Numerical Analysis of the Projection Dynamics and Their Associated Mean Field Control

Projection dynamic is an evolutionary game with a nonsmooth transition rate. Projection dynamic has been less studied compared to major evolutionary game models such as the replicator and logit dynamics due to its lower regularity, which is more challenging to theoretically address. We propose a regularized version of the dynamic, called regularized projection dynamic (RPD), where the transition rate is Lipschitz continuous, making its solution a time-dependent probability measure in a suitable Banach space. This regularization not only enables us to derive a more tractable model, but also leads to a mean field game (MFG) whose formal large-discount limit is the RPD, resulting in a forward–backward generalization of the RPD. The vanishing regularization limit of the MFG leads to an essentially unbounded control, making the incorporation of regularization essential for its analysis. We present finite difference methods that can handle the RPD and MFG, where the regularization guarantees nonnegativity of their probability densities.