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

2025-12-11

AI Summary

Deciding concavity and monotonicity in multi-player games is NP-hard. Method: We propose a hierarchy of sum-of-squares (SOS) certification frameworks that transforms the original NP-hard verification problem into a sequence of polynomial-time solvable semidefinite programs. Contribution/Results: This work establishes the first SOS-based hierarchical decision procedure for concavity and monotonicity in games. We define the classes of SOS-concave and SOS-monotone games, enabling optimal structural approximation of arbitrary games. We prove that almost all concave or monotone games are exactly identified at some finite level of the hierarchy. The framework is successfully applied to canonical settings—including imperfect-information extensive-form games—supporting efficient structural verification of Nash equilibrium existence and uniqueness, as well as approximate equilibrium reconstruction. By unifying algebraic certification with game-theoretic structure, our approach introduces a new paradigm for automated, rigorous certification of game-theoretic properties.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Certifying concavity and monotonicity in multiplayer games is NP-hard
• Developing polynomial-time sum-of-squares hierarchies to certify these properties
• Approximating concave/monotone games via SOS-concave/monotone games efficiently

Innovation

• Sum-of-squares hierarchies certify concavity and monotonicity in games
• Polynomial-time solvable levels approximate concave and monotone games
• Compute closest SOS-concave/monotone games efficiently via optimization