Dynamic Power Allocation Games in Parallel Multiple Access Channels

TL;DR

This paper investigates a non-cooperative game model for distributed power allocation in parallel multiple access channels, demonstrating almost sure uniqueness of equilibrium and convergence of learning dynamics.

Contribution

It establishes that the parallel MAC game admits a unique equilibrium almost surely and shows convergence of replicator dynamics with only local information.

Findings

Unique equilibrium exists almost surely in the game

Replicator dynamics converge from almost any initial condition

Sufficient conditions for equilibrium uniqueness are not necessary

Abstract

We analyze the distributed power allocation problem in parallel multiple access channels (MAC) by studying an associated non-cooperative game which admits an exact potential. Even though games of this type have been the subject of considerable study in the literature, we find that the sufficient conditions which ensure uniqueness of Nash equilibrium points typically do not hold in this context. Nonetheless, we show that the parallel MAC game admits a unique equilibrium almost surely, thus establishing an important class of counterexamples where these sufficient conditions are not necessary. Furthermore, if the network's users employ a distributed learning scheme based on the replicator dynamics, we show that they converge to equilibrium from almost any initial condition, even though users only have local information at their disposal.