Optima and Equilibria for a Model of Traffic Flow

Abstract

The paper is concerned with the Lighthill-Whitham model of traffic flow, where the density of cars is described by a scalar conservation law. A cost functional is introduced, depending on the departure time and on the arrival time of each driver. Under natural assumptions, we prove the existence of a unique globally optimal solution, minimizing the total cost to all drivers. This solution contains no shocks and can be explicitly described. We also prove the existence of a Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time. A characterization of the Nash solution is provided, establishing its uniqueness. Some explicit examples are worked out, comparing the costs of the optimal and the equilibrium solution. The analysis also yields a strategy for optimal toll pricing.

Nash equilibria for a model of traffic flow with several groups of drivers

“…The function u → f (u) = ρ is obtained by inverting the function ρ → ρv(ρ) = u, over the interval [0, ρ * ] where ∂ ∂ρ [ρv(ρ)] ≥ 0. As in [2], we extend f to a function f : R → R∪{+∞}, by setting (see fig. 1)…”

“…As in [2], we consider a cost ϕ(x) for early departure and a cost ψ(x) for late arrival. The basic assumptions will be: …”

“…In the case all drivers have the same starting and arrival costs, under the assumptions (A1)-(A2) the existence and uniqueness of the Nash equilibrium solution were established by Theorem 3 in [2]. We remark that the functions ϕ, ψ in (3.2) may not satisfy all the assumptions stated in (A2).…”

“…These are solutions where no single driver can lower his own cost by changing his own departure time. In the case n = 1, where all drivers share the same cost, the existence, uniqueness, and a characterization of Nash solutions were recently established in [2].…”

“…A solution of (1.1) is determined by assigning the incoming flow ρ(0, t) v(ρ(0, t)) =ū(t) (1.3) at the entrance of the freeway. As remarked in [2], it is more convenient to switch the roles of the independent variables t, x, and rewrite the boundary value problem (1.1), (1.3) as a Cauchy problem for the flux of cars u = ρv(ρ), namely Here u → f (u) = ρ is defined as a partial inverse of the map ρ → ρ v(ρ) = u, as in Fig. 1.…”