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.

翻译：经典Bradley-Terry模型具有传递性，即选手A战胜B的概率与B战胜C的概率共同决定了A战胜C的概率，且这些概率由每位选手的技能参数决定。此类传递模型无法解释不同选手对之间存在的不同策略，由此引发{\it非传递性}。现有多种非传递参数化模型被提出，但它们在涵盖$n$名选手间的不同策略方面缺乏灵活性——用$O(n)$个参数建模$O(n^2)$个非传递性取值，且当非传递性结构简单时不够简约。我们通过将每对选手分配至随机数量的$K$个非传递性水平（每个水平代表一种不同策略）来克服这一适应性不足。在技能参数建模方面，我们提出新方法：将$n$名选手分配至随机数量的$A < n$个不同技能水平，以提高效率并避免虚假排名。尽管对于$(A,K)$可能需要估计高达$O(n^2)$个未知参数，但我们预期在实际场景中$A+K < n$。通过贝叶斯分层模型，将$(A,K)$视为未知参数，并采用可逆跳转马尔可夫链蒙特卡洛（RJMCMC）算法进行推断。该半参数模型在$(A=n-1, K=0)$时退化为Bradley-Terry模型，在仿真数据和美国联盟棒球数据的样本外测试中，其拟合优度优于Bradley-Terry模型及现有非传递性模型。本文补充材料可在线获取。