Identification and Asymptotic Theory for the Empirical Bayesian Price of Anarchy in Atomic Congestion Games under Partial Observability

Empirical estimates of the Price of Anarchy on real road networks have, to our knowledge, treated drivers as identical and fully informed, disconnecting them from the finite, privately informed driver populations that road networks actually carry. No asymptotic theory exists for empirical Bayesian Price of Anarchy (BPoA) estimation under the partial observability that probe-vehicle sampling induces, with only a selected subset of trips observed and value-of-time types latent. This paper supplies that foundation. We characterize the atomic Bayesian congestion game, giving sufficient conditions for a unique mean-field (aggregative) Bayes-Nash equilibrium on a parallel-route restricted class. We formalize probe-data observability as missing-at-random selection on the driver population and give conditions under which the BPoA is point-identified, together with Manski-style partial-identification bounds when those conditions fail. We establish that the plug-in BPoA estimator is consistent and root-n asymptotically normal, with a Godambe sandwich variance valid under non-constant probe penetration, and we derive Wald and bootstrap confidence intervals; directional inference covers the boundary regime where the equilibrium map loses full differentiability. A Monte Carlo study calibrated to equilibrium-constrained recursive-logit fits from a Japanese arterial network confirms the root-n rate, asymptotic normality, and conservative interval coverage. A companion empirical paper applies a production implementation of the estimator to winter-disruption data, reporting a BPoA of 1.112; the theory developed here makes that estimate interpretable with quantified uncertainty. This record contains the preprint of the methodology paper (Methodology_paper.pdf) and its supplementary material with full proofs (Supplement.pdf). Funding: This work was supported by the Ministry of Land, Infrastructure, Transport and Tourism's SBIR Phase 3 Fund Project.