14.15J/6.207J Networks: Applications of Game Theory to Networks

Networks: Lecture 12Wardrop EquilibriaMore General Traffic Model: NotationLet us start with a single origin-destination pair.Directed network N = (V , E ).P denotes the set of paths between origin and destination.x p denotes the flow on path p ∈ P.Each link i ∈ E has a latency function l i (x i ), wherex i = x p .{p∈P|i∈p}Here the notation p ∈ P|i ∈ p denotes the paths p that traverse linki ∈ E .The latency function captures congestion effects. Let us assume forsimplicity that l i (x i ) is nonnegative, differentiable, and nondecreasing.We normalize total traffic to 1 and in the context of the gametheoretic formulation here, I = [0, 1]. We also assume that all trafficis homogeneous. Each motorist wishes to minimize delay.7