MNL-Bandit: A Dynamic Learning Approach to Assortment Selection

Abstract

We consider a dynamic assortment selection problem, where in every round the retailer offers a subset (assortment) of N substitutable products to a consumer, who selects one of these products according to a multinomial logit (MNL) choice model. The retailer observes this choice and the objective is to dynamically learn the model parameters, while optimizing cumulative revenues over a selling horizon of length T . We refer to this exploration-exploitation formulation as the MNL-Bandit problem. Existing methods for this problem follow an explore-then-exploit approach, which estimate parameters to a desired accuracy and then, treating these estimates as if they are the correct parameter values, offers the optimal assortment based on these estimates. These approaches require certain a priori knowledge of "separability," determined by the true parameters of the underlying MNL model, and this in turn is critical in determining the length of the exploration period. (Separability refers to the distinguishability of the true optimal assortment from the other sub-optimal alternatives.) In this paper, we give an efficient algorithm that simultaneously explores and exploits, without a priori knowledge of any problem parameters. Furthermore, the algorithm is adaptive in the sense that its performance is near-optimal in both the "well separated" case, as well as the general parameter setting where this separation need not hold.

On Asymptotically Tight Tail Bounds for Sums of Geometric and Exponential Random Variables

“…We make several remarks comparing the above results from [AAGZ17a] with our tail bounds. First, the case of µ ≥ 1 and δ > 1 is missing in the above tail bounds, while our Theorem 1 and Proposition 1 cover all cases of µ, δ > 0.…”

“…In this section we give several approximations of H(λ, µ) with simpler forms. We also compare our result with existing tail bounds for sums of geometric random variables, mostly from [Jan18] and [AAGZ17a], showing our bound is tighter in several cases and easier to use overall.…”

“…In Appendix D of [AAGZ17a], the following were established for sums of i.i.d. geometrically distributed random variables: for δ > 0,…”

“…The objective of this paper is to characterize the tail probabilities Pr[X n ≥ λµ] for λ ∈ (1, ∞) or Pr[X n ≤ λµ] for λ ∈ (0, 1). Such tail bounds are important in statistics, computer science and operations research, and have recently found interesting applications in assortment selection problems in operations management [AAGZ17a,AAGZ17b,CWZ18].…”

Dynamic pricing and assortment under a contextual MNL demand

“…Dynamic assortment under a MNL choice model has been recently considered (see for instance Rusmevichientong et al (2010), Sauré and Zeevi (2013) and Wang et al (2018)). In particular, UCB and Thompson sampling-based policies are proposed in Agrawal et al (2017a) and Agrawal et al (2017b). These policies achieve optimal regrets of O( √ N T ) in the non-feature based setting.…”