A game formulation of duopoly market with coexistence of SoftSim and regular users

The stable subscription relationships between users and Wireless Service Providers (WSPs) are established on the long-term service contracts and SIM (Subscriber Identity Module) cards. However, Apple's 'Soft Sim' plan will break the subscription relationships. With SoftSim devices, users can flexibly switch WSPs at fine time scales, which provides more choices to users and benefits them. Thus more and more users will turn to SoftSim devices. But due to many challenges unsettled, it is likely that SoftSim users will coexist with regular users who are bound to certain WSPs for a long time. How to form best strategies to attract more demand, fully utilize network capacity and thus improve the revenue in the market with heterogeneous users remains a challenge for WSPs. In this paper, we address the problem of WSPs' revenue maximization in a duopoly market with the coexistence of regular and SoftSim users. We formulate the scenario as a non-cooperative two-stage game where the WSPs first announce their prices and then users decide the subscription and demand response. We derive the unique Nash Equilibrium (NE) under our framework and also give a convergence algorithm based on best response functions. The simulation results verify our theoretic conclusions.