1260 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 26, NO. 7, SEPTEMBER 2008 Inefficient Noncooperation in Networking Games of Common-Pool Resources Hisao Kameda, Fellow, IEEE and Eitan Altman, Senior Member, IEEE Abstract—We study in this paper a noncooperative approach for sharing resources of a common pool among users, wherein each user strives to maximize its own utility. The optimality notion is then a Nash equilibrium. First, we present a general framework of systems wherein a Nash equilibrium is Pareto inefficient, which are similar to the ‘tragedy of the commons’ in economics. As examples that fit in the above framework, we consider noncooperative flow-control problems in communication networks where each user decides its throughput to optimize its own utility. As such a utility, we first consider the power which is defined as the throughput divided by the expected end-to-end packet delay, and then consider another utility of additive costs. For both utilities, we establish the non-efficiency of the Nash equilibria. Index Terms—Braess paradox, common-pool resource, com- munication networks, flow control, Nash equilibrium, noncoop- erative game, Pareto inefficiency, power criterion, tragedy of the commons. I. I T HERE exist many systems where multiple independent users, or players, may strive to optimize each own utility unilaterally, which can be modeled as noncooperative games. As examples of the noncooperative games, communication networks like the Internet are joined by a number of inde- pendent users or organizations, like Internet service providers, that make decisions independently. Given users’ decisions, the utilities of all users are determined. We call a situation where the decisions of all users are determined an allocation. The al- location where each user attains its own optimum coincidently is a Nash equilibrium. It is natural that these independent users seek their own benefits or utilities noncooperatively. Thus, such systems are regarded as noncooperative games. Nash equilibria may be Pareto inefficient (or, simply, inef- ficient), that is, there may exist another allocation of a system where no users have less benefits and some have more benefits than in the Nash equilibrium of the system. In particular, we call an allocation of a system strongly Pareto inefficient if all users have more benefits in another allocation. Dubey [1] has shown that Nash equilibria may generally be Pareto inefficient based on the difference between the conditions to be satisfied Manuscript received on August 7, 2007; revised March 8, 2008. The work of the first author was supported in part by the Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science. The work of the second author was supported in part by the BIONETs European Contract. H. Kameda is with the Department of Computer Science, Univer- sity of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan (e-mail: kameda@cs.tsukuba.ac.jp). E. Altman is with INRIA Sophia Antipolis, B.P. 93, 06902 Sophia Antipolis Cedex, France (e-mail: Eitan.Altman@sophia.inria.fr). Digital Object Identifier 10.1109/JSAC.2008.080922. by Nash equilibria and those to be satisfied by Pareto optima. It appears, however, to be difficult to obtain concrete cases of inefficient Nash equilibria from his result. For communication and transportation networks, examples of such strong Pareto inefficiency have been shown with respect to noncooperative routing, first by Braess [2], and a number of related studies followed [3]–[10]. As for the nonco- operative load balancing in distributed computer systems, the existence of paradoxes that are similar to that of Braess but that appear only in the case of a finite number of (atomic) players and not in the case of infinitesimal (nonatomic) players, in the same environment, has been shown [11], [12]. It is natural to think of noncooperative flow control and of the Nash equilibrium concept therein. It appears, however, that few studies have addressed the issue of Pareto inefficiency of Nash equilibria in noncooperative flow control. This article examines mainly this issue as examples. In addition, we note that the Nash equilibrium concept has been discussed with respect to the power control in wireless communications [13]– [15]. These kinds of problems are regarded as those in which players compete for common-pool resources. The examples of such problems as considered here have been studied in social science under the name, ‘Tragedy of the Commons’ (see, Hardin [16], Roemer [17], Roemer and Silvester [18], Funaki and Yamamoto [19], etc.). This article first shows a fairly general framework of strongly Pareto-inefficient Nash equilibria. The framework may cover many examples of noncooperative games including noncooperative flow-control problems, noncooperative power- control problems in wireless networks, and a general prob- lem named, the ‘tragedy of commons’ in social science as mentioned above. Therefore, the framework characterizes a class of noncooperative games that may be spread over various fields but may have a mutually similar structure. As an example of the general framework, this article consid- ers flow-control problems for communication networks with multiple ports of entry and of exit, where each user decides its throughput, that is, the rate of its packets to inject into a network so as to optimize its own performance objective unilaterally. As such an objective, we firstly consider the power that is defined as the throughput divided by the expected delay (the expected delay is the expected time for a packet to pass through the network) [20]. This unilaterally optimized allocation is a Nash equilibrium, the existence of which is proved here. We show that the Nash equilibrium is always strongly Pareto inefficient, and we identify an allocation that is Pareto superior to it. (We note that Korilis and Lazar showed 0733-8716/08/$25.00 c  2008 IEEE Authorized licensed use limited to: University of Houston. Downloaded on April 20, 2009 at 16:36 from IEEE Xplore. Restrictions apply.