TY - JOUR

T1 - Modeling Intransitivity in Pairwise Comparisons with Application to Baseball Data

AU - Spearing, Harry

AU - Tawn, Jonathan

AU - Irons, David

AU - Paulden, Tim

PY - 2023/10/2

Y1 - 2023/10/2

N2 - 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 (Formula presented.) values of intransitivity modeled using (Formula presented.) 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 (Formula presented.) distinct skill levels, to improve efficiency and avoid false rankings. Although we may have to estimate up to (Formula presented.) unknown parameters for (Formula presented.) we anticipate that in many practical contexts (Formula presented.). Our semiparametric model, which gives the Bradley-Terry model when (Formula presented.), 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.

AB - 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 (Formula presented.) values of intransitivity modeled using (Formula presented.) 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 (Formula presented.) distinct skill levels, to improve efficiency and avoid false rankings. Although we may have to estimate up to (Formula presented.) unknown parameters for (Formula presented.) we anticipate that in many practical contexts (Formula presented.). Our semiparametric model, which gives the Bradley-Terry model when (Formula presented.), 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.

KW - Baseball

KW - Bayesian hierarchical modeling

KW - Bradley-Terry

KW - Clustering

KW - Intransitivity

KW - Pairwise Comparisons

KW - Ranking

KW - Reversible jump Markov chain Monte Carlo

KW - Tournament structure

U2 - 10.1080/10618600.2023.2177299

DO - 10.1080/10618600.2023.2177299

M3 - Journal article

VL - 32

SP - 1383

EP - 1392

JO - Journal of Computational and Graphical Statistics

JF - Journal of Computational and Graphical Statistics

SN - 1061-8600

IS - 4

ER -