奇异方向上两人零和博弈的近似纳什均衡求解

Approximate Nash Equilibrium Solution of Two Person Zero Sum Game in Singular Direction

摘要: 纳什均衡的近似求解是博弈论中基础且重要的话题，其中基于特征向量的方法提供了一个新的视角，因此对博弈矩阵非方阵情形的探讨是值得期待的。本文以两人零和博弈为例，探讨了纳什均衡求解体系与非方支付矩阵的奇异方向的关系。关于纳什均衡计算的两个定理表明：当支付矩阵存在非负奇异值，且其对应的左奇异向量和右奇异向量元素都非负时，则该奇异向量分别对应博弈双方的纳什均衡解。以局中人的策略选择数目分别为3个和2个的情况为算例，验证了该定理的适用性，并对大规模非方矩阵博弈的纳什均衡的近似求解提供了一个新的方向。

Abstract: The approximate solution of Nash equilibrium is a basic and important topic in game theory, in which the method based on the eigenvector provides a new perspective, so it is worth looking forward to the discussion of the non-square matrix of the game. Taking a two-person zero-sum game as an example, this paper discusses the relationship between Nash equilibrium solution system and the singular direction of the non-square payment matrix. Two theorems about Nash equilibrium calculation show that when the payoff matrix has a non-negative singular value, and its corresponding left singular vector and right singular vector elements are all non-negative, then the singular vectors correspond to the Nash equilibrium solutions of both players in the game. Taking the cases where the number of strategy choices of the players in the game is 3 and 2 respectively, the applicability of the theorem is verified, and a new direction is provided for the approximate solution of Nash equilibrium for large-scale non-square matrix games.