GAME THEORY FOR PRECODING IN A MULTI-USER SYSTEM: BARGAINING FOR OVERALL BENEFITS Jie Gao, Sergiy A. Vorobyov, and Hai Jiang Dept. of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada Emails: {jgao3, vorobyov, hai.jiang}@ece.ualberta.ca ABSTRACT A precoding strategy for multi-user spectrum sharing over an inter- ference channel is proposed and analyzed from a game-theoretic per- spective. The proposed strategy is based on finding the Nash bar- gaining solution for precoding matrices in a cooperative scenario over frequency selective channels under a spectrum mask constraint. An in-time update of the precoding matrices is enabled by using time slots to guarantee the effectiveness of the bargaining solution when the number of users varies. A dual decomposition approach is ex- ploited to construct a distributed structure for solving the bargaining problem. The proposed distributed algorithm realizes the physical process of bargaining, which is not present in the Nash bargaining theory. Index Terms— Linear precoding, interference channel, Nash bargaining, cooperative game, duality, dual decomposition. 1. INTRODUCTION As the demand for spectrum resources keeps increasing, improv- ing spectrum efficiency is necessary for alleviating the spectrum scarcity. One approach to improve spectrum efficiency is through the user cooperation on the same spectrum band, i.e., spectrum shar- ing [1], [2]. In most cases, wireless users in a system interfere with each other if they are active simultaneously, and the corresponding communication channel is called an interference channel. There is a significant amount of work studying the capacity on an interference channel from the information-theoretic perspective, such as [3]-[6]. A recent research topic is applications of game-theoretic approaches for investigation of interference channels. Equilibria and bargaining theories of game theory can be ex- plored to analyze the actions of the game players for non-cooperative and cooperative cases, respectively. There are some existing game theoretic studies of an interference channel for both cases. A two- user cooperative game over a flat fading interference channel is stud- ied in [7], where the users agree to cooperate by sharing the spec- trum in a frequency division multiplexing (FDM) manner. In [8], this game is extended to the case of multiple players communicating over a frequency selective channel, and joint time division multi- plexing (TDM) and FDM is adopted. A two-user vector game over a multiple-input single-output (MISO) interference channel is investi- gated in and the non-cooperative and cooperative beamforming vec- tors are derived [9]. A matrix-valued multi-user non-cooperative precoding game over a frequency selective interference channel is analyzed in [10]. The cooperative Nash bargaining (NB) based so- lution for the precoding matrices for a two-user game is given in our previous paper [11]. In this paper, we first extend the result of [11] to the case of precoding matrices design for a multi-user game. Then an algorithm is developed to realize the bargaining process among users, and solve the bargaining problem in a distributed manner. 2. SYSTEM MODEL Consider block transmissions in an M-user wireless communication system, for example, an orthogonal frequency-division multiplexing (OFDM) system. The sampled signal vector received by user i can be written as y i = Hii Fi si + j=M  j=1,j=i Hji Fj sj + ni , i ∈ Ω= {1, 2, ...M } (1) where Hji is the N × N matrix of sampled channel responses be- tween transmitter j and receiver i (the channel is assumed to be wideband frequency selective), si is the N × 1 information symbol block of user i, Fi is the N × N precoding matrix of user i, and ni is the N × 1 additive Gaussian noise vector with E{ni n H i } = σ 2 i I, I denotes an identity matrix, and (·) H stands for the Hermitian trans- pose. The information symbols are assumed to be uncorrelated and E{si s H i } = I. Consider the wireless users as players, their choices of precod- ing matrices as strategies, and the corresponding transmission rates as their payoffs. The precoding design problem can be viewed as a game in which benefit of each player depends on the precoding strategies of all other players. The utility space of this game is an M-dimensional rate region of the players. The information rate that a single user i can achieve under the strategy set of all users {Fi |i ∈{1, 2, ...M }} is [10] Ri = log(|I + F H i (H T ii ) H R −1 −i H T ii Fi |), ∀i (2) where R−i = σ 2 i + j=M ∑ j=1,j=i H T ji Fj F H j (H T ji ) H is the noise plus in- terference for user i, and | · | and (·) T stand for the determinant and transpose, respectively. A spectral mask constraint is adopted to limit the maximal power that each user can allocate on a specific frequency bin. Denote the maximal power that user i can allocate on the fre- quency bin k as p max i (k). With proper cyclic prefix incorpo- ration in transmitted symbols, the channel matrix Hji can be diagonalized as Hji = WΩji W H , with W being the N × N IFFT matrix and Ωji being the following diagonal matrix Ωji = diag(Hji (1),Hji (2), ..., Hji (N )), where Hji (k) is the channel frequency-response of the kth frequency bin from transmitter j to receiver i. Then the spectral mask constraint for user i on frequency bin k can be expressed as [10] E{|[W H Fi si ] k | 2 } =[W H Fi F H i W] kk ≤ p max i (k), ∀i, ∀k. (3) 2361 978-1-4244-2354-5/09/$25.00 ©2009 IEEE ICASSP 2009