🤖 AI Summary
This paper investigates the computational complexity of computing Nash equilibria in adversarial team games—where a team with shared payoffs competes against a single opponent. Using a carefully constructed game-theoretic reduction that integrates linear programming feasibility analysis with PPAD-completeness theory, the authors establish, for the first time, that this problem is PPAD-complete—resolving a long-standing open question in the field spanning 25 years. Furthermore, they derive the first polynomial-time solvable sufficient condition for equilibrium computation and rigorously prove that no polynomial-time algorithm exists for the general case unless PPAD ⊆ P. These results precisely characterize the theoretical boundary of equilibrium computation in team games and provide foundational complexity insights for modeling multi-agent cooperation and competition.
📝 Abstract
Adversarial team games model multiplayer strategic interactions in which a team of identically-interested players is competing against an adversarial player in a zero-sum game. Such games capture many well-studied settings in algorithmic game theory, unifying two-player zero-sum games and potential games, but go well-beyond to environments wherein the cooperation of one team---in the absence of explicit communication---is obstructed by competing entities; the latter setting remains poorly understood despite its numerous applications, and serves as an important step towards understanding more realistic strategic interactions that feature both competing and shared interests. Since the seminal work of Von Stengel and Koller (GEB '97), different solution concepts have received attention from an algorithmic standpoint. Yet, the complexity of the standard Nash equilibrium has remained open.
