  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
$\epsilon$-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.
