Statistics in the Candidates
The Candidates Tournament is coming, and the field is closer than ever. Carlsen says picking a favorite “would be impossible, and [he thinks] it’s going to be very close probably and be decided in the last round.” But how close do the players in the tournament actually are to each other?
The largest difference in the live ratings is 43.3, between Caruana (2794.3) and Svidler (2751). And as you probably already know, the ratings are very good indicators of actual strength, and are pretty accurate predictors of performance.
One of the advantages of the ELO system is that it allows us to get at the expected scores a player would have if paired against another player. For example, If Caruana (2794.3) plays against Svidler (2751), the expected score would be .56 for Caruana and .44 for Svidler. That is, in a match Caruana is expected to get 56% of the points, and Svidler 44%.
The problem is, expected scores tell us how many points a player is expected to get against another player, but they don’t tell us how many of those points would be due to wins or due to draws. For example, in a 100 game match, Caruana could win 56 games and lose 44, or he could win 18 and draw 76 (wins are worth 1 point and draws .5).
To get around this problem, I used each payers’ historical record against each another to get to their draw ratio (total draws/total number of games played against one another). In the case of Caruana and Svidler, they have drawn 12 times out of 17 games played. Thus, their draw ratio is .71.
Then, if the expected score for Caruana is .56 and their draw ratio is .71, to solve for the probability of Caruana winning I solved .56(1) -- .71(.5) =.209. Thus, the probability of Caruana winning against Svidler, using their live ratings and their historical draw ratio, is 20.9%, while they have a 71% chance of drawing and Caruana has a 8.5% chance of losing.
I followed this procedure to pair all of the players against one another and got the following charts:
After getting the average number of wins per player, I solved for the expected value per game per player, which is the number of points a certain player is expected to get per game. Then, I multiplied this by the number of games they will play in total (14) to reach at the number of points each player is expected to have at the end of the tournament. Finally, I divided the expected number of points by the total expected number of points to get to the probability each player has of winning the Candidates.
Of course, the probabilities do not consider physiological factors, or the specific strategy a player will follow during the tournament. However, because ratings are very good predictors of who will win between two players, I believe these are decent estimates of actual probabilities.
As you can see, Carlsen is right. The players’ chances of winning are actually very close.
Who do you think will win the candidates? The probabilities favor Caruana.