In many retail and service environments, randomly arriving customers can request service from multiple service providers and choose the one that can provide 'quicker'' or 'better'' service. Specifically, we consider Emergency Medical Services (EMS) in emerging economies, where these services are mostly still decentralized. Such customer behavior plays off the providers deliberately against each other; and they, in turn, strategically define their service regions to choose whether or not to respond to a certain customer. To the best of our knowledge, this setting, where customers pit service providers against each other dynamically and temporally, has not been studied before, and we present the first paper on this topic.
We consider a simplified problem with two service providers, each located at either endpoint of a line with one server each. Demand occurs according to a temporally Poisson, spatially uniform, process. Providers' strategies involve choosing their service regions to maximize their utility taking into account the stochasticity and the associated opportunity costs. We analyze this simplified model, which turns out to be quite complex with interesting and novel properties.
We model this problem as a non-cooperative game between the providers. We observe that the server behavior is coupled, that is, while the marginal probabilities of server availability are straightforward, their joint idle probability is not independent. Analyzing the dynamics and joint idle probability using renewal theory, we show that such dependence between the servers is 'limited'. This leads us to a close and significantly more tractable approximation of the joint idle probability. We use these to derive three types of utility (profit) functions, each with a different service cost structure, as a function of the players' strategies.
We show that the utility function of a player, for a given fixed strategy of the other player, are neither convex nor concave, but they are unimodal. Also, we show this game admits a symmetric pure strategy Nash equilibrium. Furthermore, we analyze the Price of Anarchy (PoA), which quantifies the loss of efficiency due to non-cooperation, and we show that the PoA is bounded by a small constant depending on the game parameters in most cases. Finally, we experimentally calculate the empirical PoA.
Our findings confirm that under certain demand regimes, providers choose service regions that overlap and compete for customers, as is seen in reality in emerging economies. We discuss how our work enables a social planner to compute the competitive equilibrium and cooperative utilities, and to use methods such as coco-decomposition to compute strategic payoffs to incentivize cooperation.