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 𝒢_t^I=σ(y_t), a generally non-nested information family. The conditional population law instead uses the observation filtration 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.

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Cite

BibTeX RIS

BibTeX

@misc{zhang2026partially,
  title = {{Partially Observed Mean Field Games Without Perfect Recall: Optimality Conditions and Equilibria}},
  author = {Xuanping Zhang and Xiao Zhang and Wang Yao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00880},
  eprint = {2609.00880},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00880},
}

RIS

TY  - UNPB
TI  - Partially Observed Mean Field Games Without Perfect Recall: Optimality Conditions and Equilibria
AU  - Xuanping Zhang
AU  - Xiao Zhang
AU  - Wang Yao
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00880
UR  - https://arxiv.org/abs/2609.00880
AB  - 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 𝒢_t^I=σ(y_t), a generally non-nested information family. The conditional population law instead uses the observation filtration 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.
ER  -

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