Bounds, Heuristics, and Prophet Inequalities for Assortment Optimization

We address two important concerns faced by assortment managers, namely constrained assortment optimization and assortment personalization. We contribute to addressing these concerns by developing bounds and heuristics based on auxiliary multinomial logit (MNL) models. More precisely, we first provide easily computable upper and lower bounds for the unconstrained assortment optimization problem (TAOP) for every regular choice model and then extend the bounds to important versions of the constrained problem. We next provide an upper bound on the expected revenue of a clairvoyant firm that offers to each consumer the most profitable product that she is willing to buy. We then use the upper bound to assess the maximum benefits of personalization relative to a firm that does not personalize assortments. The standard prophet inequality is then used to show that the ratio is at most 2 for discrete choice models with {\em independent} value gaps. For random utility models with dependent value gaps the ratio can be as large as the number of products. We find sufficient conditions to show that the prophet inequality holds for the $\alpha$-shaken multinomial logit ($\alpha$-MNL), a generalization of the MNL introduced here, that has the MNL and the generalized attraction model (GAM) as special cases. The prophet inequality also holds for the some versions of the Nested Logit model. For the latent-class MNL, the ratio is at most 1.5 when the coefficient of variation of the utilities goes to infinity. We show that consumers do not necessarily suffer under a clairvoyant firm and in fact their surplus may improve. On the other hand, when the clairvoyant firm has pricing power it can extract all of the consumers' surplus. We show that for the MNL model the clairvoyant can make up to $e$ times more than its non-clairvoyant counterpart