Zero-Sum Games between Mean-Field Teams: Reachability-Based Analysis under Mean-Field Sharing

This work studies the behaviors of two large-population teams competing in a discrete environment. The team-level interactions are modeled as a zero-sum game while the agent dynamics within each team is formulated as a collaborative mean-field team problem. Drawing inspiration from the mean-field literature, we first approximate the large-population team game with its infinite-population limit. Subsequently, we construct a fictitious centralized system and transform the infinite-population game to an equivalent zero-sum game between 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 ϵ-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.