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

TL;DR

The paper tackles the identifiability problem in inverse game theory when the entropy-regularized rationality parameter $\tau$ is unknown, causing a scale ambiguity with rewards $\theta$. It introduces Blind-IGT, which uses a normalization constraint $\|\theta^*\|_2=1$ to disentangle scale and recovers both $\theta$ and $\tau$ via a Normalized Least Squares estimator, achieving the optimal $\mathcal{O}(N^{-1/2})$ convergence rate. The authors establish necessary and sufficient identifiability conditions (rank and non-uniformity) and provide partial identification via confidence sets when those conditions fail, with extensions to Markov games where transition dynamics may be unknown. Empirical results show convergence rates matching theory, demonstrate the necessity of the blind approach, and reveal robustness to misspecified normalization and unknown dynamics. Overall, Blind-IGT offers a rigorous statistical foundation for recovering objectives and bounded rationality in entropy-regularized competitive environments, enabling reliable reward decoding and interpretability in multi-agent systems.

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 $τ$) 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.

Figures (5)

• Figure 1: Convergence analysis of the NLS algorithm in Matrix Games. Log-log plots demonstrate that the estimation errors for both the reward parameters ($\theta$, Left) and the unknown temperature ($\tau$, Right) decrease at the optimal parametric rate of $\mathcal{O}(N^{-1/2})$ (dashed line), confirming the statistical efficiency of the Blind-IGT framework.
• Figure 2: Robustness to Misspecified Normalization Constant $C$. (Left) The directional error of the reward parameters remains low and stable. (Right) The estimated temperature scales linearly with the misspecification ratio, matching the theoretical expectation (dashed red line).
• Figure 3: Convergence analysis in Markov Games (Theorem \ref{['thm:mg_guarantees']}). The empirical slopes for Q-parameters ($\theta$), temperature ($\tau$), and the recovered reward function ($r$) closely match the theoretical $\mathcal{O}(K^{-1/2})$ rate.
• Figure 4: Robustness to Unknown Dynamics in Markov Games. The reward recovery error when estimating $P$ (Estimated-P) closely tracks the error when $P$ is known (Known-P). Both achieve near optimal $\mathcal{O}(K^{-1/2})$ rate (empirical slopes $-0.48$ and $-0.50$, respectively).
• Figure 5: Robustness to Feature Misspecification (omitting 1 of 5 features). Both the directional error (measured as $1-\text{cosine similarity}$) (Left) and the behavioral error (Right) decrease with sample size N, suggesting that Blind-IGT remains effective for behavioral modeling even under mild misspecification.

Theorems & Definitions (26)

• Remark 1: Robustness to Misspecification
• Theorem 1: Identifiability of Blind-IGT
• proof : Proof Sketch (Appendix \ref{['app:proof_identifiability']})
• Theorem 2: Finite Sample Bounds for Blind-IGT
• proof : Proof Sketch (Appendix \ref{['app:proof_finite_sample']})
• Proposition 1: Coverage of the Confidence Set
• proof : Proof Sketch (Appendix \ref{['app:proof_confidence_set']})
• Proposition 2: Identifiability in Markov Games
• Theorem 3: Sample Complexity for MGs
• proof : Proof Sketch (Appendix \ref{['app:proof_mg']})