Modelling intransitivity in pairwise comparisons with application to baseball data

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 {\it 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 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 $O(n^2)$ unknown parameters for $(A,K)$ we anticipate that in many practical contexts $A+K < n$. Using a Bayesian hierarchical model, $(A,K)$ are treated as unknown, and inference is conducted via a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. 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. Supplementary materials for the article are available online.