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3/29/2013 – In his last statistical evaluation of the tournament Jeff Sonas gave Magnus Carlsen a 71% chance of winning the tournament and becoming the challenger. Levon Aronian had a 12% chance and Vladimir Kramnik 9%. In his latest article we see how the values have changes. In addition, if you were wondering how the tiebreak system works in case of a tie, here's a precise explanation.

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.

Candidates – tiebreaks, chances of winning

Levon Aronian's chances to win the 2013 Candidates tournament took a serious blow when he lost his Round 11 game to Peter Svidler. With three rounds remaining, Magnus Carlsen is now the overwhelming favorite to win, as he is in possession of a half-point lead, the highest rating, two games still with the white pieces, and superior tiebreak scores:

After Round 11, Magnus Carlsen appears to have an 87% chance to

win the tournament and become the World Championship Candidate.

By defeating Teimour Radjabov in their individual game, Vladimir Kramnik took over sole possession of second place, half a point behind Carlsen. Going into Round 11, I said that Aronian (at the time, half a point behind Carlsen) had a 21% chance to win the tournament, so you might think that Kramnik (now half a point behind Carlsen) would now have comparable chances. According to my simulations, this is not true:

After Round 11, the odds are 13 to 1 against Vladimir Kramnik winning

the tournament and becoming the World Championship Candidate.

Why is this? Well first of all, time is running out for those chasing Carlsen, and the Round 11 game against Alexander Grischuk was the most likely one for Carlsen to lose, given that he only has one other game left with the black pieces, against the struggling Teimour Radjabov. So Carlsen's draw against Grischuk eliminates all of the various scenarios where Carlsen loses to Grischuk and thereby allows someone else to win the tournament.

But more importantly, remember the tiebreak rules if there is a shared first place:

#1 Head-to-Head

#2 Number of Wins

#3 Sonneborn-Berger Score

#4 Rapids

If someone won the head-to-head "mini-matches", that is the first tiebreak condition. Since Carlsen drew all his games with Kramnik and Aronian, it is extremely unlikely (more than 300-to-1 against) that we will see a tied finish where the head-to-head scores are decisive - the only feasible way this can happen is if Aronian defeats Kramnik and then both those players catch or pass Carlsen – a very unlikely possibility.

So that takes us to the second tiebreak condition: Number of Wins. Going into Round 11, it looked like there was a very real scenario where Carlsen (undefeated) could finish the tournament in a tie with Aronian (having one loss). If this were to happen, it logically follows that Aronian must have had one more win than Carlsen, and so Aronian would win the tournament on tiebreak. And of course, Aronian was only half a point back (before Round 11) and had a substantial possibility to win outright. Now he has less than a 1% chance to win outright, and so his best hope is to stay on two losses (obviously), and to finish in a tie with Carlsen (having zero or one losses) or Kramnik (having zero or one loss). But it is not very likely:

After Round 11, the odds are 17 to 1 against Levon Aronian winning the tournament and become the World Championship Candidate. And there is no practical chance for any of the other five players to win - almost 12,000 to 1 against.

But we are still trying to understand why Kramnik's chances are not higher, despite being only half a point behind Carlsen with still three rounds left.

Obviously Kramnik can pass Carlsen and finish in first outright - but there is barely a 5% chance of this, given Carlsen's strength and extra remaining game with White.

The real problem from Kramnik's perspective is that if he and Carlsen share first place, they probably are both undefeated at that point, and so since they drew their two games with each other, and have the same number of wins, it falls to the third tiebreak condition - Sonneborn-Berger score. This is calculated by taking the total tournament score of the players you beat, plus half of the total tournament score of the players you drew with. Or we can simplify it and just look at the total score of the players you defeated.

Both Carlsen and Kramnik have defeated Svidler once and Grischuk once, so we can simplify even further and ignore those games. This leaves Carlsen's two victories over Gelfand (currently on -1) compared to Kramnik's victory over Radjabov (on -4). If Kramnik catches Carlsen by defeating Gelfand (-1) or Ivanchuk (-3), then Kramnik's Sonneborn-Berger score will be significantly worse than Carlsen's, unless there is lots of movement among the lower ranks during the final three rounds. The only real way to pull this off for Kramnik in a tiebreak is by defeating Aronian (currently on +2). That may be enough to give Kramnik a superior Sonneborn-Berger score (compared to Carlsen), or it may be exactly enough to draw even with Carlsen on all three tiebreaks and thereby advance to rapids. There are lots of possibilities, impossible to calculate directly (especially since it depends on the outcomes of all the other games, especially those involving Gelfand or Radjabov), and so you have to turn to the simulations and see what they yield:

Based on the results so far, I can estimate that if Carlsen and Kramnik have the same number of wins and finish in shared first, there is a 79% chance that Carlsen wins with the superior Sonneborn-Berger score, and a 14% chance that Kramnik wins with a superior Sonneborn-Berger score, and a 7% chance that they will still be tied, and it will go to rapid games. This means that from Carlsen's perspective, a shared first with Kramnik is almost as good as outright first - remember that it is Aronian that Carlsen doesn't want to share first place with! We should pay close attention to the remaining results by Gelfand and Radjabov, because even if their games don't impact their own chances of winning the title, they could have a very strong impact on the Sonneborn-Berger score for Kramnik (who beat Radjabov) and Carlsen (who beat Gelfand twice). It is true that Svidler and Grischuk also lost to Carlsen and Kramnik, but they cancel out because they actually lost to both of the leaders once, and so the final scores for Svidler and Grischuk (wherever they land) will impact the Sonneborn-Berger scores for Carlsen and Kramnik equally. The only feasible way to make a shared first acceptable to Kramnik is if he has defeated Aronian; otherwise Kramnik essentially must finish in clear first in order to win the tournament. A lot of the time in chess, the value of a draw is about halfway between the value of a win, and the value of a loss. Because of the Sonneborn-Berger tiebreak situation, that is not really true in the case of the Aronian-Kramnik Round 12 game. It is a rare case of a game that is still three rounds from the end, with a real impact on the tournament winning odds, in which the optimal tournament strategy for both players is absolutely to go all out for a win.



If Kramnik loses to Aronian, of course that is disastrous for Kramnik, and his chances to win the tournament drop to 1%. But even if he draws with Aronian (normally a great result with Black against such a strong opponent), that likely cements an inferior Sonneborn-Berger score for Kramnik, and his chances to win the tournament drop to 4%. But a victory with Black against Aronian would push Kramnik's chances to win the tournament up to 31%. So really a draw for Kramnik is almost as useless as a loss, in terms of his likelihood to win the tournament. In this special case a draw is only worth about 1/11th of a point to Kramnik, rather than the usual half a point. And for Aronian this is also true; from Aronian's mathematical perspective, a draw against Kramnik is only worth about 1/7th of a point rather than half a point, because he really needs to win his final three games (especially with White) unless Carlsen can only break even over the final three rounds. The optimal tournament strategy at this point for Magnus Carlsen (other than "keep doing what you've been doing!") is a difficult one, because so much depends on the outcome of the Aronian-Kramnik game. Remember that Carlsen will likely lose any tiebreakers to Aronian, but will likely win any tiebreakers against Kramnik. So it is almost like Carlsen's lead over Aronian is currently between half a point and a full point, and his lead over Kramnik is currently between half a point and a full point. If Aronian wins and leapfrogs over Kramnik, then once again Carlsen has to avoid a shared first, since Aronian (with more victories) would almost certainly win the tiebreak based on the Number of Wins condition. Whereas if Kramnik keeps his lead over Aronian, then Carlsen can be content to maintain at least a share of first place, which would very likely give him tournament victory. It all depends on that next game between Aronian and Kramnik; once Round 12 is complete.

Here are the updated odds after Round 11:

Name Score Games remaining Odds of winning Magnus Carlsen +4 2 white / 1 black 87% chance Vladimir Kramnik +3 1 white / 2 black 13 to 1 against Levon Aronian +2 2 white / 1 black 17 to 1 against Peter Svidler even 1 white / 2 black 12,000 to 1 against Boris Gelfand -1 2 white / 1 black 5 million to 1 against Alexander Grischuk -1 1 white / 2 black no chance Vassily Ivanchuk -3 1 white / 2 black no chance Teimour Radjabov -4 2 white / 1 black no chance

And here are the chances of the various final outcomes, based on 10 million simulations of the final three rounds:

Scenario Likelihood Carlsen wins outright 77% chance Carlsen wins from tiebreaker #3 (Sonneborn-Berger) 11 to 1 against Kramnik wins outright 19 to 1 against Aronian wins from tiebreaker #2 (most wins) 22 to 1 against Carlsen wins from tiebreaker #2 (most wins) 60 to 1 against Kramnik wins from tiebreaker #3 (Sonneborn-Berger) 66 to 1 against Aronian wins outright 116 to 1 against Rapids are required 137 to 1 against Aronian wins from tiebreaker #3 (Sonneborn-Berger) 248 to 1 against Kramnik wins from tiebreaker #2 (most wins) 302 to 1 against Aronian wins from tiebreaker #1 (head-to-head) 334 to 1 against Kramnik wins from tiebreaker #1 (head-to-head) 5,400 to 1 against Any other outcome 11,000 to 1 against

Summary

Carlsen has to avoid sharing first with Aronian, and Kramnik has to avoid sharing first with Carlsen, and Kramnik’s only real hope for a shared first being acceptable is if he has beaten Aronian, which means that Kramnik ought to go all out for a win next round with Black against Aronian, who should also go all out for a win against Kramnik. Oh, and if Radjabov moves way up in the standings, and/or Gelfand moves way down in the standings, then perhaps Kramnik’s tiebreak score will increase enough to catch Carlsen's tiebreak score. And anyone else that Kramnik or Carlsen beats in the next 3 rounds will become just as important. This means that essentially all 12 remaining games have a very real impact on the tournament outcome!

Current standings

Schedule and results

Round 1 March 15 at 14:00 Levon Aronian ½-½ Magnus Carlsen Boris Gelfand ½-½ Teimour Radjabov Vassily Ivanchuk ½-½ Alexander Grischuk Peter Svidler ½-½ Vladimir Kramnik Playchess commentary: GM Daniel King Round 2 March 16 at 14:00 Magnus Carlsen ½-½ Vladimir Kramnik Alexander Grischuk ½-½ Peter Svidler Teimour Radjabov 1-0 Vassily Ivanchuk Levon Aronian 1-0 Boris Gelfand Playchess commentary: GM Chris Ward Round 3 March 17 at 14:00 Boris Gelfand 0-1 Magnus Carlsen Vassily Ivanchuk 0-1 Levon Aronian Peter Svidler 1-0 Teimour Radjabov Vladimir Kramnik ½-½ Alexander Grischuk Playchess commentary: GM Yasser Seirawan Round 4 March 19 at 14:00 Magnus Carlsen 1-0 Alexander Grischuk Teimour Radjabov ½-½ Vladimir Kramnik Levon Aronian ½-½ Peter Svidler Boris Gelfand ½-½ Vassily Ivanchuk Playchess commentary: GM Daniel King Round 5 March 20 at 14:00 Vassily Ivanchuk ½-½ Magnus Carlsen Peter Svidler ½-½ Boris Gelfand Vladimir Kramnik ½-½ Levon Aronian Alexander Grischuk ½-½ Teimour Radjabov Playchess commentary: GM Yasser Seirawan Round 6 March 21 at 14:00 Peter Svidler 0-1 Magnus Carlsen Vladimir Kramnik ½-½ Vassily Ivanchuk Alexander Grischuk ½-½ Boris Gelfand Teimour Radjabov 0-1 Levon Aronian Playchess commentary: GM Chris Ward Round 7 March 23 at 14:00 Magnus Carlsen ½-½ Teimour Radjabov Levon Aronian ½-½ Alexander Grischuk Boris Gelfand ½-½ Vladimir Kramnik Vassily Ivanchuk ½-½ Peter Svidler Playchess commentary: GM Alejandro Ramirez Round 8 March 24 at 14:00 Magnus Carlsen ½-½ Levon Aronian Teimour Radjabov 0-1 Boris Gelfand Alexander Grischuk 1-0 Vassily Ivanchuk Vladimir Kramnik 1-0 Peter Svidler Playchess commentary: GM Alejandro Ramirez Round 9 March 25 at 14:00 Vladimir Kramnik ½-½ Magnus Carlsen Peter Svidler ½-½ Alexander Grischuk Vassily Ivanchuk 1-0 Teimour Radjabov Boris Gelfand 1-0 Levon Aronian Playchess commentary: GM Maurice Ashley Round 10 March 27 at 14:00 Magnus Carlsen 1-0 Boris Gelfand Levon Aronian 1-0 Vassily Ivanchuk Teimour Radjabov ½-½ Peter Svidler Alexander Grischuk 0-1 Vladimir Kramnik Playchess commentary: GM Yasser Seirawan Round 11 March 28 at 14:00 Alexander Grischuk ½-½ Magnus Carlsen Vladimir Kramnik 1-0 Teimour Radjabov Peter Svidler 1-0 Levon Aronian Vassily Ivanchuk ½-½ Boris Gelfand Playchess commentary: GM Chris Ward Round 12 March 29 at 14:00 Magnus Carlsen - Vassily Ivanchuk Boris Gelfand - Peter Svidler Levon Aronian - Vladimir Kramnik Teimour Radjabov - Alexander Grischuk Playchess commentary: GM Daniel King Round 13 March 31 at 14:00 Teimour Radjabov - Magnus Carlsen Alexander Grischuk - Levon Aronian Vladimir Kramnik - Boris Gelfand Peter Svidler - Vassily Ivanchuk Playchess commentary: GM Daniel King Round 14 April 1 at 14:00 Magnus Carlsen - Peter Svidler Vassily Ivanchuk - Vladimir Kramnik Boris Gelfand - Alexander Grischuk Levon Aronian - Teimour Radjabov Playchess commentary: GM Maurice Ashley

The games start at 14:00h = 2 p.m. London time = 15:00h European time, 17:00h Moscow, 8 a.m. New York. You can find your regional starting time here. Note that Britain and Europe switch to Summer time on March 31, so that the last two rounds will start an hour earlier for places that do not swich or have already done so (e.g. USA). The commentary on Playchess begins one hour after the start of the games and is free for premium members.