The Elo rating system, a logistic model for ranking players in competitive games, is shown to possess a hidden mathematical structure akin to relativistic rapidity in special relativity. By expressing the expected score as a hyperbolic tangent of scaled rating differences, we derive a quantity V whose transitivity obeys Einstein's velocity addition law. This formal analogy reveals a deep connection between a probabilistic ranking system and the hyperbolic geometry of relativistic kinematics, offering a novel mathematical reinterpretation of Elo ratings.
Fabrice PFAFF (Fri,) studied this question.