A Game Theoretic Framework for joint Routing and Pricing in Networks with Elastic Demands * Eitan Altman INRIA Sophia Antipolis 2004 Route des Lucioles B.P. 93, 06902 Sophia Antipolis Cedex, France Eitan.Altman@sophia.inria.fr Jocelyne Elias Department of Electronics and Information Politecnico di Milano 34/5 Via Ponzio Milano 20133, Italy elias@elet.polimi.it Fabio Martignon Department of Information Technology and Mathematical Methods University of Bergamo Dalmine (BG) 24044, Italy fabio.martignon@unibg.it ABSTRACT In this paper, we study the economic interactions between network users and providers. Each user must ship his traffic from a source to a destination node, splitting it over multi- ple paths, each owned by an independent network provider. Users are charged a fixed price per unit of bandwidth used, and face both access and transport costs. The transmis- sion rate of each user is assumed to be function of network congestion (like for TCP traffic) and the price per band- width unit. Network providers compete among themselves to cover network users, and set transport prices to maximize their revenue. We provide sufficient conditions for the existence and the uniqueness of the Nash equilibrium under a variety of cost functions, and we derive optimal price and routing settings. Finally, we analyze and discuss several numerical examples that provide insights into the models’ solution. Keywords Routing, Pricing, Stackelberg Game, Elastic Traffic 1. INTRODUCTION The complexity of modern, large-scale networks calls for de- centralized control algorithms, where routing, pricing, net- work design and control decisions are made by each network entity independently, according to its own individual perfor- mance objectives. * Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. VALUE- TOOLS 2009, October 20-22, 2009 - Pisa, Italy. Copyright 2009 ICST 978-963-9799-70-7/00/0004 $5.00. The design and operation of such networks is not handled by a central authority, but arises from the interactions of a high number of self-interested agents. This is the case of the Internet, where connectivity is ensured by Autonomous Systems’ agreements, but also of overlay networks built on top of the Internet, where a large number of independent Service Providers seek to selfishly optimize the quality and cost of their own network, while covering the largest set of customers, to increase their revenue. These networks are henceforth called noncooperative, and game theory provides the systematic framework to study and understand their behavior. Competitive Routing with selfish users in the context of telecommunication networks has been the focus of several works [1, 2, 3]. Noncooperative games in the context of com- petitive routing and pricing were initially studied in the area of transportation networks [4, 5, 6, 7]. Nonetheless, the user considered in such networks controls just an infinitesimally small portion of the network flow, whereas, in this work, we are concerned with users that control non-negligible portions of flow. The joint problem of routing and price setting is also tackled in [5, 6, 8, 9], making at least one of the following limiting assumptions: (1) infinitesimally small users are con- sidered and (2) a monopolist service provider manages the whole network. In this paper, we overcome these limitations by proposing a novel game theoretic model that solves the joint problem of noncooperative routing and price setting considering both multiple Service Providers (SPs), which set prices for their links in the network, and a given set of users which are characterized by elastic traffic demands that must be routed over one or multiple links. We model the interaction between the SPs and the users as a Stackelberg game [10]. SPs set their prices and the users respond by presenting a certain amount of flow to the network. The users do not cooperate among themselves, thus leading to a Nash game. Then, we extend such model taking into account the access costs incurred by the users to ship their flows on the transit links through a set of access nodes.