Upper Bounds for Bayesian Ranking & Selection

We consider the Bayesian ranking and selection problem, with independent normal prior, independent samples, and a sampling cost. While several procedures have been developed for this problem in the literature, the gap between the best existing procedure and the Bayes-optimal one remains unknown, because computing the Bayes-optimal procedure using existing methods requires solving a stochastic dynamic program whose dimension increases with the number of alternatives. In this paper, we give a tractable method for computing an upper bound on the value of the Bayes-optimal procedure, which uses a decomposition technique to break a high-dimensional dynamic program into several low-dimensional ones, avoiding the curse of dimensionality. This allows calculation of an optimality gap, giving information about how much additional benefit we may obtain through further algorithmic development. We apply this technique to several problem settings, finding some in which the gap is small, and others in which it is large.