The Impact of Sojourn Time Distributions on the Price of Anarchy

Mean Field Games (MFGs) provide a powerful framework for modeling strategic interactions in large-scale multi-agent systems. This paper studies continuous-time MFGs with finite action spaces and general sojourn-time distributions, extending beyond the classical exponential assumption. We establish the existence of stationary equilibria under mild regularity conditions and show that stateless continuous-time MFGs are equivalent to population games. Our main contribution is a systematic efficiency analysis via the Price of Anarchy (PoA) that highlights the critical role of sojourn-time distributions. We characterize extremal PoA values across different distributional families and identify distributions that achieve them. In particular, we prove a uniform lower bound of two on the PoA that holds across all feasible families of sojourn-time distributions satisfying a mild mean-richness condition. This result shows that inefficiency is intrinsic to decentralized decision-making and cannot be eliminated by distributional modeling choices alone. Together, our findings demonstrate that distributional heterogeneity in delays is a first-order determinant of efficiency in continuous-time multi-agent systems.