Repeated games for power control in wireless communications: Equilibrium and regret

Game theory, as a powerful conceptual framework, has in the past decade or so been widely applied to wireless communications in a variety of contexts. One such context is power control, where a game-theoretic formulation is particularly well-motivated since the distributed power control paradigm makes it natural to treat each wireless link (consisting of a transmitter and a receiver) as a player equipped with its own incentives. However, much of the work on game-theoretic studies of power control has been focused on one-shot games, where the emphasis is placed on either characterizing the existence and/or uniqueness of a Nash equilibrium or dynamics for reaching that Nash equilibrium In this paper, we consider a repeated game framework for power control in wireless communications that captures the salient features of the repeated interactions between different wireless links. We present a unified presentation of both finitely and infinitely repeated games and discuss the interesting information structure special to the power control setting. We then consider in depth two classes of solution concepts: Subgame Perfect Nash Equilibrium (SPNE) and no-regret strategy. For the former, we characterize the existence and uniqueness/multiplicity of SPNE; for the latter, we take an online convex optimization approach and design a power control scheme that is no regret for a link (irrespective of what the other links do), with an explicit finite time regret bound. These two solution concepts not only serve the normative role (from an economic-theoretical standpoint) of explaining how the wireless links transmit power in repeated interactions, but also induce distributed power control schemes that enjoy desirable properties from an engineering standpoint. Finally, we study a special case (one with “good” channel conditions) where we give a different power control strategy that both enjoys a more refined regret bound and converges to the unique Nash equilibrium of the stage game.