Stochastic approximation and user adaptation in a competitive resource sharing system

This paper is motivated by the work of Altman and Shimkin (1998). Customers arrive at a service center and must choose between two types of service; a channel that is shared by all currently in it, and a dedicated line. The mean service cost (or time) for any customer entering the shared resource depends on the decisions of all future arrivals up to the time of departure, and so has a competitive aspect. The system keeps a record of the (discounted) mean sample service costs (or times) for the shared resource, as a function of the number there when a new arrival joins, and the arriving customers use this to make their decisions. The decision rule of each arriving customer is based on its own immediate self interest, given the available data. They select the service with the smallest estimated service cost. But, if the current estimate of the cost for the shared resource equals that of the dedicated line, any decision is possible. The procedure is a type of learning algorithm. The long term behavior of the arrivals and its effect on the system averages is of interest. The convergence problem is one in asynchronous stochastic approximation, where the ODE has a set-valued right hand side. The set value arises due to the arbitrariness of the decision at certain values of the current estimates. It is shown that, asymptotically, the performance of the learning system is that for the symmetric Nash strategy, despite the allowed arbitrariness and lack of coordination