A Markov Chain Approximation to Choice Modeling

Assortment planning is an important problem that arises in many industries such as retailing and airlines. One of the key challenges in an assortment planning problem is to identify the “right ” model for the substitution behavior of customers from the data. Error in model selection can lead to highly sub-optimal decisions. In this paper, we present a new choice model that is a simultaneous approximation for all random utility based discrete choice models including the multinomial logit, the probit, the nested logit and mixtures of multinomial logit models. Our model is based on a new primitive for substitution behavior where substitution from one product to another is modeled as a state transition of a Markov chain. We show that the choice probabilities computed by our model are a good approximation to the true choice probabilities for any random utility based choice model under mild conditions. Moreover, they are exact if the underlying model is a Multinomial logit model. We also show that the assortment optimization problem for our choice model can be solved efficiently in polynomial time. In addition to the theoretical bounds, we also conduct numerical experiments and observe that the average maximum relative error of the choice probabilities of our model with respect to the true probabilities for any offer set is less than 3 % where the average is taken over different offer sets. Therefore, our model provides a tractable data-driven approach to choice modeling and assortment optimization that is robust to model selection errors. Moreover, the state transition primitive for substitution provides interesting insights to model the substitution behavior in many real-world applications.