Congestion Games and Selfish Routing

In a congestion game, multiple users (here, aircraft) share a network of edges -- sectors, corridors, or arcs -- and the travel time on each edge depends on how many users select it. The travel time on edge $e$ when its flow is $x_e$ is called the latency of $e$, written

$$\ell_e(x_e).$$

A standard model is the affine latency

$$\ell_e(x_e) = a_e\, x_e + b_e,$$

where $b_e \geq 0$ is the free-flow travel time and $a_e \geq 0$ measures congestion sensitivity.

The latency of a path $P$ at flow $\mathbf{x}$ is the sum of its edge latencies:

$$\ell_P(\mathbf{x}) = \sum\limits_{e \in P} \ell_e(x_e).$$

For example, if an aircraft traverses two sectors with latencies $\ell_{S_1}(x) = x + 3$ and $\ell_{S_2}(x) = 2x$, and currently $x_{S_1} = 2$, $x_{S_2} = 4$, then the path latency is

$$\ell_P = (2 + 3) + (2 \cdot 4) = 5 + 8 = 13.$$