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

Vincent Leon, Iosif Sakos, Ryann Sim, Antonios Varvitsiotis

TL;DR

The paper investigates how to certify concavity and monotonicity in polynomial games with semialgebraic strategy sets, showing that the decision problem is NP-hard. It then develops two sum-of-squares hierarchies that yield polynomial-time SDP certificates at each level, with provable convergence and the property that almost all games become certifiable at a finite level. It introduces SOS-concave/monotone relaxations as global approximators to the exact classes and proves these approximations are dense and effective for practical use, including computing the closest SOS-structured game to any given game. The framework is applied to extensive-form games with imperfect recall, demonstrating how the approach can certify or closely approximate desirable equilibrium properties and even produce nearest SOS-monotone versions of games that lack Nash equilibria. The experiments illustrate the method on canonical examples and discuss scalability and potential enhancements like DSOS/SDSOS to tackle larger instances.

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.

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

TL;DR

The paper investigates how to certify concavity and monotonicity in polynomial games with semialgebraic strategy sets, showing that the decision problem is NP-hard. It then develops two sum-of-squares hierarchies that yield polynomial-time SDP certificates at each level, with provable convergence and the property that almost all games become certifiable at a finite level. It introduces SOS-concave/monotone relaxations as global approximators to the exact classes and proves these approximations are dense and effective for practical use, including computing the closest SOS-structured game to any given game. The framework is applied to extensive-form games with imperfect recall, demonstrating how the approach can certify or closely approximate desirable equilibrium properties and even produce nearest SOS-monotone versions of games that lack Nash equilibria. The experiments illustrate the method on canonical examples and discuss scalability and potential enhancements like DSOS/SDSOS to tackle larger instances.

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.

Paper Structure

This paper contains 36 sections, 12 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our Techniques and Contributions.
  3. Preliminaries
  4. Polynomial Games over Semialgebraic Sets
  5. Sum-of-Squares Optimization
  6. Concave & Monotone Games
  7. Certifying Concavity and Monotonicity in Polynomial Games
  8. Sum-of-Squares Certificates for Concavity & Monotonicity
  9. SOS-Concave & SOS-Monotone Games
  10. Modifications for the Certification of Concavity
  11. Application: Extensive-Form Games with Imperfect Recall
  12. Experimental Methodology & Results
  13. Example 1: The absent-minded taxi driver in Figure \ref{['fig:taxidriver']}.
  14. Example 2: A game with no Nash equilibria in Figure \ref{['fig:forgetfulpenaltyshootout']}.
  15. Example 3: A degree-4 strictly monotone general-sum game.
  16. ...and 21 more sections

Key Result

Theorem 3.1

Let $\mathscr{G}(\llbracket n\rrbracket, \mathcal{X}, u)$ be a polynomial game over a compact convex . If for some player $i$, $u_{i}$ is a polynomial of degree at least $3$ with respect to $x_{i} \in \mathcal{X}_{i}$, verifying whether $\mathscr{G}$ is concave or monotone is strongly -hard.

Figures (4)

  • Figure 1: Connections and inclusions among the game classes we study.
  • Figure 2: A Game with No Nash Equilibria
  • Figure 3: The Absent-minded Taxi Driver
  • Figure 4: Another Zero-sum Game with No Nash Equilibria

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2: Concave Games
  • Definition 2.3: Monotone Games
  • Definition 2.4: Strictly Concave/Monotone Games
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Definition 4.1: ℓ-SOS-Monotone Game
  • Theorem 4.2
  • Theorem 4.3
  • ...and 20 more