{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T20:53:39Z","timestamp":1771102419362,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T00:00:00Z","timestamp":1651104000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we quantify the non-transitivity in chess using human game data. Specifically, we perform non-transitivity quantification in two ways\u2014Nash clustering and counting the number of rock\u2013paper\u2013scissor cycles\u2014on over one billion matches from the Lichess and FICS databases. Our findings indicate that the strategy space of real-world chess strategies has a spinning top geometry and that there exists a strong connection between the degree of non-transitivity and the progression of a chess player\u2019s rating. Particularly, high degrees of non-transitivity tend to prevent human players from making progress in their Elo ratings. We also investigate the implications of non-transitivity for population-based training methods. By considering fixed-memory fictitious play as a proxy, we conclude that maintaining large and diverse populations of strategies is imperative to training effective AI agents for solving chess.<\/jats:p>","DOI":"10.3390\/a15050152","type":"journal-article","created":{"date-parts":[[2022,4,28]],"date-time":"2022-04-28T12:06:01Z","timestamp":1651147561000},"page":"152","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Measuring the Non-Transitivity in Chess"],"prefix":"10.3390","volume":"15","author":[{"given":"Ricky","family":"Sanjaya","sequence":"first","affiliation":[{"name":"Department of Computer Science, University College London, London WC1E 6EA, UK"}]},{"given":"Jun","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University College London, London WC1E 6EA, UK"}]},{"given":"Yaodong","family":"Yang","sequence":"additional","affiliation":[{"name":"Institute for Artificial Intelligence, Peking University, Beijing 100191, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1177\/0306312711424596","article-title":"Is chess the drosophila of artificial intelligence? 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