Zero-Sum Games between Mean-Field Teams: A Common Information and Reachability based Analysis

This work studies the behaviors of two competing teams in a discrete environment, where the team-level interactions are captured as a zero-sum game while the dynamics within each team is formulated as a mean-field team problem. Following the ideas from the mean-field game literature, we first approximate the large-population team game with its infinite-population limit. We then introduce two fictitious coordinators and transform the infinite-population game into an equivalent zero-sum game between the two coordinators. We study the optimal coordination strategies for each team via a novel reachability analysis and later translate them back to decentralized strategies that the original agents deploy. We prove that the strategies are $\epsilon$-optimal for the original finite-population team game, and we further show that the suboptimality diminishes when team size approaches infinity. The theoretical guarantees are verified by numerical examples