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 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 \(G(x,m) = \int \varphi (x-z){\textrm{d}}m(z)\) $G(x,m)=\int \phi (x-z)\text{d}m(z)$ where \(\varphi \) $\phi$ 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 \(\varphi \) $\phi$ . We discuss the potential implications of this result for real-world opinion dynamics and political systems.