Abstract—This paper considers the noncooperative maximization of mutual information in the Gaussian interference channel in a fully distributed fashion via game theory. This problem has been studied in a number of papers during the past decade for the case of frequency-selective channels. A variety of conditions guaranteeing the uniqueness of the Nash Equilibrium (NE) and convergence of many different distributed algorithms have been derived. In this paper we provide a unified view of the state-ofthe-art results, showing that most of the techniques proposed in the literature to study the game, even though apparently different, can be unified using our recent interpretation of the waterfilling operator as a projection onto a proper polyhedral set. Based on this interpretation, we then provide a mathematical framework, useful to derive a unified set of sufficient conditions guaranteeing the uniqueness of the NE and the global convergence of waterfilling based asynchronous distributed algorithms. The proposed mathematical framework is also instrumental to study the extension of the game to the more general MIMO case, for which only few results are available in the current literature. The resulting algorithm is, similarly to the frequency-selective case, an iterative asynchronous MIMO waterfilling algorithm. The proof of convergence hinges again on the interpretation of the MIMO waterfilling as a matrix projection, which is the natural generalization of our results obtained for the waterfilling mapping in the frequency-selective case. Index Terms—Game Theory, MIMO Gaussian interference channel, Nash equilibrium, totally asynchronous algorithms, waterfilling. I.

Abstract — [1], [2] have shown that dynamic spectrum management (DSM) using the market competitive equilibrium (CE), which sets a price for transmission power on each channel, leads to better system performance in terms of the total data transmission rate (by reducing cross talk), than using the Nash equilibrium (NE). But how to achieve such a CE is an open problem. We show that the CE is the solution of a linear complementarity problem (LCP) and can be computed efficiently. We propose a decentralized tâtonnement process for adjusting the prices to achieve a CE. We show that under reasonable conditions, any tâtonnement process converges to the CE. The conditions are that users of a channel experience the same noise levels and that the cross-talk effects between users are low-rank and weak. Index Terms—Radio spectrum management, dynamic spectrum management (DSM), linear complementarity problem (LCP), competitive equilibrium I.

Abstract—In this paper, we address the problem of spectrum sharing where competitive operators coexist in the same frequency band. First, we model this problem as a strategic non-cooperative game where operators simultaneously share the spectrum according to the Nash Equilibrium (N.E). Given a set of channel realizations, several Nash equilibria exist which renders the outcome of the game unpredictable. For this reason, the spectrum sharing problem is reformulated as a Stackelberg game where the first operator is already being deployed and the secondary operator follows next. The Stackelberg equilibrium (S.E) is reached where the best response of the secondary operator is taken into account upon maximizing the primary operator’s utility function. Finally, we assess the goodness of the proposed distributed approach by comparing its performance to the centralized approach. I.

In this paper we address the problem of spectrum allocation in cognitive radio networks in which licensed users allow unlicensed users to make use of their allocated capacity, provided a set of interference constraints on the receivers of licensed users are satisfied. In this scenario, the traditional distributed dynamical spectrum allocation approach is unable to enforce such interference constraints. To address this problem, we propose to allocate the spectrum to unlicensed users by pricing the interference constraints. In such a scheme, a total charge for each unlicensed user will be determined based upon their contribution to the total interference as measured by each licensed user. We also formulate the notion of interference equilibrium, which is a state of the network where i) all interference constraints are satisfied and ii) no unlicensed user has an incentive to alter its own transmission power levels. We propose a distributed algorithm and apply it to two cognitive radio network configurations. For each configuration we give sufficient conditions to guarantee convergence of the unlicensed users ’ power profiles and ensure interference prices converge to an interference equilibrium. I.

Consider a scenario in which K users and a jammer share a common spectrum of N orthogonal tones. Both the users and the jammer have limited power budgets. The goal of each user is to allocate its power across the N tones in such a way that maximizes the total sum rate that he/she can achieve, while treating the interference of other users and the jammer’s signal as additive Gaussian noise. The jammer, on the other hand, wishes to allocate its power in such a way that minimizes the utility of the whole system; that being the total sum of the rates communicated over the network. For this non-cooperative game, we propose a generalized version of the existing iterative water-filling algorithm whereby the users and the jammer update their power allocations in a greedy manner. We study the existence of a Nash equilibrium of this non-cooperative game as well as conditions under which the generalized iterative water-filling algorithm converges to a Nash equilibrium of the game. The conditions that we derive in this paper depend only on the system parameters, and hence can be checked a priori. Simulations show that when the convergence conditions are violated, the presence of a jammer can cause the, otherwise convergent, iterative water-filling algorithm to oscillate.

In this paper 1, we study the discrete power allocation game for the fast fading multiple-input multiple-output multiple access channel. Each player or transmitter chooses its own transmit power policy from a certain finite set to optimize its individual transmission rate. First, we prove the existence of at least one pure strategy Nash equilibrium. Then, we investigate two learning algorithms that allow the players to converge to either one of the NE states or to the set of correlated equilibria. At last, we compare the performance of the considered discrete game with the continuous game in [7]. 1.

This paper discusses the issue of dynamic resource allocation (DRA) in the context of cognitive radio (CR) networks. We present a general framework adopting generalized transmitter and receiver signalexpansion functions, which allow us to join DRA with waveform adaptation, two procedures that are currently carried out separately. Moreover, the proposed DRA can handle many types of expansion functions or even combinations of different types of functions. An iterative game approach is adopted to perform multi-player DRA, and the best-response strategies of players are derived and characterized using convex optimization. To reduce the implementation costs of having too many active expansion functions after optimization, we also propose to combine DRA with sparsity constraints for dynamic function selection. Generally, it incurs little rate-performance loss since the effective resources required by a CR are in fact sparse. Index Terms — cognitive radio, dynamic resource allocation, game theory, waveform adaptation, sparsity 1.

Abstract—In this note, the problem of rate control in multiuser multiple-input multiple-output (MIMO) interference systems is formulated as a multicriteria optimization (MCO) problem. The Pareto rate region of the MCO problem is first characterized by giving a sufficient condition for the convexity of the Pareto rate region. Second, various rate region convexification approaches including a multistage interference cancellation and a full projection (FP)-based interference avoidance scheme are analyzed. An achievable rate region based on FP is also given for MIMO interference systems. Third, Nash bargaining (NB) is applied to transform the MCO problem into a single-objective problem. The characteristics of the NB over MIMO interference systems such as the uniqueness and optimality of different type NB solutions are investigated. A method to determine the optimality of FP- and time-division multiplexing-based NB solutions is presented as well. Finally, the convexity of the rate region and the existence of the FP-based NB solution for MIMO interference systems are examined numerically. Index Terms—MIMO interference channel, multicriteria optimization (MCO), Nash bargaining, Pareto region. I.

The problem of decentralized power allocation for competitive rate maximization in a frequency-selective Gaussian interference channel is considered. In the absence of perfect knowledge of channel state information (CSI), a distribution-free robust game is formulated. A robust-optimization equilibrium (RE) is proposed where each player formulates a best response to the worst-case interference. The conditions for existence, uniqueness and convergence of the RE are derived. It is shown that the convergence reduces as the uncertainty increases. Simulations show an interesting phenomenon where the proposed RE moves closer to a Pareto-optimal solution as the CSI uncertainty bound increases, when compared to the classical Nash equilibrium under perfect CSI. Thus, the robust-optimization equilibrium successfully counters bounded channel uncertainty and increases system sum-rate due to users being more conservative about causing interference to other users. Index Terms — Rate maximization, robust games, waterfilling, decentralized power control, Gaussian interference channel

We address the problem of spectrum sharing where competitive operators coexist in the same frequency band. First, we model this problem as a strategic non-cooperative game where operators simultaneously share the spectrum according to the Nash Equilibrium (NE). Given a set of channel realizations, several Nash equilibria exist which renders the outcome of the game unpredictable. Then, in a cognitive context with the presence of primary and secondary operators, the inter-operator spectrum sharing problem is reformulated as a Stackelberg game using hierarchy where the primary operator is the leader. The Stackelberg Equilibrium (SE) is reached where the best response of the secondary operator is taken into account upon maximizing the primary operator’s utility function. Moreover, an extension to the multiple operators spectrum sharing problem is given. It is shown that the Stackelberg approach yields better payoffs for operators compared to the classical water-filling approach. Finally, we assess the goodness of the proposed distributed approach by comparing its performance to the centralized approach. Copyright © 2009 Mehdi Bennis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.