Wardrop Equilibrium

A flow assignment in which every used path has equal cost and no unused path has lower cost.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

A flow $f$ on a directed graph with latency functions $\ell_e(f_e)$ is a Wardrop equilibrium if for each O-D pair, all used paths $p$ have equal latency $L_p(f)=\sum_{e\in p}\ell_e(f_e)$ and no unused path has strictly lower latency.

Beckmann-McGuire-Winsten (1956): Wardrop equilibrium minimizes the potential $\sum_e\int_0^{f_e}\ell_e(t)\,dt$ .

Intuition

Drivers distribute themselves so that no individual can reduce their travel time by switching routes.

It is the Nash equilibrium of a nonatomic congestion game (continuum of infinitesimal agents).

Worked example

Pigou's two-link network: one fast road ( $\ell(x)=1$ ) and one that congests ( $\ell(x)=x$ ). Wardrop splits flow $1/2$ - $1/2$ ; socially optimal flow is $1/4$ - $3/4$ giving PoA $=4/3$ .

Braess's paradox: adding a link can increase equilibrium travel time for all.

The math

Variational inequality formulation: find $f^*$ such that $\sum_e\ell_e(f_e^*)(f_e-f_e^*)\ge 0$ for all feasible $f$ .

Equivalent to convex optimization: $\min \sum_e\int_0^{f_e}\ell_e(t)\,dt$ subject to flow conservation.

Where it is used

Transportation planning, Internet routing (BGP/OSPF), and communication network design.

Foundational in algorithmic game theory and the Price of Anarchy literature.

More in Game theory

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.