Non-Potential Congestion Games with a Continuum of Players

Abstract

We study a class of congestion games with a continuum of heterogeneous players and a finite set of resources, without assuming the existence of a potential function. Players differ in their individual characteristics and may contribute differently to congestion, which is represented through general aggregate features of the population choosing each resource. We characterize the conditions under which such a game admits a potential and show that this structure generally fails when players have heterogeneous impacts on congestion. To analyze the non-potential case, we introduce the concept of Stochastic Variational Nash Equilibrium, in which players observe idiosyncratic cost shocks before selecting a resource. We show that equilibrium choices are pure conditional on players' characteristics and shock realizations, whereas the induced aggregate allocation is probabilistic. Under Gumbel-distributed shocks, the equilibrium is equivalent to a continuum-player logit Quantal Response Equilibrium. Using a fixed-point formulation and the Schauder--Tychonoff theorem, we establish existence for every positive noise level and recover a pure Nash equilibrium in the zero-noise limit. Under a coordinate-wise monotonicity condition on aggregate interaction costs, we further prove uniqueness and convergence of the regularized allocations to the pure equilibrium. An application to electric vehicle charging illustrates how heterogeneous energy demands jointly affect station choice, congestion, and market shares.