Lemma 2.2 — a feasible flow is at Nash equilibrium iff every used route is a minimum-latency route

Consider a routing instance: edges $e$ with latency functions $\ell_e$ that are nonnegative, nondecreasing and continuous on $[0,\infty)$; routes $P$ given by $0/1$ edge–route incidence columns, each serving one commodity $i$ (the set $\mathcal P_i$); and positive rates $r_i$. For a flow $f$ write $\ell_P(f)=\sum_{e\in P}\ell_e(f_e)$ for the latency of route $P$.

Lemma 2.2. A flow $f$ feasible for the instance is at Nash equilibrium (Definition 2.1) if and only if, for every commodity $i$ and all $P_1,P_2\in\mathcal P_i$,

The right-hand side is the Wardrop condition: all routes of a commodity that carry flow have the same latency, and none is longer than an unused route. It is the form of the equilibrium used in every later argument of the paper.

Formalization Note The right-hand side is the published KellyStochasticNetworks.IsWardropEquilibrium, which includes feasibility, so both sides presuppose a feasible flow as the page does. Latency hypotheses are imposed on $[0,\infty)$ only, where flows live; continuity replaces the paper's differentiability, which footnote 3 (p. 7) allows. Routes are arbitrary $0/1$ incidence columns rather than simple paths, a disclosed generalization. Definition 2.1 is read with $P_1\ne P_2$ and $f_{P_1}>0$ (see the definition item).