Optimal Linear Precoding Strategies for Wideband Noncooperative Systems Based on Game Theory—Part I: Nash Equilibria

Abstract

In this two-parts paper we propose a decentralized strategy, based on a game-theoretic formulation, to find out the optimal precoding/multiplexing matrices for a multipoint-to-multipoint communication system composed of a set of wideband links sharing the same physical resources, i.e., time and bandwidth. We assume, as optimality criterion, the achievement of a Nash equilibrium and consider two alternative optimization problems: 1) the competitive maximization of mutual information on each link, given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies; and 2) the competitive maximization of the transmission rate, using finite order constellations, under the same constraints as above, plus a constraint on the average error probability. In Part I of the paper, we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies. Then, we prove that the optimal precoding/multiplexing scheme for both games leads to a channel diagonalizing structure, so that both matrix-valued problems can be recast in a simpler unified vector power control game, with no performance penalty. Thus, we study this simpler game and derive sufficient conditions ensuring the uniqueness of the Nash equilibrium. Interestingly, although derived under stronger constraints, incorporating for example spectral mask constraints, our uniqueness conditions have broader validity than previously known conditions. Finally, we assess the goodness of the proposed decentralized strategy by comparing its performance with the performance of a Pareto-optimal centralized scheme.To reach the Nash equilibria of the game, in Part II, we propose alternative distributed algorithms, along with their convergence conditions.

Game theory for precoding in a multi-user system: Bargaining for overall benefits

“…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] …”

“…A two-user vector game over a multiple-input single-output (MISO) interference channel is investigated in and the non-cooperative and cooperative beamforming vectors 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 solution for the precoding matrices for a two-user game is given in our previous paper [11].…”

Game‐theoretic beamforming techniques for multiuser multi‐cell networks under mixed quality of service constraints

“…The latter work was extended to a multi-carrier data network in [6]. Noncooperative games were used in [7][8][9][10] to develop distributed algorithms. Demonstrating the inefficiency of the Nash equilibrium with respect to Pareto optimality, a pricing mechanism was introduced in [7,8] to obtain efficient solutions; hence, this required cooperation among the players.…”

“…A multiple-input multiple-output interference channel setup was considered in [9][10][11]. Both the non-cooperative and the cooperative game-theoretic methods were considered in [10].…”

“…Both the non-cooperative and the cooperative game-theoretic methods were considered in [10]. The cooperative games proposed in [12][13][14][15][16] were able to achieve Pareto optimal solutions.…”