Considerable effort has also been devoted to developing highly accurate models for forecasting match outcomes (Wunderlich and Memmert, 2021). Tennis is a sport well-suited to predictive modelling, with dense tournament schedules generating extensive historical data. The official ranking systems of the Association of Tennis Professionals (ATP) and Women's Tennis Association (WTA) have been shown to exhibit some predictive power for match outcomes (Clarke and Dyte, 2000; Klaassen and Magnus, 2003), but there are notable limitations: for example, ranking points accumulate over a 52-week period, without decay, which can mask recent changes in player form; while match-specific factors, such as surface type, tournament progression difficulty, and margin of victory in individual matches, are overlooked. Some well-known methods have been applied to tennis and modified to accommodate these factors, such as a Bradley-Terry model with surface-specific adjustments (McHale and Morton, 2011) or Elo rating systems that incorporate margin of victory (Kovalchik, 2020; Angelini et al., 2022). Bookmakers are considered the most accurate predictors of match outcomes (Kovalchik, 2016), with sophisticated models that adjust odds based on betting patterns and proprietary methods. Yet, despite the multi-billion dollar betting industry, one limitation that persists is the poor consideration of intransitivity (van Ours, 2025). Intransitivity is analogous to rock-paper-scissors. In tennis, it occurs where player A tends to defeat B, B defeats C, yet C defeats A, violating the assumption of transitive dominance.

In competitive games, strength ratings like Elo are widely used to quantify player skill and support matchmaking by accounting for skill disparities better than simple win rate statistics. However, scalar ratings cannot handle complex intransitive relationships, such as counter strategies seen in Rock-Paper-Scissors. To address this, recent work introduced Neural Rating Table and Neural Counter Table, which combine scalar ratings with discrete counter categories to model intransitivity. While effective, these methods rely on neural network training and cannot perform real-time updates. In this paper, we propose an online update algorithm that extends Elo principles to incorporate real-time learning of counter categories. Our method dynamically adjusts both ratings and counter relationships after each match, preserving the explainability of scalar ratings while addressing intransitivity. Experiments on zero-sum competitive games demonstrate its practicality, particularly in scenarios without complex team compositions.

Most statistical models for pairwise comparisons, including the Bradley-Terry (BT) and Thurstone models and many extensions, make a relatively strong assumption of stochastic transitivity. This assumption imposes the existence of an unobserved global ranking among all the players/teams/items and monotone constraints on the comparison probabilities implied by the global ranking. However, the stochastic transitivity assumption does not hold in many real-world scenarios of pairwise comparisons, especially games involving multiple skills or strategies. As a result, models relying on this assumption can have suboptimal predictive performance. In this paper, we propose a general family of statistical models for pairwise comparison data without a stochastic transitivity assumption, substantially extending the BT and Thurstone models. In this model, the pairwise probabilities are determined by a (approximately) low-dimensional skew-symmetric matrix. Likelihood-based estimation methods and computational algorithms are developed, which allow for sparse data with only a small proportion of observed pairs. Theoretical analysis shows that the proposed estimator achieves minimax-rate optimality, which adapts effectively to the sparsity level of the data. The spectral theory for skew-symmetric matrices plays a crucial role in the implementation and theoretical analysis. The proposed method's superiority against the BT model, along with its broad applicability across diverse scenarios, is further supported by simulations and real data analysis.

Intransitivity is a critical issue in pairwise preference modeling. It refers to the intransitive pairwise preferences between a group of players or objects that potentially form a cyclic preference chain and has been long discussed in social choice theory in the context of the dominance relationship. However, such multifaceted intransitivity between players and the corresponding player representations in high dimensions is difficult to capture. In this paper, we propose a probabilistic model that jointly learns each player's d-dimensional representation (d>1) and a dataset-specific metric space that systematically captures the distance metric in Rd over the embedding space. Interestingly, by imposing additional constraints in the metric space, our proposed model degenerates to former models used in intransitive representation learning. Moreover, we present an extensive quantitative investigation of the vast existence of intransitive relationships between objects in various real-world benchmark datasets. To our knowledge, this investigation is the first of this type. The predictive performance of our proposed method on different real-world datasets, including social choice, election, and online game datasets, shows that our proposed method outperforms several competing methods in terms of prediction accuracy.

Positions of chess players in intransitive (rock-paper-scissors) relations are considered. Namely, position A of White is preferable (it should be chosen if choice is possible) to position B of Black, position B of Black is preferable to position C of White, position C of White is preferable to position D of Black, but position D of Black is preferable to position A of White. Intransitivity of winningness of positions of chess players is considered to be a consequence of complexity of the chess environment -- in contrast with simpler games with transitive positions only. The space of relations between winningness of positions of chess players is non-Euclidean. The Zermelo-von Neumann theorem is complemented by statements about possibility vs. impossibility of building pure winning strategies based on the assumption of transitivity of positions of chess players. Questions about the possibility of intransitive positions of players in other positional games are raised.

Playing human games such as chess and Go has long been considered to be a major benchmark of human capabilities. Computer programs have become robust chess players and, since the late 1990s, have been able to beat even the best human chess champions; though, for a long time, computers were unable to beat expert Go players -- the game of Go has proven to be especially difficult for computers. However, in 2016, a new program called AlphaGo finally won a victory over a human Go champion, only to be beaten by its subsequent versions (AlphaGo Zero and AlphaZero). AlphaZero proceeded to beat the best computers and humans in chess, shogi and Go, including all its predecessors from the Alpha family [1]. Core to AlphaZero's success is its use of a deep neural network, trained through reinforcement learning, as a powerful heuristic to guide a tree search algorithm (specifically Monte Carlo Tree Search). The recent successes of machine learning are good reason to consider the limitations of learning algorithms and, in a broader sense, the limitations of AI. In the context of a particular competition (or'game'), a natural question to ask is whether an absolute winner AI might exist -- one that, given sufficient resources, will always achieve the best possible outcome.