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Operator Splitting for Learning to Predict Equilibria in Convex Games

Daniel McKenzie, Howard Heaton, Qiuwei Li, Samy Wu Fung, Stanley Osher, Wotao Yin

TL;DR

This paper addresses learning to predict equilibria in context-rich convex games where the underlying game dynamics are unknown, by introducing Nash Fixed Point Networks (N-FPNs) that directly output equilibria. N-FPNs employ a constraint decoupling scheme to handle complex agent action sets without expensive projections and are trained using Jacobian-Free Backpropagation, enabling implicit-network training that is faster and easier than prior approaches. Empirical results demonstrate that N-FPNs scale to problem sizes far beyond previous learned-game solvers and maintain performance in context-driven settings. The work advances practical equilibrium prediction by combining a structurally aligned neural architecture with efficient training and broad scalability, offering a compelling tool for context-aware decision-making in economics, traffic, and policy contexts.

Abstract

Systems of competing agents can often be modeled as games. Assuming rationality, the most likely outcomes are given by an equilibrium (e.g. a Nash equilibrium). In many practical settings, games are influenced by context, i.e. additional data beyond the control of any agent (e.g. weather for traffic and fiscal policy for market economies). Often the exact game mechanics are unknown, yet vast amounts of historical data consisting of (context, equilibrium) pairs are available, raising the possibility of learning a solver which predicts the equilibria given only the context. We introduce Nash Fixed Point Networks (N-FPNs), a class of neural networks that naturally output equilibria. Crucially, N- FPNs employ a constraint decoupling scheme to handle complicated agent action sets while avoiding expensive projections. Empirically, we find N-FPNs are compatible with the recently developed Jacobian-Free Backpropagation technique for training implicit networks, making them significantly faster and easier to train than prior models. Our experiments show N-FPNs are capable of scaling to problems orders of magnitude larger than existing learned game solvers.

Operator Splitting for Learning to Predict Equilibria in Convex Games

TL;DR

This paper addresses learning to predict equilibria in context-rich convex games where the underlying game dynamics are unknown, by introducing Nash Fixed Point Networks (N-FPNs) that directly output equilibria. N-FPNs employ a constraint decoupling scheme to handle complex agent action sets without expensive projections and are trained using Jacobian-Free Backpropagation, enabling implicit-network training that is faster and easier than prior approaches. Empirical results demonstrate that N-FPNs scale to problem sizes far beyond previous learned-game solvers and maintain performance in context-driven settings. The work advances practical equilibrium prediction by combining a structurally aligned neural architecture with efficient training and broad scalability, offering a compelling tool for context-aware decision-making in economics, traffic, and policy contexts.

Abstract

Systems of competing agents can often be modeled as games. Assuming rationality, the most likely outcomes are given by an equilibrium (e.g. a Nash equilibrium). In many practical settings, games are influenced by context, i.e. additional data beyond the control of any agent (e.g. weather for traffic and fiscal policy for market economies). Often the exact game mechanics are unknown, yet vast amounts of historical data consisting of (context, equilibrium) pairs are available, raising the possibility of learning a solver which predicts the equilibria given only the context. We introduce Nash Fixed Point Networks (N-FPNs), a class of neural networks that naturally output equilibria. Crucially, N- FPNs employ a constraint decoupling scheme to handle complicated agent action sets while avoiding expensive projections. Empirically, we find N-FPNs are compatible with the recently developed Jacobian-Free Backpropagation technique for training implicit networks, making them significantly faster and easier to train than prior models. Our experiments show N-FPNs are capable of scaling to problems orders of magnitude larger than existing learned game solvers.

Paper Structure

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem