Wardrop equilibrium

A Wardrop equilibrium assigns positive traffic only to routes with minimum delay for their source-sink pair. With fixed demands and continuous increasing link delays, it minimizes the Beckmann potential. Strictly increasing delays make link throughputs unique, while route flows can remain nonunique.

In an elastic-demand Wardrop equilibrium, demand $f_s$ responds to the minimum route delay ${\lambda }_s$ through $f_s=B_s({\lambda }_s)$. An inverse demand function supplies the utility term that must be subtracted from the Beckmann potential.

A strictly decreasing demand function has a decreasing inverse on its range. The utility primitive $\int P_s(u)du$ is a concave function. A positive interior reference value avoids assuming that an improper integral from zero is finite.

The Beckmann potential integrates each link's delay function. Its gradient with respect to route flows is the corresponding vector of route delays, making Wardrop equilibrium a convex optimization problem.

This matrix has $H_{sr}=1$ when route $r$ serves source-sink pair $s$, and zero otherwise. Thus $Hx=f$ aggregates route flows into source-sink demands.