The ELO rating system in chess is a means of comparing the relative strengths of chess players, devised by Professor Arpad Elo. Players gain or lose rating points depending on the ELO rating of their opponents. If a player wins a game of chess in a rated tournament, they gain a number of rating points that increases in proportion to the difference between their rating and their opponent's rating.

The usefulness of the system lies in the conversion of these ratings into winning or losing probabilities in a tournament situation, which allows tournaments to be sectioned and qualifying thresholds to be set. Also, chess titles are awarded on the basis of reaching certain ELO rating levels.

The central statistical assumption of the ELO system is that any player's tournament performances, spread over a long enough career, will follow a normal distribution. A detailed description of the formulae and theory behind the system can be found at http://home.clear.net.nz/pages/petanque/ratings/descript.htm

A chess rating is a number that, in essence, provides a numerical comparison between a particular player and other chess players. The number is based upon the player's performance in tournaments based upon the performance against a sum of each specific opponent and that opponent's rating. This system, in the end, allows players to judge their chess strength relative to other rated chess players.

The method of chess ratings is very general and can also be applied to other games or sports in which there are a large number of individuals competing. Besides chess, for instance, Magic: the Gathering uses a nearly identical system. Having each player play each other player under a variety of conditions is logistically impossible, so in order to have a fair way to grade players, a numerical scheme that constantly compares player performances is necessary, and thus the chess rating system was developed by FIDE and USCF.

The chess rating system is calculated as follows. When a player first becomes registered with a governing body of the game, he or she is assigned a base rating of 1600. This is considered to be the average score of all players, so you start off at the fiftieth percentile among all rated players (in theory; in practice, many players with lower ratings quit the game, so among active players, your percentile is likely much lower).

Once you have your rating and the rating of your opponent, you then calculate your win expectancy (We) against that opponent using the following formula:

```               1                We = win expectancy
We = --------------------       R1 = rating of the player
10^((R2-R1)/400) + 1       R2 = rating of the opponent
```

Given the win expectancy, then the result of the match is taken into consideration, and each player is awarded a score. A win gets a score of 1, a draw gets a score of 0.5, and a loss gets a score of 0. Also, each match is given a particular K value. K is a number that tells how many points are at stake in a given match; more important matches have a higher K value. Common K values are 24 and 32 for most low-level tournaments.

We then use the score in the match (W), the old rating (Ro), the K value, and the win expectancy (We) and combine them all together to calculate the new rating (Rn), as shown below:

```Rn = Ro + (K * (W - We))
```

The rating is recalculated at the end of each match.

Let's take an example. Let's say we have a new player, Bobby. He's given a rating of 1600, and his first match is against Bill, who has a rating of 1700, in a tournament with a K value of 24. Let's calculate the We:

```                1
We = ------------------------ = 0.359935
10^((1700-1600)/400) + 1
```

Now, let's say Bobby beats Bill. His new rating is as follows:

```Rn = 1600 + (24 * (1 - 0.359935)) = 1615.36 = 1615
```

So after Bobby's first big win, he now has a rating of 1615. Now, let's see how Bill's rating changes due to the big loss:

```                1
We = ------------------------ = 0.640065
10^((1600-1700)/400) + 1

Rn = 1700 + (24 * (0 - 0.640065)) = 1684.65 = 1685
```

Given enough people and enough matches, the ratings eventually provide a rather accurate ranking of how good a player is in comparison to his or her competition. Often, ratings are used to rank the players in comparison to one another to determine tournament seedings and invitations.

The Elo rating system is a numerical rating system in chess to compare the performance of individual players. It is a common misconception that the letters "ELO" in the Elo-rating system are some sort of abbreviation; the system was named after the Hungarian-American Physics Professor Arpad Elo.

Chess was only one of the many hobbies of Dr. Elo, although he was quite a respected player at the Master Level. He won over forty tournaments, including eight Wisconsin State Championships. But Dr. Elo was also involved with the chess community in other ways; he was the president of the (old) American Chess Federation from 1935 to 1937, and he was a co-founder of the United States Chess Federation (USCF) in 1939.

Before the adoption of the Elo rating system, there were several other rating systems in use, but they were not considered to be very accurate. The USCF was using a rating system developed by Kenneth Harkness. In this system, 1500 points marked an average player; 2000 points a strong club player and 2500 points a grandmaster player. Dr. Elo more or less retained the existing level-range, but he provided a much sounder statistical basis for comparing the individual player scores.

The Elo rating system was adopted by the USCF in 1960, and in 1970 by the World Chess Federation, FIDE. Until 1980, Dr. Elo was in charge of all the calculating all the ratings for FIDE, using nothing more than a Hewlett-Packard calculator. The concept of the Elo ratings proved to be quite useful, and it has been adopted to other sports as well (e.g. tennis, golf)

The Elo rating is based on the statistical concept of win expectancy. The outcome of a chess game (or any sporting event) is not a constant, but it exhibits a certain distribution around an average (think of a an athlete competing in the long jump; not every jump will be the same distance). The Elo rating number represents a certain probability for a player to win against another player. Or to be more precise, the difference in Elo ratings between two players is a measure of the expected outcome of a match between the two.

The concept is best explained with an example; Garry Kasparov's current Elo rating is 2838. Nigel Short's Elo rating is 2675. The difference in Elo ratings is 2838-2675=163 Elo points. This difference corresponds to a win expectancy of 72% for Kasparov. If Short and Kasparov would play a match consisting of 10 games, the expected outcome of the match would be close to 7-3 in favor of Kasparov (7.2-2.8 to be exact)

The win expectancies for the Elo rating were designed to follow the Gaussian Distribution. Every rated player has an Elo number that represents an average playing strength, with an associated (but fixed) standard deviation. The win expectancies as a function of Elo difference points can be found in the following table.

```Win expectancies (Exp.) as a function of Elo difference points (Diff.)
between two rated players
---------------------------------------------------------------
Diff.  Exp. |   Diff.   Exp. |  Diff.   Exp. |  Diff.   Exp.
---------------------------------------------------------------
0-3   .50  |   92-98   .63  | 198-206  .76  | 345-357  .89
4-10  .51  |   99-106  .64  | 207-215  .77  | 358-374  .90
11-17  .52  |  107-113  .65  | 216-225  .78  | 375-391  .91
18-25  .53  |  114-121  .66  | 226-235  .79  | 392-411  .92
26-32  .54  |  122-129  .67  | 236-245  .80  | 412-432  .93
33-39  .55  |  130-137  .68  | 246-256  .81  | 433-456  .94
40-46  .56  |  138-145  .69  | 257-267  .82  | 457-484  .95
47-53  .57  |  146-153  .70  | 268-278  .83  | 485-517  .96
54-61  .58  |  154-162  .71  | 279-290  .84  | 518-559  .97
62-68  .59  |  163-170  .72  | 291-302  .85  | 560-619  .98
69-76  .60  |  171-179  .73  | 303-315  .86  | 620-735  .99
77-83  .61  |  180-188  .74  | 316-328  .87  | > 735   1.0
84-91  .62  |  189-197  .75  | 329-344  .88  |
--------------------------------------------------------------
```

Of course chess tournaments and matches usually don't end up exactly like the statistics would predict. Otherwise there wouldn't be any point in playing the matches in the first place! This is where the adjustments to the Elo ratings come into play: players are rated on the outcome of their matches against other players. Getting back to the example of the Kasparov-Short match; suppose these players finish the match, with an outcome of 6-4 for Kasparov. Even though Kasparov won the match, he didn't score as high as was predicted by the Elo difference. As a result, Kasparov's Elo rating will drop. And even though Short has lost the Match, his Elo rating will increase.

In a single match between two players, the rating change is:

ΔR = K (W - We)

ΔR is the rating change for each player. K is called the Development Coefficient; this factor determines how much an Elo rating is adjusted, based on the outcome of the match. The value for K=25 for new players (played in matches with a total of less than 30 games), K=15 for players with an Elo rating below 2400, and K=10 for players with an Elo rating at or above 2400. W is the score achieved, and WE is the expected score.

In the Kasparov-Short example, Kasparov scored only 6 points (W), where he was expected to win by 7.2 points (We). Kasparov's rating changes by:

ΔR = 10 (6 - 7.2) = -12 points

Similarly, Short was expected to lose by 2.8 points, thus his rating changes by:

ΔR = 10 (4 - 2.8) = +12 points

So after the match, Kasparov's Elo rating drops to 2826 points, and Short's Elo rating increases to 2687 points. Please note that there are additional regulations, for instance for playing against unrated players, and for tournament play. Also note that the FIDE updates player ratings every six months, so the outcome of one single match will not affect the Elo rating immediately. Of course, the Elo ratings do not supply any information on the individual aspects of a chess player's capabilities; it doesn't rate the individual style of a player, or how well his defense and end game are. It was in fact Dr. Arpad Elo himself who recognized the limitations of any rating system and the difficulties to objectively quantify player strength:

Often people who are not familiar with the nature and limitations of statistical methods tend to expect too much of the rating system. Ratings provide merely a comparison of performances, no more and no less. The measurement of the performance of an individual is always made relative to the performance of his competitors and both the performance of the player and of his opponents are subject to much the same random fluctuations. The measurement of the rating of an individual might well be compared with the measurement of the position of a cork bobbing up and down on the surface of agitated water with a yard stick tied to a rope and which is swaying in the wind. -- Dr. Arpad Elo, Chess Life (1962).

Nevertheless, the Elo rating system has proved to be a relatively accurate measure for predicting the outcome of chess matches, based on a quantified figure of the strenght of individual chess players.

factual sources:
http://handbook.fide.com/handbook.cgi?level=B&level=02&level=10& (official FIDE rules)
http://www.bio.vu.nl/vakgroepen/microb/reijnders/elo.html