The Mathematics of Elo Ratings

Calculating the relative skill of players in zero-sum games

Chess-players are rated according to how well they perform when facing other players. For example, reigning World Chess Champion and fellow Norwegian Magnus Carlsen is as of September 2019 rated by the International Chess Federation FIDE (Fédération Internationale des Échecs) at 2876 points, the highest in the world yet six points shy of his own peak rating of 2882 in 2014 when he was 24 years old.

The rating system used by FIDE and nearly all other chess federations is called the Elo rating system. It was introduced in 1960 by the United States Chess Federation at the suggestion of a Hungarian-American chess master whose name it has carried since, Arpad Elo. Elo’s system tracks the relative performance of players in zero-sum games such as chess. Based on the assumptions of

  1. Performance behaving as a random variable

  2. Performance conforming to a bell-curve shaped probability distribution

  3. Mean performance of players increasing slowly

the system allows one to sort a group of players by relative performance, and so make educated, probabilistic guesses about expected outcomes of games and variation in player performance over time. Continue reading


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