Real World Games Look Like Spinning Tops

This paper investigates the geometrical properties of real world games (e.g. Tic-Tac-Toe, Go, StarCraft II). We hypothesise that their geometrical structure resembles a spinning top, with the upright axis representing transitive strength, and the radial axis representing the non-transitive dimension, which corresponds to the number of cycles that exist at a particular transitive strength. We prove the existence of this geometry for a wide class of real world games by exposing their temporal nature. Additionally, we show that this unique structure also has consequences for learning - it clarifies why populations of strategies are necessary for training of agents, and how population size relates to the structure of the game. Finally, we empirically validate these claims by using a selection of nine real world two-player zero-sum symmetric games, showing 1) the spinning top structure is revealed and can be easily reconstructed by using a new method of Nash clustering to measure the interaction between transitive and cyclical strategy behaviour, and 2) the effect that population size has on the convergence of learning in these games.