# The Price of Anarchy in Routing Games as a Function of the Demand

## TL;DR

This paper analyzes how the efficiency loss in routing games, measured by the price of anarchy, varies with demand levels, revealing piecewise linear behavior and the significance of break points in affine cost scenarios.

## Contribution

It characterizes the price of anarchy as a function of demand in nonatomic routing games, especially identifying the structure and finiteness of break points for affine costs.

## Key findings

- Price of anarchy is piecewise linear with demand.
- Number of break points is finite for affine costs.
- Maximum price of anarchy occurs at break points.

## Abstract

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.