Wardrop User Equilibrium

In a congestion game, each user picks a route to minimize their own travel time. The flow that arises when no user can save time by unilaterally switching routes is called Wardrop user equilibrium.

A flow $(x_1, x_2, \ldots, x_n)$ on a network with route latencies $\ell_1, \ldots, \ell_n$ is at Wardrop user equilibrium (UE) if there exists a common value $L$ such that, for every route $i$:

$$\ell_i(x_i) = L \quad \text{when } x_i > 0,$$ $$\ell_i(0) \geq L \quad \text{when } x_i = 0.$$

In words: every used route shares the same travel time $L$, and every unused route would be at least as slow. No flight can save time by switching.

Suppose 10 flights/hr fly between two airports along a northern corridor $N$ or a southern corridor $S$, with latencies (in minutes)

$$\ell_N(x_N) = 2x_N + 4, \qquad \ell_S(x_S) = x_S + 12.$$

If the flow splits as $(x_N, x_S) = (6, 4)$, then

$$\ell_N(6) = 16, \qquad \ell_S(4) = 16.$$

Both used routes share latency $L = 16$, so this flow is at user equilibrium.