While Nash equilibria are guaranteed to exist, they may exhibit dense support, making them difficult to understand and execute in some applications. In this paper, we study $k$-sparse commitments in games where one player is restricted to mixed strategies with support size at most $k$. Finding $k$-sparse commitments is known to be computationally hard. We start by showing several structural properties of $k$-sparse solutions, including that the optimal support may vary dramatically as $k$ increases. These results suggest that naive greedy or double-oracle-based approaches are unlikely to yield practical algorithms. We then develop a simple approach based on mixed integer linear programs (MILPs) for zero-sum games, general-sum Stackelberg games, and various forms of structured sparsity. We also propose practical algorithms for cases where one or both players have large (i.e., practically innumerable) action sets, utilizing a combination of MILPs and incremental strategy generation. We evaluate our methods on synthetic and real-world scenarios based on security applications. In both settings, we observe that even for small support sizes, we can obtain more than $90\%$ of the true Nash value while maintaining a reasonable runtime, demonstrating the significance of our formulation and algorithms.

翻译：尽管纳什均衡的存在性已得到保证，但其可能具有稠密支撑集，导致在某些应用中难以理解和执行。本文研究$k$-稀疏承诺博弈，其中一方参与者被限制在支撑集大小至多为$k$的混合策略中。已知寻找$k$-稀疏承诺在计算上是困难的。我们首先展示$k$-稀疏解的几个结构特性，包括最优支撑集可能随$k$增大而发生剧烈变化。这些结果表明，基于朴素贪心或双重预言机的方法难以产生实用算法。随后我们针对零和博弈、一般和斯塔克尔伯格博弈及多种结构化稀疏形式，提出基于混合整数线性规划（MILP）的简洁方法。针对单方或双方参与者具有大规模（即实际不可枚举）行动集的情形，我们结合MILP与增量策略生成技术提出实用算法。我们在基于安全应用的合成场景和现实场景中评估所提方法。两种实验均表明，即使对于较小支撑集规模，我们仍能在保持合理运行时间的前提下获得超过$90\%$的真实纳什值，这验证了我们提出的建模框架与算法的重要意义。