Optimal Rates for Feasible Payoff Set Estimation in Games

2026-02-04

AI Summary

This study addresses the problem of estimating, with high probability, the set of payoff functions compatible with observed (approximate) Nash equilibrium behaviors in two-player bimatrix games, without prior knowledge of the game structure. Building on inverse game theory, the work proposes a set-estimation approach grounded in statistical learning theory and quantifies estimation accuracy via the Hausdorff metric. It establishes, for the first time, minimax optimal convergence rates for estimating the feasible payoff set under both exact and approximate equilibria, covering both zero-sum and general-sum settings. These results provide a rigorous theoretical foundation and optimal learning guarantees for payoff inference in multi-agent systems.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Multiagent Systems: Mechanism Design
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study a setting in which two players play a (possibly approximate) Nash equilibrium of a bimatrix game, while a learner observes only their actions and has no knowledge of the equilibrium or the underlying game. A natural question is whether the learner can rationalize the observed behavior by inferring the players'payoff functions. Rather than producing a single payoff estimate, inverse game theory aims to identify the entire set of payoffs consistent with observed behavior, enabling downstream use in, e.g., counterfactual analysis and mechanism design across applications like auctions, pricing, and security games. We focus on the problem of estimating the set of feasible payoffs with high probability and up to precision $\epsilon$ on the Hausdorff metric. We provide the first minimax-optimal rates for both exact and approximate equilibrium play, in zero-sum as well as general-sum games. Our results provide learning-theoretic foundations for set-valued payoff inference in multi-agent environments.

Problem

• feasible payoff set
• inverse game theory
• Nash equilibrium
• bimatrix game
• payoff inference

Innovation

• inverse game theory
• feasible payoff set estimation
• minimax-optimal rates
• Nash equilibrium
• Hausdorff metric