Final published version
Licence: CC BY: Creative Commons Attribution 4.0 International License
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Modelling intransitivity in pairwise comparisons with application to baseball data
AU - Spearing, Harry
AU - Tawn, Jonathan
AU - Irons, David
AU - Paulden, Tim
PY - 2023/3/23
Y1 - 2023/3/23
N2 - The seminal Bradley-Terry model exhibits transitivity, i.e., 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(n2) values of intransitivity modelled using O(n) parameters, whilst 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(n2) unknown parameters for (A;K) we anticipate that in many practical contexts A + K < n. Our semi-parametric 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.
AB - The seminal Bradley-Terry model exhibits transitivity, i.e., 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(n2) values of intransitivity modelled using O(n) parameters, whilst 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(n2) unknown parameters for (A;K) we anticipate that in many practical contexts A + K < n. Our semi-parametric 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.
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
JO - Journal of Computational and Graphical Statistics
JF - Journal of Computational and Graphical Statistics
SN - 1061-8600
ER -