We consider the Bayesian formulation of the ranking and selection problem, with an independent normal prior, independent samples, and a cost per sample. While a number of 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 computation of 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 a number of low-dimensional ones, avoiding the curse of dimensionality. This allows calculation of the optimality gap for any given problem setting, 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.
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