Distributed Learning Policies for Power Allocation in Multiple Access Channels

TL;DR

This paper studies distributed power allocation in multiple access channels using game theory, showing convergence of learning algorithms to equilibrium in both static and stochastic channel conditions.

Contribution

It introduces a potential game framework for distributed power control and demonstrates exponential convergence of replicator dynamics in static channels and convergence in stochastic channels.

Findings

Unique equilibrium exists in static channels.

Distributed learning converges exponentially fast.

Convergence persists under stochastic channel fluctuations.

Abstract

We analyze the problem of distributed power allocation for orthogonal multiple access channels by considering a continuous non-cooperative game whose strategy space represents the users' distribution of transmission power over the network's channels. When the channels are static, we find that this game admits an exact potential function and this allows us to show that it has a unique equilibrium almost surely. Furthermore, using the game's potential property, we derive a modified version of the replicator dynamics of evolutionary game theory which applies to this continuous game, and we show that if the network's users employ a distributed learning scheme based on these dynamics, then they converge to equilibrium exponentially quickly. On the other hand, a major challenge occurs if the channels do not remain static but fluctuate stochastically over time, following a stationary ergodic process. In that case, the associated ergodic game still admits a unique equilibrium, but the learning analysis becomes much more complicated because the replicator dynamics are no longer deterministic. Nonetheless, by employing results from the theory of stochastic approximation, we show that users still converge to the game's unique equilibrium. Our analysis hinges on a game-theoretical result which is of independent interest: in finite player games which admit a (possibly nonlinear) convex potential function, the replicator dynamics (suitably modified to account for nonlinear payoffs) converge to an eps-neighborhood of an equilibrium at time of order O(log(1/eps)).