Best-Response Dynamics and Iterated Routing

In iterated routing, players (individual flights or fleets) take turns updating their chosen path while every other player's choice stays fixed. The new path chosen by player $i$ is its best response: the path that minimizes the cost player $i$ would incur after the move.

For a single flight choosing among parallel routes with load-dependent latencies $c_j(x_j),$ the cost player $i$ would pay on route $k$ is the latency evaluated at the load that results after the move:

$$\text{post-move cost on route } k \;=\; c_k\!\left(x_k + 1 - [\,i \text{ is currently on } k\,]\right).$$

If $i$ switches to route $k,$ the load there goes up by one. If $i$ stays on its current route, the load is unchanged. The best response is the route with the lowest post-move cost.

To illustrate, suppose two parallel routes have latencies $c_1(x) = 2 + x$ and $c_2(x) = 5 + \tfrac{1}{2}x,$ with current loads $x_1 = 5$ and $x_2 = 2.$ A flight currently on route $1$ compares

$$c_1(5) = 7 \quad (\text{stay}), \qquad c_2(3) = 6.5 \quad (\text{switch}).$$

Route $2$ has the lower post-move cost, so the best response is to switch to route $2.$