Bounds, Heuristics, and Prophet Inequalities for Assortment Optimization

We introduce odds-ratios in discrete choice models and utilize them to formulate bounds instrumental to the development of heuristics for the assortment optimization problem subject to totally unimodular constraints, and to the assess the benefit of personalized assortments. These heuristics, which only require the first and last-choice probabilities of the underlying discrete choice model, are broadly applicable, efficient, and come with worst-case performance guarantees. We propose a clairvoyant firm model to assess, in the limit, the potential benefits of personalized assortments. Our numerical study indicates that when the mean utilities of the products are heterogeneous among the consumer types, and the variance of the utilities is small, then firms can gain substantial benefits from personalized assortments. We support these observations, and others, with theoretical findings. For regular DCMs, we show that a clairvoyant firm can generate up to $n$ times more in expected revenues than a traditional firm. For discrete choice models with independent value gaps, we demonstrate that the clairvoyant firm can earn at most twice as much as a traditional firm. Prophet inequalities are also shown to hold for a variety of DCMs with dependent value gaps, including the MNL and GAM. While the consumers' surplus can potentially be larger under personalized assortments, clairvoyant firms with pricing power can extract all surplus, and earn arbitrarily more than traditional firms that optimize over prices but do not personalize them. For the price-aware MNL, however, a clairvoyant firm can earn at most $\exp(1)$ more than a traditional firm.