How bad is selfish routing?

TLDR

Routing traffic in congested networks aims to minimize total travel time, but regulating traffic to achieve optimal assignments is often infeasible, and selfish routing by users typically leads to higher overall latency. This article quantifies the performance loss caused by unregulated, selfish routing. The authors model a network with given traffic rates and edge latency functions, assume each user selects the minimum‑latency path based on current congestion, and analyze the resulting total latency for linear and general continuous nondecreasing latency functions. They prove that with linear latency functions, selfish routing increases total latency by at most a factor of 4/3, whereas for general continuous nondecreasing latencies the inefficiency can be arbitrarily large but is bounded by the latency of optimally routing twice the traffic.

Abstract

We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times---the total latency---is minimized.In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance.In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total latency of the routes chosen by unregulated selfish network users may be arbitrarily larger than the minimum possible total latency; however, we prove that it is no more than the total latency incurred by optimally routing twice as much traffic.