Optimal Linear Precoding Strategies for Wideband Non-Cooperative Systems based on Game Theory-Part I: Nash Equilibria

TL;DR

This paper introduces a game-theoretic approach to determine optimal linear precoding strategies for wideband multi-link systems, ensuring decentralized solutions that are efficient and comparable to centralized schemes.

Contribution

It demonstrates that optimal precoding schemes are channel diagonalizing and simplifies the problem to a vector power control game with proven uniqueness of Nash equilibria.

Findings

Optimal precoding leads to channel diagonalization.

The solution set of the game is always nonempty with pure strategies.

Derived conditions ensure the uniqueness of 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.