Corollary 2.7 — if $x\,\ell_e(x)\le\alpha\int_0^x\ell_e$, a Nash flow costs at most $\alpha$ times any feasible flow

Consider a routing instance with latency functions $\ell_e$ that are nonnegative, nondecreasing and continuous on $[0,\infty)$, routes given by $0/1$ edge–route incidence columns, and positive rates $r_i$. Write $C(f)=\sum_P\ell_P(f)f_P$ for the cost of a flow. Let $\alpha\ge1$ be a constant such that

Corollary 2.7. If $f$ is a flow at Nash equilibrium and $f^*$ is any feasible flow, then

In particular $\rho(G,r,\ell)=C(f)/C(f^*)\le\alpha$ for an optimal flow $f^*$.

The hypothesis says that the latencies are "not too steep": the Beckmann potential $\sum_e\int_0^{f_e}\ell_e$, which a Nash flow minimizes, is within a factor $\alpha$ of the true cost.

Formalization Note The paper's conclusion $\rho(G,r,\ell)\le\alpha$ is stated as the inequality $C(f)\le\alpha C(f^*)$ for every feasible $f^*$, which is what the proof's chain of inequalities (pp. 10–11) establishes and which implies the ratio bound for an optimal $f^*$; this avoids dividing by a cost that may be $0$. The hypothesis is quantified over $x>0$ only, as printed, and $\alpha\ge1$ is kept as printed. Continuity on $[0,\infty)$ makes $\int_0^x\ell_e$ an honest integral. Routes are $0/1$ incidence columns (a generalization of simple paths); continuity replaces differentiability (footnote 3, p. 7).