The rank of a bimatrix game is defined as the rank of the sum of the payoff matrices of the two players. The rank of a game is known to impact both the most suitable computation methods for determining a solution and the expressive power of the game. Under certain conditions on the payoff matrices, we devise a method that reduces the rank of the game without changing the equilibrium of the game. We leverage matrix pencil theory and the Wedderburn rank reduction formula to arrive at our results. We also present a constructive proof of the fact that in a generic square game, the rank of the game can be reduced by 1, and in generic rectangular game, the rank of the game can be reduced by 2 under certain assumptions.
翻译:比马特里克游戏的等级被定义为两个玩家报酬矩阵的总和的等级。 一个游戏的等级已知会影响最合适的计算方法来决定一个解决方案和游戏的表达力。 在报酬矩阵的某些条件下,我们会设计一种方法来降低游戏的等级而不改变游戏的平衡。 我们利用矩阵铅笔理论和韦德伯恩降级公式来得出我们的结果。 我们还提供了一个建设性的证据,证明在通用平方游戏中,游戏的等级可以减少1,而在通用矩形游戏中,在某些假设下,游戏的等级可以减少2。