# Measuring the Non-Transitivity in Chess

arXiv: https://arxiv.org/abs/2110.11737 · HTML: https://mlsanity.com/paper/2110.11737/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 40 references that are in the map.
- First posted on arXiv 22 Oct 2021.

## Builds on (most cited first, 13 of 13)

- [Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm](https://mlsanity.com/paper/1712.01815/) (2017)
- [OpenSpiel: A Framework for Reinforcement Learning in Games](https://mlsanity.com/paper/1908.09453/) (2019)
- [Open-ended Learning in Symmetric Zero-sum Games](https://mlsanity.com/paper/1901.08106/) (2019)
- [Game-Theoretic Multiagent Reinforcement Learning](https://mlsanity.com/paper/2011.00583/) (2020)
- [Aligning Superhuman AI with Human Behavior: Chess as a Model System](https://mlsanity.com/paper/2006.01855/) (2020)
- [Real World Games Look Like Spinning Tops](https://mlsanity.com/paper/2004.09468/) (2020)
- [Re-evaluating Evaluation](https://mlsanity.com/paper/1806.02643/) (2018)
- [Modelling Behavioural Diversity for Learning in Open-Ended Games](https://mlsanity.com/paper/2103.07927/) (2021)
- [Diverse Auto-Curriculum is Critical for Successful Real-World Multiagent Learning Systems](https://mlsanity.com/paper/2102.07659/) (2021)
- [MALib: A Parallel Framework for Population-based Multi-agent Reinforcement Learning](https://mlsanity.com/paper/2106.07551/) (2021)
- [On the Complexity of Computing Markov Perfect Equilibrium in General-Sum Stochastic Games](https://mlsanity.com/paper/2109.01795/) (2021)
- [Online Double Oracle](https://mlsanity.com/paper/2103.07780/) (2021)
- [Unifying Behavioral and Response Diversity for Open-ended Learning in Zero-sum Games](https://mlsanity.com/paper/2106.04958/) (2021)

## Built on by (most cited first, 10 of 10)

- [On Transforming Reinforcement Learning by Transformer: The Development Trajectory](https://mlsanity.com/paper/2212.14164/) (2022)
- [Offline Pre-trained Multi-Agent Decision Transformer: One Big Sequence Model Tackles All SMAC Tasks](https://mlsanity.com/paper/2112.02845/) (2021)
- [On the Limitations of Elo: Real-World Games, are Transitive, not Additive](https://mlsanity.com/paper/2206.12301/) (2022)
- [Diversifying AI: Towards Creative Chess with AlphaZero](https://mlsanity.com/paper/2308.09175/) (2023)
- [A Game-Theoretic Approach for Improving Generalization Ability of TSP Solvers](https://mlsanity.com/paper/2110.15105/) (2021)
- [Policy Space Response Oracles: A Survey](https://mlsanity.com/paper/2403.02227/) (2024)
- [Ranking Unraveled: Recipes for LLM Rankings in Head-to-Head AI Combat](https://mlsanity.com/paper/2411.14483/) (2024)
- [An Empirical Study on Google Research Football Multi-agent Scenarios](https://mlsanity.com/paper/2305.09458/) (2023)
- [AI in Human-computer Gaming: Techniques, Challenges and Opportunities](https://mlsanity.com/paper/2111.07631/) (2021)
- [Sample-Efficient Policy Space Response Oracles with Joint Experience Best Response](https://mlsanity.com/paper/2602.06599/) (2026)

## Abstract (arXiv)

It has long been believed that Chess is the 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 transitive dimension (e.g., A beats B, B beats C, A beats C), and the radial axis representing the 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 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.