  It has long been believed that Chess is the \emph{Drosophila} of Artificial
Intelligence (AI). Studying Chess can productively provide valid knowledge
about complex systems. Although remarkable progress has been made on solving
Chess, the geometrical landscape of Chess in the strategy space is still
mysterious. Judging on AI-generated strategies, researchers hypothesised that
the strategy space of Chess possesses a spinning top geometry, with the upright
axis representing the \emph{transitive} dimension (e.g., A beats B, B beats C,
A beats C), and the radial axis representing the \emph{non-transitive}
dimension (e.g., A beats B, B beats C, C beats A). However, it is unclear
whether such a hypothesis holds for real-world strategies. In this paper, we
quantify the non-transitivity in Chess through real-world data from human
players. Specifically, we performed two ways of non-transitivity
quantifications -- Nash Clustering and counting the number of
Rock-Paper-Scissor cycles -- on over one billion match data from Lichess and
FICS. Our findings positively indicate that the strategy space occupied by
real-world Chess strategies demonstrates a spinning top geometry, and more
importantly, there exists a strong connection between the degree of
non-transitivity and the progression of a Chess player's rating. In particular,
high degrees of non-transitivity tend to prevent human players from making
progress on their Elo rating, whereas progressions are easier to make at the
level of ratings where the degree of non-transitivity is lower. Additionally,
we also investigate the implication of the degree of non-transitivity for
population-based training methods. By considering \emph{fixed-memory Fictitious
Play} as a proxy, we reach the conclusion that maintaining large-size and
diverse populations of strategies is imperative to training effective AI agents
in solving Chess types of games.
