Blind Inverse Game Theory: Jointly Decoding Rewards and Rationality in Entropy-Regularized Competitive Games

TL;DR

This paper introduces Blind-IGT, a novel framework for jointly recovering reward parameters and rationality levels in entropy-regularized competitive games, addressing the scale ambiguity problem when the rationality parameter is unknown.

Contribution

Blind-IGT is the first statistical method to jointly estimate rewards and rationality in inverse game theory, with theoretical guarantees and extensions to Markov games.

Findings

Achieves optimal convergence rate of O(N^{-1/2}) for joint parameter estimation.

Provides conditions for unique identification of parameters.

Demonstrates strong empirical performance even with unknown transition dynamics.

Abstract

Inverse Game Theory (IGT) methods based on the entropy-regularized Quantal Response Equilibrium (QRE) offer a tractable approach for competitive settings, but critically assume the agents' rationality parameter (temperature $\tau$) is known a priori. When $\tau$ is unknown, a fundamental scale ambiguity emerges that couples $\tau$ with the reward parameters ($\theta$), making them statistically unidentifiable. We introduce Blind-IGT, the first statistical framework to jointly recover both $\theta$ and $\tau$ from observed behavior. We analyze this bilinear inverse problem and establish necessary and sufficient conditions for unique identification by introducing a normalization constraint that resolves the scale ambiguity. We propose an efficient Normalized Least Squares (NLS) estimator and prove it achieves the optimal $\mathcal{O}(N^{-1/2})$ convergence rate for joint parameter recovery. When strong identifiability conditions fail, we provide partial identification guarantees through confidence set construction. We extend our framework to Markov games and demonstrate optimal convergence rates with strong empirical performance even when transition dynamics are unknown.