  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.
