Optimal Rates for Feasible Payoff Set Estimation in Games

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 $ε$ 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.

💡 Research Summary

**
The paper addresses a fundamental inverse problem in game theory: given only observations of the actions taken by two players who are repeatedly playing a (possibly approximate) Nash equilibrium of an unknown bimatrix game, can a learner recover the entire set of payoff matrices that are consistent with the observed equilibrium behavior? Rather than estimating a single payoff matrix or imposing restrictive parametric assumptions, the authors aim to infer the feasible payoff set—the collection of all payoff pairs (A, B) (or a single matrix A in zero‑sum games) for which the observed mixed strategies (x, y) constitute an α‑Nash equilibrium. The quality of the estimate is measured in the Hausdorff distance (using the ℓ∞ norm) and the goal is to achieve an ε‑accurate reconstruction with probability at least 1 − δ while using as few samples as possible.

Problem setting.

• The game is either a general‑sum game specified by two matrices A, B ∈