Comparison of top chess players throughout history

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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[]

Main articles: Elo rating system and List of chess players by peak FIDE rating

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.[1] 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:

  • 2725: José Raúl Capablanca
  • 2720: Mikhail Botvinnik, Emanuel Lasker
  • 2700: Mikhail Tal
  • 2690: Alexander Alekhine, Paul Morphy, Vasily Smyslov

(Though published in 1978, Elo's list did not include five-year averages for Bobby Fischer and Anatoly Karpov. It did list January 1978 ratings of 2780 for Fischer and 2725 for Karpov.)[2]

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.[3]

As of December 2015, there were 101 chess players in history who broke 2700 and thirteen of them exceeded 2800.[4][5][6][7][8] Particularly notable are the peak ratings of Fischer, Karpov, and Kasparov, who achieved their peak ratings in earlier years (1972, 1994, and 1999 respectively).

Table of top 20 rated players of all-time, with date their best ratings were first achieved
Rank Rating Player Date Age
011 2882 Magnus Carlsen 2014-05May 2014 23 years, 5 months
022 2851 Garry Kasparov 1999-07July 1999 36 years, 2 months
033 2844 Fabiano Caruana 2014-10October 2014 22 years, 2 months
044 2830 Levon Aronian 2014-03March 2014 31 years, 4 months
055 2822 Wesley So 2017-02February 2017 23 years, 3 months
066 2820 Shakhriyar Mamedyarov 2018-09September 2018 33 years, 4 months
077 2819 Maxime Vachier-Lagrave 2016-08 August 2016 25 years, 9 months
088 (tie) 2817 Viswanathan Anand 2011-03March 2011 41 years, 2 months
088 (tie) 2817 Vladimir Kramnik 2016-10October 2016 41 years, 3 months
1010 (tie) 2816 Veselin Topalov 2015–07July 2015 40 years, 3 months
1010 (tie) 2816 Hikaru Nakamura 2015-10October 2015 27 years, 9 months
1010 (tie) 2816 Ding Liren 2018-11November 2018 26 years
1313 2810 Alexander Grischuk 2014-12December 2014 31 years, 1 month
1414 2798 Anish Giri 2015-10October 2015 21 years, 3 months
1515 2793 Teimour Radjabov 2012-11November 2012 25 years, 7 months
1616 2792 Ian Nepomniachtchi 2021-05May 2021 30 years, 9 months
1717 (tie) 2788 Alexander Morozevich 2008-07July 2008 30 years, 11 months
1717 (tie) 2788 Sergey Karjakin 2011-07July 2011 21 years, 5 months
1919 2787 Vassily Ivanchuk 2007–10October 2007 38 years, 6 months
2020 2785 Bobby Fischer 1972-04April 1972 29 years

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.[9] 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.[10]

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".[11]

Chessmetrics[]

Further information: 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."[12]

Sonas, like Elo, claims that it is impossible to compare the strength of players from different eras, saying:

Of course, a rating always indicates the level of dominance of a particular player against contemporary peers; it says nothing about whether the player is stronger/weaker in their actual technical chess skill than a player far removed from them in time. So while we cannot say that Bobby Fischer in the early 1970s or José Capablanca in the early 1920s were the "strongest" players of all time, we can say with a certain amount of confidence that they were the two most dominant players of all time. That is the extent of what these ratings can tell us.[13]

Nevertheless, Sonas' website does compare players from different eras. Including data until December 2004, the ratings were:

Rank 1-year peak[14] 5-year peak[15] 10-year peak[16] 15-year peak[17] 20-year peak[18]
1 Bobby Fischer, 2881 Garry Kasparov, 2875 Garry Kasparov, 2863 Garry Kasparov, 2862 Garry Kasparov, 2856
2 Garry Kasparov, 2879 Emanuel Lasker, 2854 Emanuel Lasker, 2847 Anatoly Karpov, 2820 Anatoly Karpov, 2818
3 Mikhail Botvinnik, 2871 José Capablanca, 2843 Anatoly Karpov, 2821 Emanuel Lasker, 2816 Emanuel Lasker, 2809
4 José Capablanca, 2866 Mikhail Botvinnik, 2843 José Capablanca, 2813 José Capablanca, 2798 Alexander Alekhine, 2781
5 Emanuel Lasker, 2863 Bobby Fischer, 2841 Bobby Fischer, 2810 Alexander Alekhine, 2794 Viktor Korchnoi, 2766
6 Alexander Alekhine, 2851 Anatoly Karpov, 2829 Mikhail Botvinnik, 2810 Mikhail Botvinnik, 2789 Vasily Smyslov, 2759

In 2005,[19] 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:[20]

Rank Rating Player
1 2895 Bobby Fischer
2 2886 Garry Kasparov
3 2885 Mikhail Botvinnik
4 2878 Emanuel Lasker
5 2877 José Capablanca
6 2860 Alexander Alekhine
7 2848 Anatoly Karpov
8 2833 Viswanathan Anand
9 2826 Vladimir Kramnik
10 2826 Wilhelm Steinitz

Warriors of the Mind[]

In contrast to Elo and Sonas's systems, Raymond Keene and Nathan Divinsky's book Warriors of the Mind[21] 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:[22]

  1. Garry Kasparov, 3096
  2. Anatoly Karpov, 2876
  3. Bobby Fischer, 2690
  4. Mikhail Botvinnik, 2616
  5. José Raúl Capablanca, 2552
  6. Emanuel Lasker, 2550
  7. Viktor Korchnoi, 2535
  8. Boris Spassky, 2480
  9. Vasily Smyslov, 2413
  10. Tigran Petrosian, 2363

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,[23] and Warriors of the Mind has been accused of arbitrarily selecting players and bias towards modern players.[24]

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.[25] 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.[26] 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.[27]

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.[28] It arrived at the following results:[29][30]

Position best year best 2-year period best 3-year period best 5-year period best 10-year period best 15-year period
1 Fischer (1968, aged 25) Fischer Fischer Fischer; Kasparov Fischer; Capablanca Capablanca
2 Kramnik (2007, aged 32) Kramnik; Capablanca;
Kasparov		 Capablanca; Kasparov Karpov; Kramnik
3 Kasparov (1998, aged 35) Capablanca Kramnik
4 Botvinnik (1939, aged 28) Smyslov Kramnik; Botvinnik Kasparov Smyslov; Kasparov
5 Capablanca (1915, aged 27) Karpov; Smyslov Botvinnik Karpov; Smyslov
6 Karpov (1983, aged 32) Kramnik Smyslov Fischer
7 Smyslov (1983, aged 62) ;
Tal (1987, aged 51)		 Botvinnik; Alekhine Karpov Karpov; Lasker Botvinnik; Spassky Botvinnik; Spassky;
Petrosian
8 Spassky; Lasker
9 Petrosian (1962, aged 33) Anand Alekhine; Anand Anand
10 Euwe (1938, aged 37) Tal; Spassky Anand Lasker; Petrosian Anand
11 Spassky (1980, aged 43) Petrosian Petrosian; Spassky Tal
12 Alekhine (1927, aged 35) ;
Anand (2006, aged 37)		 Lasker; Euwe Tal; Alekhine Tal; Alekhine Alekhine; Lasker
13 Euwe; Tal
14 Lasker (1909, aged 41) Petrosian Euwe Euwe Euwe
15 Morphy (1858, aged 21) Morphy Morphy Steinitz Steinitz Steinitz
16 Steinitz (1886, aged 50) Steinitz Steinitz — — —

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).[31]

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.[32] 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, Anand, and then Fischer.[33]

Markovian Model[]

In an article[34] published by the ICGA Journal in 2017, 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)[35] 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,[36] starting from move number 10.[37] 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.

Head to head match result predictions between different World Champions (by their best year)
Name Year Ca Kr Fi Ka An Kh Sm Pe Kp Ks Bo Po La Sp To Ca Ta Eu Al St
Carlsen 2013 — 52 54 54 57 58 57 58 56 60 61 59 60 61 61 64 66 69 70 82
Kramnik 1999 49 — 52 52 55 56 56 57 55 59 60 58 60 60 60 63 65 68 70 83
Fischer 1971 47 49 — 51 53 57 56 57 56 59 60 60 61 61 62 64 68 70 73 85
Kasparov 2001 47 49 50 — 53 54 54 54 53 57 58 56 56 58 58 60 62 66 68 82
Anand 2008 44 46 48 48 — 54 52 53 53 57 56 57 57 59 59 62 64 69 71 86
Khalifman 2010 43 45 44 47 47 — 50 51 52 53 54 55 55 56 56 60 62 64 67 79
Smyslov 1983 43 45 45 47 49 51 — 50 51 53 55 54 54 54 55 59 63 64 68 82
Petrosian 1962 43 44 45 47 49 50 51 — 52 53 54 54 55 55 56 59 63 63 67 80
Karpov 1988 44 46 45 48 48 49 50 49 — 51 52 52 52 52 52 56 58 60 63 76
Kasimdzhanov 2011 41 43 42 45 45 48 48 48 50 — 52 52 52 54 53 56 60 62 65 80
Botvinnik 1945 40 41 41 44 45 48 46 48 49 49 — 50 54 52 52 56 60 60 64 80
Ponomariov 2011 42 43 41 45 44 47 47 47 49 49 51 — 51 52 52 55 58 59 62 77
Lasker 1907 41 41 40 45 44 46 47 46 49 49 48 50 — 51 50 54 58 59 63 78
Spassky 1970 40 41 40 43 42 45 47 46 48 47 49 49 50 — 51 53 58 57 61 75
Topalov 2008 40 41 39 44 42 45 46 45 49 48 49 49 50 51 — 54 57 57 61 75
Capablanca 1928 37 38 37 41 39 42 42 42 45 45 45 47 47 48 47 — 53 54 59 76
Tal 1981 35 36 34 39 37 39 39 38 43 41 41 43 43 43 44 48 — 49 54 72
Euwe 1941 32 33 32 36 32 37 37 38 41 39 41 42 43 44 44 47 52 — 56 75
Alekhine 1922 31 31 29 34 30 35 33 35 38 36 37 39 38 40 40 43 47 45 — 69
Steinitz 1894 20 19 17 20 16 22 19 22 25 22 22 25 24 27 27 26 30 27 33 —

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.[38][39] He considered Morphy to be "perhaps the most accurate", writing: "In a set match he would beat anyone alive today."[40]

In 1970, Fischer named Morphy, Steinitz, Capablanca, Botvinnik, Petrosian, Tal, Spassky, Reshevsky, Svetozar Gligorić and Bent Larsen the greatest chess players in history.[41]

Irving Chernev (1974)[]

In 1974, popular chess author Irving Chernev published an article titled Who were the greatest? in the English magazine CHESS.[42] 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.[43]

Miguel Quinteros (1992)[]

In a 1992 interview GM Miguel Quinteros gave the opinion:[44] "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.[45]

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.[46]

In 2012, Anand stated that he considered Fischer the best player and also the greatest, because of the hurdles he faced.[47]

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.[48]

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.[49]

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."[50]

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".[51]

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.[52]

Levon Aronian (2012 and 2015)[]

In a 2012 interview, Levon Aronian stated that he considers Alexander Alekhine the best player of all time.[53]

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.[54]

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.[55]

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.[56]

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."[57] He made a similar claim in 2021, saying "Garry Kasparov, in my opinion, the greatest player there's ever been..."[58]

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.

Champion Total Undisputed FIDE Classical Years as
Undisputed
Champion		 Years as
FIDE/Classical
Champion		 Total reign
Emanuel Lasker 6 6 27 27
Garry Kasparov 6 4 2 8 7 15
Anatoly Karpov 6 3 3 10 6 16
Mikhail Botvinnik 5 5 13 13
Viswanathan Anand 5 4 1 6 2 8
Alexander Alekhine 4 4 17 17
Wilhelm Steinitz 4 4 8 8
Magnus Carlsen 4 4 7 7
Vladimir Kramnik 3 1 2 1 6 7
Tigran Petrosian 2 2 6 6
José Raúl Capablanca 1 1 6 6
Boris Spassky 1 1 3 3
Bobby Fischer 1 1 3 3
Max Euwe 1 1 2 2
Vasily Smyslov 1 1 1 1
Mikhail Tal 1 1 1 1
Ruslan Ponomariov 1 1 2 2
Alexander Khalifman 1 1 1 1
Rustam Kasimdzhanov 1 1 1 1
Veselin Topalov 1 1 1 1

See also[]

References[]

  1. ^ Arpad E. Elo, The Rating of Chessplayers, Past and Present, Arco, 1978. ISBN 0-668-04721-6.
  2. ^ Arpad Emre Elo – 100th anniversary, Chessbase, 2003
  3. ^ This table is based on one created by Przemek Jahr of Poland, which was reported in Chessbase 2-7-2005. FIDE ratings were officially compiled and released quarterly, in January, April, July, and October until July 2009. Starting in July 2009 they were released on a two-monthly basis, and since July 2012 FIDE publishes its ratings monthly.
  4. ^ Administrator. "Highest-Rated Chess Players of All Time".
  5. ^ "All time Top 100 by Highest Elo Rating". Archived from the original on 6 March 2014. Retrieved 7 August 2014.
  6. ^ Highest "Live ratings" of All Time
  7. ^ "Live Chess Ratings – 2700chess.com".
  8. ^ "Live Elo" covers rating achieved after every single chess game played, what constitutes live performance of players in between of FIDE's regular monthly rating lists.
  9. ^ World Top chess players and Statistics at FIDE.com
  10. ^ "ChessBase News | Rating inflation – its causes and possible cures". Chessbase.com. 27 July 2009. Retrieved 21 October 2013.
  11. ^ Chess Life, 1962.
  12. ^ The Greatest Chess Player of All Time – Part I, Jeff Sonas, at Chessbase
  13. ^ About the Chessmetrics Rating System Archived 15 May 2012 at the Wayback Machine, by Jeff Sonas
  14. ^ "Peak Average Ratings: 1 year peak range". Archived from the original on 9 March 2012.
  15. ^ "Peak Average Ratings: 5 year peak range". Archived from the original on 9 March 2012.
  16. ^ "Peak Average Ratings: 10 year peak range". Archived from the original on 9 March 2012.
  17. ^ "Peak Average Ratings: 15 year peak range". Archived from the original on 9 March 2012.
  18. ^ "Peak Average Ratings: 20 year peak range". Archived from the original on 9 March 2012.
  19. ^ Sonas, J. (2005). "The Greatest Chess Player of All Time – Part IV". Chessbase. Part IV gives links to the 3 earlier parts
  20. ^ Sonas, J. (2005). "The Greatest Chess Player of All Time – Part II". Chessbase.
  21. ^ Warriors of the Mind, Raymond Keene and Nathan Divinsky, (1989)
  22. ^ "Divinsky-Keene rankings". Archived from the original on 26 November 2009.
  23. ^ Hooper, David; Whyld, Kenneth (1992). The Oxford Companion to Chess. Oxford University Press. ISBN 0-19-280049-3.
  24. ^ Winter, Edward (1996). Chess Explorations. Cadogan. ISBN 1-85744-171-0.
  25. ^ Computers choose: who was the strongest player?, Chessbase, 2006
  26. ^ Review of "Computer Analysis of World Chess Champions", by Søren Riis, Chessbase, 2006
  27. ^ Chess Play Quality Index, Chess-db.com, 2013
  28. ^ "Compare the World Champions!", by Charles Sullivan, TrueChess, 2007
  29. ^ "by Charles Sullivan". Truechess.com. Retrieved 21 October 2013.
  30. ^ "by Charles Sullivan". Truechess.com. Retrieved 21 October 2013.
  31. ^ https://www.scribd.com/doc/132380754/Chess-Player-Analysis-by-Rybka-3-14ply
  32. ^ (DanielRensch), Daniel Rensch (24 October 2016). "Better Than Ratings? Chess.com's New 'CAPS' System". Chess.com. Retrieved 17 April 2017.
  33. ^ (DanielRensch), Daniel Rensch (3 January 2017). "Who Was The Best World Chess Champion In History?". Chess.com. Retrieved 3 February 2017.
  34. ^ Who is the master?, ICGA Journal, 39–1, April 2017
  35. ^ Who is the master?, ICGA Journal, 39–1, April 2017 (2.1 Evaluation of the ELO strength of the program used)
  36. ^ Who is the master?, ICGA Journal, 39–1, April 2017 (1 Introduction)
  37. ^ Who is the master?, ICGA Journal, 39–1, April 2017 (2.3 The experimental settings)
  38. ^ Bobby Fischer, "The Ten Greatest Masters in History", Chessworld, Vol. 1, No. 1 (Jan.-Feb. 1964), pp. 56–61.
  39. ^ "Fischer's Top 10". Archived from the original on 6 February 2009.
  40. ^ the Even More Complete Chess Addict, by Mike Fox and Richard James, 1993, pp. 129–30
  41. ^ CHESS magazine, November 1970, p. 70
  42. ^ CHESS magazine, April 1974, pp. 201–202
  43. ^ Twelve Great Chess Players and Their Best Games, Irving Chernev, 1995 (reprint of 1976 edition).
  44. ^ Seirawan, Yasser; Stefanovic, George (1992). "Belgrade; Interview with GM Miguel Quinteros". No Regrets • Fischer–Spassky 1992. International Chess Enterprises. p. 255. ISBN 1-879479-09-5.
  45. ^ "The Grandmaster on his ten greatest chess players". Archived from the original on 20 November 2003.
  46. ^ "He (Fischer) and Kasparov were the greatest in history, but I judge Kasparov as a little ahead. Fischer was a phenomenon from 1970 to 1972 while Kasparov was on top for many years." – Morelia-Linares Super-GM starts today Archived 6 February 2009 at the Wayback Machine, Chessbase, 15 February 2008
  47. ^ Anand takes a dig at Kasparov, Viswanathan Anand, The Hindu
  48. ^ Gligorić, Svetozar (2002). Shall We Play Fischerandom Chess?. B.T. Batsford Ltd. p. 8. ISBN 0-7134-8764-X.
  49. ^ Edmonds, David; Eidinow, John (2004). Bobby Fischer Goes to War. HarperCollins Publishers Inc. p. 310. ISBN 0-06-051024-2.
  50. ^ "The most important interviews by GM Vladimir Kramnik, World Chess Champion 2000–2007". Kramnik.com. Archived from the original on 12 May 2008. Retrieved 21 October 2013.
  51. ^ "Vladimir Kramnik on Chess, Anand, Topalov, and his future". 31 August 2011. Archived from the original on 23 June 2012. Retrieved 28 May 2012.
  52. ^ "Most experts place him (Fischer) the second or third best ever, behind Kasparov but probably ahead of Karpov." – Obituary of Bobby Fischer, Leonard Barden, The Guardian, 19 January 2008
  53. ^ "Aronian names Alekhine best player of all time". WhyChess. 22 August 2012. Archived from the original on 19 November 2012. Retrieved 21 October 2013.
  54. ^ https://www.youtube.com/watch?v=WGdywyQlo2E "Well, in my opinion Garry is the strongest player of all time"
  55. ^ "Magnus Carlsen: – Jeg tar verdensrekorden – VG Nett om Sjakk". Vg.no. 1 January 1970. Retrieved 21 October 2013.
  56. ^ Chess prodigy Magnus Carlsen: 'Bobby Fischer is my dream opponent'., 2015-12-16, CNN
  57. ^ Magnus Carlsen: ‘You need to be very fortunate to be No 1 in fantasy football’, The Guardian, 10 Jan 2020
  58. ^ Magnus Carlsen ranks Garry Kasparov, chess24 YouTube channel, 6 May 2021
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