Competition between wireless service providers sharing a radio resource

We present a model of competition on prices between two telecommunication service providers sharing an access resource, which can for example be the same WiFi spectrum. We obtain a two-level game corresponding to two time scales of decisions: at the smallest time scale, users play an association game by choosing their provider (or none) depending on price, provider reputation and congestion level, and at the largest time scale, providers compete on prices. We show that the association game always has an equilibrium, but that several can exist. The pricing game is then solved by assuming that providers are risk- averse and try to maximize the minimal revenue they can get at a user equilibrium. We illustrate what can be the outcome of this game and that there are situations for which providers can co-exist.