Stationary Equilibrium of Mean Field Games with Congestion-dependent Sojourn Times

We consider stationary equilibria of mean-field games between agents which follow continuous time semi-Markov decision processes with finite states and actions, when congestion affects their state-sojourn times but not the reward and transition structure. Games of this type arise in situations where selfish agents either traverse or circulate a network of congestible resources, as in routing games and models of driver mobility in ride-hailing platforms. A variational characterization of equilibria is employed to establish existence and uniqueness of average rewards. In contrast to ordinary routing games, where the price of anarchy can be unbounded, the latter equals 2 when agents never exit.