Stationary Mean Field Games on networks with sticky transition conditions
Jules Berry, Fabio Camilli
TL;DR
We address stationary Mean Field Games on finite networks with sticky vertex transitions, where the diffusion spends positive time both on edges and at vertices. The authors formulate a coupled HJB-FP system on the network and show that the stationary distribution decomposes as $m\,\mathscr{L} + \sum_{\varv} \eta_\varv T_\varv[m] \delta_\varv$, with the FP part carrying Dirac masses at vertices and the HJB part satisfying a generalized Kirchhoff condition. Existence and uniqueness of solutions to the MFG system are established via a Schauder fixed-point argument and monotonicity assumptions on the coupling; a verification theorem connects the HJB solution to a bounded-control optimal control problem for the sticky diffusion, providing a constructive way to obtain optimal policies. Overall, the work extends MFG analysis to networks with vertex congestion via sticky dynamics, offering rigorous tools for long-time average cost problems on graphs and preserving HJB-FP duality in this setting.
Abstract
We study stochastic Mean Field Games on networks with sticky transition conditions. In this setting, the diffusion process governing the agent's dynamics can spend finite time both in the interior of the edges and at the vertices. The corresponding generator is subject to limitations concerning second-order derivatives and the invariant measure breaks down into a combination of an absolutely continuous measure within the edges and a sum of Dirac measures positioned at the vertices. Additionally, the value function, solution to the Hamilton-Jacobi-Bellman equation, satisfies generalized Kirchhoff conditions at the vertices.