Pairwise Comparisons with Flexible Time-Dynamics

Maystre, Lucas; Kristof, Victor; Grossglauser, Matthias

doi:10.1145/3292500.3330831

Cited by 12 publications

(8 citation statements)

References 29 publications

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“…(2) Bounding ρ 0 : Recall that ρ 0 = N 2 M max {max i α i , max j β j } with α i and β j defined in (16). Next, we will first bound max i α i and max j β j in sequence.…”

Section: Discussionmentioning

confidence: 99%

“…Meanwhile, individuals' tastes can change over time; candidates in an election adapt their platforms; companies change the price and features of their products over time, etc. All of these factors can cause pairwise comparison matrices to change over time [16][17][18]. Additionally, data may arrive in streaming fashion, noisy and incomplete.…”

Section: Introductionmentioning

confidence: 99%

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Recovery Guarantees for Time-varying Pairwise Comparison Matrices with Non-transitivity

Li,

Wakin

2021

Preprint

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Pairwise comparison matrices have received substantial attention in a variety of applications, especially in rank aggregation, the task of flattening items into a one-dimensional (and thus transitive) ranking. However, non-transitive preference cycles can arise in practice due to the fact that making a decision often requires a complex evaluation of multiple factors. In some applications, it may be important to identify and preserve information about the inherent non-transitivity, either in the pairwise comparison data itself or in the latent feature space. In this work, we develop structured models for non-transitive pairwise comparison matrices that can be exploited to recover such matrices from incomplete noisy data and thus allow the detection of non-transitivity. Considering that individuals' tastes and items' latent features may change over time, we formulate time-varying pairwise comparison matrix recovery as a dynamic skew-symmetric matrix recovery problem by modeling changes in the low-rank factors of the pairwise comparison matrix. We provide theoretical guarantees for the recovery and numerically test the proposed theory with both synthetic and real-world data.

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“…(2) Bounding ρ 0 : Recall that ρ 0 = N 2 M max {max i α i , max j β j } with α i and β j defined in (16). Next, we will first bound max i α i and max j β j in sequence.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Recovery Guarantees for Time-varying Pairwise Comparison Matrices with Non-transitivity

Li,

Wakin

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Ranking analysis with temporal variants has also become increasingly important because of the growing needs for models and methods to handle time-dependent data. A series of results in this direction can be found in Glickman (1993), Glickman and Stern (1998), Cattelan et al (2013), Lopez et al (2018), Maystre et al (2019), Bong et al (2020), Karlé and Tyagi (2021) and references therein. Much of the aforementioned literature on time-varying BTL model postulates that temporal changes in the model parameters are smooth functions of time and thus occur gradually on a relatively large time scale.…”

Section: Introductionmentioning

confidence: 94%

Detecting Abrupt Changes in Sequential Pairwise Comparison Data

Li¹,

Wang²,

Rinaldo³

2022

Preprint

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The Bradley-Terry-Luce (BTL) model is a classic and very popular statistical approach for eliciting a global ranking among a collection of items using pairwise comparison data. In applications in which the comparison outcomes are observed as a time series, it is often the case that data are non-stationary, in the sense that the true underlying ranking changes over time. In this paper we are concerned with localizing the change points in a high-dimensional BTL model with piecewise constant parameters. We propose novel and practicable algorithms based on dynamic programming that can consistently estimate the unknown locations of the change points. We provide consistency rates for our methodology that depend explicitly on the model parameters, the temporal spacing between two consecutive change points and the magnitude of the change. We corroborate our findings with extensive numerical experiments and a real-life example.Preprint. Under review.

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“…This dynamic setting has received significantly less attention than its static counterpart and has mostly been studied with a focus on applications, such as sports tournaments [5,11,21]. It has rarely been analysed theoretically in the past however, although some results exist for a state-space generalization of the BTL model [9,10,19] and for a Bayesian framework [10,16].…”

Section: Introductionmentioning

confidence: 99%

Dynamic Ranking with the BTL Model: A Nearest Neighbor based Rank Centrality Method

Karlé¹,

Tyagi²

2021

Preprint

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Many applications such as recommendation systems or sports tournaments involve pairwise comparisons within a collection of n items, the goal being to aggregate the binary outcomes of the comparisons in order to recover the latent strength and/or global ranking of the items. In recent years, this problem has received significant interest from a theoretical perspective with a number of methods being proposed, along with associated statistical guarantees under the assumption of a suitable generative model.While these results typically collect the pairwise comparisons as one comparison graph G, however in many applications -such as the outcomes of soccer matches during a tournamentthe nature of pairwise outcomes can evolve with time. Theoretical results for such a dynamic setting are relatively limited compared to the aforementioned static setting. We study in this paper an extension of the classic BTL (Bradley-Terry-Luce) model for the static setting to our dynamic setup under the assumption that the probabilities of the pairwise outcomes evolve smoothly over the time domain [0, 1]. Given a sequence of comparison graphs (G t ) t ∈T on a regular grid T ⊂ [0, 1], we aim at recovering the latent strengths of the items w t ∈ R n at any time t ∈ [0, 1]. To this end, we adapt the Rank Centrality method -a popular spectral approach for ranking in the static case -by locally averaging the available data on a suitable neighborhood of t. When (G t ) t ∈T is a sequence of Erdös-Renyi graphs, we provide non-asymptotic 2 and ∞ error bounds for estimating w * t which in particular establishes the consistency of this method in terms of n, and the grid size |T |. We also complement our theoretical analysis with experiments on real and synthetic data.

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Pairwise Comparisons with Flexible Time-Dynamics

Cited by 12 publications

References 29 publications

Recovery Guarantees for Time-varying Pairwise Comparison Matrices with Non-transitivity

Recovery Guarantees for Time-varying Pairwise Comparison Matrices with Non-transitivity

Detecting Abrupt Changes in Sequential Pairwise Comparison Data

Dynamic Ranking with the BTL Model: A Nearest Neighbor based Rank Centrality Method

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