In this contribution, the performance of a multiuser system is analyzed in the context of frequency selective fading channels. Using game theoretic tools, a useful framework is provided in order to determine the optimal power allocation when users know only their own channel (while perfect channel state information is assumed at the base station). This scenario illustrates the case of decentralized schemes, where limited information on the network is available at the terminal. Various receivers are considered, namely the matched filter, the MMSE filter and the optimum filter. The goal of this paper is to extend previous work, and to derive simple expressions for the non-cooperative Nash equilibrium as the number of mobiles becomes large and the spreading length increases. To that end two asymptotic methodologies are combined. The first is asymptotic random matrix theory which allows us to obtain explicit expressions of the impact of all other mobiles on any given tagged mobile. The second is the theory of non-atomic games which computes good approximations of the Nash equilibrium as the number of mobiles grows.
Game theory provides a wealth of tools that can be applied to the design and operation of communications systems. In this article, we provide a brief introduction to game theory. We then present applications of game theory to problems in random access and power control. In the case of random access, we examine the behavior of selfish users in a simplified Aloha system; surprisingly, rational selfish users do not implement the "always transmit" strategy that one might expect. In the case of power control, we show that game theoretic techniques can yield an optimal operating point without the intervention of an external controller. With cellular phones mass-market consumer items, the next frontier is mobile multimedia communications. This situation raises the question of how to perform power control for information sources other than voice. To explore this issue, we use the concepts and mathematics of microeconomics and game theory. In this context, the quality of service of a telephone call is referred to as the "utility" and the distributed power control problem for a CDMA telephone is a "noncooperative game." The power control algorithm corresponds to a strategy that has a locally optimum operating point referred to as a "Nash equilibrium." The telephone power control algorithm is also "Pareto efficient," in the terminology of game theory. When we apply the same approach to power control in wireless data transmissions, we find that the corresponding strategy, while locally optimum, is not Pareto efficient. Relative to the telephone algorithm, there are other algorithms that produce higher utility for at least one terminal, without decreasing the utility for any other terminal. This article presents one such algorithm. The algorithm includes a price function proportional to transmitter power. When terminals adjust their power levels to maximize the net utility (utility-price), they arrive at lower power levels and higher utility than they achieve when they individually strive to maximize utility.