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Preference-Based Dynamic Ranking Structure Recognition

Nan Lu, Jian Shi, Xin-Yu Tian

TL;DR

The paper addresses the problem of uncovering evolving ranking structures from time-varying preference data by developing a dynamic Bradley-Terry framework that leverages kernel-smoothed spectral estimation. It jointly identifies dynamic ranking groups using an adaptive group-lasso objective and detects group-change points via a separable, dynamic-programming approach, with theoretical guarantees for group-consistency, asymptotic distribution of group-level estimates, and change-point consistency. Key contributions include an unconstrained reformulation enabling efficient optimization, a refit step to mitigate shrinkage, and rigorous consistency results, complemented by synthetic experiments and an empirical NBA study demonstrating interpretability and robustness. The work provides a principled, scalable toolkit for time-varying ranking analysis with practical impact on sports analytics, recommender systems, and related domains where comparison results evolve over time.

Abstract

Preference-based data often appear complex and noisy but may conceal underlying homogeneous structures. This paper introduces a novel framework of ranking structure recognition for preference-based data. We first develop an approach to identify dynamic ranking groups by incorporating temporal penalties into a spectral estimation for the celebrated Bradley-Terry model. To detect structural changes, we introduce an innovative objective function and present a practicable algorithm based on dynamic programming. Theoretically, we establish the consistency of ranking group recognition by exploiting properties of a random `design matrix' induced by a reversible Markov chain. We also tailor a group inverse technique to quantify the uncertainty in item ability estimates. Additionally, we prove the consistency of structure change recognition, ensuring the robustness of the proposed framework. Experiments on both synthetic and real-world datasets demonstrate the practical utility and interpretability of our approach.

Preference-Based Dynamic Ranking Structure Recognition

TL;DR

The paper addresses the problem of uncovering evolving ranking structures from time-varying preference data by developing a dynamic Bradley-Terry framework that leverages kernel-smoothed spectral estimation. It jointly identifies dynamic ranking groups using an adaptive group-lasso objective and detects group-change points via a separable, dynamic-programming approach, with theoretical guarantees for group-consistency, asymptotic distribution of group-level estimates, and change-point consistency. Key contributions include an unconstrained reformulation enabling efficient optimization, a refit step to mitigate shrinkage, and rigorous consistency results, complemented by synthetic experiments and an empirical NBA study demonstrating interpretability and robustness. The work provides a principled, scalable toolkit for time-varying ranking analysis with practical impact on sports analytics, recommender systems, and related domains where comparison results evolve over time.

Abstract

Preference-based data often appear complex and noisy but may conceal underlying homogeneous structures. This paper introduces a novel framework of ranking structure recognition for preference-based data. We first develop an approach to identify dynamic ranking groups by incorporating temporal penalties into a spectral estimation for the celebrated Bradley-Terry model. To detect structural changes, we introduce an innovative objective function and present a practicable algorithm based on dynamic programming. Theoretically, we establish the consistency of ranking group recognition by exploiting properties of a random `design matrix' induced by a reversible Markov chain. We also tailor a group inverse technique to quantify the uncertainty in item ability estimates. Additionally, we prove the consistency of structure change recognition, ensuring the robustness of the proposed framework. Experiments on both synthetic and real-world datasets demonstrate the practical utility and interpretability of our approach.

Paper Structure

This paper contains 30 sections, 5 theorems, 58 equations, 6 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Ranking Structure Recognition
  3. Dynamic Bradley-Terry Model
  4. Recognition of Dynamic Ranking Groups
  5. Recognition of Group Changes
  6. Statistical Learning Theory
  7. Consistency of Ranking Group Estimation
  8. Asymptotic Distribution of Item Ability Estimates
  9. Consistency of Group Changes Detection
  10. Computational Experiments
  11. Recognition of Dynamic Ranking Groups
  12. Recognition of Group Changes
  13. Empirical Analysis: Ranking Structure Recognition of NBA Teams
  14. Conclusion
  15. Notations
  16. ...and 15 more sections

Key Result

Theorem 3.3

Let Assumptions asmp:pi and asmp:K hold. When $Mh\rightarrow \infty$, $n\rightarrow\infty$ and $nMh^5\rightarrow 0$, if 1. $\max\{\delta,\frac{1}{m},\sqrt{\frac{B}{n^3Mh}}\}=o(\delta_{1})$ and $\delta_{2}=o(\sqrt{ \frac{1+\cos\frac{(n-B)\bm{\pi}}{n-B+1}}{B(n-B)} \frac{1}{n^2Mh} })$; 2. $\sqrt{\f

Figures (6)

  • Figure 1: Workflow of the proposed method for dynamic ranking structure recognition. The left panel depicts the data format, representing preference outcomes obtained from pairwise comparisons. The middle panel illustrates the score evolution of each group over a given interval. The right panel shows the structure detection procedure, where: (1) each node can be decomposed into subproblems represented by its child nodes; (2) each child node corresponds to a subproblem and a grouping problem on intervals; (3) subproblems at level C reuse results from nodes preceding their parent node at level B, thereby avoiding redundant computation.
  • Figure 2: The winning percentage of Team A over Team B.
  • Figure 3: Estimation of team strengths using KRC (left) and our method (right).
  • Figure 4: $\bm{\pi}^{*}(t)$ of each group in simulations.
  • Figure 5: $\bm{\pi}^{*}(t)$ in the first setting.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.6
  • Theorem 3.9
  • Corollary 3.10
  • ...and 1 more