chess ratings

Chess players are rated by the associations that govern competition in chess, so that players of comparable ability will meet in matches. International players are rated by the rules of the Fédération International des Échecs (FIDE), while within the United States the rules of the U.S. Chess Federation¹ are observed. Both systems derive from a method developed in the 1950's by Arpad Emrick Elo (1903–1992), a professor of physics at Marquette University, but they differ somewhat in detail.

In any good rating system, if two players with the same rating played a large number of games, one would expect each to win half of the games that were not a draw. As the difference in their ratings increases, the probability that the higher-rated player will win increases. In the U. S. system the difference in ratings at which the better player will win 90.9% of the time is arbitrarily set at 400. A player with a rating of 1100 will win 91% of his games with a player with a rating of 700, and a player with a rating of 2000 will win 91% of her games with a player with a rating of 1600.

For any particular match, it should be possible to calculate from the difference in the player's ratings the probability that one of the players will win. Taking “W_e” to be the “win expectancy” and “ΔR” the difference in the players' ratings,

an equation. W sub e equals the reciprocal of 10 to the delta r over 400 plus 1 th power

For example, using this formula, if two players differ by, say, 90 rating points, the probability of a win for the higher-rated player is 0.627, and for the lower-rated player, 0.373. If the results of a series of games bear out this expectation, the players' ratings are “correct,” and shouldn't change. Players' ratings change only when the results of a match are not what the difference in their ratings led one to expect, and the extent of the change in ratings is based on how far off the expectation was.

Although players often calculate the effect a tournament has had on their ratings, they can only approximate the answer, because, for example, the rating their opponent thought he had going into the match may not be the rating he actually had after the Federation made various adjustments. Real ratings, “published ratings,” are all calculated by the Federation centrally.

The U.S. Chess Federation defines three separate rating systems.

One is only for matches conducted by correspondence.
A second is for matches and tournaments in which the players must, on average, move within two minutes or less (events with time controls of G/10 – G/15). These are called “Quick ratings,” and in almost all respects are calculated in the same way as the third system.
The third system applies to regular tournament play (with time controls of G/30 or slower).

The third rating system, that for tournament play, is described below.

Provisional ratings

A player first entering tournament play has no rating, but is assigned matches with rated players on the basis of officials' estimates of the new player's ability. Until the player has completed 20 games, he or she is given a “provisional rating” using the assumptions that

every game lost shows the new player's unknown rating is 400 points less than his opponent's
every game won shows his rating is 400 points higher than his opponent's
every draw shows his rating is the same as his opponent's

For example, imagine the results of the player's first four games in tournament play are:

wins against a player rated 1110, so	1110 + 400 = 1510
loses against a player rated 1785, so	1785 − 400 = 1385
draws against a player rated 1318, so	1318
loses against a player rated 1810, so	1810 − 400 = 1410

The presumptive ratings from all the games the player has played so far are averaged, and 1406 is the result. Provisional ratings are usually published in the form “1406/4”, the “4” after the slash indicating the rating is the result of playing 4 games. The greater the number of games played, the more reliable the rating, but all provisional ratings are notoriously unreliable.

Established rating

Once players have completed 20 games of tournament play, they receive an “established rating,” and thereafter their rating is calculated in a different and more complicated manner. With equally-matched players, as many points as a game adds to the winner's rating are subtracted from the loser's.

In each match, each player scores 1 game point for a win, 0.5 for a draw, and zero for a loss; call this S, for score. (Note that these are not rating points!) There is also a K factor that determines the number of rating points that can change hands as the result of a single match, and that depends to some extent on the player's rating: 32 for ratings from 0–2099, 24 for 2100–2399, and 16 for 2400 and up. (There are also so-called ½K and even ¼K events where the number of points that can change hands is reduced as the fractions suggest, that is, 16, 12 and 8, and 8, 6 and 4 respectively.) A player's new rating as the result of a match can then be calculated from

The number of points that changes hands depends on the probability of the game's result as predicted by the number of points difference in the players' ratings going into the game. The formula applies equally well to a series of matches; for S, sum the scores of the individual games, and for W_e, sum the player's win expectancies for each of the games.

For example, a player has a win expectancy of 0.749 against a player with a rating 190 points below his. Roughly speaking, he is expected to win slightly less than 3 out of 4 games. If the players in fact perform this way their ratings are “correct” and a 4-game match in which the stronger player wins 3 of the games should result in almost no change in either's rating. And here's how it works out:

The better player, his new rating is 2000 + 32 ( (1 + 1 + 1 + 0)- (.749 + .749 + .749 + .749)), which rounds off to 2000. The other player's win expectancy is 0.251; work it out and you will see his rating doesn't change either.

A further complication is that if a player's rating would rise or fall by more than the K factor for the tournament, it is adjusted according to the following schedule:

Pre-tournament rating	Post-tournament rating before adjustment	Rating after adjustment
0–2099	2100–2399	2100 + 0.75 × (post-t − 2100)
2100–2399	0–2099	2100 + 1.33 × (post-t − 2100
2100–2399	2400–3000	2400 + 0.66 × (post-t − 2400)
2400–3000	2100–2399	2400 + 1.5 × (post-t − 2400)

When the Federation calculates the effect a tournament has had on the players' ratings, it calculates first the rating of the previously unrated players, then the provisionally rated players, and finally those with established ratings.

A player's rating can also be changed as the result of individual matches outside of a tournament, but restrictions apply, among them:

the players' ratings before the match cannot differ by more than 400 points
the resulting rating change cannot be more than 50 points
no player's rating can change by more than 200 points during a three-year period solely as a result of individual matches.

Like other sorting systems based on probability and statistics, such as the Scholastic Assessment Tests, small differences between ratings are not significant. The chance that a player with a rating 1 point higher than another's is actually a better player is vanishingly small. For chess ratings, an individual's rating should be interpreted to mean that he or she falls within a range of plus or minus 56 points of the rating given (that is the standard deviation).

The U. S. Chess Federation divides players into classes on the basis of their ratings:

Name	Range	Percentile
Senior Master	above 2399	99
Master	2200–2399	97–98
Expert	2000–2199	89–96
Class A	1800–1999	77–88
Class B	1600–1799	59–76
Class C	1400–1599	41–58
Class D	1200–1399	22–40
Class E	below 1200	1–21

Regardless of the numbers, a player can earn a Master rating only in tournament competition against Masters or Experts, and a Senior Master rating only in tournament competition against Senior Masters and Masters.

In the Fédération International des Échecs's Elo ratings, world champions earn ratings around 2700, persons with ratings around 2600 are considered suitable candidates for matches with world champions, and the area around 2500 is populated by plain old grandmasters. In 1992, Gary Kasparov held an “elo” of 2790, the highest rating ever achieved. Unlike almost all U.S. Chess Federation titles, FIDE's “International Master” and “International Grand-master” titles are bestowed for the life of the awardee.

1.
U. S. Chess Federation
186 Route 9W
New Windsor, NY 12553.