Optimal Rates for Feasible Payoff Set Estimation in Games

TL;DR

This paper establishes the first minimax-optimal rates for estimating the set of feasible payoffs in game-theoretic settings, enabling more accurate inverse game theory analysis from observed actions.

Contribution

It introduces the first minimax-optimal rates for feasible payoff set estimation in both zero-sum and general-sum games, advancing the theoretical foundation of inverse game theory.

Findings

Provides minimax-optimal rates for payoff set estimation

Applies to both exact and approximate equilibrium play

Works for zero-sum and general-sum games

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.