Modeling Intransitivity in Pairwise Comparisons with Application to Baseball Data

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