n 2018, Fabiano Caruana challenged Magnus Carlsen, the defending world chess champion for his title. The format of the match was twelve classical games.  If there was no winner at the end of the twelve classical games, a series of rapid, blitz, and Armageddon chess games were to follow and serve as the tie breakers.  It is beyond the scope of this article to discuss in detail what all of these variations from classical chess are, but suffice it to say each provides the players with less and less time to make all of their moves. All twelve classical games were drawn, the first time that has happened in a world championship match.  Carlsen subsequently won three straight rapid games to retain his title, which he has held since 2013. 

This was not unexpected.  The chess world employs several versions of the Elo rating system, a way to rank the relative strength of players in zero-sum games such as chess.  Again, without explaining the inner workings of the  Elo system, Carlsen was rated at 2843 (number 1 in the world) at the beginning of 2018, while Caruana was rated 2784 (number seven in the world). A player with a 100 point advantage is expected to win 62% of the time. A player with a 200 point rating advantage is expected to win 76% of the time. 

During the last classical game of the match Carlsen had a superior but possibly not a winning position.  He offered Caruana a draw, which was promptly accepted by the Challenger. Many chess players did not understand this course of action, but anyone that has taken a decision analysis class would. 

 Grandmaster Andrew Soltis explained the decision In his outstanding column “Chess to Enjoy”, published in the magazine Chess Life (August 2021 edition): “In rapid games, Caruana was outranked by some 100 rating points. That means Carlsen had about a 64% chance of winning in each game.  That adds up. Since there were four playoff games, the likelihood of a Carlsen victory in the mini-match was about 75%”

Carlsen made a decision about how to proceed based on probability and his knowledge of his opponent.  In classical games, Carlsen’s rating advantage was less than a hundred points, almost a statistical dead heat.  By moving into the rapid chess playoffs, Carlsen believed his rating advantage would be decisive.  It obviously was since he won the first three rapid games to retain his title. 

While Grandmaster Soltis got the overall reason right, his mathematics was a little fuzzy.  The expected value ( a concept known to students of statistics, finance, and decision analysis) of Carlsen winning one game was 64%.  The expected value of Carlsen winning a four game match is still 64%.  Put another way, the 4 games times 64% would lead to an expected value of 2.56.  In chess, where a win counts for 1 point, a draw as a half point, and loss as no points (except in Armageddon chess, but that is a story for another day) Carlsen would be expected to either (1)  win one game and draw three or (2) win two games and draw one.  I believe Grandmaster Soltis just misread the Elo table, mistakenly reading 75% instead of 62%.  Sometimes even the Greats make mistakes. 

Lest you think I am nitpicking the comments of a great chess player ( believe me, he has a vastly superior Elo rating than me), here is my letter to Chess Life and Grandmaster Soltis:

And here is a great presentation from Grandmaster Soltis:

Anyone who wants to learn more about chess would do well to buy one of his books!

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