Strategic Service Logistics Games with Customer-Induced Competition
41 Pages Posted: 21 Jul 2018
Date Written: June 30, 2018
In many retail and service environments, randomly arriving customers can request service from multiple service providers while they choose the one that provides "quicker" or "better" service. This 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 version of this problem with two service providers, each located at either endpoints of a line with one server each. Demand occurs according to a Poisson process at a location uniformly distributed on the line. The providers strategically determine their service regions to maximize their utility while taking into account the stochasticity and the associated opportunity costs. We thoroughly analyze this simplified model, which turns out to be quite complex with interesting and novel properties.
We model this problem setting as a non-cooperative game between the providers who are the players of the game, with their strategies associated with the service regions they choose. Using the Erlang-B formula and a Markov chain, we first obtain the stationary probabilities of the servers being idle. The joint probability turns out to be not independent. Using these probabilities, we derive three types of utility functions, with different service costs, of each player as a function of their strategies. We show that the utility functions of a player, for a given fixed strategy of the other player, are neither convex nor concave, but they are unimodal. Surprisingly, this game admits a unique symmetric pure Nash equilibrium (SPNE). We note that the existence and uniqueness of SPNE is essential for the game-theoretic solution concepts to be plausible outcomes of a competitive setting in practice.
Furthermore, we analyze the Price of Non-Cooperation (PoNC), which quantifies the loss of efficiency due to non-cooperation, and we show that the PoNC is bounded by a small constant depending on the game parameters in most cases. Finally, we conduct experimental studies to calculate the empirical PoNC.
Keywords: Service Logistics Games, Symmetric Pure Nash Equilibrium, Uniqueness of Equilibrium, Price of Non-Cooperation
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