3/26/2013 – Last night an old friend – a very strong player – asked us to estimate the chances of Magnus Carlsen finishing this tournament as the winner. He thought it was four to one. Today we received exact calculations from statistician Jeff Sonas. They contain some remarkable information. Did you know that for Carlsen finishing tied for first is almost as bad as finishing second? Read and believe.

From March 14 to April 1, 2013, FIDE and AGON – the World Chess Federation’s commercial partner – are staging the 2013 Candidates Tournament for the World Chess Championship 2013. It will be the strongest tournament of its kind in history. The venue is The IET, 2 Savoy Place, London. The Prize Fund to be shared by the players totals €510,000. The winner of the Candidates will become the Challenger to Viswanathan Anand who has reigned as World Champion since 2007. The main sponsor for the Candidates is State Oil Company of the Azerbaijan Republic SOCAR, which has sponsored elite events chess in the past.

Who will win the Candidates Tournament?

By Jeff Sonas

Based on the current standings of the fantastic London FIDE Candidates tournament, it is clear that Magnus Carlsen is the most likely winner. His pre-tournament rating made Carlsen the clear statistical favorite coming in, and if anything he has slightly outperformed that rating. With five rounds remaining, Carlsen is currently alone in first place with a +3 score.

After Round 9, Magnus Carlsen appears to have a 75% chance

to win the tournament and become the World Championship Candidate.

However, a close inspection of the crosstable and the tiebreaking rules, reveals that there are still many possibilities. If two players finish in a tie after Round 14, it is very unlikely to be resolved by rapid games, as there are first several tiebreak conditions that need to be evaluated. In fact there is a very real scenario (about 10% likely) where Carlsen (currently at +3 with three wins and no losses) and Levon Aronian (currently at +2 with three wins and one loss) each go undefeated the rest of the way, and Aronian makes up half a point across the final five rounds and wins on tiebreak thanks to having one more win than Carlsen.

After Round 9, Levon Aronian appears to have a 21% chance

to win the tournament and become the World Championship Candidate.

In third place is former World Champion Vladimir Kramnik, who came very close to defeating Carlsen in Round 9 and achieving a tie for first place, but instead had to settle for a draw. Before that game, Kramnik had an estimated 7% chance to win the tournament, and defeating Carlsen would have increased those chances to 25%. But having only two more games with the white pieces (Carlsen and Aronian each have three games with White left), and sitting a full point behind Carlsen, Kramnik (currently at +1) will need significant help from others to win the tournament.

After Round 9, the odds against Vladimir Kramnik winning the tournament and becoming the World Championship Candidate are 38-to-1 against. And the remaining five players (Boris Gelfand, Alexander Grischuk, Peter Svidler, Vassily Ivanchuk, and Teimour Radjabov) have less than a 1% combined chance to win:

These calculations are based on a Monte Carlo simulation model that I have developed and refined over the years for chess predictions. I didn't make use of the players' current FIDE ratings; instead, I actually utilized my own invented Chessmetrics formula, which I have shown in the past to be a better predictor of future results than the Elo rating. Nevertheless those ratings were quite similar and yielded the same favorites in both formulas:

Name FIDE rating Chessmetrics rating Magnus Carlsen 2872 2870 Levon Aronian 2809 2809 Vladimir Kramnik 2810 2808 Teimour Radjabov 2793 2787 Vassily Ivanchuk 2757 2766 Alexander Grischuk 2764 2762 Peter Svidler 2747 2747 Boris Gelfand 2740 2737

And when I used my normal approach for simulating the tournament before it started, these ratings yielded the following pre-tournament odds of winning:

Name Pre-tournament odds of winning Magnus Carlsen 42% chance Levon Aronian 15% chance Vladimir Kramnik 15% chance Teimour Radjabov 10% chance Vassily Ivanchuk 17 to 1 against Alexander Grischuk 17 to 1 against Peter Svidler 26 to 1 against Boris Gelfand 32 to 1 against

It is important to realize, however, that a chess rating is not a precise measurement of a player's ability. It is an overall measurement of their past results, and (at the same time) an estimate of their current ability. That estimate has a larger error than most people realize - something like plus-or-minus 40 or 50 for these top players. So although Carlsen has had the greatest success in the recent past, the statistics still admitted the small possibility that other players might actually be stronger. In fact, as the tournament progresses, and we get more and more evidence of their current strength, a Bayesian approach allows us to actually revise our estimate of each player's strength, in order to more precisely calculate their chances in the remaining games.

But unfortunately for the rest of the field, Carlsen and Aronian seem to be performing even higher than their ratings would suggest, with Carlsen approximately at a 2890-level and Aronian still about 60 points behind him. Some other players, notably Teimour Radjabov and Vassily Ivanchuk, have definitely underperformed and this impacts their predicted results for the last few rounds. My simulations incorporate a mid-tournament adjustment to our estimate of everyone's strength:

Name Chessmetrics

rating Expected score

after 9 rounds Actual score

after 9 rounds Adjustment to

Chessmetrics rating Carlsen 2870 +2 +3 2887 (17 higher) Aronian 2809 Even or +1 +2 2828 (19 higher) Kramnik 2808 Even or +1 +1 2815 (7 higher) Radjabov 2787 Even -3 2745 (42 lower) Ivanchuk 2766 Even -2 2748 (18 lower) Grischuk 2762 -1 or Even Even 2768 (6 higher) Svidler 2747 -1 -1 2744 (3 lower) Gelfand 2737 -1 Even 2750 (13 higher)

And if we use those updated estimates, and then simulate the remaining five rounds a million different times, we can look at how frequently each player wins the tournament. We find, of course, that Magnus Carlsen has improved his chances considerably. He was given a 42% chance to win the tournament originally, and now those chances have risen to 75%, with Levon Aronian hanging on to a 21% chance, and the odds not looking very good for anyone else:

Name Current score W/B games remaining Odds of winning tournament Carlsen +3 3 white / 2 black 75% chance Aronian +2 3 white / 2 black 21% chance Kramnik +1 2 white / 3 black 38 to 1 against Grischuk Even 3 white / 2 black 148 to 1 against Gelfand Even 2 white / 3 black 486 to 1 against Svidler -1 2 white / 3 black 3,000 to 1 against Ivanchuk -2 2 white / 3 black 17,000 to 1 against Radjabov -3 3 white / 2 black 100,000 to 1 against

It is worth mentioning the tiebreak rules at this point. There are four levels of tiebreak:

Direct encounter (i.e. head-to-head) Greatest number of victories Sonneborn-Berger score Rapids

Currently there is an 83% chance that someone will win the tournament outright, and less than a 1% chance that it will be decided by tiebreak condition #1 (direct encounter). This is mostly because Carlsen has already drawn his mini-matches against Aronian and Kramnik, and has already defeated the next three players in the crosstable, so nobody gets the head-to-head tiebreak advantage against Carlsen, even if one of those three players (Gelfand, Grischuk, or Svidler) does manage to beat him.

But on the other hand, Carlsen doesn't really need his current head-to-head tiebreak advantages against Gelfand, Grischuk, or Svidler, because they probably can't catch him without beating him. So Carlsen almost has to win the tournament outright – if the final round finishes with two or more players tied, it is 95% likely that the head-to-head comparison will not resolve anything, and it will then fall to who has the most wins. And if someone catches Carlsen, it will almost certainly be thanks to picking up a couple more wins during the remaining rounds.

So in fact, by far the most likely tiebreaker outcome is the scenario where Aronian, currently tied with Carlsen for the most wins, manages to surpass him by an extra win, and they finish tied but the tiebreaker goes to Aronian (as the head-to-head would be a tie, so the second tiebreak – most wins – goes to Aronian. Another way to summarize the remaining possibilities is like this:

Scenario Likelihood Carlsen wins outright 70% chance Aronian wins outright 10% chance Aronian wins from tiebreaker #2 (most wins) 10% chance Carlsen wins from tiebreaker #3 (Sonneborn-Berger) 30 to 1 against Kramnik wins outright 59 to 1 against Carlsen wins from tiebreaker #2 (most wins) 102 to 1 against Rapids are required 201 to 1 against Kramnik wins from tiebreaker #3 (Sonneborn-Berger) 208 to 1 against Aronian wins from tiebreaker #3 (Sonneborn-Berger) 214 to 1 against Any other outcome 45 to 1 against

So while things look pretty good for Carlsen right now, that can change in a hurry if Aronian catches up to him with a win while Carlsen draws. On the other hand, if Aronian catches up to him with a draw while Carlsen loses, then they have the same number of wins, and Carlsen will likely win the Sonneborn-Berger since he did so well against the middle of the pack. This suggests that Aronian would do well to be very aggressive in his games, since playing it safe and drawing the rest of the games and hoping Carlsen loses one, would likely not win the title for Aronian.

And for Carlsen, the equation is pretty simple – finishing tied for first is almost as bad as finishing second. He is not very likely to win a tiebreak scenario (if the tournament does finish in a tie for first, it is almost 75% likely that someone other than Carlsen will win) so he needs to finish in first outright, which means hanging on aggressively to his tournament lead. We can all be very thankful for the great show that the organizers have put on, and to the players for their very entertaining play thus far!

Some people may be interested in more details of my methodology, and to compare and contrast it against the predictive approach suggested by Peter Zhdanov a couple of weeks ago. I will be describing this methodology in a later article, after the Candidates tournament has concluded.