Rank aggregation systems collect ordinal preferences from individuals to produce a global ranking that represents the social preference. Rank-breaking is a common practice to reduce the computational complexity of learning the global ranking. The individual preferences are broken into pairwise comparisons and applied to efficient algorithms tailored for independent paired comparisons. However, due to the ignored dependencies in the data, naive rank-breaking approaches can result in inconsistent estimates. The key idea to produce accurate and consistent estimates is to treat the pairwise comparisons unequally, depending on the topology of the collected data. In this paper, we provide the optimal rank-breaking estimator, which not only achieves consistency but also achieves the best error bound. This allows us to characterize the fundamental tradeoff between accuracy and complexity. Further, the analysis identifies how the accuracy depends on the spectral gap of a corresponding comparison graph.
We visit the following fundamental problem: For a 'generic' model of consumer choice (namely, distributions over preference lists) and a limited amount of data on how consumers actually make decisions (such as marginal preference information), how may one predict revenues from offering a particular assortment of choices? This problem is central to areas within operations research, marketing and econometrics. We present a framework to answer such questions and design a number of tractable algorithms (from a data and computational standpoint) for the same. The need to rank items based on user input arises in many practical applications such as elections, group-decision making and recommendation systems. The primary challenge in such scenarios is to decide a global ranking based on partial preferences provides by users. The standard approach to address this challenge is to ask users to provide explicit numerical ratings (cardinal information) of a subset of items. The main appeal of such an approach is the ease of aggregation. However, the rating scale as well as the individual ratings are often arbitrary and may not be consistent from one user to another. A more natural alternative to numerical ratings requires users to compare pairs of items (ordinal information). In contrast to cardinal information, such comparisons provide an “absolute” indicator of the user's preference. However, it is often hard to combine or aggregate comparisons to obtain a consistent global ranking. In this work, we provide a tractable framework for utilizing comparison data as well as first-order marginal information for the purpose of ranking. We treat the available information as partial samples from an unknown distribution over permutations. Using the Principle of Maximum Entropy, we devise a concise parameterization of distribution consistent with observations using only O(n2) parameters, where n is the number of items in question. We propose a distributed, iterative algorithm for estimating the parameters of the distribution. We establish the correctness of the algorithm as well as identify the rate of convergence explicitly. Using the learnt distribution, we provide efficient approach to (a) learn the mode of the distribution using ‘maximum weight matching’, (b) identification of top k items, and (c) an aggregate ranking of all n items. Through evaluation of our approach on real-data, we verify effectiveness of our solutions as well as the scalability of the algorithm. Choice models today are ubiquitous across a range of applications in operations and marketing. Real-world implementations of many of these models face the formidable stumbling block of simply identifying the “right” model of choice to use. Because models of choice are inherently high-dimensional objects, the typical approach to dealing with this problem is positing, a priori, a parametric model that one believes adequately captures choice behavior. This approach can be substantially suboptimal in scenarios where one cares about using the choice model learned to make fine-grained predictions; one must contend with the risks of mis-specification and overfitting underfitting. Thus motivated, we visit the following problem: For a “generic” model of consumer choice namely, distributions over preference lists and a limited amount of data on how consumers actually make decisions such as marginal information about these distributions, how may one predict revenues from offering a particular assortment of choices? An outcome of our investigation is a nonparametric approach in which the data automatically select the right choice model for revenue predictions. The approach is practical. Using a data set consisting of automobile sales transaction data from a major U.S. automaker, our method demonstrates a 20 improvement in prediction accuracy over state-of-the-art benchmark models; this improvement can translate into a 10 increase in revenues from optimizing the offer set. We also address a number of theoretical issues, among them a qualitative examination of the choice models implicitly learned by the approach. We believe that this paper takes a step toward “automating” the crucial task of choice model selection. This paper was accepted by Yossi Aviv, operations management. There has been much interest recently in the problem of rank aggregation from pairwise data. A natural question that arises is: under what sorts of statistical assumptions do various rank aggregation algorithms converge to an 'optimal' ranking? In this paper, we consider this question in a natural setting where pairwise comparisons are drawn randomly and independently from some underlying probability distribution. We first show that, under a 'time-reversibility' or Bradley-Terry-Luce (BTL) condition on the distribution, the rank centrality (PageRank) and least squares (HodgeRank) algorithms both converge to an optimal ranking. Next, we show that a matrix version of the Borda count algorithm, and more surprisingly, an algorithm which performs maximum likelihood estimation under a BTL assumption, both converge to an optimal ranking under a 'low-noise' condition that is strictly more general than BTL. Finally, we propose a new SVM-based algorithm for rank aggregation from pairwise data, and show that this converges to an optimal ranking under an even more general condition that we term 'generalized low-noise'. In all cases, we provide explicit sample complexity bounds for exact recovery of an optimal ranking. Our experiments confirm our theoretical findings and help to shed light on the statistical behavior of various rank aggregation algorithms. 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. There are various parametric models for analyzing pairwise comparison data, including the Bradley-Terry-Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this work, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations. We consider revenue management problems when customers choose among the offered products according to the Markov chain choice model. In this choice model, a customer arrives into the system to purchase a particular product. If this product is available for purchase, then the customer purchases it. Otherwise, the customer transitions to another product or to the no purchase option, until she reaches an available product or the no purchase option. We consider three classes of problems. First, we study assortment problems, where the goal is to find a set of products to offer to maximize the expected revenue obtained from each customer. We give a linear program to obtain the optimal solution. Second, we study single resource revenue management problems, where the goal is to adjust the set of offered products over a selling horizon when the sale of each product consumes the resource. We show how the optimal set of products to offer changes with the remaining resource inventory. Third, we study network revenue ...