# Real World Games Look Like Spinning Tops

arXiv: https://arxiv.org/abs/2004.09468 · HTML: https://mlsanity.com/paper/2004.09468/

## Key facts

- Cited by 66 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Apr 2020.

## Built on by (most cited first, 30 of 66)

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- [Modelling Behavioural Diversity for Learning in Open-Ended Games](https://mlsanity.com/paper/2103.07927/) (2021)
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- [Adversarial Policies Beat Superhuman Go AIs](https://mlsanity.com/paper/2211.00241/) (2022)
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- [Diverse Auto-Curriculum is Critical for Successful Real-World Multiagent Learning Systems](https://mlsanity.com/paper/2102.07659/) (2021)
- [On the Limitations of Elo: Real-World Games, are Transitive, not Additive](https://mlsanity.com/paper/2206.12301/) (2022)
- [Quantifying the effects of environment and population diversity in multi-agent reinforcement learning](https://mlsanity.com/paper/2102.08370/) (2021)
- [General Intelligence Requires Rethinking Exploration](https://mlsanity.com/paper/2211.07819/) (2022)
- [TiZero: Mastering Multi-Agent Football with Curriculum Learning and Self-Play](https://mlsanity.com/paper/2302.07515/) (2023)
- [A social path to human-like artificial intelligence](https://mlsanity.com/paper/2405.15815/) (2024)
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- [Online Double Oracle](https://mlsanity.com/paper/2103.07780/) (2021)
- [Evolving Diverse Red-team Language Models in Multi-round Multi-agent Games](https://mlsanity.com/paper/2310.00322/) (2023)
- [Policy Space Diversity for Non-Transitive Games](https://mlsanity.com/paper/2306.16884/) (2023)
- [Diversifying AI: Towards Creative Chess with AlphaZero](https://mlsanity.com/paper/2308.09175/) (2023)
- [Unifying Behavioral and Response Diversity for Open-ended Learning in Zero-sum Games](https://mlsanity.com/paper/2106.04958/) (2021)
- [A Game-Theoretic Approach for Improving Generalization Ability of TSP Solvers](https://mlsanity.com/paper/2110.15105/) (2021)
- [NeuPL: Neural Population Learning](https://mlsanity.com/paper/2202.07415/) (2022)
- [Complex Momentum for Optimization in Games](https://mlsanity.com/paper/2102.08431/) (2021)
- [Measuring the Non-Transitivity in Chess](https://mlsanity.com/paper/2110.11737/) (2021)
- [Towards Multi-Agent Reinforcement Learning driven Over-The-Counter Market Simulations](https://mlsanity.com/paper/2210.07184/) (2022)
- [Policy Space Response Oracles: A Survey](https://mlsanity.com/paper/2403.02227/) (2024)
- [Efficient Policy Space Response Oracles](https://mlsanity.com/paper/2202.00633/) (2022)

## Abstract (arXiv)

This paper investigates the geometrical properties of real world games (e.g. Tic-Tac-Toe, Go, StarCraft II). We hypothesise that their geometrical structure 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, 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 re-constructed 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 in these games.