How unfair is optimal routing

Abstract

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 overall traffic, so that feasible assignments of traffic to paths in network can be modeled as network flows. We also assume that time needed to traverse a single link of network is load-dependent, that is, common latency suffered by all traffic on link increases as link becomes more congested.We consider two types of traffic assignments. In first, we measure quality of an assignment by total latency incurred by network users; an optimal assignment is a feasible assignment that minimizes total latency. On other hand, it is often difficult in practice to impose optimal routing strategies on traffic in a network, leaving network users free to act according to their own interests. We assume that, in absence of network regulation, users act in a selfish manner. Under this assumption, we can expect network traffic to converge to 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 optimal assignment really a better assignment than an assignment at Nash equilibrium? While optimal assignment obviously dominates one at Nash equilibrium from 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 latency of edge e1 is always 2(1 - e) (independent of edge congestion, where e > 0 is a very small number), and that latency of edge e2 is same as edge congestion (i.e., if x units of traffic are on edge e2, then all of this flow incurs x units of latency). In assignment at Nash equilibrium, all traffic is on second link; in minimum-latency assignment, 1 - e units of traffic use edge e2 while remaining e units of traffic use edge e1. Roughly, a small fraction of traffic is sacrificed to slower edge because it improves overall social welfare (by reducing congestion experienced by overwhelming majority of network users); needless to say, these martyrs may not appreciate a doubling of their travel time in 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 next section (all of which are classical; see [2] for historical references), we provide an exact solution to this problem under weak hypotheses on class of allowable latency functions.