Assortment Optimization under the General Luce Model

This paper studies the assortment optimization problem under the General Luce Model (GLM), a discrete choice introduced by Echenique and Saito (2015) that generalizes the standard multinomial logit model (MNL). The GLM does not satisfy the Independence of Irrelevant Alternatives (IIA) property but it ensures that each product has an intrinsic utility and uses a dominance relation between products. Given a proposed assortment S, consumers first discard all dominated products in S before using a MNL model on the remaining products. The General Luce Model may violate the traditional regularity condition, which states that the probability of choosing a product cannot increase if the offer set is enlarged. As a result, the model can model behaviour that cannot be captured by any discrete choice model based on random utilities. The paper proves that the assortment problem under the GLM is polynomially-solvable. Moreover, it proves that the capacitated assortment optimization problem under the General Luce Model (GLM) is NP-hard and presents polynomial-time algorithms for the cases where (1) the dominance relation is utility-correlated and (2) its transitive reduction is a forest. The proofs exploit a strong connection between assortments under the GLM and independent sets in comparability graphs.