On the Nash equilibrium for a model of traffic flow
The paper consider a conservation law model of traffic flow on a highway, where each driver chooses an optimal departure time $τ^d$, based on a departure cost $φ(τ^d)$ and an arrival cost $ψ(τ^a)$. Nash equilibria are studied, where no driver can reduce his own cost by changing his departure time. The first results provide continuity estimates on the arrival time, depending on the distribution of departures. In terms of the ${\bf L}^1$ distance, we show that the corresponding map is Hölder continuous with exponent $1/2$ in Lagrangian variables (labelling individual drivers), and Lipschitz continuous in Eulerian space variables. Concerning the regularity of the a Nash equilibrium, our results show that, when the total number of drivers is sufficiently large, a queue alway forms at the entrance of the highway and the corresponding traffic density contains a shock. Moreover, for any number of drivers (possibly small), we prove that for generic $\mathcal C^2$ cost functions $φ, ψ$ the Nash equilibrium solution still contains a shock. In the last section of the paper, assuming that the initial departure rate is not a Nash equilibrium, we consider an evolution equation for the departure rate, where each driver keeps changing his daily departure time in order to reduce his own cost. It is proved that this nonlocal integro-differential evolution equation admits a unique solution, globally in time. This provides some theoretical support to earlier numerical simulations, which indicated the dynamic instability of the Nash equilibrium and the possible existence of a chaotic attractor.