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.

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.

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 paper, we consider the problem of pricing the spectrum usage in a cognitive radio network. In such a network, where licensed/primary users (who has the right to use the spectrum) and the unlicensed/secondary users coexist, a secondary spectrum market can be established where the primary network service provider charges the secondary users for the usage of the spectrum, while the secondary users also compete with each other for the access. Several important questions arise regarding the operation of such spectrum market: 1) how the spectrum should be priced; 2) how the secondary users should distributedly access the spectrum based on the spectrum price. To answer these questions, we interpret the available spectrum in the network as the aggregated interference tolerable at the receivers of the primary users, and introduce the notion of a market equilibrium in which the prices of the spectrum/interference are set correctly such that 1) the supply of the spectrum equals the demand of the spectrum, and 2) the secondary network is stable. We also developed an algorithm to distributedly compute such market equilibrium, and proved its convergence. I.

In this paper, we consider a wireless amplify-and-forward relay network with multiple source-relay-destination pairs transmitting concurrently over parallel channels and investigate the distributed power allocation problem within the framework of non-cooperative game theory. In order to combat the interference effect, each source node iteratively maximizes its own rate based on local information by allocating its power across different subchannels, subject to its power constraint, while treating the signals from the other users as additive noise. First, by focusing on the low signal to interference plus noise ratio (SINR) region, we propose a modified iterative water-filling algorithm. The existence of Nash equilibrium (NE) is guaranteed and the sufficient condition to reach a NE is determined. Then, we consider medium and high SINR regions and propose distributed algorithms based on both best (or optimal) and sub-optimal responses. The proposed algorithm based on the sub-optimal response is mathematically tractable and easier to compute, and exhibits a negligible performance loss, compared to the best-response based algorithm. Furthermore, the sub-optimal-response algorithm can be reduced to the classic Gaussian interference channel model, for which analytical sufficient conditions for the convergence to the unique NE can be readily obtained. The results show that, in low SINR regions, the proposed modified iterative water-filling algorithm yields a higher average sum rate than two simplified algorithms, i.e., the equal power allocation scheme and the conventional time-division based protocol, while in medium and high SINR regions, both the best-response and sub-optimal-response based algorithms outperform these two simplified algorithms in terms of the average sum rate.

Abstract—This paper considers the non-cooperative maximization of mutual information in the vector Gaussian interference channel in a fully distributed fashion via game theory. This problem has been widely studied in a number of works during the past decade for frequency-selective channels, and recently for the more general MIMO case, for which the state-of-the art results are valid only for nonsingular square channel matrices. Surprisingly, these results do not hold true when the channel matrices are rectangular and/or rank deficient matrices. The goal of this paper is to provide a complete characterization of the MIMO game for arbitrary channel matrices, in terms of conditions guaranteeing both the uniqueness of the Nash equilibrium and the convergence of asynchronous distributed iterative waterfilling algorithms. Our analysis hinges on new technical intermediate results, such as a new expression for the MIMO waterfilling projection valid (also) for singular matrices, a mean-value theorem for complex matrix-valued functions, and a general contraction theorem for the multiuser MIMO watefilling mapping valid for arbitrary channel matrices. The quite surprising result is that uniqueness/convergence conditions in the case of tall (possibly singular) channel matrices are more restrictive than those required in the case of (full rank) fat channel matrices. We also propose a modified game and algorithm with milder conditions for the uniqueness of the equilibrium and convergence, and virtually the same performance (in terms of Nash equilibria) of the original game. Index Terms—Game Theory, MIMO Gaussian interference channel, Nash equilibrium, totally asynchronous algorithms, waterfilling. I.