什均衡点之间建立一一对应关系垂直张量互补问题的博弈及求解。