A Mathematical Programming Approach to Computing and Learning Berk--Nash Equilibria in Infinite-Horizon MDPs

We study sequential decision-making when the agent's internal model class is misspecified. Within the infinite-horizon Berk-Nash framework, stable behavior arises as a fixed point: the agent acts optimally relative to a subjective model, while that model is statistically consistent with the long-run data endogenously generated by the policy itself. We provide a rigorous characterization of this equilibrium via coupled linear programs and a bilevel optimization formulation. To address the intrinsic non-smoothness of standard best-response correspondences, we introduce entropy regularization, establishing the existence of a unique soft Bellman fixed point and a smooth objective. Exploiting this regularity, we develop an online learning scheme that casts model selection as an adversarial bandit problem using an EXP3-type update, augmented by a novel conjecture-set zooming mechanism that adaptively refines the parameter space. Numerical results demonstrate effective exploration-exploitation trade-offs, convergence to the KL-minimizing model, and sublinear regret.