1094 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 16, NO. 5, OCTOBER 2008 The Analysis of Nash Equilibria of the One-Shot Random-Access Game for Wireless Networks and the Behavior of Selfish Nodes Hazer Inaltekin, Member, IEEE, and Stephen B. Wicker, Senior Member, IEEE Abstract—We address the fundamental question of whether or not there exist stable operating points in a network in which selfish nodes share a common channel, and if they exist, how the nodes behave at these stable operating points. We begin with a wireless communication network in which nodes (agents), which might have different utility functions, contend for access on a common, wireless communication channel. We characterize this distributed multiple-access problem in terms of a one-shot random-access game, and then analyze the behavior of the nodes using the tools of game theory. We give necessary and sufficient conditions on nodes for the complete characterization of the Nash equilibria of this game for all . We show that all centrally controlled optimal solutions are a subset of this game theoretic solution, and almost all (w.r.t. Lebesgue measure) transmission probability assignments chosen by a central authority are sup- ported by the game theoretic solution. We analyze the behavior of the network throughput at Nash equilibria as a function of the costs of the transmitters incurred by failed transmissions. Finally, we conclude the paper with the asymptotic analysis of the system as the number of transmitters goes to infinity. We show that the asymptotic distribution of the packet arrivals converges in distri- bution to a Poisson random variable, and the channel throughput converges to with being the cost of failed transmissions. We also give the best possible bounds on the rates of convergence of the packet arrival distribution and the channel throughput. Index Terms—Channel throughput, game theory, Nash equilib- rium, random access control, slotted ALOHA. I. INTRODUCTION A DENSE wireless network is defined to be a wireless net- work whose domain is fixed but the number of nodes lying inside the network domain approaches infinity. Main- stream ongoing research on this type of networks can be cat- egorized into two basic areas: 1) characterizing the upper and lower bounds on capacity as a function of and 2) developing scalable network protocols. Papers such as [1] and [2], follow the first mainstream, while [3] and [4], follow the second one. Manuscript received November 11, 2005; revised April 4, 2007. First pub- lished March 12, 2008; current version published October 15, 2008. Approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor R. Mazumdar. This work was supported in part by the National Science Foundation (NSF) TRUST Science and Technology Center and the NSF Nets-NOSS and ITR programs. H. Inaltekin is with the School of Electrical and Computer Engineering, Cor- nell University, Ithaca, NY 14853 USA and also with the Department of Elec- trical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: hi27@ece.cornell.edu; hinaltek@princeton.edu). S. B. Wicker is with the School of Electrical and Computer Engineering, Cor- nell University, Ithaca, NY 14853 USA (e-mail: wicker@ece.cornell.edu). Digital Object Identifier 10.1109/TNET.2007.909668 There are two types of bottlenecks associated with dense wireless networks. The first bottleneck arises from their phys- ical limitations. When a node transmits, it causes interference to the other nodes in the network. This interference, in turn, hinders the nodes in the vicinity of the transmitter from re- ception. As a result, if the network domain is fixed, per-node throughput goes to zero as . In [1], Gupta and Kumar showed that there exist two positive constants such that the wireless network capacity lies in the closed interval . The works [1] and [2] determine the rate at which the network capacity goes to zero as a function of . The second type of bottleneck derives from the need for scal- able protocols that show little performance degradation as . It is almost impossible to have a protocol which exhibits equally good performance in terms of a chosen metric for all values of . Thus, it turns out that different protocols and rules should be imposed on nodes for different values of node density. In addition, it is hard to guarantee that all the nodes in a dense wireless network run the same algorithm because users might be willing to modify their communication nodes in order to im- prove their network performance. This might lead to a further degradation in the overall network performance. The works [3] and [4] propose some scalable routing strategies, but they as- sume that all the nodes run the same algorithm. In this work, we focus primarily on the second type of bottle- neck. We push all of the decision making mechanism to the indi- vidual nodes, and consider them as selfish agents who are trying to improve their network performance. The only assumption we make about the nodes is that they are rational and intelligent en- tities. A rational node is aware of its alternatives, forms expec- tations about unknown parameters, and chooses its action after some process of optimization to maximize its expected utility. An intelligent node can analyze a conflict situation and takes its knowledge or expectations of other nodes’ behavior into ac- count while making a decision. The tool that we chose to an- alyze the wireless networks with selfish nodes is game theory. Game theory is the formal analysis of strategic settings in which all the agents nvolved are intelligent, rational decision-makers. The involved questions that this work addresses with the help of game theory are whether or not there exist stable operating points of the network at which all selfish nodes agree to operate, and how the selfish transmitters behave at these stable operating points for any . Game theory has an important role to play in design of large networks in general, and that of sensor networks in particular (see [5]). Game theory and the related field of mechanism design bring key tools into play—equilibrium concepts, utility func- tions, and signaling/bargaining—that are critical to the design 1063-6692/$25.00 © 2008 IEEE