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.
This section is dedicated to present some of the works that use game theory for power control. We remind that a Nash equilibrium is a stable solution, where no player has an incentive to deviate unilaterally, while a Pareto equilibrium is a cooperative dominating solution, where there is no way to improve the performance of a player without harming another one. Generally, both concepts do not coincide. Following the general presentation of power allocation games in @cite_11 @cite_31 , an abundance of works can be found on the subject.
@cite_31 @cite_11
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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 address the problem of spectrum pricing in a cognitive radio network where multiple primary service providers compete with each other to offer spectrum access opportunities to the secondary users. By using an equilibrium pricing scheme, each of the primary service providers aims to maximize its profit under quality of service (QoS) constraint for primary users. We formulate this situation as an oligopoly market consisting of a few firms and a consumer. The QoS degradation of the primary services is considered as the cost in offering spectrum access to the secondary users. For the secondary users, we adopt a utility function to obtain the demand function. With a Bertrand game model, we analyze the impacts of several system parameters such as spectrum substitutability and channel quality on the Nash equilibrium (i.e., equilibrium pricing adopted by the primary services). We present distributed algorithms to obtain the solution for this dynamic game. The stability of the proposed dynamic game algorithms in terms of convergence to the Nash equilibrium is studied. However, the Nash equilibrium is not efficient in the sense that the total profit of the primary service providers is not maximized. An optimal solution to gain the highest total profit can be obtained. A collusion can be established among the primary services so that they gain higher profit than that for the Nash equilibrium. However, since one or more of the primary service providers may deviate from the optimal solution, a punishment mechanism may be applied to the deviating primary service provider. A repeated game among primary service providers is formulated to show that the collusion can be maintained if all of the primary service providers are aware of this punishment mechanism, and therefore, properly weight their profits to be obtained in the future. So far, most equilibrium concepts in game theory require that the rewards and actions of the other agents are known and or observed by all agents. However, in real life problems, agents are generally faced with situations where they only have partial or no knowledge about their environment and the other agents evolving in it. In this context, all an agent can do is reasoning about its own payoffs and consequently, cannot rely on classical equilibria through deliberation, which requires full knowledge and observability of the other agents. To palliate to this difficulty, we introduce the satisfaction principle from which an equilibrium can arise as the result of the agents' individual learning experiences. We define such an equilibrium and then we present different algorithms that can be used to reach it. Finally, we present experimental results that show that using learning strategies based on this specific equilibrium, agents will generally coordinate themselves on a Pareto-optimal joint strategy, that is not always a Nash equilibrium, even though each agent is individually rational, in the sense that they try to maximize their own satisfaction. In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria.