Generalized Top-k Mallows Model for Ranked Choices

TL;DR

This work extends Mallows modeling to generalized top-$k$ lists by introducing TopKGMM, a weighted, $p$-parameterized distribution over partial rankings. It delivers three practical tools: PRIM for efficient sampling, DyPChiP for exact choice-probability computation under TopKGMM, and BuCChoi (with FindTop) for active-learning of the center from choice data, all supported by rigorous analysis. Empirical results on sushi preferences show that TopKGMM achieves higher predictive accuracy than the Multinomial Logit, and synthetic experiments confirm favorable runtime and sample-complexity characteristics. The framework enables scalable analysis and prediction in decision-making contexts where users reveal only top-$k$ preferences, with implications for recommender systems, assortment optimization, and active data collection.

Abstract

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.

Figures (6)

• Figure 1: Test error of the top-$k$ Mallows model compared to MNL. Parameter $\beta$ and $p$ have been selected to derive highest accuracy. \ref{['table:allparams_onecluster', 'table:testerror-allbetas']} show the test error for all choices of $p$ and $\beta$.
• Figure 2: performance of algorithms DyPChiP, FindTop and BuCChoi on synthetic data.
• Figure 3: Runtime of DyPChiP (in seconds) for various choices of $n,k$. Here $\beta=0.6$, $p=0.5$ and $w=2\vec{1}$, size of assortment $r=k-2$.
• Figure 4: Sample complexity of FindTop for a wide range of parameters.
• Figure 5: Sample complexity of BuCChoi for a wide range of parameters. y axis shows the Kendall's Tau distance between true and learned center while x axis shows sample complexity

Theorems & Definitions (18)

• Definition 2.1: Inversion Vectors of a Top-$k$ list
• Theorem 3.1: Sampling from TopKGMM
• Definition 3.2: top-k Profile
• Lemma 3.3
• Theorem 4.1: Calculation of choice probabilities
• Theorem 5.1
• Lemma 5.2
• Lemma : Restatement of \ref{['lem:profile_prob']}
• proof
• proof : Proof of \ref{['thm:sampling']}