Partially Observed Mean Field Games Without Perfect Recall: Optimality Conditions and Equilibria

Abstract

This paper studies partially observed mean field games without perfect recall (WPR). The representative agent observes a noisy signal, but the control at time \(t\) uses only \(\mathcal G_t^I=\sigma(y_t)\), a generally non-nested information family. The conditional population law instead uses the observation filtration \(\mathbb F^Y\). These coupled levels rely on different information scales and are difficult to close within one construction. We parameterize the environment by a deterministic compatible joint law of state, driving variables, and random mean field term, thereby preserving its dependence structure without enlarging the agent's control information. For a fixed law, a reference measure and Girsanov's theorem yield a WPR stochastic maximum principle; the selected response is represented by the conditional Hamiltonian and WPR belief measure. The joint path posterior of hidden state and mean field term gives a weak Kushner-Stratonovich representation of the conditional population law. A recursive response map is continuous on a compact convex set of compatible laws, so Schauder-Tychonoff yields a weak WPR equilibrium. For a fixed equilibrium law and feedback, a compatible Yamada-Watanabe theorem lifts pathwise uniqueness to a strong realization. Finally, a linear-quadratic interbank lending example compares perfect recall (PR) with WPR. The PR response follows the Kalman-Bucy feedback, whereas the WPR response solves a Fredholm-Volterra equation and is affine in the current observation under Gaussianity. The numerical experiment illustrates how equilibrium behavior differs between PR and WPR.