Blind Inverse Game Theory: Jointly Decoding Rewards and Rationality in Entropy-Regularized Competitive Games
This work addresses a fundamental identifiability issue in inverse game theory for competitive settings, enabling more accurate modeling of agent behavior in applications like robotics and economics.
The paper tackles the problem of jointly recovering reward parameters and rationality levels in competitive games when the rationality parameter is unknown, which causes statistical unidentifiability. It introduces Blind-IGT, a framework that resolves this ambiguity with a normalization constraint, achieving optimal convergence rates and strong empirical performance.
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 $τ$) is known a priori. When $τ$ is unknown, a fundamental scale ambiguity emerges that couples $τ$ with the reward parameters ($θ$), making them statistically unidentifiable. We introduce Blind-IGT, the first statistical framework to jointly recover both $θ$ and $τ$ 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.