Certifying Concavity and Monotonicity in Games via Sum-of-Squares Hierarchies

Concavity and its refinements underpin tractability in multiplayer games, where players independently choose actions to maximize their own payoffs which depend on other players’ actions. In concave games, where players’ strategy sets are compact and convex, and their payoffs are concave in their own actions, strong guarantees follow: Nash equilibria always exist and decentralized algorithms converge to equilibria. If the game is furthermore monotone, an even stronger guarantee holds: Nash equilibria are unique under strictness assumptions. Unfortunately, we show that certifying concavity or monotonicity is NP-hard, already for games where utilities are multivariate polynomials and compact, convex basic semialgebraic strategy sets – an expressive class that captures extensive-form games with imperfect recall. On the positive side, we develop two hierarchies of sum-of-squares programs that certify concavity and monotonicity of a given game, and each level of the hierarchies can be solved in polynomial time. We show that almost all concave/monotone games are certified at some finite level of the hierarchies. Subsequently, we introduce SOS-concave/monotone games, which globally approximate concave/monotone games, and show that for any given game we can compute the closest SOS-concave/monotone game in polynomial time. Finally, we apply our techniques to canonical examples of imperfect recall extensive-form games.

💡 Research Summary

This paper addresses the computational challenge of verifying two fundamental structural properties in continuous games: concavity and monotonicity. These properties, central to the seminal work of Rosen, guarantee the existence (and under strict conditions, uniqueness) of Nash equilibria and the convergence of decentralized learning dynamics. The authors focus on the expressive class of polynomial games, where players’ utility functions are multivariate polynomials and their strategy sets are compact, convex basic semialgebraic sets.

The paper first establishes a negative result: certifying whether a given polynomial game is concave or monotone is strongly NP-hard (Theorem 3.1). This hardness holds even when utility functions are cubic polynomials in a player’s own action, presenting a significant barrier to leveraging the strong theoretical guarantees of these game classes in practice.

In response, the authors develop a positive, optimization-based framework using Sum-of-Squares (SOS) programming. They construct two hierarchies of sufficient conditions for concavity and monotonicity. For concavity, the condition that the Hessian of each player’s utility is negative semidefinite is reformulated as the non-negativity of a certain polynomial p_i(x, y) over the product of the strategy set and the unit ball. The ℓ-th level of the hierarchy checks whether this polynomial admits a Putinar-type SOS representation of degree at most 2ℓ. This check can be formulated as a semidefinite program (SDP) solvable in polynomial time in the problem size. A parallel hierarchy is constructed for monotonicity by considering the negative semidefiniteness of the symmetrized Jacobian of the game’s pseudogradient.

The authors prove the asymptotic completeness of these hierarchies. They show that every strictly concave or strictly monotone game is certified at some finite level ℓ. More powerfully, they prove that for “almost all” concave/monotone games (in a measure-theoretic sense), a certificate exists at some finite level (Theorem 3.3), meaning the SOS conditions are not only sufficient but also generically necessary.

Subsequently, the paper defines new subclasses: ℓ-SOS-concave and ℓ-SOS-monotone games (Definition 4.1), which are those games whose properties are certifiable at the ℓ-th level. They demonstrate that these classes globally approximate the original classes of concave/monotone games. A key practical contribution is Theorem 4.3, which shows that for any given polynomial game, the closest ℓ-SOS-concave (or monotone) game—with distance measured in the coefficient space of the utility functions—can be found by solving a single SDP. This provides a method to approximate an intractable game with a nearby, tractable one.

Finally, the authors apply their framework to canonical examples of extensive-form games with imperfect recall, demonstrating how the hierarchies can be used to verify properties and compute the closest SOS-based approximation. The work thus bridges game theory and computational algebraic geometry, offering a principled and computationally feasible approach to identifying and approximating games with desirable equilibrium properties.