From Altruism to Non-Cooperation in Routing Games

arXiv:0808.4079v3 [cs.GT] 14 Oct 2008 From Altruism to Non-Cooperation in Routing Games Amar Prakash Azad∗+ , Eitan Altman∗and R. El-Azouzi+ ∗Maestro group, INRIA, 2004 Route des Lucioles, Sophia Antipolis, France +LIA, University of Avignon, 339, chemin des Meinajaries, Avignon, France {amar.azad,eitan.altman}@sophia.inria.fr, rachid.elazouzi@univ-avignon.fr. Abstract— The paper studies the routing in the network shared by several users. Each user seeks to optimize either its own performance or some combination between its own performance and that of other users, by controlling the routing of its given flow demand. We parameterize the degree of cooperation which allows to cover the fully non-cooperative behavior, the fully cooperative behavior, and even more, the fully altruistic behavior, all these as special cases of the parameter’s choice. A large part of the work consists in exploring the impact of the degree of cooperation on the equilibrium. Our first finding is to identify multiple Nash equilibria with cooperative behavior that do not occur in the non-cooperative case under the same conditions (cost, demand and topology). We then identify Braess like paradox (in which adding capacity or adding a link to a network results in worse performance to all users) in presence of user’s cooperation. We identify another type of paradox in cooperation scenario: when a given user increases its degree of cooperation while other users keep unchanged their degree of cooperation, this may lead to an improvement in performance of that given user. We then pursue the exploration and carry it on to the setting of Mixed equilibrium (i.e. some users are non atomic-they have infinitesimally small demand, and other have finite fixed demand). We finally obtain some theoretical results that show that for low degree of cooperation the equilibrium is unique, confirming the results of our numerical study. I. INTRODUCTION Non-cooperative routing has long been studied both in the framework of road-traffic as well as in the framework of telecommunication networks. Such frameworks allow to model the flow configuration that results in networks in which routing decisions are made in a non-cooperative and dis- tributed manner between the users. In the case of a finite (not very large) number of agents, the resulting flow configuration corresponds to the so called Nash equilibrium [17] defined as a situation in which no agent has an incentive to deviate unilaterally. The Nash equilibrium has been extensively used in telecommunications, see e.g. [2], [6]. The authors in [2] studied a routing games in which each user has a given amount of flow to ship and has several paths through which he may split that flow. Such a routing game may be handled by models similar to [8] in the special case of a topology of parallel links. This type of topology is studied in detail in the first part of [2] as well as in [9]. However, the model of [8] does not extend directly to other topologies. Indeed, in more general topologies, the delay over a path depends on how much traffic is sent by other users on any other path that shares common links. Routing games with general topologies have been studied, for example, in the second part of [2], as well as in [9]. A related model was studied thirty years ago by Rosenthal in [10], yet in a discrete setting. It is shown that in such a model there always exists a pure strategy Nash equilibrium. He introduces a kind of discrete potential function for computing the equilibrium. Nevertheless if a player has more than 1 unit to ship such an equilibrium doesn’t always exist. In this work, we embark on experimental investigation of the impact of cooperation in the context of routing games. In particular we consider parallel links and load balancing network topology for investigation, originally presented in [2] and [7] in the context of selfish users. The experimentation is mainly aimed at exploring some strange behaviors which appears in presence of user’s partial cooperation (Cooperation in Degree), which is further strengthened with some theoretical results. Firstly, we identify loss of uniqueness of Nash equilibria. We show by a simple example of parallel links and load balancing network that there may exist several such equi- libria. Moreover, even the uniqueness of link utilization at equilibria may fail even in the case of simple topology. A similar example of parallel links, in absence of the cooperation between users there would be a single equilibrium [2]. Beyond Nash equilibrium we investigate further in the setting of Mixed users i.e. where there are two types of users, Group user and Individual users. Group users seek Nash equilibrium while the Individual users seek equilibrium with Wardrop conditions. Strengthening our earlier finding, we observe loss of uniqueness with partial cooperation against the unique solutions shown in [15] for selfish users. However in the latter section (Sec. V), we show theoretically that there exis

…(Full text truncated)…

This content is AI-processed based on ArXiv data.