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 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. In particular, we consider a Markov chain where there is a state for each product, and model the substitution behavior as follows: a customer arrives in the state corresponding to his most preferred product. If that product is not available, he/she transitions to other product states according to the transition probabilities of the Markov chain. Therefore, the preferences of the customers are approximated by Markovian transitions in this choice model.

We show that the choice probabilities computed by our model are a good approximation to the true choice probabilities of any random utility discrete based choice model under mild conditions. Moreover, they are exact if the underlying model is a Multinomial logit model. We also give a data-driven procedure to estimate the parameters of the Markov chain model that does not require any knowledge of the latent choice model. Furthermore, we show that the assortment optimization problem under our choice model can be solved efficiently in polynomial time. In an assortment optimization problem, the goal is to find an assortment (or offer set) that maximizes the total expected revenue. The result is quite surprising as we can not even express the choice probabilities for different offer sets using a simple functional form in the Markov chain model.

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% (the average being 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.