Abstract: There is at least one mixed strategy Nash equilibrium in continuous game, but there are little related research results in existing literature about infinite mixed strategy Nash equilibrium and uncertainty game problem. The uncertainty game refers to the game equilibrium problem that the players’ strategy set or benefit function are uncertainty. This paper creates an approximation algorithm of infinitely mixed strategy Nash equilibrium, using the advantages of particle swarm optimization, fewer parameters, simple coding and not strictly required to objective function. This paper also proposes the concept of rough game theory, and gives a method converting rough game to a classic game theory based on rough set and vague set theory. This paper provides a theoretical basis for game problem when the strategy sets and benefit function problem are fuzzy. The examples show that the approximation algorithm of infinite mixed strategy Nash equilibrium based on improved particle swarm algorithm and rough game theory solution are feasible and effective.

Key words: rough set, particle swarm optimization, mixed strategy Nash equilibrium, Vague set

摘要： 连续博弈中至少存在一个混合策略Nash均衡，但是关于无限策略混合策略Nash均衡的解法，以及局中人的策略集或是效益函数是不确定性博弈均衡问题，国内外相关的研究成果还比较少。运用粒子群算法对目标函数没有严格要求，参数较少，编码简单的优势,创立了一种计算无限策略混合策略的近似算法；并在此基础上提出了粗糙博弈论的概念，以粗糙集和Vague集的理论为基础，发现了一种粗糙博弈论转化为经典博弈论的方法。无限策略混合策略Nash均衡的近似算法和粗糙博弈论的研究为策略集和效益函数不确定时的博弈问题提供了理论依据。算法示例结果表明，基于改进的粒子群算法的无限策略混合策略Nash均衡近似算法和粗糙博弈论的解法是有效可行的。

关键词: 粗糙集, 粒子群算法, 混合策略纳什均衡, Vague集