There is agreement on a statistical system to rate the strengths of current players, called the Elo system, but disagreement about methods used to compare players from different generations who never competed against each other The most well known statistical method was devised by Arpad Elo in 1960 and elaborated on in his 1978 book The Rating of Chessplayers, Past and Present.
According to this system the highest ratings achieved were: Though published in 1978, Elo's list did not include five-year averages for later players Bobby Fischer and Anatoly Karpov.
[3] Many people believe that this rise is mostly due to an anomaly known as ratings inflation, making it impractical to compare players of different eras.
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".
He also published the following list of the highest ratings ever attained according to calculations done at the start of each month:[15] In contrast to Elo and Sonas's systems, Raymond Keene and Nathan Divinsky's book Warriors of the Mind[16] 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).
A computer-based process of analyzing chess abilities across history came from Matej Guid and Ivan Bratko at the University of Ljubljana, Slovenia, in 2006.
[21] 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.
[22] In 2017, Jean-Marc Alliot of the Toulouse Computer Science Research Institute (IRIT) presented a new method,[23] based on a Markovian interpretation of a chess game.
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.
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.
Larry Kaufman published an article in 2023 estimating the ratings of chess players throughout history by comparing their games with the choices of top engines, using Chess.com accuracy scores.
He estimated ratings for 47 players, with the following top ten:[24] (Kaufman noted however that Fabiano Caruana could not be considered by his methodology because his World Championship match had all classical games drawn.
)[24] Kaufman concluded that the quality of play rose steadily by about 2.5 Elo points per year from 1900 to 2023, though he conceded that the rate was greater in the 19th century and thus that Paul Morphy "might have rivaled Fischer for the top spot if we could properly correct for these factors".
In 1964, Bobby Fischer listed his top 10 in Chessworld magazine: Morphy, Staunton, Steinitz, Tarrasch, Chigorin, Alekhine, Capablanca, Spassky, Tal, and Reshevsky.
"[27] In 1970, Fischer named Morphy, Steinitz, Capablanca, Botvinnik, Petrosian, Tal, Spassky, Reshevsky, Svetozar Gligorić, and Bent Larsen the greatest chess players in history.
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.
[40] 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.
At the end of the video, Hikaru said he "can live with" the top 5 as: Kasparov, Carlsen, Fischer, Capablanca and Karpov but he would put from 6 through 10: Anand, Kramnik, Botvinnik, Lasker, Morphy.
During Game 6 of World Chess Championships 2023, as he was commenting on the game, Hikaru mentioned Magnus Carlsen, Garry Kasparov, Vishy Anand, Bobby Fischer, Anatoly Karpov, José Raúl Capablanca and Vladimir Kramnik as the top chess players of all time in order.