Asymptotically efficient adaptive allocation schemes for controlled Markov chains: finite parameter space

The authors consider a controlled Markov chain whose transition probabilities and initial distribution are parametrized by an unknown parameter θ belonging to some known parameter space Θ. There is a one-step reward associated with each pair of control and the following state of the process. The objective is to maximize the expected value of the sum of one-step rewards over an infinite horizon. The loss associated with a control scheme at a parameter value is the function of time giving the difference between the maximum reward that could have been achieved if the parameter were known and the reward achieved by the scheme. Since it is impossible to minimize the loss uniformly for all parameter values, the authors define uniformly good adaptive control schemes and restrict attention to these schemes. They develop a lower bound on the loss associated with any uniformly good control scheme. They construct an adaptive control scheme whose loss equals the lower bound for every parameter value and is therefore asymptotically efficient