Which Is the Worst-Case Nash Equilibrium?

Abstract

A Nash equilibrium of a routing network represents a stable state of the network where no user finds it beneficial to unilaterally deviate from its routing strategy. In this work, we investigate the structure of such equilibria within the context of a certain game that models selfish routing for a set of n users each shipping its traffic over a network consisting of m parallel links. In particular, we are interested in identifying the worst-case Nash equilibrium-the one that maximizes social cost. Worst-case Nash equilibria were first introduced and studied in the pioneering work of Koutsoupias and Papadimitriou [9]. More specifically, we continue the study of the Conjecture of the Fully Mixed Nash Equilibrium, henceforth abbreviated as FMNE Conjecture, which asserts that the fully mixed Nash equilibrium, when existing, is the worst-case Nash equilibrium. (In the fully mixed Nash equilibrium, the mixed strategy of each user assigns (strictly) positive probability to every link.) We report substantial progress towards identifying the validity, methodologies to establish, and limitations of, the FMNE Conjecture.

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.…”

Facets of the Fully Mixed Nash Equilibrium Conjecture

“…In fact, our estimation techniques significantly extend those for the case (ii) above in [17]; due to the increased complexity of the Quadratic Maximum Social Cost function (over Maximum Social Cost), far more involved estimations have been required in the present proof. Counterexamples to the FMNE Conjecture appeared (i) for the case of unrelated links in [17], and (ii) for the case of weighted users in [5]. In the context of selfish routing, the fully mixed Nash equilibrium and the FMNE Conjecture have attracted a lot of interest and attention; they both have been studied extensively in the last few years for a wide variety of theoretical models of selfish routing and Social Cost measures -see, e.g., [2,4,9,10,12,16,18].…”

“…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]. We now calculate the Quadratic Maximum Social Cost of the fully mixed Nash equilibrium φ.…”

Cost Sharing Mechanisms for Fair Pricing of Resource Usage

“…The fully mixed Nash equilibrium was originally proposed by Mavronicolas and Spirakis [15]; its various existence and uniqueness properties were subsequently studied very extensively; see, e.g., [4,5,6,7,13,14].…”