# When is selfish routing bad? The price of anarchy in light and heavy traffic

## TL;DR

This paper investigates how the inefficiency of selfish routing, measured by the price of anarchy, varies with traffic levels, showing convergence to optimality in many cases and providing explicit convergence rates.

## Contribution

It provides theoretical conditions under which the price of anarchy approaches 1 in congestion games, including explicit convergence rates for polynomial cost functions.

## Key findings

- Price of anarchy can stay away from 1 for all traffic levels in some networks.
- For polynomial costs, the price of anarchy converges to 1 in both light and heavy traffic.
- The convergence follows a power law with explicitly computable degree.

## Abstract

This paper examines the behavior of the price of anarchy as a function of the traffic inflow in nonatomic congestion games with multiple origin-destination (O/D) pairs. Empirical studies in real-world networks show that the price of anarchy is close to 1 in both light and heavy traffic, thus raising the question: can these observations be justified theoretically? We first show that this is not always the case: the price of anarchy may remain a positive distance away from 1 for all values of the traffic inflow, even in simple three-link networks with a single O/D pair and smooth, convex costs. On the other hand, for a large class of cost functions (including all polynomials), the price of anarchy does converge to 1 in both heavy and light traffic, irrespective of the network topology and the number of O/D pairs in the network. We also examine the rate of convergence of the price of anarchy, and we show that it follows a power law whose degree can be computed explicitly when the network's cost functions are polynomials.