Choosing an n-Pack of Substitutable Products

We develop a framework to model the shopping and consumption decisions of forward-looking consumers. Assuming that the consumer’s future utility for each product alternative can be characterized by a standard random utility model, we use dynamic programming to determine the optimal consumption policy and the maximum expected value of consuming any n substitutable products selected while shopping (an n-pack). We propose two models. In the first (canonical) model, we assume that an alternative is consumed on each successive consumption occasion and obtain a closed-form optimal policy and a closed-form value function. Given a consumer's preferences for the product alternatives in an assortment, we then show how to identify that consumer's optimal n-pack using a simple swapping algorithm that converges in at most n swaps. In the second (generalized) model, we introduce an outside option so that a product alternative need not be consumed on each consumption occasion. We obtain a closed-form value function for the generalized model and show that its optimal n-pack is related to that of the canonical model using a special type of majorization. Additional structural properties and implications of each model are explored, as are other applications