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Candidates – tiebreaks, chances of winning

3/31/2013 – Round 12 of the Candidates Tournament for the World Chess Championship was a thrilling affair, and it ripped a huge hole in the sail of Magnus Carlsen, who had been coasting to victory. Now the odds have been overturned and Kramnik is the statistical favourite to win. But it all depends on the tiebreak, as Jeff Sonas explains. Anyone for an expert lesson in statistics and chess?
 

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 statistics going into round 13

After a thrilling Round 12 victory with the black pieces against Levon Aronian, Vladimir Kramnik has jumped into the sole lead at +4, and suddenly is the tournament favorite! There are very few scenarios where Kramnik shares first place and wins on tiebreak, so his best hope is to win the tournament outright, which is now quite likely:

After Round 12, Vladimir Kramnik appears to have a 65% chance to win
the tournament and become the World Championship Candidate.

By itself, Kramnik's unlikely defeat of Aronian with Black was not devastating to Magnus Carlsen's chances. If it had been coupled with a Carlsen-Ivanchuk draw, then Carlsen and Kramnik would be sharing first place at +4, and since Carlsen's Sonneborn-Berger score is superior to Kramnik's, a shared first with Kramnik would be almost as good as clear first ahead of Kramnik (from Carlsen's perspective).

And by itself, Carlsen's unlikely loss to Vassily Ivanchuk with White would not have been devastating to Carlsen's chances. If it had been coupled with an Aronian-Kramnik draw, then Carlsen and Kramnik would be sharing first place at +3, and Carlsen sharing first with an undefeated Kramnik gives Carlsen the tiebreak 100% of the time. That last one is worth repeating – now that Carlsen has lost a game, if you remember just one thing from this article, remember this:

Carlsen sharing first with an undefeated Kramnik gives Carlsen
the tiebreak (and tournament victory) 100% of the time.

But the two unlikely outcomes, combined with each other, are devastating to Carlsen's chances. In one day, the chances of Kramnik winning clear first place have jumped from 5% to 60%, and now Carlsen's only consolation is that he still holds a superior tiebreak against Kramnik, and will likely win the tournament if he can just grab a share of first place:

After Round 12, Magnus Carlsen appears to have a 35% chance to win
the tournament and become the World Championship Candidate.

Thanks to his head-to-head loss against Kramnik, Levon Aronian must now finish ahead of Kramnik in order to win the tournament, and so there is a very long list of things that must happen in order for Aronian to triumph.  First of all, Aronian must win his last two games (ending at +3) and Kramnik must lose his last two games (ending at +2).  Further, Carlsen must finish at +3 or worse (he is currently at +3). If Aronian and Carlsen finish in a two-way tie for first at +3, then Aronian wins the tiebreak due to having more wins.  With so many requirements, Aronian essentially has no chance to win the tournament anymore.

Peter Svidler could theoretically finish in shared first at +2, with Kramnik losing twice to finish at +2 as well, and Carlsen and Aronian both no higher than +2, but it turns out that due to the head-to-head considerations (Kramnik having defeated Svidler and Aronian now) there is no group that could finish in shared first at +2 (including Svidler and Kramnik) where Svidler survives the tiebreak against Kramnik. So that is why Svidler literally has no chance of winning the tournament, despite a microscopic chance of sharing first place.

After Round 12, the odds are 7,100 to 1 against Levon Aronian winning the tournament and becoming the World Championship Candidate, and none of the other five players has any chance at all.  However, even the upcoming games not involving Kramnik or Carlsen may nevertheless have vital importance, since they affect the Sonneborn-Berger tiebreaker score for Kramnik and Carlsen in the event that Kramnik loses at least once and finishes in shared first place with Carlsen.

So if we set aside all those other six players from consideration, it is just a question now of Kramnik and Carlsen. Here are the key findings from my Monte Carlo analysis, considering the myriad possible ways those two players could finish in shared first, including how the final scores of the other six players would impact the Sonneborn-Berger scores of Carlsen and Kramnik,

  1. If Carlsen catches Kramnik by winning one more game than Kramnik during Rounds 13/14, and neither of them lose any more games, then once-beaten Carlsen would share first with an undefeated Kramnik and Carlsen would win the tournament due to the tiebreaker for Number of Wins.

  2. If Carlsen catches Kramnik because Kramnik loses one of his next two games, then it will come down to the Sonneborn-Berger scores, which depend on all the total scores of the various players that they beat or drew with or lost to. It appears that if it comes to this, there is a 63% chance that Carlsen's SB-score would be higher, a 24% chance that Kramnik's SB-score would be higher, and a 13% chance their scores would be identical, leading to a need for rapid games.

So, if Kramnik and Carlsen share first place, there is a 92% chance that Carlsen wins the tiebreak based on Number of Wins or Sonneborn-Berger score, and only a 5% chance that Kramnik wins it based on Sonneborn-Berger score, and a 3% chance that their tiebreaker scores are completely equal and therefore rapid games are required. 

Therefore a shared first place for Magnus Carlsen is likely going to be good enough to win the tournament.

  1. Simulations tell us that Kramnik is about 80% likely to finish at +4 or +5, and Carlsen is about 80% likely to finish at +3 or +4.  That gives four possible combinations that are very plausible - and in three of those four, Kramnik wins outright.  In the fourth - where they both finish at +4 - it is Carlsen who almost certainly wins the tournament, since… repeat after me…

Carlsen sharing first with an undefeated Kramnik gives Carlsen the tiebreak
(and tournament victory) 100% of the time.

And here is a complete list of the possible remaining scenarios, sorted by their likelihood:

Kramnik wins outright
63% chance
Carlsen shares first with Kramnik and wins from tie-breaker #2: most wins
20% chance
Carlsen wins outright
11% chance
Carlsen shares first with Kramnik and wins from tiebreaker #3: SB-score
28 to 1 against
Kramnik shares first with Carlsen and wins from tiebreaker #3: SB-score
73 to 1 against
Kramnik and Carlsen share first and have exactly the same
tiebreaker scores, so rapids are required
140 to 1 against
Kramnik shares first with Aronian (and possibly Carlsen) and wins
from tiebreaker #1:head-to-head
318 to 1 against
Aronian shares first with Carlsen and wins from tiebreaker #2: most wins
7,900 to 1 against
Aronian wins outright
75,000 to 1 against

If Kramnik finishes unbeaten but Carlsen catches him by winning more games during the last two rounds, then Carlsen automatically wins the tournament. It is only the less likely shared first, the one where Carlsen catches Kramnik because Kramnik loses a game, that will consider Sonneborn-Berger score (since in that scenario they would have the same number of wins). This is relatively unlikely, as you can tell from the small slivers in that pie chart that mention "tiebreaker #3: SB-score", but it is still worth investigating, since it is the hardest to conceptualize.  It was much more likely to matter 24 hours ago, before Carlsen had lost a game, but we will still look at it.

Again, remember that Sonneborn-Berger is calculated as the total score of the people you beat, plus half the total score of the people you drew. For this tournament, we can also think of it as the sum of scores of the people you beat, minus the sum of scores of the people you lost to. And you can also ignore people that both players did equally well against - they both beat Svidler and Grischuk, so we can ignore their scores.  Carlsen beat Boris Gelfand twice, so (for the purpose of SB-score) he really wants Gelfand to do well in his remaining games (particularly the one against Kramnik!).  Whereas Kramnik beat Aronian and Radjabov, so (for the purpose of SB-score) he wants them to do well (particularly Radjabov's game against Carlsen!).

The two simplest examples where Sonneborn-Berger will matter are the ones where Kramnik loses one more game and otherwise Kramnik and Carlsen draw their remaining games to finish on shared +3. This might mean Kramnik losing with White to Gelfand in Round 13, or with Black to Ivanchuk in Round 14.

  1. If Kramnik loses with White to Gelfand in Round 13, then it is a disaster for Kramnik's Sonneborn-Berger score, and essentially means that Carlsen will always win the tiebreaker against Kramnik if they share first place.  Their relative SB-scores would be very sensitive to Gelfand's final score because Carlsen would have beaten Gelfand twice, whereas Kramnik would have lost to him.

  2. On the other hand, if Kramnik loses with Black to Ivanchuk in Round 14, and otherwise Kramnik and Carlsen draw their remaining games to finish on shared +3, then their SB-scores could swing one way or the other based on what happens during Alexander Grischuk's games in the final two rounds.  This is because Kramnik beat Aronian, so he would want Aronian to do well against Grischuk in Round 13 and boost Aronian's total score, whereas Carlsen defeated Gelfand twice, so (again) he would really want Gelfand to do well against Grischuk in Round 14 and boost Gelfand's total score.

It is an interesting counterpoint to the idea of using rapid games as a tiebreak – using Sonneborn-Berger first does make it much less likely that rapid games are needed, but on the other hand it potentially places very large importance on the outcome of games between players who no longer have any chance to win the tournament.

It is a bit depressing that there are only two rounds left, and all of this is still so complex, but perhaps this is a good way to characterize the tiebreak situation with regard to Sonneborn-Berger score:

  1. The better that Boris Gelfand is doing in his games, the more likely it is that Carlsen is happy to share first place with Kramnik. 

  2. The better that Alexander Grischuk is doing during his Round 13 game against Levon Aronian, the more likely it is that Carlsen is happy to share first place with Kramnik.

Standings after twelve rounds

Pairings for the next rounds

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

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
0-1
Vassily Ivanchuk
Boris Gelfand
½-½
Peter Svidler
Levon Aronian
0-1
Vladimir Kramnik
Teimour Radjabov
½-½
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.

Links

The games will be broadcast live on the official web site and on the chess server Playchess.com. If you are not a member you can download a free Playchess client there and get immediate access. You can also use ChessBase 12 or any of our Fritz compatible chess programs.

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Topics Candidates

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