gelfand is a boss at blowing draws.

Yeah openings are not everything.

Went to the tournament today. Great day's play. The setup there is excellent with tablets everywhere that let you follow just about anything.

So now that we're three rounds in, it's time to revisit the odds. I re-ran the simulation, with rounds 1-3 set in stone, to see what things look like at this point. Before I share updated odds, here are the obligatory methodology notes. First of all, obviously, all simulations now have round 1-3 results locked, and are only randomizing results for rounds 4-14. Secondly, I have updated everyone's ELO rating using the current liveratings at 2700chess.com. This means Aronian has not only gained equity by performing above expectations in the first three rounds, but also by increasing his expectations in the remaining rounds by bumping his rating up by about 9 points thusfar. Third, I added the 12 games played so far to the dataset from which expected draw rates and white advantages are being calculated. This increased the sample size by roughly 2%, and did have some relevant if minor tweaks to expectations as well.

So what's changed? Aronian is obviously in MUCH better shape, picking up a lot of win equity, and has turned this into a two-horse race. Carlsen is still the favorite, but by a much less convincing margin. Svidler's win today is nice, and his odds are better when we started, but really it changed little, as his ELO deficit is still too large for the one win to give him much realistic hope of winning it all in today's estimations. So without further ado, here are the updated odds of tourney results:

PLAYER - Odds of winning - Expected score:
Carlsen - 58.8% - 9.07
Aronian - 31.5% - 8.55
Kramnik - 4.5% - 7.39
Radjabov - 2.3% - 6.99
Svidler - 1.8% - 6.84
Grischuk - 1.0% - 6.62
Ivanchuk - 0.03% - 5.30
Gelfand - 0.02% - 5.25

Other updated odds: White's expected score for the event is +12.9 =36.3 -6.8, so the 5 decisive games in 3 rounds lowered the overall draw expectation by about two draws over the course of the event, and with 2 of the 5 wins so far, black is still expected to account for about 34% of the overall wins that occur, just like before.

The odds of a sole champion after round 14 remain exactly 81.7%, but the odds of a two-way tie are up to 15.8% (guess who those two are most likely to be), while the collective odds of a 3+ way tie have dropped accordingly.

And finally, with ratings, draw rates, etcetera all updated, here are the odds on tomorrow's games:

Carlsen (W) vs. Grischuk (B):
Carslen wins - 42.9%
Grischuk wins - 3.0%
Draw - 54.1%

Radjabov (W) vs. Kramnik (B):
Radjabov wins - 13.6%
Kramnik wins - 6.3%
Draw - 80%

Aronian (W) vs. Svidler (B):
Aronian wins - 33.5%
Svidler wins - 3.5%
Draw - 63%

Gelfand (W) vs. Ivanchuk (B):
Gelfand wins - 19.6%
Ivanchuk wins - 12.8%
Draw - 67.6%

Er, I called them "tomorrow's games", but should have said "round four's games", as tomorrow is a rest day.

Tournament slowing taking shape...only surprise about it being "who can beat Gelfand and Ivanchuk the most?" is that Svidler isn't on that list.

Curious to see how many decisive games come from the six played among Carlsen/Aronian/Kramnik. (I love the simulations, by the way, BJJ...is it easy to see how many are expected to that question?)

So... Aronian ftw? He's got momentum now and white in the next round.

That would be easy to answer. Actually we could do it precisely. We're only talking about 6 total games, and 2 of them have already been played (to draws), and by narrowing the results we care about to two options (draw or decisive) there's only 16 possible ways it can play out. I can do that calculation by hand. I'll do the math on my lunch break at work tomorrow, and get back to you with an answer tomorrow night

Well, the newest sim already accounts for his momentum (in the form of using his liverating - 9 elo points do for your expected results over 11 games) and is fully aware of his having white next round (see his awesome odds for that game in my previous post). His odds of winning the event have more than doubled since the tourney began - but right now he's still an underdog to Carlsen. He'd be a lot more "ftw" right now if Gelfand had held the draw today

The amusing thing is that if Carlsen was Anand he would have offered/taken a draw about 40 moves earlier. I love Carlsen's play as it seems some people believe modern technique is just vastly better than that of past, justifying the short Anand-esque draws as what's the point? Far from it, it seems modern GMs just got lazier!

By the way, speaking of Aronian's awesome odds for his round 4 game: I'm curious what people's thoughts on those game-by-game predictions are, particularly the Aronian and Carlen's games. I'm a little uncomfortable with claiming that I genuinely believe Carlsen has a 43% chance to win, while Grischuk will only win 3% of the time. It's hard to imagine Grischuk (or anyone) beating Carlsen with black - but it's not impossible. As great as Carlsen is, it's certainly possible for him to blunder - and Grischuk is still extremely strong - he should be able to convert a win out of a true blunder. Is it really only a three percent chance that that happens?

There's no way to make the math work on the games where the rating difference gets too large without some arbitrary adjustment of some kind. Since I'm giving white an ELO boost, when Carlsen gets white against Grischuk, we're looking at a rating difference of around 150 - which per the ELO formula is supposed to translate to a score of 70% for Carlsen. Even if we assume Carlsen NEVER loses, we still can't assume anything worse than +40% =60%, as drawing 6 times out of 10 gets Grischuk his full 30% allotment. Now, Carlsen's draw rate with white and Grischuk's draw rate with black average out to higher than 60% - so something is broken initially. We have to make one of two assumptions.

1) The ELO > expected score formula is accurate, and the draw rate is therefore too high. So we arbitrarily reduce the draw rate to 60% or below, and calculate the possibilities based on the original expected score.

or 2) The draw rate is accurate, and regardless of what the ELO formula may say, a player like Grischuk is simply too capable of drawing if he wants to for Carlsen's expectation to truly be 70%. So we would arbitrarily reduce Carlsen's expected score to the maximum allowed by the draw rate - or lower if we want to give Grischuk credit for a few potential wins as well as the draws.

I went with 1, on the grounds that if the ELO formula weren't accurate and top players could draw Carlsen with black if they wanted to, then we'd see them doing just that and his rating wouldn't be so high in the first place. There's a reason he has the lowest draw rate (with white) of any of the participants - it's because he can beat people with white at a better rate than other players, since he's that much better. Further, since I didn't want to claim that it's *impossible* for the underdog to steal a win, I went with an adjustment factor of 1.8, rather than 2. So in this case, Grischuk's expected score of 30% is multiplied by 1.8 to get a draw rate of 54%. That leaves him room to win 3% of the games to round out his score, with Carlsen taking the full point the other 43% of the time.

It's an arbitrary fudge factor - and there's no way around it. The question is, is it the best one I could use? Should I set the multiplier lower, like 1.5, so that we'd only expect 45% draws, and Grischuk would get a 7.5% chance of winning, but Carlsen would now be winning at a 52.5% clip? Or should I use option 2 instead? And how much should I reduce the favorite's expected score beyond the draw rate, if so? Or is this the best option?

There's no definitive answer to this question that I can see, but a good test is what people's first thoughts were when they saw those odds. Did you laugh and think it was ridiculous? Or did you think "looks about right" and not worry about it?

I haven't thought through the details, but does it make sense to think that the tension is really between the color bonus (for white) and the adjustments to draw rates?

Elo seems pretty robust even at the highest levels as far as I have seen, and the predictions seem in the ballpark.

(The next step if for you to start tracking the prediction error and then optimize these parameters to minimize error on cross-validation samples...)

The stuff you guys are doing is pretty cool and making it more fun to sweat the tourney. Thanks BJJ

Agree with this, thank you BJJ for all of the simulation work. Very interesting stuff.

I don't think it has so much to do with the color bonus as just with the draw rates themselves. I mean if this algorithm is sound, it should work for more than just this event, and I need a way to handle what happens when Magnus plays a ~2600 player. Say Tata Steel earlier this year, I would have wanted to be able to predict Hou Yifan's results as well.

Fundamentally, there's just a major challenge in accurately predicting draw rates. There are clearly two main factors - the specific players involved, and the rating difference between them. Individual players will have stylistic differences - some players of the same rating as each other will have much higher draw rates, while other players will have much lower draw rates, with more wins and more losses balancing each other out to achieve the same scores. At the same time, as the ELO gap grows, draw rates will shrink. A 2000 rated player who stylistically has an unusually high draw percentage against similarly rated opponents still isn't likely to garner many draws against GMs.

I had initially hoped that all these players would be close enough in skill that I could ignore that second factor, but that's just not the case. The average draw rates of this field, when going up against each other, are too high to be reasonable for a >100 ELO differential, and that happens in several of Carlsen's games even without applying a color adjustment. So I had to make an effort to model that second factor, and figure out how much I should reduce draw rates as the ELO differential grows. I used the 1.8 multiplier I described before as a simple way to try to do something with it, but I'd like to come up with something better (or else to find evidence to support the idea that what I'm doing is in fact the best option).

I hope you haven't covered this above and I missed it, but is it possible that draw rates increase with rating regardless of (or in addition to) ELO differential? For example perhaps a 2950-rated player will draw more often against a 3100-rated player than a 2800 vs a 2950 and so on, simply because as players get further along on a scale zero->chess perfection that they are armed with more drawing resources and that the stronger player is "asymptoting" a bit more and isn't therefore acquiring extra winning resources at the same rate?

I hope that makes sense to someone outside my head (the concept seems simple but I don't know how to words) and that you didn't already answer it clearly in your last two posts, although I think I am just saying "maybe 2"? from your earlier post. ;p The sample size for ELO in Carlsen's range IRL is tiny and it doesn't seem unlikely that it works less well at the margins.

BJJ -- makes sense, although I have no intuition for this; it feels to me like a purely empirical question (and at lower ratings, something where there probably is enough data).

Playing through the last round, I don't understand Radjabov's choice to give up two pieces for rook and pawn. Since this is (in the abstract) a really elementary error, what is the specific reason why this looked reasonable? (Bxc3 also seems odd...I'm really missing the point of the entire sequence.)


17...Nxb2 18. Qxb2 Nd3 19. Qd2 Nxc1 20. Rxc1 Bxc3

I'm pretty sure he was concerned about: 1. .. h6 2. Bf4 Rfb8 3. Bxe5!?

Oddly enough the annotations on chessbase don't even consider this critical idea. 3. .. Nxe5 4. f4 Ng4 is forced. White has a pawn, the initiative and a million fun ways to start messing with black's awkward knight on g4. I think black's compensation is questionable at best. So Radjabov is down a pawn, facing an attack in a difficult position and also down by about an hour on the clock.

1. .. Rb8 2. b3 Na3 3. f4 is just a better version (for white) of the same idea.

I'm sure he realized Nxb2 was tantamount to suicide but it's very easy to play black's position and he still has some chances if Svidler gets even a little sloppy. Unfortunately for Radjabov, Svidler didn't.

Thanks, that makes sense.

I did do some empirical research on the question of how overmatched players do with black yesterday. I looked at Carlsen, Kramnik, and Aronian's performances with white over the last three years when their opponent was rated 50 or more points below them. Conclusion #1 is that Carlsen is an absolute beast, scoring 76% in such cases over an 80 game sample, far above even his expected results and implying a white advantage of over 100 points. Considering that "players rated 50 or more points below him" is now everyone in the world, it looks like there's some good reason to think that Carlsen's rating may not yet have plateaued.

That said, the important part for this discussion is that when he DID surrender points to overmatched opponents, it was in the form of 29 draws and 4 losses, while Kramnik allowed 21 draws and 2 losses, and Aronian surrendered 5 losses and 26 draws. Most of the games in this sample were against players 2700+, so I think the results are relevant to high level play. Overall, I have reached the following conclusion:

If a player at the Super-GM level is unable to achieve his normal stylistic draw rate due to his opponent's superiority (after adjusting for color of pieces), then he can be expected to score whatever points he does achieve with a ratio of roughly 7 draws for every upset win. Therefore the expected draw rate of such a game is the underdog's expected score multiplied by 1.56.

I adjusted my formula appropriately, so Grischuk and Svidler both have better winning chances than I listed above (where my multiplier was 1.8, having assumed too many draws and not enough upset wins). The new formula will be applied to all future simulations.

In the meantime, we're half an hour away from more chess, so let's put the math away and let the games begin! (I'd love to see the super super-unlikely result of Aronian and Carlsen both losing, just to make things more interesting as the tournament progresses. Three viable contenders would be more interesting than two!)

DiR, in the 1...h6 2.Bf4 Rfb8 3.Bxe5 Nxe5 4.f4 line, is black really doing that badly? It looks like a Benko gambit in which black's g4 knight is awkward, but so is white's g3 knight. Plus, the white rooks are poorly positioned to defend the queenside pawns and black's rooks are already pressuring the a and b pawns. Also, black has already traded a pair of knights which is usually good for him in the Benko, and the g7 bishop is unblocked and unopposed. This looks like a decent Benko for black to me after capturing with 3.Bxe5.

Ivanchuk with the Chigorin, uh oh for him. Svidler also on some less beaten paths

Is he 'really' doing that badly? No idea. Best bet is to run it through houdini for the answer to that. I just know that that position, to me, looks incredibly tough for black to play - something I'm just about certain Radjabov agreed with! The generalities don't mean that much as the position is becoming very tactical with e5 and other ideas imminent. It's not just a kingside attack either - check the c5 pawn. I also quite like white's g3 knight, it supports f5 and Ne4 will be a very nice square.

I can't even begin to imagine how black ought reasonably organize a defense.

In Svidler - Radjabov, the key line mentioned on a few Russian sites that I saw was h6 Bh4 with the idea of g5 Nf5 and White recovers the Knight with a crushing advantage. And if Black doesn't play for g5, then inserting h6 Bh4 just weakens Black's Kingside (and takes the h6 square away from the Knight, though I don't know if this really matters).

I think Black's big mistake was Bxc3 in this game. If Black keeps the dark-squared Bishop I think he would have better chances to hold than in the game.

Oooo now 1. .. h6 2. Bh4!? is just a plain sexy looking idea. What were the sites? Chessbase commentary has really kind of gone down the crapper.