Deciphering the Chess ELO: The Language of Chess Skill : Covered From WikiPedia
The Elo rating system is a method for calculating the relative skill levels of players in zero-sum games such as chess. It is named after its creator Arpad Elo, a Hungarian-American physics professor.
The Elo system was invented as an improved chess-rating system over the previously used Harkness system,but is also used as a rating system in association football, American football, baseball, basketball, pool, various board games and esports, and more recently large language models.
The difference in the ratings between two players serves as a predictor of the outcome of a match. Two players with equal ratings who play against each other are expected to score an equal number of wins. A player whose rating is 100 points greater than their opponent's is expected to score 64%; if the difference is 200 points, then the expected score for the stronger player is 76%.
History
Arpad Elo was a master-level chess player and an active participant in the United States Chess Federation (USCF) from its founding in 1939.[4] 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, Elo devised a new system with a more sound statistical basis.[5] At about the same time, György Karoly and Roger Cook independently developed a system based on the same principles for the New South Wales Chess Association.
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 derive a number to represent that player's skill. Performance can only be inferred from wins, draws and losses. Therefore, if a player wins a game, they are assumed to have performed at a higher level than their opponent for that game. Conversely, if the player loses, they are 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. Actually, there is a probability of a draw that is dependent on the performance differential, so this latter is more of a confidence interval than any deterministic frontier. And while he thought it was likely that players might have different standard deviations to their performances, 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 players would be expected to win based on comparisons of their ratings to those of their opponents. The ratings of a player who won more games than expected would be adjusted upward, while those of a player who won fewer than expected would be adjusted downward. Moreover, that adjustment was to be in linear proportion to the number of wins by which the player had exceeded or fallen short of their expected number.
From a modern perspective, Elo's simplifying assumptions are not necessary because computing power is inexpensive and widely available. Several people, most notably Mark Glickman, have proposed using more sophisticated statistical machinery to estimate the same variables. On the other hand, the computational simplicity of the Elo system has proven to be one of its greatest assets. With the aid of a pocket calculator, an informed chess competitor can calculate to within one point what their next officially published rating will be, which helps promote a perception that the ratings are fair.
Implementing Elo's scheme
The USCF implemented Elo's suggestions in 1960,[8] and the system quickly gained recognition as being both fairer and more accurate than the Harkness rating system. Elo's system was adopted by the World Chess Federation (FIDE) in 1970.[9] Elo described his work in detail in The Rating of Chessplayers, Past and Present, first published in 1978.
The development of the Percentage Expectancy Table (table 2.11) is described in more detail by Elo as follows:
The normal probabilities may be taken directly from the standard tables of the areas under the normal curve when the difference in rating is expressed as a z score. Since the standard deviation σ of individual performances is defined as 200 points, the standard deviation σ' of the differences in performances becomes σ√2 or 282.84. The z value of a difference then is D / 282.84. This will then divide the area under the curve into two parts, the larger giving P for the higher rated player and the smaller giving P for the lower rated player.
For example, let D = 160. Then z = 160 / 282.84 = .566. The table gives .7143 and .2857 as the areas of the two portions under the curve. These probabilities are rounded to two figures in table 2.11.
The table is actually built with standard deviation 200(10/7) as an approximation for 200√2.[citation needed]
The normal and logistic distributions are, in a way, arbitrary points in a spectrum of distributions which would work well. In practice, both of these distributions work very well for a number of different games.
Different ratings systems
The phrase "Elo rating" is often used to mean a player's chess rating as calculated by FIDE. However, this usage may be confusing or misleading because Elo's general ideas have been adopted by many organizations, including the USCF (before FIDE), many other national chess federations, the short-lived Professional Chess Association (PCA), and online chess servers including the Internet Chess Club (ICC), Free Internet Chess Server (FICS), Lichess, Chess.com, and Yahoo! Games. Each organization has a unique implementation, and none of them follows Elo's original suggestions precisely.
Instead one may refer to the organization granting the rating. For example: "As of August 2002, Gregory Kaidanov had a FIDE rating of 2638 and a USCF rating of 2742." The Elo ratings of these various organizations are not always directly comparable, since Elo ratings measure the results within a closed pool of players rather than absolute skill.
FIDE ratings
For top players, the most important rating is their FIDE rating. FIDE has issued the following lists:
i.From 1971 to 1980, one list a year was issued.
ii.From 1981 to 2000, two lists a year were issued, in January and July.
iii.From July 2000 to July 2009, four lists a year were issued, at the start of January, April, July and October.
iv.From July 2009 to July 2012, six lists a year were issued, at the start of January, March, May, July, September and November.
Since July 2012, the list has been updated monthly.
The following analysis of the July 2015 FIDE rating list gives a rough impression of what a given FIDE rating means in terms of world ranking:
5,323 players had an active rating in the range 2200 to 2299, which is usually associated with the Candidate Master title.
2,869 players had an active rating in the range 2300 to 2399, which is usually associated with the FIDE Master title.
1,420 players had an active rating between 2400 and 2499, most of whom had either the International Master or the International Grandmaster title.
542 players had an active rating between 2500 and 2599, most of whom had the International Grandmaster title.
187 players had an active rating between 2600 and 2699, all of whom had the International Grandmaster title.
40 players had an active rating between 2700 and 2799.
4 players had an active rating of over 2800. (Magnus Carlsen was rated 2853, and 3 players were rated between 2814 and 2816).
The highest ever FIDE rating was 2882, which Magnus Carlsen had on the May 2014 list. A list of the highest-rated players ever is at Comparison of top chess players throughout history.
Performance rating
Performance rating or special rating is a hypothetical rating that would result from the games of a single event only. Some chess organizations [16]: p. 8 use the "algorithm of 400" to calculate performance rating. According to this algorithm, performance rating for an event is calculated in the following way:
1.For each win, add your opponent's rating plus 400,
2.For each loss, add your opponent's rating minus 400,
3.And divide this sum by the number of played games.
Example: 2 wins (opponents w & x), 2 losses (opponents y & z)
This is a simplification, but it offers an easy way to get an estimate of PR (performance rating).
FIDE, however, calculates performance rating by means of the formula
performance rating=average of opponents' ratings+dp,
Live ratings
FIDE updates its ratings list at the beginning of each month. In contrast, the unofficial "Live ratings" calculate the change in players' ratings after every game. These Live ratings are based on the previously published FIDE ratings, so a player's Live rating is intended to correspond to what the FIDE rating would be if FIDE were to issue a new list that day.
Although Live ratings are unofficial, interest arose in Live ratings in August/September 2008 when five different players took the "Live" No. 1 ranking.
The unofficial live ratings of players over 2700 were published and maintained by Hans Arild Runde at the Live Rating website until August 2011. Another website, 2700chess.com, has been maintained since May 2011 by Artiom Tsepotan, which covers the top 100 players as well as the top 50 female players.
Rating changes can be calculated manually by using the FIDE ratings change calculator.[18] All top players have a K-factor of 10, which means that the maximum ratings change from a single game is a little less than 10 points.
United States Chess Federation ratings
The United States Chess Federation (USCF) uses its own classification of players:
1. 2400 and above: Senior Master
2. 2200–2399: National Master2200–2399 plus 300 games above 2200: Original Life Master
3. 2000–2199: Expert or Candidate Master
4. 1800–1999: Class A
5. 1600–1799: Class B
6. 1400–1599: Class C
7. 1200–1399: Class D
8. 1000–1199: Class E
9. 800–999: Class F
10. 600–799: Class G
11. 400–599: Class H
12. 200–399: Class I
13. 100–199: Class J
The K-factor used by the USCF
The K-factor, in the USCF rating system, can be estimated by dividing 800 by the effective number of games a player's rating is based on (Ne) plus the number of games the player completed in a tournament.
Rating floors
The USCF maintains an absolute rating floor of 100 for all ratings. Thus, no member can have a rating below 100, no matter their performance at USCF-sanctioned events. However, players can have higher individual absolute rating floors, calculated using the following formula:
Higher rating floors exist for experienced players who have achieved significant ratings. Such higher rating floors exist, starting at ratings of 1200 in 100-point increments up to 2100 (1200, 1300, 1400, ..., 2100). A rating floor is calculated by taking the player's peak established rating, subtracting 200 points, and then rounding down to the nearest rating floor. For example, a player who has reached a peak rating of 1464 would have a rating floor of 1464 − 200 = 1264, which would be rounded down to 1200. Under this scheme, only Class C players and above are capable of having a higher rating floor than their absolute player rating. All other players would have a floor of at most 150.
There are two ways to achieve higher rating floors other than under the standard scheme presented above. If a player has achieved the rating of Original Life Master, their rating floor is set at 2200. The achievement of this title is unique in that no other recognized USCF title will result in a new floor. For players with ratings below 2000, winning a cash prize of $2,000 or more raises that player's rating floor to the closest 100-point level that would have disqualified the player for participation in the tournament. For example, if a player won $4,000 in a 1750-and-under tournament, they would now have a rating floor of 1800.
Mathematical details
Performance is not measured absolutely; it is inferred from wins, losses, and draws against other players. Players' ratings depend on the ratings of their opponents and the results scored against them. The difference in rating between two players determines an estimate for the expected score between them. Both the average and the spread of ratings can be arbitrarily chosen. The USCF initially aimed for an average club player to have a rating of 1500 and Elo suggested scaling ratings so that a difference of 200 rating points in chess would mean that the stronger player has an expected score of approximately 0.75.
A player's expected score is their probability of winning plus half their 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. In practice, since the true strength of each player is unknown, the expected scores are calculated using the player's current ratings as follows.
When a player's actual tournament scores exceed their expected scores, the Elo system takes this as evidence that player's rating is too low, and needs to be adjusted upward. Similarly, when a player's actual tournament scores fall short of their expected scores, that player's rating is adjusted downward. Elo's original suggestion, which is still widely used, was a simple linear adjustment proportional to the amount by which a player over-performed or under-performed their expected score. The maximum possible adjustment per game, called the K-factor, was set at K = 16 for masters and K = 32 for weaker players.
This update can be performed after each game or each tournament, or after any suitable rating period.
An example may help to clarify:
Suppose player A has a rating of 1613 and plays in a five-round tournament. They lose to a player rated 1609, draw with a player rated 1477, defeat a player rated 1388, defeat a player rated 1586, and lose to a player rated 1720. The player's actual score is (0 + 0.5 + 1 + 1 + 0) = 2.5. The expected score, calculated according to the formula above, was (0.51 + 0.69 + 0.79 + 0.54 + 0.35) = 2.88.
Therefore, the player's new rating is [1613 + 32·(2.5 − 2.88)] = 1601, assuming that a K-factor of 32 is used. Equivalently, each game the player can be said to have put an ante of K times their expected score for the game into a pot, the opposing player does likewise, and the winner collects the full pot of value K; in the event of a draw, the players split the pot and receive 1 2 K points each.
Note that while two wins, two losses, and one draw may seem like a par score, it is worse than expected for player A because their opponents were lower rated on average. Therefore, player A is slightly penalized. If player A had scored two wins, one loss, and two draws, for a total score of three points, that would have been slightly better than expected, and the player's new rating would have been [1613 + 32·(3 − 2.88)] = 1617.
This updating procedure is at the core of the ratings used by FIDE, USCF, Yahoo! Games, the Internet Chess Club (ICC) and the Free Internet Chess Server (FICS). However, each organization has taken a different approach to dealing with the uncertainty inherent in the ratings, particularly the ratings of newcomers, and to dealing with the problem of ratings inflation/deflation. New players are assigned provisional ratings, which are adjusted more drastically than established ratings.
The principles used in these rating systems can be used for rating other competitions—for instance, international football matches.
Elo ratings have also been applied to games without the possibility of draws, and to games in which the result can also have a quantity (small/big margin) in addition to the quality (win/loss).
Suggested modification
In 2011 after analyzing 1.5 million FIDE rated games, Jeff Sonas demonstrated according to the Elo formula, two players having a rating difference of X actually have a true difference of around X(5/6). Likewise, one can leave the rating difference alone and divide by 480 instead of 400. Since the Elo formula is overestimating the stronger player's win probability, stronger players are losing points against weaker players despite playing at their true strength. Likewise, weaker players gain points against stronger players. When the modification is applied, observed win rates deviate by less than 0.1% away from prediction, while traditional Elo can be 4% off the predicted rate.
Most accurate distribution model
The first mathematical concern addressed by the USCF was the use of the normal distribution. They found that this did not accurately represent the actual results achieved, particularly by the lower rated players. Instead they switched to a logistic distribution model, which the USCF found provided a better fit for the actual results achieved.[29][citation needed] FIDE also uses an approximation to the logistic distribution.
Most accurate K-factor
The second major concern is the correct "K-factor" used. The chess statistician Jeff Sonas believes that the original K = 10 value (for players rated above 2400) is inaccurate in Elo's work. If the K-factor coefficient is set too large, there will be too much sensitivity to just a few, recent events, in terms of a large number of points exchanged in each game. And if the K-value is too low, the sensitivity will be minimal, and the system will not respond quickly enough to changes in a player's actual level of performance.
Elo's original K-factor estimation was made without the benefit of huge databases and statistical evidence. Sonas indicates that a K-factor of 24 (for players rated above 2400) may be both more accurate as a predictive tool of future performance and be more sensitive to performance.
Certain Internet chess sites seem to avoid a three-level K-factor staggering based on rating range. For example, the ICC seems to adopt a global K = 32 except when playing against provisionally rated players.
The USCF (which makes use of a logistic distribution as opposed to a normal distribution) formerly staggered the K-factor according to three main rating ranges:
Currently, the USCF uses a formula that calculates the K-factor based on factors including the number of games played and the player's rating. The K-factor is also reduced for high rated players if the event has shorter time controls.
FIDE uses the following ranges:
The gradation of the K-factor reduces rating change at the top end of the rating range, reducing the possibility for rapid rise or fall of rating for those with a rating high enough to reach a low K-factor.
In theory, this might apply equally to online chess players and over-the-board players, since it is more difficult for all players to raise their rating after their rating has become high and their K-factor consequently reduced. However, when playing online, 2800+ players can more easily raise their rating by simply selecting opponents with high ratings – on the ICC playing site, a grandmaster may play a string of different opponents who are all rated over 2700.[34] In over-the-board events, it would only be in very high level all-play-all events that a player would be able to engage that number of 2700+ opponents. In a normal, open, Swiss-paired chess tournament, frequently there would be many opponents rated less than 2500, reducing the ratings gains possible from a single contest for a high-rated player.
Formal derivation for win/loss games
The above expressions can be now formally derived by exploiting the link between the Elo rating and the stochastic gradient update in the logistic regression.
If we assume that the game results are binary, that is, only a win or a loss can be observed, the problem can be addressed via logistic regression, where the games results are dependent variables, the players' ratings are independent variables, and the model relating both is probabilistic: the probability of the player A winning the game is modeled as
and, using the stochastic gradient descent the log loss is minimized as follows:
Formal derivation for win/draw/loss games
Since the very beginning, the Elo rating has been also used in chess where we observe wins, losses or draws and, to deal with the latter a fractional score value, S A = 0.5 , is introduced. We note, however, that the scores S A = 1 and S A = 0 are merely indicators to the events when the player A wins or loses the game. It is, therefore, not immediately clear what is the meaning of the fractional score. Moreover, since we do not specify explicitly the model relating the rating values R A and R B to the probability of the game outcome, we cannot say what the probability of the win, the loss, or the draw is.
To address these difficulties, and to derive the Elo rating in the ternary games, we will define the explicit probabilistic model of the outcomes. Next, we will minimize the log loss via stochastic gradient.
Since the loss, the draw, and the win are ordinal variables, we should adopt the model which takes their ordinal nature into account, and we use the so-called adjacent categories model which may be traced to the Davidson's work
Using the ordinal model defined above, the log loss is now calculated as
Then, the stochastic gradient descent applied to minimize the log loss yields the following update for the rating
Practical issues
Game activity versus protecting one's rating
In some cases the rating system can discourage game activity for players who wish to protect their rating.In order to discourage players from sitting on a high rating, a 2012 proposal by British Grandmaster John Nunn for choosing qualifiers to the chess world championship included an activity bonus, to be combined with the rating.
Beyond the chess world, concerns over players avoiding competitive play to protect their ratings caused Wizards of the Coast to abandon the Elo system for Magic: the Gathering tournaments in favour of a system of their own devising called "Planeswalker Points"
Selective pairing
A more subtle issue is related to pairing. When players can choose their own opponents, they can choose opponents with minimal risk of losing, and maximum reward for winning. Particular examples of players rated 2800+ choosing opponents with minimal risk and maximum possibility of rating gain include: choosing opponents that they know they can beat with a certain strategy; choosing opponents that they think are overrated; or avoiding playing strong players who are rated several hundred points below them, but may hold chess titles such as IM or GM. In the category of choosing overrated opponents, new entrants to the rating system who have played fewer than 50 games are in theory a convenient target as they may be overrated in their provisional rating. The ICC compensates for this issue by assigning a lower K-factor to the established player if they do win against a new rating entrant. The K-factor is actually a function of the number of rated games played by the new entrant.
Therefore, Elo ratings online still provide a useful mechanism for providing a rating based on the opponent's rating. Its overall credibility, however, needs to be seen in the context of at least the above two major issues described—engine abuse, and selective pairing of opponents.
The ICC has also recently introduced "auto-pairing" ratings which are based on random pairings, but with each win in a row ensuring a statistically much harder opponent who has also won x games in a row. With potentially hundreds of players involved, this creates some of the challenges of a major large Swiss event which is being fiercely contested, with round winners meeting round winners. This approach to pairing certainly maximizes the rating risk of the higher-rated participants, who may face very stiff opposition from players below 3000, for example. This is a separate rating in itself, and is under "1-minute" and "5-minute" rating categories. Maximum ratings achieved over 2500 are exceptionally rare.
Ratings inflation and deflation
The term "inflation", applied to ratings, is meant to suggest that the level of playing strength demonstrated by the rated player is decreasing over time; conversely, "deflation" suggests that the level is advancing. For example, if there is inflation, a modern rating of 2500 means less than a historical rating of 2500, while the reverse is true if there is deflation. Using ratings to compare players between different eras is made more difficult when inflation or deflation are present. (See also Comparison of top chess players throughout history.)
Analyzing FIDE rating lists over time, Jeff Sonas suggests that inflation may have taken place since about 1985.[42] Sonas looks at the highest-rated players, rather than all rated players, and acknowledges that the changes in the distribution of ratings could have been caused by an increase of the standard of play at the highest levels, but looks for other causes as well.
The number of people with ratings over 2700 has increased. Around 1979 there was only one active player (Anatoly Karpov) with a rating this high. In 1992 Viswanathan Anand was only the 8th player in chess history to reach the 2700 mark at that point of time.[43] This increased to 15 players by 1994. 33 players had a 2700+ rating in 2009 and 44 as of September 2012. Only 14 players have ever broken a rating of 2800.
One possible cause for this inflation was the rating floor, which for a long time was at 2200, and if a player dropped below this they were struck from the rating list. As a consequence, players at a skill level just below the floor would only be on the rating list if they were overrated, and this would cause them to feed points into the rating pool.[42] In July 2000 the average rating of the top 100 was 2644. By July 2012 it had increased to 2703.
Using a strong chess engine to evaluate moves played in games between rated players, Regan and Haworth analyze sets of games from FIDE-rated tournaments, and draw the conclusion that there had been little or no inflation from 1976 to 2009.
In a pure Elo system, each game ends in an equal transaction of rating points. If the winner gains N rating points, the loser will drop by N rating points. This prevents points from entering or leaving the system when games are played and rated. However, players tend to enter the system as novices with a low rating and retire from the system as experienced players with a high rating. Therefore, in the long run a system with strictly equal transactions tends to result in rating deflation.
In 1995, the USCF acknowledged that several young scholastic players were improving faster than the rating system was able to track. As a result, established players with stable ratings started to lose rating points to the young and underrated players. Several of the older established players were frustrated over what they considered an unfair rating decline, and some even quit chess over it.
Combating deflation
Because of the significant difference in timing of when inflation and deflation occur, and in order to combat deflation, most implementations of Elo ratings have a mechanism for injecting points into the system in order to maintain relative ratings over time. FIDE has two inflationary mechanisms. First, performances below a "ratings floor" are not tracked, so a player with true skill below the floor can only be unrated or overrated, never correctly rated. Second, established and higher-rated players have a lower K-factor. New players have a K = 40, which drops to K = 20 after 30 played games, and to K = 10 when the player reaches 2400.[31] The current system in the United States includes a bonus point scheme which feeds rating points into the system in order to track improving players, and different K-values for different players.[46] Some methods, used in Norway for example, differentiate between juniors and seniors, and use a larger K-factor for the young players, even boosting the rating progress by 100% for when they score well above their predicted performance.
Rating floors in the United States work by guaranteeing that a player will never drop below a certain limit. This also combats deflation, but the chairman of the USCF Ratings Committee has been critical of this method because it does not feed the extra points to the improving players. A possible motive for these rating floors is to combat sandbagging, i.e., deliberate lowering of ratings to be eligible for lower rating class sections and prizes.
Ratings of computers
Human–computer chess matches between 1997 (Deep Blue versus Garry Kasparov) and 2006 demonstrated that chess computers are capable of defeating even the strongest human players. However, chess engine ratings are difficult to quantify, due to variable factors such as the time control and the hardware the program runs on, and also the fact that chess is not a fair game. The existence and magnitude of the first-move advantage in chess becomes very important at the computer level. Beyond some skill threshold, an engine with White should be able to force a draw on demand from the starting position even against perfect play, simply because White begins with too big an advantage to lose compared to the small magnitude of the errors it is likely to make. Consequently, such an engine is more or less guaranteed to score at least 25% even against perfect play. Differences in skill beyond a certain point could only be picked up if one does not begin from the usual starting position, but instead chooses a starting position that is only barely not lost for one side. Because of these factors, ratings depend on pairings and the openings selected.[48] Published engine rating lists such as CCRL are based on engine-only games on standard hardware configurations and are not directly comparable to FIDE ratings.
