Some properties of batch value of information in the selection problem

Given a set of items of unknown utility, we need to select one with a utility as high as possible ("the selection problem"). Measurements (possibly noisy) of item values prior to selection are allowed, at a known cost. The goal is to optimize the overall sequential decision process of measurements and selection. Value of information (VOI) is a well-known scheme for selecting measurements, but the intractability of the problem typically leads to using myopic VOI estimates. Other schemes have also been proposed, some with approximation guarantees, based on submodularity criteria. However, it was observed that the VOI is not submodular in general. In this paper we examine theoretical properties of VOI for the selection problem, and identify cases of submodularity and supermodularity. We suggest how to use these properties to compute approximately optimal measurement batch policies, with an example based on a "wine selection problem".