Quantal Response Equilibrium as a Measure of Strategic Sophistication: Theory and Validation for LLM Evaluation

arXiv:2603.10029v1 Announce Type: new
Abstract: Theory of Mind benchmarks for large language models typically produce aggregate scores without theoretical grounding, making it unclear whether high performance reflects strategic reasoning or surface-level heuristics. We introduce a game-theoretic evaluation framework grounded in quantal response equilibrium (QRE). We derive closed-form equilibria for four strategic games, each targeting a distinct cognitive capability. We estimate QRE rationality parameters lambda that place model behavior on a continuous scale calibrated against human data (lambda_human in [1.0, 2.5]), and establish finite-sample convergence bounds via martingale concentration. Validation across 1,855 games with seven frontier models (plus four expansion models) confirms predictions: bluff rates converge to within 4% of equilibrium, lambda estimates range from 0.05 to 1.10 across games and models with substantial cross-model variation, and capability profiles differ across cognitive axes. Robustness analyses reveal high sensitivity to prompt framing and version instability in QRE rankings, highlighting the need for standardized protocols.