Games - Elo rating system

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	Games - Elo rating system

The Elo rating system is a method for calculating the relative skill levels of players in two-player games such as chess and Go. It is also used as a rating system for competitive multi-player play in a number of computer games. It was originally invented as an improved chess rating system. "Elo" is often written in capital letters (ELO), but it is not an acronym. It is the family name of the system's creator, Árpád Élő (1903-1992), a Hungarian-born American physics professor. Professor Élő spelled his own name "Elo" after he left Hungary, a common anglicization.

A statistical system, not a reward system

Árpád Élő was a master-level chess player and an active participant in the United States Chess Federation (USCF) from its founding in 1939. The USCF used a numerical ratings system, devised by Kenneth Harkness, to allow members to track their individual progress in terms other than tournament wins and losses. The Harkness system was reasonably fair, but in some circumstances gave rise to ratings which many observers considered inaccurate. On behalf of the USCF, Élő devised a new system with a statistical flavor.

It was (and still is) daring to substitute statistical estimation for a system of competitive rewards. Rating systems for many sports award points in accordance with subjective evaluations of the greatness of certain achievements. For example, winning an important golf tournament might be worth five times as many rating points as winning a lesser tournament, and taking third place might be worth half the points of taking first place, etc.

A statistical endeavor, in contrast, postulates a model of some aspect of reality, and seeks to mathematically estimate, based on observation, the variables in that model. Competitors may still feel that they are being rewarded and punished for good and bad results, but the lofty claim of a statistical system is that it estimates real unknowns, and thus mirrors some hidden truth.

Élő's specific assumptions about the nature of real chess performance are open to doubt, but chess fans praise the accuracy of ELO ratings with a fervor unheard of in other sports. For example, professional tennis ratings are purely rewards based on tournament results. (Statistically rating tennis players would be complicated by variables chess doesn't have, particularly the playing surface, but the rating organizations don't even try for predictive accuracy.) As a result, it is routine for tennis fans to consider the higher-rated player an underdog in a given match. In chess the higher-rated player is regarded as the favorite in almost every case.

Élő's rating system model

Élő's central assumption was that the chess "performance" of each player in each game is a normally distributed random variable. Although a player might perform significantly better or worse from one game to the next, Élő assumed that the mean value of the performances of any given player changes only slowly over time. Élő thought of the mean of a player's performance random variable as that player's true skill.

A further assumption is necessary, because chess performance in the above sense is still not measurable. One cannot look at a sequence of moves and say, "That performance is 2039." Performance can only be inferred from wins, draws and losses. Therefore, if a player wins a game, he is assumed to have performed at a higher level than his opponent for that game. Conversely if he loses, he is assumed to have performed at a lower level. If the game is a draw, the two players are assumed to have performed at nearly the same level.

Élő waved his hands at several details of his model. For example, he did not specify exactly how close two performances ought to be to result in a draw rather than a decisive result. And while he thought it likely that each player might have a different standard deviation to his performance, he made a simplifying assumption to the contrary.

To simplify computation even further, Élő proposed a straightforward method of estimating the variables in his model —i.e. the true skill of each player. One could calculate relatively easily, from tables, how many games a player is expected to win based on a comparison of his rating to the ratings of his opponents. If a player won more games than he was expected to win, his rating would be adjusted upward, while if he won fewer games than expected his rating would be adjusted downward. Moreover, that adjustment was to be in exact linear proportion to the number of wins by which the player had exceeded or fallen short of his expected number of wins.

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