Happy Tiebreak Day everyone. This is as fun as it gets for chess fans.

Kramnik's amazing with these Be3 Grünfeld lines

What an epic round that was!

Wei Yi goes with the Evan's Gambit in his must-win game as white. I didn't think I could like this kid any more than I already did! Mamedyarov declined it though. Boo hiss.

With round three completed, I did some more detailed math, projecting the entire rest of the tournament out (using the laughable assumption that the "expected result" you get when you compare ELO ratings is exactly what the odds are of each player winning any specific mini-match.) Ignoring that my method for putting odds on any given mini-match is rather suspect, this is otherwise a 100% probabilistic model - it fully accounts for brackets, and factors in the respective odds of each possible matchup as it predicts each player's odds of advancing in a given round. And it uses live ratings, not published ratings, to theoretically represent current playing strength as well as possible, and to at least give some credence to results so far in the event. So here are my predicted odds for each of the 16 players remaining in the field (split into two groups reflecting which half of the bracket they're in):

As you can see, Kramnik is the most likely champion at this point, and his most likely opponent in the finals is Karjakin, so until round four blows everything up, those two are my temporarily projected finalists (and Candidates qualifiers). Also of note is that the bottom half of the bracket (with the #2, #3, and #6 seeds all still alive) is obviously a bit stronger than the top half (where #1, #4, and #7 seeds have all been eliminated.) The Kramnik/Caruana/Nakamura side of the bracket has a 53.1% chance of producing the World Cup winner.

http://www.chessbase.com/Home/TabId/...ut-200813.aspx "Pictures provided by Paul Truong in Tromsø". Bleh Truong, I still dislike him back from chess.net days (and I think his scumminess/craziness was further proven in the uscf mess he was in iirc)

Is that Susan Polgar's husband? The name sounds familiar for some reason.

Yep.

Gelfand seems to have done his homework

Interestingly, in my math on trying to project these results, it appears that by drawing with the black pieces AS FAVORITES, Gelfand and Karjakin both added almost 10 percentage points to their overall equity in winning the round, however Korobov, drawing with the black pieces AS AN UNDERDOG gained almost zero equity in his mini-match.

So it looks like as a favorite with white in the first game, it's actually pretty safe to take an early draw, because the threat of your opponent beating you (even with white) is pretty low, and you still have the edge in tiebreaks as well. And of course as the favorite with black in the first game, a draw is a great result, as much of your opponents equity is tied up in the relatively unlikely hope of stealing a win with white, and when you squelch that, now you get a chance to put it away with white.

I guess this isn't that interesting after all. It basically just means that being the stronger player is beneficial, and that while draws always benefit black to some extent, they also benefit the stronger player, and those two factors can cancel each other out if the rating gap is large enough.

Kamsky is playing an incredibly brilliant game vs. Mamedyarov. He needs to finish it with little time on his clock but it looks good right now.

What a masterpiece by Kamsky! Without a doubt a game of the tournament.

Looks like Ivanchuk just blundered, and now Kramnik will likely pick up the third decisive result of the day. And with white tomorrow, that should leave him pretty much a lock to advance to the quarterfinals.

yep Sure looked like he'd hold it

Vlad post-game analysis on now

How valuable is a win in game one of the classical mini-match? Entering today I had the collective odds of one of Kamsky/Caruana/Kramnik ultimately winning the World Cup at 24.9%. Now I have it at 40.2%. Kramnik is the favorite to reach the finals out of his half of the bracket, and Kamsky is the favorite to reach the finals from his half (the only way this changes is if Morozevich swindles a win from Tomashevsky in the one game still being played, since Moro has the added virtue that Kamsky lacks of being rated higher than his opponent, which makes his win that much more likely to hold up.)

Kramnik postgame interviews are always awesome.

Your Korobov result doesn't make sense if you're just using rating differentials and not some kind of player-specific results with colors or skewing the tiebreaks way more to the favorite than the classical games, which would seem odd.

The key is my draw rate that I'm using. Solely on rating differential, Nakamura should score 59.2% against Korobov. Therefore I assumed that Nakamura will win the mini-match 59.2% of the time, and also that if it goes to tiebreaks, Nakamura will win the tiebreaks 59.2% of the time.

After the draw, we now have one classical game left, with Korobov as white. I use a value of white being worth 40 ELO points, so in that one game, Korobov's expectation is a bit better. Specifically he should score 46.1% (rather than just 40.8%). Since I assume that 55.8% of games are drawn (based on results so far in the tournament), this means that Korobov should win tomorrow 18.2% of the time, and Nakamura should win 26% of the time, with a 55.8% chance that it goes to tiebreaks.

Therefore IF there is a decisive game tomorrow, there is a 58.8% chance that it goes in Naka's favor. And if there is NOT a decisive game, then we go to tiebreaks, which go in Naka's favor 59.2% of the time. It's coincidence that Naka outrates Korobov by just the right amount, to where Naka's winning odds with black (excluding draws) line up so perfectly with his overall score expectation (including draws).

So there is a skew for color, but there is not any sort of player-specific skew.

Edit: The one other factor is that I did update the live ratings mid-way through the calculations, and the 0.8 rating point gain for Korobov, and corresponding drop for Naka, does affect the numbers at the tenth-of-a-percent level. It has nothing to do with why the odds are *essentially* unchanged, though, that's as described above.

Yeah, that's just wrong (would Naka also only win 59.2% of the time if it were a 24-game match?).

If you take game results as independent, that would make game 1 (approximately) 36% score for korobov, meaning 55.8% draw, 8.1% win, 36.1% loss.

So Koro wins the minimatch with

8.1% * 55.8 + 18.2 * 55.8 + 18.2*8.1 = 4.5% +10.2% +1.5% = 16.2%

Naka wins it with

26%*55.8 + 36.1*55.8 + 26*36.1 = 14.5% + 20.1 + 9.4 = 44%

and it draws with

55.8*55.8 + 8.1*26 + 18.2*36 = 31.1 + 2.1 + 6.6 = 39.8

and that adds up to 100 so I probably did it right. Splitting the tiebreaks 59/41 (that's more reasonable in quicker games in general I'd think) means Naka wins 44% + .59*39.8% = 67.5% and Koro 16.2% + .41*39.8% = 32.5%.

So Koro picked up 8%ish with his draw as black.

Well yes. It is wrong. And if you've been following all of my posts about odds on this event, I referred to it as a "laughable assumption" from the beginning. The problem is that while it's relatively easy to break down current rounds in full detail, it's harder to build a full bracket that calculates those odds for all possible future matchups. Not impossible, but it would have been more effort than I felt I had time for. And it would still have the flaw of assuming that the two games are independent results, so it would be a whole lot of extra work for maybe(?) more accuracy.

At some point, I will put more work into building a model for predicting matches of varying lengths based on relative ELOs, that accounts for match strategies (in the form of varying draw rates in different situations). Once I have that model built, the next time I project the World Cup I'll be able to do so more accurately. For now, I'm just sticking with my quick and dirty assessment.

Obviously any odds I post are for entertainment only.

I hadn't read everything about the calculation, but yeah, game 2 should have a lower draw probability after a decisive result, I'd think, so my number shouldn't match reality either. But the weaker player drawing game 1 with black is always going to be a significantly good result.

Everything else you said is dead on, but I've been wondering about this piece for a while. Obviously if both players are eager for a draw (see Dubov/Ponomariov) then the draw rates go up (and the expected score for the favorite should drop, since decisive games are where the favorite's equity must come from, by definition).

What happens, though, in must win situations? When one person is more willing than usual to accept a draw, that should increase the draw rate. But when the other player is less willing to accept one, that should decrease the draw rate. I don't know if this tournament has produced a large enough sample size to prove anything, but I'm very curious whether the draw rate actually changes when someone is 1-0. And if so - do draws become more or less common?

Edit: Note how neatly I shift the topic of conversation away from the flaws in my methodology?

I don't know. If white has to win, which happens more often, I'd guess more decisive results. If black has to win, maybe white can just kill play more than usual. I could see that splitting opposite ways. And you have the rating considerations where the guy down 1-0 and losing again might offer/accept a match-conceding draw to try to save a few points, although those shouldn't change the minimatch win% appreciably.

Interesting, I just pulled the data so far, and it looks like when the first game IS decisive, the draw rate in game two remains nearly unchanged, but once the first game is drawn, both players become reluctant to risk everything on one game, shell up for tiebreaks, and the draw rate skyrockets.

Here are the numbers:
First game: 56 draws in 111 games = 50.5% draw rate
Second game (after decisive result): 28 draws in 55 games = 50.9% draw rate
Second game (after draw): 43 draws in 56 games = 76.8% draw rate

And the decisive games broken down by color:
When black must win (game 1 was won by black, the player up 1-0 now has white), black scores +3-9=7
When white must win, white scores +7-8=21

So when a player faces a must-win situation, they're obviously in trouble, and only succeed in winning 16% of the time with black, or 19% with white, but when they fail with black they're much more likely to fail and lose (draw rate only 37%), while when they fail with white they're much more likely to hold the draw (draw rate 58%). Of course those must-win sample sizes are pretty small and thus inconclusive - especially the black-must-win set of 19 games...