Sequential Decomposition of Correlated Mean-Field Games

In this paper, we study \emph{mean-field correlated equilibria} (MFCE), a class of correlated equilibria in mean-field games (MFGs) that enables decentralized coordination through a shared correlation device. Unlike standard mean-field equilibria (MFE), where agents act independently, MFCE allows agents to receive statistically correlated strategy recommendations, improving efficiency and fairness without direct communication. Using a sequential decomposition approach, we provide a backward-recursive characterization of MFCE for finite-horizon discrete-time games. The recursion incorporates incentive compatibility constraints, ensuring that no agent benefits from deviating from the recommended strategies while maintaining mean-field consistency. We illustrate MFCE through numerical examples a public investment game with heterogeneous costs. In both cases, MFCE achieves higher social efficiency, better steady-state outcomes, and increased continuation values for high-cost or high-risk agents compared to MFE. Our results establish MFCE as a tractable and conceptually meaningful extension of MFE, offering a practical framework for coordinating large populations of decentralized agents in settings where independent strategies are inefficient.