Near-Optimal Mixed Strategy for Zero-Sum Differential Games

Synthesizing near-optimal mixed strategies for zero-sum differential games (ZSDGs) has been a longstanding challenge. Existing research mainly focuses on characterizing the theoretical value function, while the practical design of executable mixed strategies remains open. To address this issue, we propose a novel weak approximation framework. The core idea is to map the original mixed-strategy game into a surrogate stochastic differential game (SDG) under pure strategies. This mapping ensures that both state distributions and cost expectations closely match the original game. Based on the solution of this auxiliary SDG, the original game value can be approximated, and near-optimal mixed strategies can be synthesized. To operationalize this framework, we develop a constructive control-space discretization algorithm for general ZSDGs. By parameterizing the infinite-dimensional measure optimization into standard probability simplices and solving local linear programs, our method efficiently synthesizes executable mixed strategies. Furthermore, we rigorously prove that the global weak approximation error is strictly of order $\mathcal{O}(\bar\pi)$ with respect to the maximum commitment delay $\bar\pi$, and derive explicit analytical upper bounds for the strategy suboptimality gaps. Numerical examples are provided to illustrate and validate our theoretical results.