  The price of anarchy has become a standard measure of the efficiency of
equilibria in games. Most of the literature in this area has focused on
establishing worst-case bounds for specific classes of games, such as routing
games or more general congestion games. Recently, the price of anarchy in
routing games has been studied as a function of the traffic demand, providing
asymptotic results in light and heavy traffic. The aim of this paper is to
study the price of anarchy in nonatomic routing games in the intermediate
region of the demand. To achieve this goal, we begin by establishing some
smoothness properties of Wardrop equilibria and social optima for general
smooth costs. In the case of affine costs we show that the equilibrium is
piecewise linear, with break points at the demand levels at which the set of
active paths changes. We prove that the number of such break points is finite,
although it can be exponential in the size of the network. Exploiting a scaling
law between the equilibrium and the social optimum, we derive a similar
behavior for the optimal flows. We then prove that in any interval between
break points the price of anarchy is smooth and it is either monotone
(decreasing or increasing) over the full interval, or it decreases up to a
certain minimum point in the interior of the interval and increases afterwards.
We deduce that for affine costs the maximum of the price of anarchy can only
occur at the break points. For general costs we provide counterexamples showing
that the set of break points is not always finite.
