Chess → Rating Calculation

The Elo rating system is a method for calculating the relative skill levels of players in two-player games such as chess. It is named after its creator Arpad Elo, a Hungarian-born American physics professor.

Elo's system replaced earlier systems of competitive rewards with a system based on statistical estimation. 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 an arbitrarily chosen five times as many points as winning a lesser tournament.

A statistical endeavor, by contrast, uses a model that relates the game results to underlying variables representing the ability of each player.

Elo'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, Elo assumed that the mean value of the performances of any given player changes only slowly over time. Elo thought of a player's true skill as the mean of that player's performance random variable.

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.

Elo did not specify exactly how close two performances ought to be to result in a draw as opposed to a win or loss. And while he thought it is 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, Elo 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.

A player's expected score is his probability of winning plus half his probability of drawing. Thus an expected score of 0.75 could represent a 75% chance of winning, 25% chance of losing, and 0% chance of drawing. On the other extreme it could represent a 50% chance of winning, 0% chance of losing, and 50% chance of drawing. The probability of drawing, as opposed to having a decisive result, is not specified in the Elo system. Instead a draw is considered half a win and half a loss.

where:

• Ea — expected score of player A;
• Ra — player A rating;
• Rb — player B rating.

Supposing Player A was expected to score Ea points but actually scored Sa points. The formula for updating his rating is:

where:

• K - K factor, which is 10 for top players (rating 2400 and above), 15 — for players with rating below 2400 and 25 — for new players (first 30 games from receiving FIDE rating);
• Sa — player A scored points (1 point for victory, 0,5 — draw and 0 — for losing);
• R'a — player A updated score.