Bradley-Terry Model Research Articles

The Bradley-Terry Model in Binary Outcome Driven Rankings: An Application in Amateur Hockey

Ranking systems serve critical roles in sport settings, most notably in determining playoff participants and seeding. Many ranking methodologies exist that are flexible enough to incorporate many input measures and produce models that are highly predictive of game outcomes. However, there are circumstances—especially for amateur sport leagues—in which more complex inputs are either unavailable or not desirable, as they may lead to adverse performance incentives. Therefore, the goal of this paper is to highlight a ranking methodology that only considers binary game outcomes, i.e., wins and losses. Specifically, we consider the efficacy of the Bradley-Terry Model to efficiently rank sport teams for playoff consideration. We apply this method as a case study to the New England Prep School Ice Hockey Association (NEPSIHA), and compare the accuracy of their current ranking system to the Bradley-Terry model using simulation methods. We show that Bradley-Terry significantly outperforms NEPSIHA’s current method, especially when teams face unbalanced strengths of schedule. This result holds under various league competitive balance distributions.

The Effect of NBA Attendance on Teams’ Performance

This paper analyzed how attendance in the NBA during the COVID-19 pandemic, specifically, in the 2020-2021 seasons, affected team performance. To answer this question, we used team data of all 30 teams from 2015 to 2022, and attendance data for the 2020-2021 season; we separated team attendance into three levels: Full (or almost full), Partial, and Empty. We used logistic models and the Bradley-Terry Model to analyze each team’s strength coefficient (compared to one baseline team) using the Win/Loss data as the determining factor. Our result showed a slight advantage for home teams when audiences were present. However, results revealed that home court advantage became less evident when there was partial or no audiences.

A spectral approach for the dynamic Bradley–Terry model

SummaryThe dynamic ranking, due to its increasing importance in many applications, is becoming crucial, especially with the collection of voluminous time‐dependent data. One such application is sports statistics, where dynamic ranking aids in forecasting the performance of competitive teams, drawing on historical and current data. Despite its usefulness, predicting and inferring rankings pose challenges in environments necessitating time‐dependent modelling. This paper introduces a spectral ranker called Kernel Rank Centrality, designed to rank items based on pairwise comparisons over time. The ranker operates via kernel smoothing in the Bradley–Terry model, utilising a Markov chain model. Unlike the maximum likelihood approach, the spectral ranker is nonparametric, demands fewer model assumptions and computations and allows for real‐time ranking. We establish the asymptotic distribution of the ranker by applying an innovative group inverse technique, resulting in a uniform and precise entrywise expansion. This result allows us to devise a new inferential method for predictive inference, previously unavailable in existing approaches. Our numerical examples showcase the ranker's utility in predictive accuracy and constructing an uncertainty measure for prediction, leveraging data from the National Basketball Association (NBA). The results underscore our method's potential compared with the gold standard in sports, the Arpad Elo rating system.

Liberals and Conservatives Rely on Very Similar Sets of Foundations When Comparing Moral Violations

Applications of moral foundations theory in political science have revealed differences in the degree to which liberals and conservatives explicitly endorse five core moral foundations of care, fairness, authority, loyalty, and sanctity. We argue that differences between liberals and conservatives in their explicit ratings of abstract and generalized moral principles do not imply that citizens with different political orientations have fundamentally different moral intuitions. We introduce a new approach for measuring the importance of the five moral foundations by asking U.K. and U.S. survey respondents to compare pairs of vignettes describing violations relevant to each foundation. We analyze responses to these comparisons using a hierarchical Bradley–Terry model which allows us to evaluate the relative importance of each foundation to individuals with different political perspectives. Our results suggest that, despite prominent claims to the contrary, voters on the left and the right of politics share broadly similar moral intuitions.

Bias-aware ranking from pairwise comparisons

Human feedback is often used, either directly or indirectly, as input to algorithmic decision making. However, humans are biased: if the algorithm that takes as input the human feedback does not control for potential biases, this might result in biased algorithmic decision making, which can have a tangible impact on people’s lives. In this paper, we study how to detect and correct for evaluators’ bias in the task of ranking people (or items) from pairwise comparisons. Specifically, we assume we are given pairwise comparisons of the items to be ranked produced by a set of evaluators. While the pairwise assessments of the evaluators should reflect to a certain extent the latent (unobservable) true quality scores of the items, they might be affected by each evaluator’s own bias against, or in favor, of some groups of items. By detecting and amending evaluators’ biases, we aim to produce a ranking of the items that is, as much as possible, in accordance with the ranking one would produce by having access to the latent quality scores. Our proposal is a novel method that extends the classic Bradley-Terry model by having a bias parameter for each evaluator which distorts the true quality score of each item, depending on the group the item belongs to. Thanks to the simplicity of the model, we are able to write explicitly its log-likelihood w.r.t. the parameters (i.e., items’ latent scores and evaluators’ bias) and optimize by means of the alternating approach. Our experiments on synthetic and real-world data confirm that our method is able to reconstruct the bias of each single evaluator extremely well and thus to outperform several non-trivial competitors in the task of producing a ranking which is as much as possible close to the unbiased ranking.

A Bayesian active learning approach to comparative judgement within education assessment

Assessment is a crucial part of education. Traditional marking is a source of inconsistencies and unconscious bias, placing a high cognitive load on the assessors. One approach to address these issues is comparative judgement (CJ). In CJ, the assessor is presented with a pair of items of work, and asked to select the better one. Following a series of comparisons, a rank for any item may be derived using a ranking model, for example, the Bradley-Terry model, based on the pairwise comparisons. While CJ is considered to be a reliable method for conducting marking, there are concerns surrounding its transparency, and the ideal number of pairwise comparisons to generate a reliable estimation of the rank order is not known. Additionally, there have been attempts to generate a method of selecting pairs that should be compared next in an informative manner, but some existing methods are known to have created their own bias within results inflating the reliability metric used within the process. As a consequence, a random selection approach is usually deployed.In this paper, we propose a novel Bayesian approach to CJ (which we call BCJ) for determining the ranks of a range of items under scrutiny alongside a new way to select the pairs to present to the marker(s) using active learning, addressing the key shortcomings of traditional CJ. Furthermore, we demonstrate how the entire approach may provide transparency by providing the user insights into how it is making its decisions and, at the same time, being more efficient. Results from our synthetic experiments confirm that the proposed BCJ combined with entropy-driven active learning pair-selection method is superior (i.e. always equal to or significantly better) than other alternatives, for example, the traditional CJ method with differing selection methods such as uniformly random, or the popular no repeating pairs where pairs are selected in a round-robin fashion. We also find that the more comparisons that are conducted, the more accurate BCJ becomes, which solves the issue the current method has of the model deteriorating if too many comparisons are performed. As our approach can generate the complete predicted rank distribution for an item, we also show how this can be utilised in probabilistically devising a predicted grade, guided by the choice of the assessor. Finally, we demonstrate our approach on a real dataset on assessing GCSE (UK school-level) essays, highlighting the advantages of BCJ over CJ.

Generalized Bradley-Terry Models for Score Estimation from Paired Comparisons

Many applications, e.g. in content recommendation, sports, or recruitment, leverage the comparisons of alternatives to score those alternatives. The classical Bradley-Terry model and its variants have been widely used to do so. The historical model considers binary comparisons (victory/defeat) between alternatives, while more recent developments allow finer comparisons to be taken into account. In this article, we introduce a probabilistic model encompassing a broad variety of paired comparisons that can take discrete or continuous values. We do so by considering a well-behaved subset of the exponential family, which we call the family of generalized Bradley-Terry (GBT) models, as it includes the classical Bradley-Terry model and many of its variants. Remarkably, we prove that all GBT models are guaranteed to yield a strictly convex negative log-likelihood. Moreover, assuming a Gaussian prior on alternatives' scores, we prove that the maximum a posteriori (MAP) of GBT models, whose existence, uniqueness and fast computation are thus guaranteed, varies monotonically with respect to comparisons (the more A beats B, the better the score of A) and is Lipschitz-resilient with respect to each new comparison (a single new comparison can only have a bounded effect on all the estimated scores). These desirable properties make GBT models appealing for practical use. We illustrate some features of GBT models on simulations.

Multiverse analysis and the Bradley-Terry model: A proposed approach for evaluating palatability and preference

The palatability of feed for dairy cows is an important consideration but is difficult to measure, particularly when considering more than 2 feeds. We outline how a combination of multiverse analysis and Bradley-Terry modeling, 2 methodological tools that have rarely been applied in dairy science, can be adapted to address this problem. Specifically, we propose to apply multiverse analysis as a way to consider a range of thresholds for how much of a mixed grass-legume (MGL) silage had to be consumed (as a percent of the total DMI) to be designated as preferred. Each threshold gives rise to a separate dataset and a corresponding fitted Bradley-Terry model. Bradley-Terry models attribute to each feed what is commonly referred to as an “ability” in the context of sports or other competitions but can be interpreted as palatability when applied to feeds. This combined approach is a way of estimating palatabilities that appropriately reflect the degree of preference cows express through their feeding behavior. It has the advantages of being transparent and relatively easy to implement. A possible disadvantage is that this method is limited to a paired comparison approach and has difficulties with main-effects statistical inference. We demonstrate the use of this methodology on an example dataset comparing MGL silages under different ensiling conditions and exposed to oxygen for different durations.

A comparative study of the emotional characteristics of violin and erhu musical excerpts: Influence of playing techniques and instrument

This study investigates the emotional characteristics of the violin and erhu in multiple musical excerpts. Participants assessed 52 musical excerpts played by both instruments using the Self-Assessment Manikin (SAM) scale. Pairwise comparisons explored the impact of playing techniques (Vibrato, Portamento, and Trill) and instrument on emotional perception. Fifty-eight participants evaluated four versions of 13 musical pieces. Significant agreement was found between the emotion categories and Valence-Arousal (VA) ratings. Ambiguity between calmness and sadness emerged, with VA tending to classify ambiguous excerpts as sad. Playing techniques enhanced the energetic qualities of both instruments. The violin consistently evoked more positive and energetic perceptions compared to the erhu, which were further enhanced when playing techniques such as vibrato were employed. Although erhu with playing techniques tended to be more negative, it still elicited greater energy and positivity than erhu without techniques. Valence-Arousal means were used to determine the emotional qualities of the 13 musical pieces. A subsequent pairwise comparison involving 33 participants revealed that versions with playing techniques had stronger emotional impact than those without techniques. Furthermore, emotional impact was higher when the piece was originally composed for the instrument being played, which was manifest in higher Bradley–Terry–Luce (BTL) values.

Asymptotics in the Bradley-Terry model for networks with a differentially private degree sequence

. The Bradley-Terry model is a common model for analyzing paired comparison data. Under differential private mechanism, there is a lack of asymptotic properties for the parameter estimator of parameters in this model. In this article, we show that the moment estimators of the parameters based on the differential private degree sequence with Laplace noise is uniformly consistent and asymptotically normal. Simulations are provided to illustrate asymptotic results.

Genomic Characterization of Acute Myeloid Leukemia with Aberrations of Chromosome 7: A Multinational Cohort of 523 Patients

Introduction: Aberrations of chromosome 7 [abn(7)] are found in ≈ 10% of newly diagnosed acute myeloid leukemia (AML) and associate with a dismal prognosis. A large-scale comprehensive exploration of the underlying genetic heterogeneity in AML with abn(7) has yet to be performed and could add essential insights into the outcome of this poorly understood patient group. Methods: We collected diagnostic samples from 523 adult AML patients (median age 59 years) with abn(7). Whole-exome sequencing (WES) was performed to discover potentially underestimated genetic lesions in 61 paired diagnostic / remission samples. Subsequently, a gene panel including 66 genes and a SNP backbone for copy-number aberration (CNA) detection was designed and applied to the remaining 471 samples. The majority of patients (78%) were diagnosed with de novo (dn) AML, whereas 22% had secondary (s) or therapy-related (t) AML. Intensive induction treatment was administered to 80% of the patients, while 36 % underwent allogenic stem cell transplantation. Apart from 43% of patients with concomitant complex karyotype (abn(7)/CK+), 24% had -7 as a sole abnormality (-7 sole) and 13% del(7q) sole. Results: A mean of 15.3 single-nucleotide variants (SNVs) and 7.5 CNAs per patient were found by WES. Here, the most frequent recurrent SNVs were identified in TP53 (29.5%), followed by mutations (mut) in genes involved in epigenetic regulation ( DNMT3A, TET2, ASXL1, IDH2), transcription factors ( RUNX1), and genes affecting RAS-signaling ( NF1, KRAS), Figure 1. Targeted sequencing revealed 1829 SNVs with a VAF≥ 2% in 64 genes (mean: 3.8 SNVs / patient). 30% of patients harbored at least one mutation in genes located within the commonly deleted region of 7q, most frequently KMT2C, EZH2CUX1, SAMD9L, SAMD9, LUC7L2 and BRAF. The number of driver gene mutations was higher in CK- than in CK+ patients (4.5 vs. 3.3 SNVs). We found KRAS (OR 3.76, CI 1.17-16.87, P= .044) and RUNX1 (OR 3.61, CI 1.62-8.78, P= .003) mutations to be enriched in -7 sole patients, and FLT3 mutations to be associated with del(7q) sole status (OR 0.33, CI 0.13-0.82, P= .019). With respect to previously unknown lesions, a high amount of KMT2C mutations (16.6%) and recurrent alterations in FAT1 and TACC2 were discovered (6.4% each; Fig. 1). For KMT2C - located on chr7q36.1 - a total of 98 SNVs with mainly missense (73.5%) and truncating (20.4%) mutations at known cancer hotspots were noted. KMT2Cmut was associated with dnAML and AML with maturation morphology. In the entire cohort, the most common co-occurring chromosomal alterations discovered by high-resolution CNA analysis were deletions in chromosomes 5, 17, and 12. Our approach enabled the identification of small fragment deletions (≤10Mb) affecting the TP53, NF1, and ETV6 loci in 5-9% of all cases, which were missed by conventional G-banding. Cancer Cell Fraction and Bradley-Terry models were used to simulate the sequential order of genomic aberrations. While mutations in TP53 and epigenetic-related genes were early events, -7 and del(7q) were often subclonal and SNVs in NRAS, KMT2C very late events in leukemogenesis. Survival analyses in intensively-treated patients (n= 414) revealed that 61% reached complete remission, 67% relapsed, and median overall survival (OS) was 11.9 months. Abn TP53 and high WBC count were independently associated with shorter relapse-free survival (RFS). For OS, besides older age and high WBC count, we identified abn TP53, PTPN11mut, TET2mut,-5, and -18 as poor prognostic factors in multivariate analysis (Table 1). In contrast, IDH2mutconferred an independent favorable prognostic effect for RFS and OS. Notably, abn TP53 outcompeted the prognostic impact of CK+ (Table 1). Compared to TP53wt, patients with abn TP53/CK+ and abn TP53/CK- had a similar poor outcome with median RFS of 6 and 4 months (CK+/CK-, P<.001) and OS of 6.8 and 8.6 months (CK+/CK-, P<.001). In contrast to other genomic studies in myelodysplastic syndrome, we found abn TP53 to be associated with poor outcome irrespective of the single- or multihit mutation status following definitions of the latest ICC classification (Blood, 2022). Conclusion: Our results offer novel insights into the genomic landscape and clonal trajectory of AML with abn(7). This work unravels formerly underestimated genetic lesions ( KMT2Cmut) and alterations with high prognostic impact (abn TP53 and IDH2mut) for better future risk stratification.

The incomplete analytic hierarchy process and Bradley–Terry model: (In)consistency and information retrieval

Several methods of preference modeling, ranking, voting and multi-criteria decision-making include pairwise comparisons. It is usually simpler to compare two objects at a time, furthermore, some relations (e.g., the outcome of sports matches) are naturally known for pairs. This paper investigates and compares pairwise comparison models and the stochastic Bradley–Terry model. It is proved that they provide the same priority vectors for consistent (complete or incomplete) comparisons. For incomplete comparisons, all filling in levels are considered. Recent results identified the optimal subsets and sequences of multiplicative/additive/reciprocal pairwise comparisons for small sizes of items (up to n=6). Simulations of this paper show that the same subsets and sequences are optimal in the case of the Bradley–Terry and the Thurstone models as well. This somehow surprising coincidence suggests the existence of a more general result. Further models of information and preference theory are subject to future investigation to identify optimal subsets of input data.

Bayesian Disaggregate and Aggregate Calibration of Path Logit Choice Models

In transport demand analysis, the calibration of a model means estimation of its (endogenous) parameters from observed data with an inference statistical estimator. Indeed, these considerations apply to any choice behaviour model, such as those derived from Random Utility Theory or any other choice modelling theory. Calibration of choice models can be carried out from disaggregate vs. aggregate data, while inference statistical estimators can be specified through Bayesian vs. Classic (or Frequentist) approaches. In this paper, the resulting Bayesian or Classic disaggregate or aggregate calibration methods are discussed, analysed in detail, and compared from the mathematical point of view. These methods are applied to calibrate Logit choice models for describing path choice behaviour at national scale on a small sample. The Logit choice model can be derived from Random Utility Theory (or be considered an instance of the Bradley–Terry model). Path choice set definition is also discussed, and specialised indicators are used for result comparison. The main contributions of this study concern the use of two different estimation approaches, Bayesian vs. Classic, adopting and introducing some indicators of goodness of estimation. The results of this work, relating to the sample of users adopted, show that the Bayesian approach provides a better estimate than the Classic approach because the calibrated parameters are more stable, the specific constants of the alternatives decrease, and the resulting models show better values of clearly right indicator.

Seeing Cleft Lip from a New Angle: Crowdsourcing to Determine whether Scar Severity or Lip Angle Matters More to the General Public.

Modern cleft lip surgery aims to restore symmetry and create a level, normal-appearing Cupid's bow. However, families' concerns often center on the degree of scarring. The authors hypothesized that the lip angle would be less influential than scar severity in layperson ratings. The authors received institutional review board approval for modifying patients' postoperative photographs to create systematic variations displaying different levels of scarring and lip angle. Each child's resulting composite images were presented in pairs to internet raters using Amazon Mechanical Turk. Users selected the simulated postoperative result they felt to be most normal. The Bradley-Terry model was used to determine raters' preferences between different levels of scarring and lip angle. Four children with primary unilateral cleft lip repair had their postoperative photographs modified (mean age, 1.4 years; mean follow-up, 1.0 years). Twelve-hundred crowdsourced pairwise ratings were collected for each patient (4800 combined ratings). For all four children, raters preferred images with more severe scarring than those with a greater lip angle, suggesting uneven lip angle has a more negative effect on perceived appearance. Online crowdsourcing postoperative lip angle had a significantly greater influence on ratings of normal appearance than does the severity of scarring. Although patients may ask about scars more often, clinically, this study suggests perceptions of a cleft lip repair result may be more likely influenced by the angle of the cleft lip repair.

Comparative Analysis of the Existence and Uniqueness Conditions of Parameter Estimation in Paired Comparison Models

In this paper, paired comparison models with stochastic background are investigated. We focus on the models that allow three options for choice and the parameters are estimated by maximum likelihood method. The existence and uniqueness of the estimator are key issues of the evaluation. In the case of two options, a necessary and sufficient condition is given by Ford in the Bradley–Terry model. We generalize this statement for the set of strictly log-concave distribution. Although in the case of three options the necessary and sufficient condition is not known, there are two different sufficient conditions that are formulated in the literature. In this paper, we generalize them; moreover, we compare these conditions. Their capacities to indicate the existence of the maximum were analyzed using a large number of computer simulations. These simulations support that the new condition indicates the existence of the maximum much more frequently than the previously known ones.

17th International Conference on Malignant Lymphoma, Palazzo dei Congressi, Lugano, Switzerland, 13 - 17 June, 2023.

Background: Emerging evidence suggests the prognostic impact of the tumor microenvironment (TME) in peripheral T-cell lymphomas (PTCLs). To better understand PTCL pathobiology, we performed an integrative multi-omics study to explore the genetic subtypes and the TME signatures of PTCLs, especially nodal T-follicular helper T-cell lymphomas (nTFHLs) and PTCL, not otherwise specified (PTCL-NOS). Method: We performed whole-exome sequencing of 92 nTFHLs and 37 PTCL-NOSs, a total of 129 cases. Ninety-two nTFHLs included 85 angioimmunoblastic T-cell lymphomas (AITLs), four nPTCL with TFH phenotypes, two follicular T-cell lymphomas, and one unclassifiable nTFHL. Genetic drivers were integrated using non-negative matrix factorization consensus clustering (NMF). We estimated the proportion of immune cell fractions from bulk RNA sequencing data (n = 57) using CIBERSORTx and classified the TME signatures using hierarchical clustering. Results: NMF clustering analysis revealed three genetic subtypes within our cohort, except for 10 cases lacking recurrent genetic abnormalities (denoted as cluster 0). C2 was characterized by TP53 mutation, CDKN2A loss, PTEN loss, and chromosomal instability, corresponding to the previously described GATA3-PTCL subtype. C1 and C3 shared TFH-related genomic alterations, but C3 had significantly higher frequencies of chromosome (Chr) 5 gain, Chr 21 gain, RHOA, IDH2, and CD28 mutations. A higher frequency of rash and higher C-reactive protein levels than the other AITLs characterized C3-AITL. Pairwise association analysis of genomic alterations revealed that multiple TET2 mutations, rather than a single TET2 mutation, significantly co-occurred with IDH2 and RHOA mutations. A Bradley-Terry model estimated that the first TET2 and DNMT3A mutations occurred earlier, and the second or later TET2, RHOA, and IDH2 mutations occurred later. Compared to C1, significantly inferior survival was observed in C2 (hazard ratio (HR), 2.52; 95% CI, 1.37–4.63) and C3(HR, 2.14; 95% CI, 1.17–3.89; Figure A). The TME signatures were divided into TME1 (B-cells and follicular helper T-cells), TME2 (activated memory CD4 T-cells, M2 macrophages, and CD8 T-cells), and TME3 (naïve CD4 T-cells and activated mast cells). mTORC1 signaling was enriched in the TME2 signature compared to TME1 and TME3 signatures. The TME2 signature was significantly associated with shorter survival than the TME3 signature (HR: 3.4, 95% CI 1.6–7.5; Figure B). C2 had significantly more TME2 signature than C1 (64.3% vs. 7.7%), while C1 had significantly more TME3 signature than C2 (80.8% vs. 28.6%, p = 0.006; Figure C). Keywords: genomics, epigenomics, and other -omics, microenvironment Encore Abstract—previously submitted to EHA 2023 The research was funded by: Japan Society for the Promotion of Science KAKENHI grants (grant numbers: 19K23879, 22K19451, and 21H02945) and Japan Agency for Medical Research and Development (AMED) grants (20ck0106544 and 21ck0106644)

마르코프 체인 몬테카를로 시뮬레이션과 베이지안 Bradley-Terry 모델을 적용한 배구 경기 결과 예측

As information and communication technology develops, sports data also continues to expand quantitatively and grow qualitatively. Korea"s professional volleyball game also provides various data on game results to satisfy the various needs of spectators or viewers. However, in order to effectively use sports data, it is necessary to extract useful information using appropriate statistical methods. For this reason, many volleyball-related researchers tried to explain and predict volleyball game-related phenomena by applying various statistical models to extract useful information suitable for the purpose through volleyball game records. In other words, the statistical model for predicting the outcome of a volleyball match must reflect the ability of the state team, and the estimated result must also include uncertainty to be reasonable. For this reason, this study estimated the ability parameters of each team by applying the Bayesian Bradley-Terry with MCMC to the volleyball game data to estimate the results including information on uncertainty as well as reflecting the opponent"s ability. Afterwards, the estimated parameters were compared with the estimates of the general Bradley-Terry model and compared with the estimates of the Bayesian Bradley-Terry with MCMC to evaluate and discuss the effectiveness and limitations.

Modeling Intransitivity in Pairwise Comparisons with Application to Baseball Data

The seminal Bradley-Terry model exhibits transitivity, that is, the property that the probabilities of player A beating B and B beating C give the probability of A beating C, with these probabilities determined by a skill parameter for each player. Such transitive models do not account for different strategies of play between each pair of players, which gives rise to intransitivity. Various intransitive parametric models have been proposed but they lack the flexibility to cover the different strategies across n players, with the O ( n 2 ) values of intransitivity modeled using O ( n ) parameters, while they are not parsimonious when the intransitivity is simple. We overcome their lack of adaptability by allocating each pair of players to one of a random number of K intransitivity levels, each level representing a different strategy. Our novel approach for the skill parameters involves having the n players allocated to a random number of A < n distinct skill levels, to improve efficiency and avoid false rankings. Although we may have to estimate up to O ( n 2 ) unknown parameters for ( A , K ) we anticipate that in many practical contexts A + K < n . Our semiparametric model, which gives the Bradley-Terry model when ( A = n − 1 , K = 0 ) , is shown to have an improved fit relative to the Bradley-Terry, and the existing intransitivity models, in out-of-sample testing when applied to simulated and American League baseball data. Supplementary materials for the article areavailable online.

Uncertainty quantification in the Bradley–Terry–Luce model

Abstract The Bradley–Terry–Luce (BTL) model is a benchmark model for pairwise comparisons between individuals. Despite recent progress on the first-order asymptotics of several popular procedures, the understanding of uncertainty quantification in the BTL model remains largely incomplete, especially when the underlying comparison graph is sparse. In this paper, we fill this gap by focusing on two estimators that have received much recent attention: the maximum likelihood estimator (MLE) and the spectral estimator. Using a unified proof strategy, we derive sharp and uniform non-asymptotic expansions for both estimators in the sparsest possible regime (up to some poly-logarithmic factors) of the underlying comparison graph. These expansions allow us to obtain: (i) finite-dimensional central limit theorems for both estimators; (ii) construction of confidence intervals for individual ranks; (iii) optimal constant of $\ell _2$ estimation, which is achieved by the MLE but not by the spectral estimator. Our proof is based on a self-consistent equation of the second-order remainder vector and a novel leave-two-out analysis.

Winning a Tournament According to Bradley-Terry Probability Model

We analyze the chances of winning a tournament under the assumption that the probabilities of winning individual matches follow Bradley-Terry model [2]. We present an exact solution and show a few examples of its use. The examples are from California volleyball tournaments, the round of sixteen in the World Cup and the Champions League, the group stage of the Association of Tennis Professionals tournament, and the volleyball SuperLega in Italy.&#x0D; The computational complexity of the solution grows exponentially fastwith the number of teams and we seek approximations via multivariate Gaussian laws.