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.

Adding a link to a nonatomic traffic flow network can increase the equilibrium travel delay. Extra routes change the incentives defining a Wardrop equilibrium, so the new equilibrium need not retain an earlier good allocation. The system-optimal delay cannot worsen when the old allocation remains feasible; the paradox concerns selfish equilibrium.

For a differentiable link delay $D_j(y)$, the toll $T_j(y)=y{D}_j^{\prime }(y)$ charges a user for the additional delay imposed on the other users. Its perceived link cost is the derivative of $y{D}_j(y)$. The resulting Beckmann potential equals total travel delay. If total delay is convex, its Wardrop equilibria minimize that delay globally. Strict increase of $D_j$ alone does not imply this convexity; optimal-flow calibrated congestion tolling avoids that extra requirement.

Let $y^{\ast }$ be a globally delay-minimizing feasible vector of link throughputs. For continuously differentiable strictly increasing delays, calibrated tollsmake the marginal social costs at $y^{\ast }$ equal the perceived route costs there. First-order route-exchange conditions make $y^{\ast }$ a tolled Wardrop equilibrium. The nonnegative nondecreasing penalty vanishes at the optimum; choosing ${\epsilon }_j>0$ makes the toll genuinely traffic-dependent. The perceived link delays remain strictly increasing, so their Beckmann potential gives unique equilibrium link loads. Every tolled Wardrop equilibrium therefore has the globally optimal $y^{\ast }$, even when the total-delay objective is not convex.

Strictly increasing link delays make Wardrop equilibrium link throughputs unique, but route decompositions of those throughputs can differ. With unit demand through two successive stages of two parallel links each and link delay $D(u)=u$, the four-route vectors $(t,1/2-t,1/2-t,t)$ for $0\le t\le 1/2$ all induce link loads $1/2$. All routes then have equal delay one.

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.

For nonnegative strictly increasing link delays and nonempty routes, let $M=B(0)$ and $L={\lim }_{\lambda \to \infty }B(\lambda )$. The inverse demand tends to infinity as flow decreases to $L$, so increasing flow slightly there gains more utility than the bounded link-cost increment. At $M$ the inverse demand is zero, whereas a used route has positive delay, so reducing flow improves the objective. A minimizer consequently has $L<f<M$ and satisfies the ordinary interior Wardrop equilibrium conditions. Lower-semicontinuous endpoint extensions permit minimization on the compact demand interval even when utility diverges at $L$.

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.

An interior reference point defines the utility of a continuous strictly decreasing demand function without requiring integrability of its inverse at zero. The inverse demand is decreasing, so the utility is strictly concave. For example $B(\lambda )=(1+\lambda {)}^{-1/2}$ has inverse $f^{-2}-1$, whose integral from zero diverges. The interior-reference utility still has the required derivative and is finite at every interior demand.

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.