Optimal Rates for Feasible Payoff Set Estimation in Games

TL;DR

The paper addresses the problem of learning the full feasible payoff set consistent with observed equilibrium behavior in an unknown bimatrix game. It develops minimax-optimal sample complexity rates for estimating this set under Hausdorff error, covering both general-sum and zero-sum games and both exact ($\alpha=0$) and approximate ($\alpha>0$) equilibria. The main approach uses a combination of set-valued mappings, linear-inequality representations, and concentration arguments, including a fractional-knapsack-inspired analysis for the approximate-equilibrium case. The results provide a principled learning-theoretic foundation for set-valued payoff inference in multi-agent environments and have downstream use in counterfactual analysis, mechanism design, and robustness to payoff ambiguity.

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 $ε$ 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.

Figures (1)

• Figure 1: Visual representations of payoff sets $\mathcal{G}^x_\alpha(\cdot, \cdot)$ for several pairs of instances $(x^0, y^0)$ (striped red region) and $(x^1, y^1)$ (solid green region). In each figure, a point $P^*$ is used to illustrate the Hausdorff distance between the two areas. The $y$ vector is $(1,0)$ in all problems. (Left): we set $\alpha=0$, $x^0=(0,1)$, $x^1=(\pi, 1-\pi)$ for any $\pi \in (0,1)$; (Center): $\alpha = 0.2$, $x^0=(0,1)$, and $x^1=(\pi, 1-\pi)$ for $\pi = 0.05$; (Right): $\alpha=0.1$, $\epsilon=0.1$, $x^0 = (\tfrac{\alpha}{2}, 1-\tfrac{\alpha}{2})$ and $x^1 =(\tfrac{\alpha(1+3\epsilon)}{2}, 1-\tfrac{\alpha(1+3\epsilon)}{2})$.

Theorems & Definitions (69)

• Remark 2.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 4.1
• Theorem 4.2
• proof
• proof : Proof of \ref{['thm:lb-alpha-0']}
• Theorem 1.1
• proof