This paper studies the computation of mixed Nash equilibria in weighted Boolean games. In weighted Boolean games, players aim to maximise the total expected weight of a set of formulas by selecting behavioural strategies, that is, randomisations over the truth assignments for each propositional variable under their unique control. Behavioural strategies thus present a compact representation of mixed strategies. Two results are algorithmically significant: (a) behavioural equilibria satisfy a specific independence property; and (b) they allow for exponentially fewer supports than mixed equilibria. These findings suggest two ways in which one can leverage existing algorithms and heuristics for computing mixed equilibria: a naive approach where we check mixed equilibria for the aforesaid independence property, and a more sophisticated approach based on support enumeration. In addition, we explore a direct numerical approach inspired by finding correlated equilibria using linear programming. In an extensive experimental study, we compare the performance of these three approaches.

本文研究了加权布尔博弈中混合纳什均衡的计算。在加权布尔游戏中，玩家的目标是通过选择行为策略来最大化一组公式的总预期权重，即在其唯一控制下对每个命题变量的真值分配进行随机化。因此，行为策略呈现出混合策略的紧凑表示。在算法上有两个结果是有意义的：（a）行为均衡满足特定的独立性；（b）与混合均衡相比，它们所允许的支持指数减少。这些发现提出了两种可以利用现有算法和启发式方法来计算混合均衡的方法：一种天真的方法，其中我们检查了混合均衡的上述独立性，以及一种基于支持枚举的更复杂的方法。此外，我们探索了一种直接的数值方法，其灵感来自使用线性规划找到相关的平衡点。在广泛的实验研究中，我们比较了这三种方法的性能。