Real World Games Look Like Spinning Tops

In this paper we investigate the geometrical properties of real world games (e.g. Tic-Tac-Toe, Go, StarCraft II). We hypothesise that they resemble a spinning top, with the upright axis representing transitive strength, and the radial axis, which corresponds to the number of cycles that exist at a particular transitive strength, representing the non-transitive dimension. 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 policies 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 9 real world two-player zero-sum symmetric games, showing 1) the spinning top structure is revealed and can be easily re-constructed by using a new method of Nash clustering to measure the interaction between transitive and cyclical policy behaviour, and 2) the effect that population size has on the convergence in these games.

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