The Price of Selfish Routing

Abstract

We study the problem of routing traffic through a congested network. We focus on the simplest case of a network consisting of m parallel links. We assume a collection of n network users; each user employs a mixed strategy, which is a probability distribution over links, to control the shipping of its own assigned traffic. Given a capacity for each link specifying the rate at which the link processes traffic, the objective is to route traffic so that the maximum (over all links) latency is minimized. We consider both uniform and arbitrary link capacities.How much decrease in global performance is necessary due to the absence of some central authority to regulate network traffic and implement an optimal assignment of traffic to links? We investigate this fundamental question in the context of Nash equilibria for such a system, where each network user selfishly routes its traffic only on those links available to it that minimize its expected latency cost, given the network congestion caused by the other users. We use the Coordination Ratio, originally defined by Koutsoupias and Papadimitriou [16], as a measure of the cost of lack of coordination among the users; roughly speaking, the Coordination Ratio is the ratio of the expectation of the maximum (over all links) latency in the worst possible Nash equilibrium, over the least possible maximum latency had global regulation been available.Our chief instrument is a set of combinatorial Minimum Expected Latency Cost Equations, one per user, that characterize the Nash equilibria of this system. These are linear equations in the minimum expected latency costs, involving the user traffics, the link capacities, and the routing pattern determined by the mixed strategies. In turn, we solve these equations in the case of fully mixed strategies, where each user assigns its traffic with a strictly positive probability to every link, to derive the first existence and uniqueness results for fully mixed Nash equilibria in this setting. Through a thorough analysis and characterization of fully mixed Nash equilibria, we obtain tight upper bounds of no worse than O(ln n/ln ln n) on the Coordination Ratio for (i) the case of uniform capacities and arbitrary traffics and (ii) the case of arbitrary capacities and identical traffics.

Network uncertainty in selfish routing

“…The problem of computing pure Nash equilibria was studied for congestion games in [3] and for weighted congestion games in [6]. The KP-Model [13] and its Nash Equilibria were studied extensively in the last years; see, for example, [2,4,6,12,15,16] and [5] for a survey. Feldmann et al [4] and Gairing et al [7] propose algorithms to transform any user strategy to a Nash equilibrium without increasing the maximum congestion.…”

“…Feldmann et al [4] and Gairing et al [7] propose algorithms to transform any user strategy to a Nash equilibrium without increasing the maximum congestion. Fully mixed Nash equilibria for the KPmodel were first studied in [16]. The fully mixed Nash equilibrium conjecture, stating that the fully mixed Nash equilibrium has the worst social cost among all Nash equilibria, was first formulated in [7] and it was verified in [14] for a social cost defined as the sum of the users latencies.…”

Facets of the Fully Mixed Nash Equilibrium Conjecture

“…We are especially interested in the fully mixed Nash equilibrium φ which is known to exist uniquely in the setting we consider [19]; it is also known that for each pair of user i ∈ [n] and a link ∈ [2], φ i ( ) = 1 2 , so that all 2 n pure profiles are equiprobable, each occurring with probability 1 2 n [19, Lemma 15]. The Maximum Social Cost of φ is given by MSC(φ) = n 2 + n 2 n n−1 n 2 −1 [17].…”

“…Originally considered by Kaplansky back in 1945 [14], fully mixed Nash equilibria have all their involved probabilities strictly positive; they were recently coined into the context of selfish routing by Mavronicolas and Spirakis [19]. Clearly, the fully mixed Nash equilibrium maximizes the randomization used in the mixed strategies of the players; so, it is a natural candidate to become a vehicle for the study of the effects of randomization on the quality of Nash equilibria.…”

Evolutionary Equilibrium in Bayesian Routing Games: Specialization and Niche Formation

“…Koutsoupias and Papadimitriou give bounds on the coordination ratio. These bounds are improved by Mavronicolas and Spirakis [13], and by Czumaj and Vöcking [3] who gave an asymptotically tight bound.…”