Tblisi Grand Prix – Second Rest Day Update

While our top story from the first rest day (the rise of Tomashevsky) has continued unabated, there is now a second key story line at play in this event: the fall of Grischuk. While Tomashevsky scored 2.5/4 and maintained his full point lead over the field, Grischuk dropped two games, scored only 1/4, and fell firmly out of contention. In this post we will be examining the deeper ramifications (both obvious and subtle) of these two players’ results, as everything else depends on them.

First the obvious: Tomashevsky’s position has improved greatly. While his lead hasn’t actually grown, it should be obvious that a full point lead over just one player, with only three games left, is a far stronger advantage than a full point lead over three different players, with seven games left. Evgeny didn’t particularly need to extend his lead, merely hold serve as others fell off the pace. In this case it was Grischuk and Giri who dropped further behind, while noone else closed the gap. How much better is Tomashevsky looking at this point, than before? In the last four rounds his odds of winning the event outright have improved from 31% to a whopping 79%! His expected score (in Grand Prix Points) is now 164 – keeping in mind that you score 170 for a clear first, and just 140 for second. Even the one time in five that he doesn’t win outright, he will almost always share first, usually with only one other player, and still bring home 155 points. His odds in the overall Grand Prix standings, meanwhile, are now up to a 42% chance of finishing top two (and qualifying for the Candidates Tournament). This is a humongous gain over his round four position, when despite his great start he had just a 17% chance, or especially over his pre-tournament odds when we evaluated him as having only a 2% chance of being a Candidate for the World Championship next year.

Here are average expected points scored at Tbilisi, and odds of winning first place outright, for all players. You may notice that it isn’t much of a race at this point.

Player Score (Out of 8) Tbilisi EV Odds of Clear 1st
 Evgeny Tomashevsky (RUS) 6 164 79%
 Dmitry Jakovenko (RUS) 5 124 5%
 Rustam Kasimdzhanov (UZB) 4.5 89 1%
 Teimour Radjabov (AZE) 4.5 87 1%
 Anish Giri (NED) 4 77 0%
 Shakhriyar Mamedyarov (AZE) 4 72 0%
 Baadur Jobava (GEO) 4 65 0%
 Leinier Dominguez (CUB) 4 57 0%
 Alexander Grischuk (RUS) 3.5 59 0%
 Maxime Vachier-Lagrave (FRA) 3 32 0%
 Peter Svidler (RUS) 3 26 0%
 Dmitry Andreikin (RUS) 2.5 20 0%

While it has little bearing on the overall Grand Prix standings, or even on the results at Tbilisi, we would like to highlight how impressively Jobava has bounced back from an atrocious start. After managing just one draw and three losses in the first four rounds, he has won three and drawn one in his last four rounds, returning to an even 50% score for the event, and bringing his live rating back above 2700.

Grischuk, on the other hand, has fared most poorly over the past four days. As the highest rated player in the field, he was expected to score well more often than not, so after round four we saw him as the most likely contender to potentially chase and catch Tomashevsky. Instead he tumbled. Where before we expected him to score an average of 113 Grand Prix Points, now we see him picking up only 59, which would not keep him alive in the overall Grand Prix standings. By virtue of his rating, he remains one of the favorites in the final event at Khanty-Mansiysk, but at this point his overall odds of reaching the Candidates Tournament via the Grand Prix are just 12%, whereas four games ago we had him at 41%. If he is going to get there, it will have to start over the final three rounds of Tbilisi. He has no realistic hope of winning this event, but if he can win a game or two and rise in the standings, it would gain him critical extra Grand Prix points that would keep him at least somewhat in contention overall, and keep his results relevant at Khanty-Mansiysk.

Here is where all 16 players in the Grand Prix stand, for average final score expectations, as well as odds of actually reaching the Candidates Tournament:

Player Baku Tashkent Tbilisi EV Khanty-Mansiysk EV TOTAL EV Top-Two Odds
 Fabiano Caruana (ITA) 155 75 100.9 330.9 68%
 Evgeny Tomashevsky (RUS) 82 164.4 60.7 307.1 42%
 Hikaru Nakamura (USA) 82 125 97.6 304.6 49%
 Alexander Grischuk (RUS) 82 59.0 97.3 238.3 12%
 Boris Gelfand (ISR) 155 15 64.1 234.1 8%
 Shakhriyar Mamedyarov (AZE) 35 125 72.0 232.0 1%
 Sergey Karjakin (RUS) 82 75 70.6 227.6 7%
 Dmitry Jakovenko (RUS) 30 123.7 62.0 215.7 5%
 Anish Giri (NED) 40 76.7 93.8 210.5 5%
 Dmitry Andreikin (RUS) 20 170 19.9 209.9 0%
 Teimour Radjabov (AZE) 50 50 86.8 186.8 0.05%
 Baadur Jobava (GEO) 75 64.7 41.2 180.9 1%
 Maxime Vachier-Lagrave (FRA) 75 31.7 74.0 180.7 1%
 Peter Svidler (RUS) 82 25.7 54.0 161.6 0.4%
 Rustam Kasimdzhanov (UZB) 35 15 88.8 138.8 0%
 Leinier Dominguez (CUB) 10 56.8 53.8 120.5 0.03%

You may notice, from the numbers above, two other major beneficiaries of Grishcuk’s collapse: Hikaru Nakamura and Fabiano Caruana. While sitting on the sidelines, these two have seen their chances improve significantly over the past four rounds, improving eight percentage points each over where they stood at the first rest day. Why? As the leaders in the overall Grand Prix standings prior to this event, they were in the best position to benefit from the rise of a relatively weak favorite at Tbilisi. While we don’t wish to take anything away from how impressive Tomashevsky has been at this event, he is still rated 50-60 ELO lower than the top rated players in the Grand Prix, and so our model projects him as less likely to repeat his success at Khanty-Mansiysk than, say, Grischuk or Giri would have been. So having more of Tbilisi’s Grand Prix points likely to be awarded to lower rated players puts Caruana and Nakamura in much better positions to maintain their leads through the final leg of the Grand Prix.

Also working in the leader’s favor has been the disappointing results for two other higher rated players: Giri and Vachier-Lagrave. While they never had as much hope originally as Grischuk did, making their drops less dramatic, they nevertheless have seen their odds of qualifying for the Candidates Tournament drop precipitously over the course of this event. Just like with Grischuk, these other drops from high rated players have also benefited Nakamura and Caruana… as well as helping Tomashevsky greatly, of course. Here is how each of the 16 players in the Grand Prix field has trended from before Tbilisi started, to the first rest day, to the second:

 Fabiano Caruana (ITA) 58% 59% 68%
 Hikaru Nakamura (USA) 39% 40% 49%
 Evgeny Tomashevsky (RUS) 2% 17% 42%
 Alexander Grischuk (RUS) 40% 41% 12%
 Boris Gelfand (ISR) 6% 6% 8%
 Sergey Karjakin (RUS) 5% 5% 7%
 Dmitry Jakovenko (RUS) 1% 1% 5%
 Anish Giri (NED) 14% 17% 5%
 Baadur Jobava (GEO) 0% 0% 1%
 Shakhriyar Mamedyarov (AZE) 6% 3% 1%
 Maxime Vachier-Lagrave (FRA) 15% 6% 1%
 Peter Svidler (RUS) 5% 2% 0%
 Teimour Radjabov (AZE) 0% 0% 0%
 Leinier Dominguez (CUB) 0% 0% 0%
 Dmitry Andreikin (RUS) 10% 3% 0%
 Rustam Kasimdzhanov (UZB) 0% 0% 0%

There is another, somewhat more subtle, ramification of Grischuk’s fall in the projected standings. It has become slightly less likely that the cutoff score needed to qualify for the Candidates Tournament will be extremely high. At the first rest day, when Grischuk (and his gaudy rating of 2810 at the time) was just one point off the lead, and still in legitimate contention, our model saw it as far more likely that he might post huge scores in both of the final two events, driving up the standards of qualification. Now of course Tomashevsky is still in a position to do just that, but with his lower rating he is projected as much less likely to follow up a great Tbilisi result with an equally great Khanty-Mansiysk result. Therefore, our projected cutoff has dropped 10 points. Our model now has the median points required to qualify for the Candidates as just 317, and now sees a (very remote) chance of someone qualifying with as little as 242 points, lower than our previously reported minimum. Of course there is still an extreme range of possible cutoffs, and noone should feel secure with a final score of, say, 320. In some of our simulations it still takes 392 points just to achieve second place! Here is a frequency graph of all the possible qualifying scores:


One final consideration that we mentioned in our last post is the rating mark of 2800. Both Grischuk and Giri have fallen below it in the current live ratings, but they still have three rounds to rebound. Both are now underdogs to get their end-of-tournament rating (presumed to become official on the next rating list) back above the 2800 line, but neither is completely eliminated. Giri would need to score 2.5/3 in the final rounds, with the black pieces in two of those three games, and our model predicts him to do just that 14% of the time. Grischuk also needs to score 2.5/3, but has the benefit of two games with white, and so has a slightly better chance at 21%. Still, it now looks likely that the official March rating list will have only two 2800+ players on it – despite there having been five at the same time rated over 2800 in the live ratings earlier this month.

All told, the results over rounds five through eight have removed much of the drama from Tbilisi. Tomashevsky is now an overwhelming favorite to win the event, and has turned himself into a legitimate contender for one of the eventual Candidates Tournament spots. Of course he still has to navigate the last three rounds: he has roughly a 90% chance of at least sharing first place, but 10% is still 10%, and he has a job to do to close out the win. The other most tangible question is whether Grischuk can bounce back, manage at least a plus score in the last two rounds, move up the standings a bit, and enter Khanty-Mansiysk as a contender. We have about 20 hours until round 9 begins. In the meantime, hopefully this update gives you something to chew on during this rest day!


Tbilisi Grand Prix – First Rest Day Update

We probably could have titled this post “The rise of Tomashevsky”. Certainly the biggest story is Evgeny Tomashevsky’s blistering 3.5/4 start, and full point lead after four rounds (out of 11). You may already know that we are tracking the Grand Prix in detail, and running simulations to determine each player’s odds of finishing in the top two of the final standings and earning a berth in the 2016 Candidates Tournament. Well four days ago, before this event began, we were awfully pessimistic about Evgeny’s chances, giving him a mere 2% chance of doing well enough both here in Tbilisi and also in Khanty-Mansiysk to finish in the top two. Now? We have his odds up to 17%! He still has an uphill climb before he convinces our ratings-based simulation model that he’s an actual “favorite”, but he has nevertheless made tremendous progress.

In honor of the first rest day (2/19/2015), we thought we’d dig a little deeper into the results of our simulations. First of all, we’ve talked a lot about the overall Grand Prix standings, but haven’t actually said a word about the Tbilisi tournament specifically. How valuable is Tomashevsky’s full-point lead, in terms of actually winning this event? Is it worth more than the higher ratings of Grischuk and Giri, each sitting on 2.5/4? Our numbers say yes! Here are each player’s odds of winning THIS tournament, along with their average Grand Prix Points earned at Tbilisi (“EV” stands for “expected value”):

Player Score (Out of 4) Tbilisi EV Odds of Clear 1st
 Evgeny Tomashevsky (RUS) 3.5 132 31%
 Anish Giri (NED) 2.5 113 16%
 Alexander Grischuk (RUS) 2.5 113 15%
 Dmitry Jakovenko (RUS) 2.5 82 4%
 Shakhriyar Mamedyarov (AZE) 2 71 2%
 Maxime Vachier-Lagrave (FRA) 1.5 65 1%
 Rustam Kasimdzhanov (UZB) 2 61 1%
 Leinier Dominguez (CUB) 2 59 1%
 Teimour Radjabov (AZE) 2 56 1%
 Dmitry Andreikin (RUS) 1.5 55 0%
 Peter Svidler (RUS) 1.5 48 0%
 Baadur Jobava (GEO) 0.5 16 0%

Poor Jobava. His slow start has been just as bad as Tomashevsky’s has been good. At this point, pretty much his best realistic hope is just to share last place with someone. You can see, though, that if there is a clear winner it will be Tomashevsky almost half the time! His lead really is quite commanding, even with seven rounds left, but certainly nothing has been clinched yet.

So if Tomashevsky does take a clear first place and earn 170 Grand Prix points, what does that mean for his overall chances of qualifying? It would give him 252 points through two events, since he earned 82 at Baku, but how many points will second place ultimately require? Well, we began tracking this in our simulations, and have determined that the average score needed for second place in the final standings is 327. So if Tomshevsky wins, he’ll still need 75 more points at Khanty-Mansiysk to reach that target. Of course there this is an oversimplification, and 327 does not magically guarantee a Candidates berth, as there is a very wide range of possibilities. Across 20,000 simulations, we saw qualifying targets as low as 255, or as high as 392! The latter value can only happen in one precise way: Grischuk must win Tbilisi outright for 170 points, then finish exactly second at Khanty-Mansiysk for 140 more, giving him 392 total, AND Caruana must win Khanty-Mansiysk outright bringing his own total to 400 points even and first place overall. Despite the fact that this is an incredibly specific series of events, our simulation shows it happening once every 200 times (0.5%). This is surprisingly often, for one exact outcome, and shows how much our model respects Caruana and Grischuk’s high ratings, in its simulations.

Here is the full range of possibilities:

Cutoff Graph

So if we know the expected average result for each player at Tbilisi, do we have it for Khanty-Mansiysk as well? Of course! Here we have every player’s actual scores for Baku and Tashkent, along with their expected scores for Tbilisi and Khanty-Mansiysk, and their average overall scores. We also included their current odds of finishing in the top two in the final standings, in order to highlight that it doesn’t necessarily track perfectly with expected scores. That is because expected scores are an average of all possible results, while top-two odds are skewed heavily towards the odds of particularly good results. Some players have a higher risk/reward factor in their remaining slate, that allows them to have higher odds of reaching the Candidates despite not having a higher expected score.

Player Baku Tashkent Tbilisi EV Khanty-Mansiysk EV TOTAL EV Top-Two Odds
 Fabiano Caruana (ITA) 155 75 99.2 329.2 59%
 Hikaru Nakamura (USA) 82 125 96.0 303.0 40%
 Alexander Grischuk (RUS) 82 112.6 104.5 299.1 41%
 Evgeny Tomashevsky (RUS) 82 131.5 56.8 270.3 17%
 Anish Giri (NED) 40 112.9 98.6 251.5 17%
 Dmitry Andreikin (RUS) 20 170 54.9 244.9 3%
 Boris Gelfand (ISR) 155 15 64.7 234.7 6%
 Shakhriyar Mamedyarov (AZE) 35 125 71.0 231.0 3%
 Sergey Karjakin (RUS) 82 75 70.0 227.0 5%
 Maxime Vachier-Lagrave (FRA) 75 65.4 77.6 218.0 6%
 Peter Svidler (RUS) 82 47.6 57.1 186.8 2%
 Dmitry Jakovenko (RUS) 30 81.6 57.6 169.2 1%
 Teimour Radjabov (AZE) 50 50 56.1 156.1 0%
 Baadur Jobava (GEO) 75 16.4 34.2 125.5 0%
 Leinier Dominguez (CUB) 10 58.6 53.8 122.3 0%
 Rustam Kasimdzhanov (UZB) 35 15 61.5 111.5 0%

And finally, we looked at one more consideration. The “2800 club”! While live ratings are something we love to follow along with, there is a certain gravitas that comes along with actual published ratings. So the question is: when the March rating list comes out, how many 2800+ players will we see? Nakamura flirted with the mark at Zurich, but fell short in the end. Caruana struggled at Zurich, but managed to stay above the 2800 plateau. Carlsen will be there of course, so that’s two. And then there are Grischuk and Giri, both of whom are playing at Tbilisi. What are their chances?

Well Grischuk is in pretty good shape. Although he loses a little bit of rating with every draw, as the highest rated player in the field, his 9.7 point cushion (current live rating of 2809.7) is enough that as long as he scores 50% the rest of the way he’ll be fine. However if he drops to a negative score, at only 3/7 or worse over the remaining rounds, his rating would fall below the magic 2800 mark. According to our simulation, his chance of scoring at minimum the needed 50% in his remaining games is 82% (as he’s a favorite in all but one of them, the lone exception being when he has the black pieces against Giri in round 8).

Giri is in a slightly tougher spot, as he has no real cushion at all, currently sitting at 2800.4 in the live ratings. Since he also loses rating points with draws most of the way, he needs at minimum a score of +1, or 4/7 the rest of the way, to keep his rating afloat above the 2800 mark. Actually, 4/7 would drop his live rating to 2799.8, but fortunately that’s good enough as FIDE would round up. Scoring +1 is a tougher task than just maintaining an even score, but Giri is favored in all 7 of his remaining games (thanks to having the white pieces, which we rate as being worth 40 rating points) against Grischuk. As such, Giri is a favorite to score at least the 4/7 he needs: we rate his chances of a published March rating of 2800+ to be 63%.

We hope you enjoyed this interlude as a palatable replacement for actual chess on this rest day. Perhaps turn your attention to the rapid games at Zurich, to keep yourself entertained. We will post another update at the second rest day, so please let us know if there are any other stats you’d like us to take a look at!

Simulating The Grand Prix – Methodology

On our main page for the 2014-15 Grand Prix, you will notice that we list each player’s odds of finishing in the top two of the final Grand Prix standings – an important mark because those top two players earn berths in the 2016 Candidates Match, with a chance at the World Championship. If you’ve seen this, perhaps you have wondered: where did those numbers come from? How are they calculated, and how accurate are they?

So here I will discuss our methodology. First of all, the overview is relatively simple. I’ll go over that portion first, and dive deeper into the details (that may less interesting to some readers) afterward. We have built a spreadsheet that estimates the odds of a white win, a black win, or a draw, for all 132 Grand Prix games left to play (66 in Tbilisi, 66 in Khanty-Mansiysk). Further, it can use those odds to randomly calculate a result in each game, AND correctly calculate how the Grand Prix points would be awarded given those results, and who would therefore be the top two finishers.

Our simulation simply re-runs the randomizer a large number of times, recording the top two finishers each time, and spits out a result for each player: what percentage of the overall simulations had that player in the top two? Those are the odds we show you. How accurate are they? Not perfect, of course. The main source of error is that the individual game odds cannot be perfect. If you care about the details read on. Otherwise, suffice it to say that our estimates are probably the best available and if you’re interested in knowing who has the best chance of reaching the Candidates Match, you’re welcome to follow along with us! We will post regular updates throughout the upcoming Tbilisi event, and will show both the “current” odds and the “pre-Tbilisi” odds so that you can see exactly how much individual players have benefited or suffered from the results up until that point.

That’s a lot of text, so before we continue with the nitty gritty, here’s our “pre-Tbilisi” projections, in case you aren’t interested in clicking through to a different page to see them:

 Fabiano Caruana (ITA) 58%
 Alexander Grischuk (RUS) 40%
 Hikaru Nakamura (USA) 39%
 Maxime Vachier-Lagrave (FRA) 15%
 Anish Giri (NED) 14%
 Dmitry Andreikin (RUS) 10%
 Shakhriyar Mamedyarov (AZE) 6%
 Boris Gelfand (ISR) 6%
 Sergey Karjakin (RUS) 5%
 Peter Svidler (RUS) 5%
 Evgeny Tomashevsky (RUS) 2%
 Dmitry Jakovenko (RUS) 1%
 Baadur Jobava (GEO) 0%
 Leinier Dominguez (CUB) 0%
 Teimour Radjabov (AZE) 0%
 Rustam Kasimdzhanov (UZB) 0%

Nevertheless, that’s our method. And once draw rate is calculated, and given that we already have an expected score from ELO (1/(1+10^(rating differential/400))), we can easily determine the needed win and loss odds to achieve the expected score. One final important note is that we DO account for colors. White scores higher than 50%, of course, so in all our calculations of rating differential (both to determine draw rate and to determine estimated score) we add 40 points to white’s ELO before calculating the differential. A “perfectly even” match, in our estimation, with full 60% draw odds, and equal winning chances for white and black, would be a game where white is rated 2720 and black is rated 2760, for instance. So that is how we estimate the odds of each game. From there, everything else is a relatively simple Monte Carlo simulation. We run it a large number of times, and get our results!The other challenge in using ELO to estimate results of specific games is that ELO only actually gives an expected score, not an expected result. Draw rate remains an unknown. We have plans to do a detailed study of draw rates in the future, but for now we’re using a simple estimation that the base draw rate for equal players is 60%, and that draws become less likely as the gap between the players’ ratings gets wider. Specifically, the draw rate is 60% – (rating difference)/1000, so ever 10 ELO reduces the draw rate by 1 percentage point. A 2800 vs. a 2700 would presumably draw 50% of the time. This is awfully unscientific, but fortunately it doesn’t make a huge impact if we’re off slightly. We tried another simulation with a baseline draw rate of 70% instead of 60%, and only one player saw their ultimate odds shift by more than one percentage point. There’s a little error in our draw assumptions, but not a huge amount. Of course in reality, draw rates probably also vary by the individual playing styles; we would figure Jobava to draw less than our formulas predict, for example.So first of all, our estimation of the win/draw/loss odds for each game are calculated based on ELO expectation, using current live ratings from 2700chess.com (one of the greatest things on the entire internet – if you’re interested enough in chess analysis to have read this far, and for some reason you don’t already have 2700chess bookmarked, go bookmark it now. We’ll wait.) Now this isn’t perfect, it works pretty well for players rated accurately, but that can never be everyone. The ELO system is pretty solid overall, but will always contain some underrated players and some overrated players at any given time. In the long run they balance out, but in the short run an underrated player will see their odds of finishing top-two in the Grand Prix badly understated by our formulas.

One other critical factor is the pairings. These are unknown until the day before play begins. Because each tournament is a 12 player Round Robin, meaning each player plays 11 games, half the field will get the black pieces 5 times, and half the field will get 6 blacks. This draw is important, given that we (correctly) factor in the white pieces as being worth an increase in expected score in a given game. When we first posted “pre-Tbilisi” odds, on Februrary 11th, we did not yet know the pairings for Tbilisi (or for Khanty-Mansiysk of course). Foolishly, we ran our simulation with static pairings (giving the same six players the favorable treatment of getting white 6/11 games). We concluded that Grischuk (to whom we had generously given six whites in BOTH upcoming events) had a 46% chance of reaching the Candidates match. When the Tbilisi pairings were released, and we re-ran our simulation with proper Tbilisi pairings, Grischuk’s odds (he will have six blacks in Tbilisi) dropped several percentage points! This shift made it clear how important the pairings are, and so we immediately re-designed our spreadsheet so that the pairings for Khanty-Mansiysk are randomly generated for each simulation. The odds now posted reflect these dynamic pairings within the simulation (and of course the actual pairings for Tbilisi, now that they are known), and we believe they are much more accurate than our previous posting.