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.
In particular, the utility generally considered in those articles is justified in @cite_4 where the author describes a widely applicable model from first principles''. Conditions under which the utility will allow to obtain non-trivial Nash equilibria (i.e., users actually transmit at the equilibrium) are derived. The utility consisting of throughput-to-power ratio (detailed in Sec. ) is shown to satisfy these conditions. In addition, it possesses a propriety of reliability in the sense that the transmission occurs at non-negligible rates at the equilibrium. This kind of utility function had been introduced in previous works, with an economic leaning @cite_13 @cite_5 .
@cite_13 @cite_5 @cite_4
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A game-theoretic model for studying power control in multicarrier code-division multiple-access systems is proposed. Power control is modeled as a noncooperative game in which each user decides how much power to transmit over each carrier to maximize its own utility. The utility function considered here measures the number of reliable bits transmitted over all the carriers per joule of energy consumed and is particularly suitable for networks where energy efficiency is important. The multidimensional nature of users' strategies and the nonquasi-concavity of the utility function make the multicarrier problem much more challenging than the single-carrier or throughput-based-utility case. It is shown that, for all linear receivers including the matched filter, the decorrelator, and the minimum-mean-square-error detector, a user's utility is maximized when the user transmits only on its "best" carrier. This is the carrier that requires the least amount of power to achieve a particular target signal-to-interference-plus-noise ratio at the output of the receiver. The existence and uniqueness of Nash equilibrium for the proposed power control game are studied. In particular, conditions are given that must be satisfied by the channel gains for a Nash equilibrium to exist, and the distribution of the users among the carriers at equilibrium is characterized. In addition, an iterative and distributed algorithm for reaching the equilibrium (when it exists) is presented. It is shown that the proposed approach results in significant improvements in the total utility achieved at equilibrium compared with a single-carrier system and also to a multicarrier system in which each user maximizes its utility over each carrier independently Power allocation across users in two adjacent cells is studied for a code-division multiple access (CDMA) data service. The forward link is considered and cells are modeled as one-dimensional with uniformly distributed users and orthogonal signatures within each cell. Each user is assumed to have a utility function that describes the user's received utility, or willingness to pay, for a received signal-to-interference-plus-noise ratio (SINR). The objective is to allocate the transmitted power to maximize the total utility summed over all users subject to power constraints in each cell. It is first shown that this optimization can be achieved by a pricing scheme in which each base station announces a price per unit transmitted power to the users, and each user requests power to maximize individual surplus (utility minus cost). Setting prices to maximize total revenue over both cells is also considered, and it is shown that, in general, the solution is different from the one obtained by maximizing total utility. Conditions are given for which independent optimization in each cell, which leads to a Nash equilibrium (NE), is globally optimal. It is shown that, in general, coordination between the two cells is needed to achieve the maximum utility or revenue. 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. We consider a multi-cell wireless network with a large number of users. Each user selfishly chooses the Base Station (BS) that gives it the best throughput (utility), and each BS allocates its resource by some simple scheduling policy. First we consider two cases: (1) BS allocates the same time to its users; (2) BS allocates the same throughput to its users. It turns out that, combined with users' selfish behavior, case (1) results in a single Nash Equilibrium (NE), which achieves system-wide Proportional Fairness. On the other hand, case (2) results in many possible Nash Equilibria, some of which are very inefficient. Next, we extend (1) to the case where the users have general concave utility functions. It is shown that the if each BS performs intra- cell optimization, the total utility of all users is maximized at NE. This suggests that under our model, the task of joining the ";correct"; BS can be left to individual users, leading to a distributed solution.