A Markov Chain Approximation to Choice Modeling

Abstract

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 suboptimal decisions. In this paper, we consider a Markov chain based choice model and show that it provides a simultaneous approximation for all random utility based discrete choice models including the multinomial logit (MNL), the probit, the nested logit and mixtures of multinomial logit models. In the Markov chain model, substitution from one product to another is modeled as a state transition in the Markov chain. We show that the choice probabilities computed by the Markov chain based 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 generalized attraction model (GAM) of which the MNL model is a special case. 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 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.

Dynamic Attribute-Level Best Worst Discrete Choice Experiments

“…Our interest is with discrete time finite horizon MDPs, that is in which is a fixed number of time periods. The rewards (or expected rewards) are maximized by the best sequential decisions over time, making MDPs a dynamic optimization tool as used in Blanchet et al (2016) to identify the right choices of substitution behaviors of the decision makers.…”

Estimation of Future Probable Maximum Precipitation in Korea Using Multiple Regional Climate Models

“…In other words, sub-daily extreme DPT was derived from future daily DPT time series using the scale-invariance method, assuming that the probability distribution of the data has invariant characteristics irrespective of the scale of the data. In fact, the scale-invariance method has been applied to extreme rainfall [23][24][25]. In this study, the scale-invariance method combined with the hydro-meteorological framework for estimating future PMPs was applied to DPT time series.…”

Discrete Choice, Permutations, and Reconstruction

“…For the problem of learning MNL mixtures, there has been some attempts at algorithms with provable guarantees: the setting of low-dimensional structure was considered in [7], the case when each MNL is "geometric" was solved in [1], and using pairwise comparisons to learn was studied in [12]. Blanchett et al [3] proposed a choice model based on Markov chains and obtain some algorithmic results for learning in this model; see also [6]. Farias et al [5] study the problem of learning the "sparsest" RUM, in the sense of fewest permutations, which is consistent with a set of observations.…”