How unfair is optimal routing

We are given a network and a rate of traffic between a source node and a destination node, and seek an assignment of traffic to source-destination paths. We assume that each network user controls a negligible fraction of the overall traffic, so that feasible assignments of traffic to paths in the network can be modeled as network flows. We also assume that the time needed to traverse a single link of the network is load-dependent, that is, the common latency suffered by all traffic on the link increases as the link becomes more congested.We consider two types of traffic assignments. In the first, we measure the quality of an assignment by the total latency incurred by network users; an optimal assignment is a feasible assignment that minimizes the total latency. On the other hand, it is often difficult in practice to impose optimal routing strategies on the traffic in a network, leaving network users free to act according to their own interests. We assume that, in the absence of network regulation, users act in a selfish manner. Under this assumption, we can expect network traffic to converge to the second type of assignment that we consider, an assignment at Nash equilibrium. An assignment is at Nash equilibrium if no network user has an incentive to switch paths; this occurs when all traffic travels on minimum-latency paths.The following question motivates our work: is the optimal assignment really a "better" assignment than an assignment at Nash equilibrium? While the optimal assignment obviously dominates one at Nash equilibrium from the viewpoint of total latency, it may lack desirable fairness properties. For example, consider a network consisting of two nodes, s and t, and two edges, e1 and e2, from s to t. Suppose further that one unit of traffic wishes to travel from s to t, that the latency of edge e1 is always 2(1 - e) (independent of the edge congestion, where e > 0 is a very small number), and that the latency of edge e2 is the same as the edge congestion (i.e., if x units of traffic are on edge e2, then all of this flow incurs x units of latency). In the assignment at Nash equilibrium, all traffic is on the second link; in the minimum-latency assignment, 1 - e units of traffic use edge e2 while the remaining e units of traffic use edge e1. Roughly, a small fraction of the traffic is sacrificed to the slower edge because it improves the overall social welfare (by reducing the congestion experienced by the overwhelming majority of network users); needless to say, these martyrs may not appreciate a doubling of their travel time in the name of "the greater good"! Indeed, this drawback of routing traffic optimally has inspired practitioners to find traffic assignments that minimize total latency subject to explicit length constraints [1], which require that no network user experiences much more latency than in an assignment at Nash equilibrium. The central question of this paper is how much worse off can network users be in an optimal assignment than in one at Nash equilibrium? After reviewing some technical preliminaries in the next section (all of which are classical; see [2] for historical references), we provide an exact solution to this problem under weak hypotheses on the class of allowable latency functions.