Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Data-Scarce Identification of Game Dynamics via Sum-of-Squares Optimization

Iosif Sakos, Antonios Varvitsiotis, Georgios Piliouras

TL;DR

The paper tackles the problem of identifying game dynamics from severely data-scarce observations in multiagent settings. It develops the Side-Information Assisted Regression (SIAR) framework, which embeds game-theoretic side information as polynomial constraints and solves a sum-of-squares optimization hierarchy to recover a polynomial vector field that governs strategy evolution. SIAR provides convergence guarantees and demonstrates strong empirical performance across standard games, chaotic dynamics, and equilibrium-selection problems, often outperforming data-hungry baselines like SINDy. The approach has practical implications for policymakers and system designers, enabling rapid, reliable insight into strategic dynamics and potential control or mechanism-design interventions under limited data.

Abstract

Understanding how players adjust their strategies in games, based on their experience, is a crucial tool for policymakers. It enables them to forecast the system's eventual behavior, exert control over the system, and evaluate counterfactual scenarios. The task becomes increasingly difficult when only a limited number of observations are available or difficult to acquire. In this work, we introduce the Side-Information Assisted Regression (SIAR) framework, designed to identify game dynamics in multiplayer normal-form games only using data from a short run of a single system trajectory. To enhance system recovery in the face of scarce data, we integrate side-information constraints into SIAR, which restrict the set of feasible solutions to those satisfying game-theoretic properties and common assumptions about strategic interactions. SIAR is solved using sum-of-squares (SOS) optimization, resulting in a hierarchy of approximations that provably converge to the true dynamics of the system. We showcase that the SIAR framework accurately predicts player behavior across a spectrum of normal-form games, widely-known families of game dynamics, and strong benchmarks, even if the unknown system is chaotic.

Data-Scarce Identification of Game Dynamics via Sum-of-Squares Optimization

TL;DR

The paper tackles the problem of identifying game dynamics from severely data-scarce observations in multiagent settings. It develops the Side-Information Assisted Regression (SIAR) framework, which embeds game-theoretic side information as polynomial constraints and solves a sum-of-squares optimization hierarchy to recover a polynomial vector field that governs strategy evolution. SIAR provides convergence guarantees and demonstrates strong empirical performance across standard games, chaotic dynamics, and equilibrium-selection problems, often outperforming data-hungry baselines like SINDy. The approach has practical implications for policymakers and system designers, enabling rapid, reliable insight into strategic dynamics and potential control or mechanism-design interventions under limited data.

Abstract

Understanding how players adjust their strategies in games, based on their experience, is a crucial tool for policymakers. It enables them to forecast the system's eventual behavior, exert control over the system, and evaluate counterfactual scenarios. The task becomes increasingly difficult when only a limited number of observations are available or difficult to acquire. In this work, we introduce the Side-Information Assisted Regression (SIAR) framework, designed to identify game dynamics in multiplayer normal-form games only using data from a short run of a single system trajectory. To enhance system recovery in the face of scarce data, we integrate side-information constraints into SIAR, which restrict the set of feasible solutions to those satisfying game-theoretic properties and common assumptions about strategic interactions. SIAR is solved using sum-of-squares (SOS) optimization, resulting in a hierarchy of approximations that provably converge to the true dynamics of the system. We showcase that the SIAR framework accurately predicts player behavior across a spectrum of normal-form games, widely-known families of game dynamics, and strong benchmarks, even if the unknown system is chaotic.
Paper Structure (26 sections, 2 theorems, 29 equations, 3 figures)

This paper contains 26 sections, 2 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Game Dynamics
  5. The SIAR Problem
  6. Side-Information Constraints
  7. Properties of Strategic Interactions in Normal-Form Games
  8. Forward-Invariance of the Strategy Space
  9. Game Symmetries
  10. Existence of Uncoupled Formulation
  11. Common Assumptions for Strategic Interactions
  12. Nash Stationarity
  13. Positive Correlation
  14. Elimination of Strictly Dominated Strategies
  15. Solving the SIAR Problem
  16. ...and 11 more sections

Key Result

lemma thmcounterlemma

For each type of the side-information constraints $S$ in sec:SideInformationConstraints, we can construct functionals $L_{S}(f) \colon \mathcal{C}_{1}(\mathcal{X}) \to \mathbb{R}$ that quantify how close some vector field of update policies $f$ satisfies $S$. In particular, $L_{S}$ satisfy the fol

Figures (3)

  • Figure 1: On the left, are the trajectories of the dynamical system as predicted by the and frameworks in the interval $[t_{*}, 10]$, where $t_{*} = 1$ denotes the evaluation time. The true trajectory, depicted in faint dashed lines, is given by the replicator dynamics for the matching pennies game initialized at $\lparen[\rparen]{(0.25, 0.75), (0.125, 0.875)}$. The two solutions were computed using a sample of five data points at times $t = 0.0, 0.2, \dots, 0.8$, depicted as dots. For the problem, we used the forward invariance of the strategy space and the positive correlation property as side-information constraints. The solution far outperforms the solution, achieving near perfect recovery of the true trajectory. On the right, are the phase plots of the two solutions, where the aforementioned trajectory is depicted in black, and the dataset as black dots. The phase plot of the solution is nearly identical to the true phase plot of the system, while the phase plot of the solution is accurate only around the given dataset.
  • Figure 2: Phase plots of the solutions for different sets of side-information constraints on the problem of identifying the replicator dynamics for the matching pennies game given a noisy sample of five data points, depicted as red dots, from a single trajectory of the system depicted in black. As we add side-information constraints, the accuracy of the solutions increase, and near perfect recovery is achieved whenever, both, the forward invariance of the strategy space and the positive correlation property are imposed to the problem.
  • Figure 3: On the left, are different Poincaré sections (projected into ($x_{1 1}, x_{2, 2}$)-plane) of the replicator dynamics for a 0.25-RPS game. In the middle, the same Poincaré sections as computed using the solution. On the right, is depicted the distance of the corresponding trajectories in Lyapunov times.

Theorems & Definitions (3)

  • remark thmcounterremark
  • lemma thmcounterlemma
  • corollary thmcountercorollary