92 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 1, JANUARY 2002 [27] , “Attitude control of underactuated spacecraft,” Euro. J. Control, vol. 6, no. 3, pp. 229–242, 2000. [28] H. Sussmann, “Lie brackets, real analyticity and geometric control,” in Differential Geometric Control Theory, R. W. Brockett, R. S. Millman, and H. Sussmann, Eds. Boston, MA: Birkhauser, 1983, pp. 1–116. [29] G. C. Walsh, R. Montgomery, and S. S. Sastry, “Optimal path planning on matrix Lie groups,” in Proc. 33rd Conf. Decision Control, vol. 2, Lake Buena Vista, FL, Dec. 1994, pp. 1258–1263. Competitive Routing in Networks With Polynomial Costs Eitan Altman, Tamer Bas ¸ar, Tania Jiménez, and Nahum Shimkin Abstract—We study a class of noncooperative general topology networks shared by users. Each user has a given flow which it has to ship from a source to a destination. We consider a class of polynomial link cost functions adopted originally in the context of road traffic modeling, and show that these costs have appealing properties that lead to predictable and efficient network flows. In particular, we show that the Nash equilibrium is unique, and is moreover efficient. These properties make the polynomial cost struc- ture attractive for traffic regulation and link pricing in telecommunication networks. We finally discuss the computation of the equilibrium in the spe- cial case of the affine cost structure for a topology of parallel links. Index Terms—Networks, noncooperative equilibria, nonzero-sum games, routing. I. INTRODUCTION We consider in this note a routing problem in networks where non- cooperating agents, to whom we refer to as players or users, wish to establish paths from agent-specific sources to agent-specific destina- tions so as to transport a fixed amount of total traffic (per agent). In the context of telecommunication networks, players correspond to dif- ferent traffic sources which have to route their traffic over a shared network. A similar setting has also been studied in the context of road traffic networks [11], where a player can be viewed as a transportation company which is to ship a flow of vehicles. A natural framework within which this class of problems can be an- alyzed is that of noncooperative game theory, and an appropriate solu- tion concept is that of Nash equilibrium (NE) [4]: a composite routing policy for the users constitutes a NE if no user can gain by unilaterally deviating from his own policy. There exists a rich literature [8], [14], [17], [19] on the analysis of equilibria in networks, particularly in the context of the Wardrop equi- librium [23], which pertains to the case of infinitesimal users (e.g., a single car in the context of road traffic). In such a framework, the impact Manuscript received October 19, 1999; revised May 17, 2000 and July 20, 2000. Recommended by Associate Editor L. Dai. The work of E. Altman was supported in part by INRIA/NSF Collaborative Research Grant. The work of T. Bas ¸ar was supported in part by Grants NSF INT 98-04950, NSF ANI 98-13710, MURI AF-DC-5-36128, and an ARO/EPRI MURI Grant. E. Altman is with the INRIA, 06902 Sophia Antipolis Cedex, France (e-mail: altman@sophia.inria.fr). T. Bas ¸ar is with the Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801-2307 USA. T. Jiménez is with the C.E.S.I.M.O., Universidad de los Andes, Mérida, Venezuela, and INRIA, 06902, Sophia Antipolis Cedex, France. N. Shimkin is with the Electrical Engineering Department, Technion—Israel Institute of Technology, Haifa 32000, Israel. Publisher Item Identifier S 0018-9286(02)01093-0. of a single user on the congestion experienced by other users is negli- gible. Our focus here, however, is competitive routing, where the net- work is shared by several users, with each one having a nonnegligible amount of flow. This concept has attracted considerable attention in re- cent years, as reflected in the works [1], [11], [12], [15], [18], which will be our starting point here. These papers have presented conditions for the existence and uniqueness of an equilibrium. This has allowed, in particular, the design of network management policies that induce efficient equilibria [15]. This framework has also been extended to the context of repeated games in [16], in which cooperation can be enforced by using policies that penalize users who deviate from the equilibrium. A desired property for an equilibrium is efficiency, i.e., social opti- mality. It is well known that Nash equilibria in routing games are gener- ally nonefficient. In particular, this nonefficiency can lead to paradox- ical phenomena (e.g., the Braess’s paradox [4]) where adding a link to the network could result in an increase of cost to all users. For spe- cific examples of nonefficient behavior and paradoxes, we cite [8] for road-traffic, [7], [15] for telecommunications, and [12], [13] for dis- tributed computing. In this note, we study the equilibria that arise in networks of gen- eral-topology under some polynomial cost functions introduced origi- nally in the context of road traffic by the US Bureau of Public Roads, and we obtain conditions for the uniqueness of the equilibrium. The uniqueness result is important, since to date there is not much known on uniqueness in the case of general topologies; only for a few special cases the uniqueness has been established, see [1], [12], [18]. We fur- ther present in the note conditions under which the NE is efficient. The fact that under a class of nonlinear costs the equilibrium is unique and efficient for general-topology networks may be especially useful for pricing purposes; indeed, if the network operator chooses prices that correspond to a unique and efficient equilibrium, it can enforce a globally optimal use of the network. In the context of communications, an example of an architecture where our routing framework (and the specific results obtained) would be of particular use is MultiProtocol Label Switching (MPLS) [2], [3]. One of the most important traffic engineering problems in MPLS, as identified in [3], is how to partition traffic into traffic trunks, or equiv- alently, how to distribute the arriving traffic over a given set of links. This problem has recently been studied in the framework of a single cri- terion that is optimized globally [10], but our framework here is more suitable for MPLS, since in practice routing decisions are made (and implemented) locally by each source, and typically in a noncooperative manner. In the next section, we introduce the general model considered in this note. In Section III, we establish conditions for the uniqueness of the NE. In Section IV, we establish the existence of an efficient NE under appropriate conditions. We then focus, in Section V, on the computation of the equilibrium for the special case of affine costs and parallel links. The note ends with the concluding remarks of Section VI. II. THE MODEL The topology is given by a directed graph where is the set of nodes, is the set of directed arcs, and where is the cost function of link , which gives the cost per unit traffic as a function of the total load on that link. We take it as (1) where are link-specific positive parameters. The special structure where does not depend on has been widely used in road traffic studies, such as [6], [20]. It is the cost adopted by the US Bureau 0018–9286/02$17.00 © 2002 IEEE