Generalized Top-k Mallows Model for Ranked Choices

The classic Mallows model is a foundational tool for modeling user preferences. However, it has limitations in capturing real-world scenarios, where users often focus only on a limited set of preferred items and are indifferent to the rest. To address this, extensions such as the top-$k$ Mallows model have been proposed, aligning better with practical applications. In this paper, we address several challenges related to the generalized top-$k$ Mallows model, with a focus on analyzing buyer choices. Our key contributions are: (1) a novel sampling scheme tailored to generalized top-$k$ Mallows models, (2) an efficient algorithm for computing choice probabilities under this model, and (3) an active learning algorithm for estimating the model parameters from observed choice data. These contributions provide new tools for analysis and prediction in critical decision-making scenarios. We present a rigorous mathematical analysis for the performance of our algorithms. Furthermore, through extensive experiments on synthetic data and real-world data, we demonstrate the scalability and accuracy of our proposed methods, and we compare the predictive power of Mallows model for top-$k$ lists compared to the simpler Multinomial Logit model.