Comparison of top chess players throughout history

This article presents a number of methodologies that have been suggested for the task of comparing the greatest chess players in history. Statistical methods offer objectivity but, while there is agreement on systems to rate the strengths of current players, there is disagreement on whether such techniques can be applied to players from different generations who never competed against each other.
Statistical methods.
Elo system.
Perhaps the best-known statistical model is that devised by Arpad Elo in 1960 and further elaborated on in his 1978 book "The Rating of Chessplayers, Past and Present". He gave ratings to players corresponding to their performance over the best five-year span of their career. According to this system the highest ratings achieved were:
In 1970, FIDE adopted Elo's system for rating current players, so one way to compare players of different eras is to compare their Elo ratings. The best-ever Elo ratings are tabulated below.
As of December 2015, there were 101 chess players in history who broke 2700 and thirteen of them exceeded 2800. Particularly notable are the peak ratings of Fischer, Karpov, and Kasparov, who achieved their peak ratings in earlier years (1972, 1994, and 1999 respectively).
Average rating over time.
The average Elo rating of top players has risen over time. For instance, the average of the top 10 active players rose from 2751 in July 2000 to 2794 in July 2014, a 43-point increase in 14 years. The average rating of the top 100 players, meanwhile, increased from 2644 to 2703, a 59-point increase. Many people believe that this rise is mostly due to a system artifact known as ratings inflation, making it impractical to compare players of different eras.
Arpad Elo was of the opinion that it was futile to attempt to use ratings to compare players from different eras; in his view, they could only possibly measure the strength of a player as compared to his or her contemporaries. He also stated that the process of rating players was in any case rather approximate; he compared it to "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".
Chessmetrics.
Many statisticians besides Elo have devised similar methods to retrospectively rate players. Jeff Sonas' rating system is called "Chessmetrics". This system takes account of many games played after the publication of Elo's book, and claims to take account of the rating inflation that the Elo system has allegedly suffered.
One caveat is that a Chessmetrics rating takes into account the frequency of play. According to Sonas, "As soon as you go a month without playing, your Chessmetrics rating will start to drop."
Sonas, like Elo, claims that it is impossible to compare the strength of players from different eras, saying:
Nevertheless, Sonas' website does compare players from different eras. Including data until December 2004, the ratings were:
In 2005, Sonas used Chessmetrics to evaluate historical annual performance ratings and came to the conclusion that Kasparov was dominant for the most years, followed by Karpov and Lasker. He also published the following list of the highest ratings ever attained according to calculations done at the start of each month:
"Warriors of the Mind".
In contrast to Elo and Sonas's systems, Raymond Keene and Nathan Divinsky's book "Warriors of the Mind" attempts to establish a rating system claiming to compare directly the strength of players active in different eras, and so determine the strongest player of all time (through December 2004). Considering games played between sixty-four of the strongest players in history, they came up with the following top ten:
These "Divinsky numbers" are not on the same scale as Elo ratings (the last person on the list, Johannes Zukertort, has a Divinsky number of 873, which would be a beginner-level Elo rating). Keene and Divinsky's system has met with limited acceptance, and "Warriors of the Mind" has been accused of arbitrarily selecting players and bias towards modern players.
Moves played compared with computer choices.
Matej Guid and Ivan Bratko.
A computer-based method of analyzing chess abilities across history came from Matej Guid and Ivan Bratko from the Department of Computer and Information Science of University of Ljubljana, Slovenia, in 2006. The basis for their evaluation was the difference between the position values resulting from the moves played by the human chess player and the moves chosen as best by the chess program Crafty. They compared the average number of errors in the player's game. Opening moves were excluded, in an attempt to negate the progress in chess opening theory.
The method received a number of criticisms, including: the study used a modified version of Crafty rather than the standard version; even the standard version of Crafty was not strong enough to evaluate the world champions' play; one of the modifications restricted the search depth to 12 half-moves, which is often insufficient. As of 2006 Crafty's Elo rating was 2657, below many historical top human players and several other computer programs.
A study by online chess data provider Chess-DB, based on an analysis of over 50,000 chess games, claims that the "strength" of a player, as determined by the method of Matej Guid and Ivan Bratko, correlates with the Elo rating strength of modern players.
Rybka.
A similar project was conducted for World Champions in 2007–08 using Rybka 2.3.2a (then-strongest chess program) and a modified version of Crafty 20.14. It arrived at the following results:
A 2008 analysis, using Rybka 3, showed that Capablanca had the smallest average error factor (i.e. the most accurate play); but after adjusting for factors such as the complexity of positions, the best player came out as Fischer, followed by Capablanca, Karpov and Kramnik. The best players had an average error of about 0.07 pawns per move (after the opening). Capablanca was the most positional player, and Anand by far the most tactical. The most complex game tested was Fischer v Spassky (1972 game 6, Fischer won) while the most accurately played game was Tal v Benko (1958, Tal won).
CAPS.
CAPS (Computer Aggregated Precision Score) is a system created by Chess.com that compares players from different eras by finding the percentage of moves that matches that of a chess engine. A score is then assigned based on percentage of matches and move value (for example, if the move was not the best, but still good, points are awarded). CAPS ignores both style and psychology. According to the system, Carlsen was the best player ever, with a CAPS score of 98.36 and a top engine match of 85.26%. He was followed closely by Kramnik, Kasparov, Fischer, and then Anand.
Markovian Model.
In an article published by the ICGA Journal in 2017, Jean-Marc Alliot of the Toulouse Computer Science Research Institute (IRIT) presents a new method, based on a Markovian interpretation of a chess game. Starting with those of Wilhelm Steinitz, all 26,000 games played since then by chess world champions have been processed by a supercomputer using Stockfish (rated between 3310 Elo at the CCRL and 3337 at the SSDF as of 10/2015, but around 3150 under the test condition according to the authors) in 62000 CPU hours, in order to create a probabilistic model for each player. For each position, the model estimates the probability of making a mistake and the magnitude of the mistake by comparing the two best moves calculated at an average of 2 minutes by move (26 plies on the average) with the move actually played, starting from move number 10. These models can then be used to compute the win/draw/lose probability for any given match between two players. These predictions have proven not only to be extremely close to the actual results when players have played concrete games against one another, but to also fare better than those based on Elo scores. The results demonstrate that the level of chess players has been steadily increasing. Magnus Carlsen (in 2013) tops the list, while Vladimir Kramnik (in 1999) is second, Bobby Fischer (in 1971) is third, and Garry Kasparov (in 2001) is fourth.
The complete results are as follows, with each player taken in his best year. The numbers represent the winning percentage of one player against his opponent.
The tables are not symmetric, since whether one starts with white pieces or black pieces impacts the outcome of the game.
The complete database of the chess games and their evaluations can be downloaded from the page presenting this work on the author's website.
Subjective lists.
Many prominent players and chess writers have offered their own rankings of players.
Bobby Fischer (1964 and 1970).
In 1964, Bobby Fischer listed his top 10 in "Chessworld" magazine: Morphy, Staunton, Steinitz, Tarrasch, Chigorin, Alekhine, Capablanca, Spassky, Tal, Reshevsky. He considered Morphy to be "perhaps the most accurate", writing: "In a set match he would beat anyone alive today."
In 1970, Fischer named Morphy, Steinitz, Capablanca, Botvinnik, Petrosian, Tal, Spassky, Reshevsky, Svetozar Gligorić and Bent Larsen the greatest chess players in history.
Irving Chernev (1974).
In 1974, popular chess author Irving Chernev published an article titled "Who were the greatest?" in the English magazine CHESS. He followed this up with his 1976 book "The Golden Dozen", in which he ranked his all-time top twelve: 1. Capablanca, 2. Alekhine, 3. Lasker, 4. Fischer, 5. Botvinnik, 6. Petrosian, 7. Tal, 8. Smyslov, 9. Spassky, 10. Bronstein, 11. Rubinstein, and 12. Nimzowitsch.
Miguel Quinteros (1992).
In a 1992 interview GM Miguel Quinteros gave the opinion: "I think Fischer was and still is the greatest chess player of all time. [...] During his absence other good chess players have appeared. But no one equals Fischer's talent and perfection."
Viswanathan Anand (2000, 2008 and 2012).
In 2000, when Karpov, Korchnoi and Kasparov were still active, Anand listed his top 10 as: Fischer, Morphy, Lasker, Capablanca, Steinitz, Tal, Korchnoi, Keres, Karpov and Kasparov.
When interviewed in 2008 shortly after Fischer's death, he ranked Fischer and Kasparov as the greatest, with Kasparov a little ahead by virtue of being on top for so many years.
In 2012, Anand stated that he considered Fischer the best player and also the greatest, because of the hurdles he faced.
"Chess Informant" readers (2001).
Svetozar Gligorić reported in his book "Shall We Play Fischerandom Chess?"  (Batsford, 2002):At the beginning of 2001 a large poll for the "Ten Greatest Chess Players of the 20th Century, selected by "Chess Informant" readers" resulted in Fischer having the highest percentage of votes and finishing as No. 1, ahead of Kasparov, Alekhine, Capablanca, Botvinnik, Karpov, Tal, Lasker, Anand and Korchnoi.
David Edmonds and John Eidinow (2004).
BBC award-winning journalists, from their book "Bobby Fischer Goes to War: How the Soviets Lost the Most Extraordinary Chess Match of All Time"  (HarperCollins, 2004): Fischer, some will maintain, was "the" outstanding player in chess history, though there are powerful advocates too for Lasker, Capablanca, Alekhine, and Kasparov. Many chess players will dismiss such comparisons as meaningless, akin to the futile attempt to grade the supreme musicians of all time. But the manner in which Fischer stormed his way to Reykjavik, his breathtaking dominance at the Palma de Majorca Interzonal, the trouncings of Taimanov, Larsen, and Petrosian—all this was unprecedented. There never has been an era in modern chess during which one player has so overshadowed all others.
Vladimir Kramnik (2005 and 2011).
In a 2005 interview, Vladimir Kramnik (World Champion from 2000 to 2007) did not name a greatest player, but stated, "The other world champions had something 'missing'. I can't say the same about Kasparov: he can do everything."
In an interview in 2011, Vladimir Kramnik said about Anand: "I always considered him to be a colossal talent, one of the greatest in the whole history of chess", "I think that in terms of play Anand is in no way weaker than Kasparov", and "In the last 5–6 years he's made a qualitative leap that's made it possible to consider him one of the great chess players".
Leonard Barden (2008).
In his 2008 obituary of Bobby Fischer, Leonard Barden wrote that most experts ranked Kasparov as the best ever player, with probably Fischer second and Karpov third.
Levon Aronian (2012 and 2015).
In a 2012 interview, Levon Aronian stated that he considers Alexander Alekhine the best player of all time.
In a 2015 interview after the 8th round of the Sinquefield Cup, Levon Aronian stated that he considers Garry Kasparov the strongest player of all time.
Magnus Carlsen (2012, 2015, 2020 and 2021).
In 2012, Magnus Carlsen said that Kasparov is the greatest player of all time, adding that while Fischer may have been better at his best, Kasparov remained at the top for much longer. 
In December 2015, he repeated his great respect for both Fischer and Kasparov when he mentioned them several times in an interview, saying he would like to play against them at their peak performance.
In January 2020, Carlsen said, "Kasparov had 20 years uninterrupted as the world No 1. And I would say for very few of those years was there any doubt that he was the best player. He must be considered as the best in history." He made a similar claim in 2021, saying "Garry Kasparov, in my opinion, the greatest player there's ever been..."
World Champions by world title reigns.
The table below organises the world champions in order of championship wins. (For the purpose of this table, a successful defence counts as a win, even if the match was drawn.) The table is made more complicated by the split between the "Classical" and FIDE world titles between 1993 and 2006.