Cluster formation in iterated Mean Field Games

We study a simple first-order mean field game in which the coupling with the mean field is only in the final time and gives an incentive for players to congregate. For a short enough time horizon, the equilibrium is unique. We consider the process of \emph{iterating} the game, taking the final population distribution as the initial distribution in the next iteration. Restricting to one dimension, we take this to be a model of coalition building for a population distributed over some spectrum of opinions. Our main result states that, given a final coupling of the form $G(x,m)=\int \phi (x-z)\text{dif}m(z)$ where $\phi$ is a smooth, even, non-positive function of compact support, then as the number of iterations goes to infinity the population tends to cluster into discrete groups, which are spread out as a function of the size of the support of $\phi$. We discuss the potential implications of this result for real-world opinion dynamics and political systems.