When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, theperfectnessconcept is probably the most famous one. It is known that weakly dominated strategies of two-player games cannot be part of a perfect equilibrium. In general, thisundominanceproperty however does not extend ton-player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class ofn-player games, verify the undominance property. Consequently, we prove that every perfect equilibrium of a polymatrix game is undominated and that every undominated equilibrium of a polymatrix game is perfect. This result is used to set a new characterization of perfect Nash equilibria for polymatrix games. We also prove that the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. In addition, we introduce a linear programming formulation to identify perfect equilibria for polymatrix games. These results are illustrated on two small game applications. Computational experiments on randomly generated polymatrix games with different size and density are provided.
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机译：当面对多个纳什均衡时，决策者必须调整选择。在所有已知的纳什均衡细化中，perfectnessconcept可能是最著名的一种。众所周知，两人游戏的弱势统治策略不能成为完美均衡的一部分。但是，一般而言，这种不可统治的性质不会扩展大量玩家的游戏（E. E. C. van Damme，1983）。在本文中，我们证明了构成特定类型的n玩家游戏的多矩阵游戏验证了非主导性。因此，我们证明了多矩阵博弈的每个完美平衡都是无穷大的，而一个多矩阵博弈的每个未平衡都是完美的。该结果用于为多矩阵游戏设定完美纳什均衡的新特征。我们还证明了多矩阵博弈的完美纳什均衡集是凸多面体的有限并集。此外，我们引入了线性规划公式来确定多矩阵游戏的完美平衡。在两个小型游戏应用程序上说明了这些结果。提供了对随机生成的具有不同大小和密度的多矩阵游戏的计算实验。
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