The price of anarchy in routing games as a function of the demand
Cominetti
R
author
Dose
V
author
Scarsini
M
author
2022
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.
Nonatomic routing games
Price of anarchy
Affine cost functions
Variable demand
WOS:000693858200001
exported from refbase (show.php?record=1468), last updated on Wed, 05 Jan 2022 14:47:27 -0300
text
10.1007/s10107-021-01701-7
Cominetti_etal2022
Mathematical Programming
Math. Program.
2022
continuing
periodical
academic journal
Early Access
0025-5610