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.