  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.
