It was recently observed that Elo ratings fail at preserving transitive relations among strategies and therefore cannot correctly extract the transitive component of a game. We provide a characterization of transitive games as a weak variant of ordinal potential games and show that Elo ratings actually do preserve transitivity when computed in the right space, using suitable invertible mappings. Leveraging this insight, we introduce a new game decomposition of an arbitrary game into transitive and cyclic components that is learnt using a neural network-based architecture and that prioritises capturing the sign pattern of the game, namely transitive and cyclic relations among strategies. We link our approach to the known concept of sign-rank, and evaluate our methodology using both toy examples and empirical data from real-world games.

Real-world competitive games, such as chess, go, or StarCraft II, rely on Elo models to measure the strength of their players. Since these games are not fully transitive, using Elo implicitly assumes they have a strong transitive component that can correctly be identified and extracted. In this study, we investigate the challenge of identifying the strength of the transitive component in games. First, we show that Elo models can fail to extract this transitive component, even in elementary transitive games. Then, based on this observation, we propose an extension of the Elo score: we end up with a disc ranking system that assigns each player two scores, which we refer to as skill and consistency. Finally, we propose an empirical validation on payoff matrices coming from real-world games played by bots and humans.