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Here, we derive the mathematics behind the Elo rating system, and talk about how they're used in Chess, Pokémon, and various digital video games. We also briefly discuss alternatives to Elo, such as Glicko, league systems (as in League of Legends or other games in its genre), and Smogon's GXE, and the advantages and distadvantages these systems may have. References: Elo, A. "The Proposed USCF Rating System, Its Development Theory, and Applications", (1967): https://uscf1-nyc1.aodhosting.com/CL-... Elo, A. "The USCF Rating System, A Progress Report", (1960): https://uscf1-nyc1.aodhosting.com/CL-... "FIDE Rating Regulations effective from 1 July 2014": https://web.archive.org/web/201407010... Zarel "Everything You Ever Wanted to Know About Ratings", (2016): https://www.smogon.com/forums/threads... Spirit of the Law "How Elo Ratings Are Calculated", (2017): • How Elo Ratings Are Calculated Glickman, M. "The Glicko system": http://www.glicko.net/glicko/glicko.pdf Glickman, M. "Example of the Glicko-2 system", (2022): http://www.glicko.net/glicko/glicko2.pdf X-Act "GXE (GLIXARE): A Much Better Way of Estimating A Player's Overall Rating Than Shoddy's CRE", (2009): https://www.smogon.com/forums/threads... Image Credits: Manim Community Edition: https://www.manim.community/ League of Legends logo, by Riot Games, Public domain, https://commons.wikimedia.org/w/index... Chess piece, by Cburnett, CC BY-SA 3.0, https://commons.wikimedia.org/w/index... Pokémon Showdown logo, by unknown, Fair use, https://play.pokemonshowdown.com/ Elo, by unknown, Fair use, https://en.wikipedia.org/w/index.php?... ELO, by TyrystorELO, Public domain, https://commons.wikimedia.org/w/index... Magnus Carlsen, by Этери Кублашвили (Eleri Kublashvili), CC BY-SA 3.0, https://commons.wikimedia.org/w/index... Bobby Fischer, by Bert Verhoeff for Anefo - Dutch National Archives, CC0, https://commons.wikimedia.org/w/index... Special Thanks: Mishi, Lucia, and Alessandro for participating in the inaugural Competition to Highlight Elo Systems Simply Michelle for coming up with a non-monotonically-increasing function