Candidates Qualifying: Average Rating

Next spring the 2016 Candidates Tournament will be held, with eight players vying for the right to face off against Magnus Carlsen later in the year in a match for the world championship. These eight Candidates will be selected through a variety of criteria, which we have been tracking here, and we offer a closer look at one particular criteria (average rating) as well. However things have begun to grow clearer as we move into the last few months of the year, so now let’s look even more closely at the ratings, and see if we can’t pin down specific odds of various players qualifying. Here are a series of key points to be aware of:

There are only four serious contenders

Let’s start here. The top six in our projected average ratings are Topalov, Giri, Grischuk, Kramnik, Aronian, and So. It is basically guaranteed that the two ratings qualification spots will go to two of these six players, with the last two themselves being extremely unlikely, and nobody else has a chance worth considering.

7th place Ding Liren could see his rating rise to 2832 tomorrow, and stay there for all three of the remaining lists, and if Wesley So’s rating didn’t change So would still finish ahead of Liren in the final average ratings (by a tenth of a point). The gap is just too big for Liren to have much chance of bridging, and that’s just to get into sixth place in the race, which won’t be good enough to actually qualify. So, currently in sixth place, is only in our list of “serious contenders” because if HE makes huge gains he could climb as high as fourth, and have a shot if two of the players ahead of him reach the finals of the World Cup, removing themselves from contention as qualifiers by rating by qualifying in a higher precedence way. For Liren to get to fourth place, without anyone else’s ratings changing, he would need to post a published rating of 2854(!) in all three remaining lists. Obviously this won’t happen. There are at most six contenders, and Aronian and So are huge longshots themselves (relying on huge rating changes AND two of their competitors to qualify at the World Cup). We will address their path, but each of them has well under a 1% chance of qualifying, it’s over 99% that both qualifiers will be from the top four: Topalov, Giri, Grischuk, and Kramnik.

October ratings are probably already known

The only tournament that any of the six possible contenders are scheduled to play in September (as far as we know) is the World Cup, and since that event doesn’t end until after the October rating list is published it will not impact the ratings on that list. World Cup results will only be included for the first time on the November list. The October list, therefore, will almost certainly be the same as the current live ratings for all the players that matter in this analysis. Maybe one of them will play an event we’re not currently aware of that will sneak in before the end of the month, but because of the World Cup none are likely to have anything planned. We expect the live ratings to turn into published ratings for October. This means that there are really only going to be two, not three, remaining opportunities for players to

Topalov will qualify if he doesn’t withdraw from the World Cup

Veselin Topalov has a live rating of 2813. If this is his published rating in each of the remaining three rating lists, his average rating through the year would end up at 2806.6, while third place Grischuk projects to finish at just 2783.7 with no rating changes. Maybe there is a small chance that Giri could pass Topalov, but even that isn’t likely. For TWO people to pass him, and prevent him from qualifying, would take an impossibly absurd collapse. Let us say, despite the previous point, that he somehow plays a couple rated games in September and his rating falls to 2800 in the October supplement (0/2 against a 2700 opponent would do it). Then let us say he goes into a complete tailspin between the World Cup and the European Club Cup (October 18-24) and sheds 50 more rating points, dropping to a published rating of 2750 on the November list. Then let’s say he somehow loses 50 MORE rating points in November, and finishes with a published 2700 rating in the December list. His average rating for the year would still finish at 2790.8 – better than we project for Giri! Of course he isn’t going to lose over 100 rating points in the next two months, but the fact that even if he did he would probably still qualify by average rating should convince you that he’s a lock.

There is only one catch: to qualify by rating a player must compete in either the Grand Prix or the World Cup. He skipped the Grand Prix, so the latter is mandatory. Now he has said that he will, and is listed as the #1 seed in the bracket, for this tournament that begins in a week, so our assumption is that he has clinched a spot in the Candidates Tournament. Technically though it’s not official until he actually plays his World Cup games. If for some unforeseen reason he withdrew without playing a game, our understanding of the rules is that he would not be eligible for the ratings qualification. So he has to compete to qualify. If he does, he’s in, and that’s all there is to it.

Giri is extremely likely to qualify as well

A solid performance at the Sinquefield Cup might have come close to clinching the second qualifying spot for Anish Giri. Through September (the first nine rating lists, out of twelve, that will be averaged together), Giri’s average rating is 2787.4, just barely ahead of Grischuk (2786.9). This would be a minuscule lead, except that Giri is now rated 2798, and Grischuk just 2774. The Sinquefield Cup was Grischuk’s last good chance to close this gap in time to remain in contention, and while he played well (gaining 3 rating points), he did not make the extraordinary move he needed to catch up, nor did Giri stumble (gaining 5 rating points of his own). Now we expect these ratings to be published for October, as we’ve discussed, and one more month of this large rating gap will create a big deficit for Grischuk to try to bridge in the final two months.

To catch up, assuming the October ratings match current live ratings as we expect, Grischuk’s November and December ratings would have to EXCEED Giri’s by a total of 29 for a tie, or 30 to actually surpass the Dutchman. If all the gains were to be made in November, this could mean “just” a 15 point rating edge, published twice (November and December), but to achieve this (given his current rating deficit) Grischuk would need to gain (and/or see Giri lose) 39 rating points net before the end of November. If Giri gets knocked out of the World Cup early (and preferably for Grischuk fans it would be by losing classical games, not losing on rapid/blitz tiebreaks) while Grischuk makes a deep run, then this could still be in play, but right now it’s pretty unlikely. We’ll certainly keep an eye on their World Cup results, and see if there’s hope for Grischuk, but really he probably should just focus on reaching the World Cup final and qualifying that way, because anything less probably won’t be enough to close the rating gap anyway.

What about other contenders? Who else might catch Giri with big gains at the World Cup? Kramnik trails Giri by a lot more, and would need to be rated 40(!) points higher than Giri by the end of November to catch up. Given that he’s currently 21 points lower rated, this seems highly improbable to say the least. Aronian is rated higher than Kramnik or Grischuk, but trails Giri by even more through the first 10 months, and would need to gain an absurd 82 points on Giri by the end of November to catch him in December. Wesley So would need to gain 94 rating points. The point is: only Grischuk can catch Giri, and it will take a strong effort before the end of October (for the November rating list).

The World Cup could make a big difference

If Grischuk is the only player with a realistic chance to catch Giri, then why are we mentioning Kramnik, Aronian, and So? Because it might not be necessary to catch Giri in order to end up with one of the two open Candidates berths by average rating. If either of the top two contenders (Topalov or Giri) reaches the finals of the World Cup, and earns a spot in the Candidates Tournament that way, what used to be a battle for third place suddenly turns into a battle for second place. Kramnik is alive for this reason: he does have chances at catching Grischuk.

While Grischuk’s average rating so far is higher than Kramnik’s, his current live rating is three points lower. That’s not enough; in order to actually finish with a higher average rating Kramnik will need to be rated 25 points higher than Grischuk on each of the last two lists (for the tie). This is a tall task, but not so unlikely as to discount entirely. He’s already ahead, so he only needs a net gain of 22 points based on results over the next two months to get there.

It’s worth noting, here, that we carefully referred to the possibility of Topalov OR Giri reaching the World Cup final. They are on the same half of the bracket (seeded #1 and #4, and slated to potentially face each other in the semifinals if they make it that far), so it’s impossible for both of them to qualify. What are the odds of at least one of them reaching the World Cup final, and opening a window for Grischuk (or maybe for Kramnik)? Our latest World Cup odds say that Topalov should reach the finals 24% of the time and Giri 18.6%. Since their chances are mutually exclusive, this is the rare situation in probability where we can just add two numbers together and get a valid result, so assuming Grischuk holds on to the #3 rating spot, he has a 43% chance of qualifying for the Candidates Tournament by virtue of Topalov or Giri qualifying on World Cup results and turning Grischuk into the #2 average rating among eligible players.

None of this yet addresses Aronian or So. Their odds of catching Grischuk for the #3 spot are lower than Grischuk’s chances of catching Giri were, as they would need to gain 43.5 and 55 net rating points (respectively) worth of ground to catch Grischuk. Unlikely. However there’s still another scenario. While Topalov and Giri can’t both reach the finals, Grischuk himself could, he is seeded #7 putting him on the other half of the bracket. And our calculations say he has about a 10% chance of reaching the final himself.

Aronian and So are in trouble, but not completely eliminated. Their hopes rest on a series of unlikely events. First they must root for Grischuk to reach the final, and for his opponent to be either Topalov or Giri. There’s slightly better than a 4% chance of this happening, but that’s about what Aronian’s odds of winning the Sinquefield Cup were when the event began! This is just the beginning though, if two of the top three players in the current average rating standings remove themselves from the pool by both making the World Cup finals, that still only improves Aronian and So to 3rd/4th in the standings themselves. They would ALSO have to catch and pass Kramnik to ultimately get in. Looking at it the same way we’ve looked at other deficits, Aronian would have to gain 22 net rating points on Kramnik to pass him, and So would need to gain 33. These are large deficits, but not completely insurmountable. It’s as likely for Aronian to pass Kramnik as it is for Kramnik to pass Grischuk, and it’s more likely for So to pass Kramnik than it is for Grischuk to pass Giri. All of these look unlikely, but none of them can be completely ruled out as impossible at this juncture.

What if there is a tie?

Right now the gaps between each player are large enough that it seems extremely unlikely to come down to a tie. Average ratings are rounded to two decimal places, so two players would have to have EXACTLY the same average rating over the 12 months for it to matter. For the record, though, if two players do tie for second place the tie break is number of standard rated games played in 2015. If anyone wants to go compile game counts for the players in the hunt and post it in the comments, that would be welcome! If after the October and November lists are published (and World Cup finalists are known) a tie begins to look more likely, we’ll drill down more closely on tie break possibilities, but right now it appears irrelevant.

What about games played in November?

All the numbers we gave above for how many rating points a player must gain (or the person they’re chasing must lose) before the November rating list is published, in order to pass someone ahead of them, naively assumed that the November and December rating lists would be the same. Of course this is not the case, the European Team Championships are played in November, and likely all of these players will compete, so December ratings will differ from November ratings. It’s just that rating changes that only apply on the December list are only counted once, while rating changes that impact the November list are counted twice, so it takes more effort to catch up in the final month.

Of course, though, when we said (for example) that Grischuk needs to gain 39 points (net) on Giri before the November list is published, he could alternatively achieve the same result by gaining 33 points (net) before the November list, then gaining 12 more before the December list is published. However far short of a goal someone falls on the November list, they have to gain twice the difference in that final month.

So what are the actual odds?

Topalov is 100% in the Candidates Tournament, but we don’t know if it will be by virtue of rating or by reaching the World Cup final. Making sure we’re clear on that point, here are each player’s estimated chances of qualifying ON RATINGS for the 2016 Candidates Tournament. These are a little bit intuitive and rough, we did not do detailed simulations of how many rating points can be made up in the last two months, we’re pretty much just guessing. Remember that since two spots are open, the odds must add to 200%:

Player Odds
Topalov 76%
Giri 75%
Grischuk 44%
Kramnik 4%
Aronian <1%
So <1%

Khanty-Mansiysk Preview (FIDE Grand Prix, FINAL LEG)

The pairings have been drawn for the fourth and final event in the 2014-2015 FIDE Grand Prix. Games begin in just over 6 hours! Here is our look at the event, to whet your appetite.

First and foremost, this event is critical because it will determine the final Grand Prix standings, which in turn will determine two of the eight candidates for the 2016 World Chess Championship match against defending champion Magnus Carlsen. The top two players in the Grand Prix standings will join six others in a Candidates Tournament in early 2016, with the winner of that event slated to take on Magnus later in the year. Without the relevance to the World Championship, it’s unlikely this event would be of quite so much global interest as it is – although it’s also certainly a wonderful showcase of 12 fantastic chess players in its own right.

Worth mentioning is that third (and possibly even fourth) place in the final Grand Prix standings could also matter. The official rules for the Candidates Tournament say that the first qualifier is the loser of the 2014 match (Viswanathan Anand), and the second and third qualifiers are the top two finishers in the 2015 World Cup, which will be held later this year. The top two Grand Prix finishers *not already qualified* earn spots four and five. It’s relatively unlikely that one player will finish top-two in both events, but it’s not impossible that someone like Fabiano Caruana might do so, and in that case the third place Grand Prix finisher would earn a spot as well. Furthermore, if any player who qualifies for the Candidates Tournament is unable to participate, or chooses to withdraw, then the first alternate candidate is the third place finisher from the Grand Prix.

How do the standings work? Each of 16 players competes in three of four tournaments. They earn Grand Prix points based on their finish in each event, ranging from 170 points for a clear first place (not tied with anyone) down to 10 points for a sole last place finish. Each player totals their points from the three events, and the player with the most total points wins the Grand Prix! Of course most audience members aren’t concerned with just the winner, we’re also watching closely to see who finishes second.

So what will happen? We can’t know of course, that’s the fun of watching. However we have designed a model to predict the results. Using each player’s current live rating from we can estimate odds of a white win, a black win, or a draw for every game that will be played. We then simulate the event 20,000 times, tally up each player’s finish (and the resulting final Grand Prix standings) after each simulation, and tabulate the results for your reading pleasure!

So enough with the prelude, who are our combatants?

The Favorites

  • Evgeny Tomashevsky – 2749 – currently ranked #15 in the world. Tomashevsky leads the Grand Prix standings with 252 points, driven by an absolutely stellar performance in the third leg held in Tbilisi in February. He entered that event as an underdog, rated just 2716, and given just a 2% chance of finishing in the top two of the final standings by our model. However he crushed that field, winning five games and drawing the rest, winning clear first by a huge margin of a point and a half, and earning 170 GP points (and 29 rating points) for his efforts. The performance was so remarkable and unexpected that it prompted us to take a deeper look at whether we should have seen it coming. Now our model gives him a much more impressive 51.8% chance of holding his lead and earning one of the coveted candidates berths. His job is not yet complete though. Even with his rating improved to 2749 our model still views Tomashevsky as more likely to finish in the middle of the pack than on top, projecting him to earn an average of 65 GP points from this final leg and giving him only a 3% chance of winning this event outright like he did the last one. He’s bucked such odds in the past, but two highly rated challengers are nipping at his heels, ready to soar past him if he should stumble this time around. One extra challenge in holding his spot: he has to play six games as black and only five as white.
  • Fabiano Caruana – 2803 – currently ranked #3 in the world. Caruana is second in the current standings (ignoring players who already completed their three events and are completely eliminated from contention for the top two spots) with 230 Grand Prix points. He got there by tying for first place in leg one, and adding a middle of the pack result in leg two. As the highest rated player in the field, and with the extra advantage of six games as white, our simulations give him a 17% chance of winning this event outright, while projecting him to earn an average of 106 GP points from the leg. To win, he would need to outscore Tomashevsky by 22 points, and we expect him to do so more often than not (with a 41 point lead in his average projected result). Of course to win the whole thing he also needs to avoid being passed by anyone else, but then again to reach the Candidates Tournament he doesn’t need to win, just finish in the top two. All told, this works out to a 71.1% chance of finishing in the top two of the final standings.
  • Hikaru Nakamura – 2799 – currently ranked #4 in the world. Nakamura is third in the (relevant portion of the) current standings with 207 points. He trails Tomashevsky by a much larger margin (45), and also benefits in our projections from a high rating (although unlike Caruana, he has the tougher draw with six black games). We expect him to score an average of 99 points in this event, with a 14% chance of winning it all, and as with Caruana he of course also could qualify for the Candidates Tournament from second place even without catching Toma. Our numbers show him as the smallest of underdogs to finish in the top two, at a 49.1% chance. Might as well flip a coin!

The Other Contenders

  • Dmitry Jakovenko – 2746 – #17. Tied for (relevant) fourth place in the current standings with 170 GP points, Jakovenko is still in the running if he can swing a good result here. While he has only a 3% chance of winning the event outright, and is only expected to average 68 points from this leg, if he does manage to finish first or second and two of the top three stumble, he’s definitely still alive. We project him to climb into the top two of the final standings and win a berth in the Candidates Tournament 8.6% of the time.
  • Boris Gelfand – 2744 – #18. Tied with Jakovenko for fourth is Gelfand with 170 points of his own. The same principles apply, and their ratings are almost dead even, however his odds are hurt by having the black pieces six times, while Jakovenko only has them five times. He has the same 3% chance of winning the whole event, but projects to average only 62 points. We have him finishing in the top two 7.0% of the time.
  • Sergey Karjakin – 2753 – #12. Next, at #6 in the standings, is Karjakin with 157 points. A slightly higher rating and a good draw (six games as white) actually make his projections marginally better than Gelfand’s, even though he trails currently. We have him averaging 72 points from this event, winning outright 5% of the time, and cracking the overall top two 7.5% of the time.
  • Alexander Grischuk – 2785 – #6. At #7 in the (relevant) standings is Grischuk with just 122 points. That’s an awfully steep hill to climb, but he is the third highest rated player in the field, with six games as white, so his projections are quite good. We expect him to earn an average of 93 GP points, and win this event outright 11% of the time, but even that clear first place result isn’t enough to guarantee he’ll reach the top two position he needs in the overall standings. We expect him to earn a Candidates berth just 3.1% of the time.

The Super Longshots

  • Anish Giri – 2776 – #10. One of three players tied for (relevant) 8th place with 115 points, Giri at least projects well thanks to his top-ten world rating. A 7% chance of winning the event, and an expected average of 82 points, are nice but he needs a lot of help even if he does win it all. To overcome the massive standings deficit and finish in the top two he needs tremendous luck, as we see it happening only 1.1% of the time.
  • Maxime Vachiere-Lagrave – 2743 – #19. The second player with 115 points, MVL is in even worse shape as due to his lower rating he only projects to score 65 points with a 3% chance to win the leg. His odds of reaching the Candidates Tournament are a meager 0.6%.
  • Baadur Jobava – 2699 – #47. The third player with 115 points is Jobava, who is also the lowest rated player in the field. We give him only a 1% chance of winning the event, and project that he’ll earn an average of only 42 points, although he’s very likely to do it in an exceedingly entertaining fashion given his reputation for playing “exciting” chess! Unfortunately for fans of his style, there is only a 0.2% chance he’ll earn the opportunity to showcase that style in the Candidates Tournament.
  • Peter Svidler – 2736 – #23. In 11th place with just 102 points is Svidler. We all see how this works by now: with a mediocre rating (at least in this field… millions of players around the world would love to be able to call 2736 “mediocre”) he projects to add just 58 points, win the event only 2% of the time, and reach the top two just 0.1% of the time. Or, as optimists everywhere would note: “So you’re saying there’s a chance!” Yes there is a chance. It is approximately 1 in 1,000.
  • Leinier Dominguez – 2737 – #22. Finally we have our last place contestant. Dominguez has just 85 points right now, and unfortunately that’s almost certainly not enough. Even if he came in first, he’d have just 255 points, meaning his best case scenario is a second place finish to Tomashevsky (who will add a minimum of 10 to his current 252). Second place is good enough in the Grand Prix, but it would also require Caruana to finish in the bottom two of the field, and Nakamura to finish in the bottom five, and none of the other contenders to pass him either. This is not technically impossible, and a previous simulation gave him a 0.003% shot of everything coming together, but our new simulation today that we ran after the official pairings were released did not see him reach the top two a single time in 20,000 times. His odds are essentially, but not technically, zero.

The Eliminated

  • Shakhriyar Mamedyarov
  • Teimour Radjabov
  • Dmitry Andreikin
  • Rustam Kasimdzhanov

These four players have already completed their Grand Prix run, playing the first three events and sitting this one out. Mamedyarov managed the highest score of the group, amassing 235 points, which is enough for him to technically be #2 in the standings at this moment, but that’s a spot it’s impossible for him to hold. It is 100% guaranteed that none of these four will reach the Candidates Tournament by finishing in the top two of the Grand Prix standings. When we referred to players’ “relative” standings above, it was ignoring these four eliminated players, which is why we said Caruana was in second place right now, when technically Caruana is currently third behind the eliminated Mamedyarov.

Other Notes

  • The eagle eyed (or obsessive) among you may have added up the stated odds of each player winning this event outright, and seen that of course they add up to just over 70%. Our simulation predicts a 28% chance that first place in this leg will be a tie between two or more players.
  • Here are projected results for all 12 players in this leg:
    Player K-M AVERAGE SCORE Odds of Clear 1st
     Fabiano Caruana (ITA) 106 17%
     Hikaru Nakamura (USA) 99 14%
     Alexander Grischuk (RUS) 93 11%
     Anish Giri (NED) 82 7%
     Sergey Karjakin (RUS) 72 5%
     Dmitry Jakovenko (RUS) 68 3%
     Evgeny Tomashevsky (RUS) 65 3%
     Maxime Vachier-Lagrave (FRA) 65 3%
     Boris Gelfand (ISR) 62 3%
     Peter Svidler (RUS) 58 2%
     Leinier Dominguez (CUB) 58 2%
     Baadur Jobava (GEO) 42 1%
  • Here is how each player’s chances at reaching the Candidates Tournament have ebbed and flowed since we began tracking them before the third leg began:
     Fabiano Caruana (ITA) 58% 70% 71%
     Evgeny Tomashevsky (RUS) 2% 49% 52%
     Hikaru Nakamura (USA) 39% 52% 49%
     Dmitry Jakovenko (RUS) 1% 2% 9%
     Sergey Karjakin (RUS) 5% 7% 8%
     Boris Gelfand (ISR) 6% 8% 7%
     Alexander Grischuk (RUS) 40% 3% 3%
     Anish Giri (NED) 14% 7% 1%
     Maxime Vachier-Lagrave (FRA) 15% 0.2% 0.6%
     Baadur Jobava (GEO) 0.5% 1% 0.2%
     Peter Svidler (RUS) 5% 0.1% 0.1%
     Leinier Dominguez (CUB) 0.3% 0.002% 0%
     Shakhriyar Mamedyarov (AZE) 6% 0% 0%
     Teimour Radjabov (AZE) 0.02% 0% 0%
     Dmitry Andreikin (RUS) 10% 0% 0%
     Rustam Kasimdzhanov (UZB) 0% 0% 0%

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 (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.

Playing With Date Arithmetic (And The 10 Youngest 2600+ Players Ever)

In my future posts, you’ll probably hear a lot about players’ ages at the time they achieve various milestones. I will present those dates as numbers with (usually two) decimal points, rather than the common Years-Months-Days format. The latter is more easily understandable: we know exactly what it means when we say that, for instance, Wei Yi is the youngest player to achieve a rating of 2700+, at the age of 15 years, 8 months, and 29 days*. Saying that he achieved that milestone at the age of 15.75 years old actually makes this a poor example, because that’s pretty clear, but if the number were 15.63 instead it would be harder for our brains to immediately process it. We’re not used to thinking of ages in terms of fractions of years (except the “big” fractions like 1/2, 1/4, 3/4). We break years down into months, not increments of 0.01 year, and we break those months down into days.

Unfortunately, the Years-Months-Days format is also less accurate. To see why, let’s consider another example. Here are the 10 youngest players to achieve a rating of 2600 or higher:

Youngest Published 2600+ Rating
Player Name Age
Wei Yi 14.42
Wesley So 14.98
Teimour Radjabov 15.06
Magnus Carlsen 15.09
Sergey Karjakin 15.22
Ruslan Ponomariov 15.23
Illya Nyzhnyk 15.59
Fabiano Caruana 15.67
Anish Giri 15.67
Peter Leko 15.81

Note that there is an apparent tie for 8th place between Caruana and Giri. Were they actually the exact same age? Well, let’s look at Year-Month-Day format first. Caruana was born 7/30/1992, and achieved this milestone on the 4/1/2008 rating list. From July 30 1992 to March 30 2008 is 15 years and 8 months. March 30 to April 1 is two days, so he was 15 years, 8 months, and 2 days old. How about Giri? Born 6/28/1994, he broke the 2600 barrier on 3/1/2010. From 6/28/1994 to 2/28/2010 is 15 years, 8 months, and we add 1 day from 2/28 to 3/1. So it would appear that Giri achieved this milestone 1 day earlier than Caruana did! The chart above placed them in the wrong order, right?

Well, no. Let’s break Caruana’s age down further. The “15 years” component from 7/30/1992 to 7/30/2007 is 15 * 365 = 5475 days, right? Nope. Leap Year exists! That particular span includes three extra days: 2/29/1996, 2/29/2000 and 2/29/2004. So “15 years” in this case means 5478 days. What about the “8 months” portion? Well the months whose last day was included in that span are June through February, meaning we get 5*31 + 2*30 + 1*29 = 244 days out of the span (remember that 2008 was also a Leap Year). Finally we add in the “2 days” portion, for which no breakdown is needed, and we see that Caruana achieved his first 2600+ rating when he was 5478 + 244 + 2 = 5724 days old.

Giri’s “15 years” include four Leap Years, not three, and his “8 months” do not include a February, which adds two days to his total. So “15 years, 8 months, and 1 day” is, in his particular case, 5725 days. Despite appearing to have been one day younger, using the more common format, it turns out that Giri was actually one day OLDER than Caruana. My chart above is correct after all.

Now of course it doesn’t matter at all which of two amazing players, currently ranked #3 and #4 in the World, got to 2600 one day faster than the other. However this example serves perfectly to demonstrate why I will not use the Years-Months-Days format to express players’ ages. In fact, behind the scenes, all my ages are simply number of days, but I don’t imagine anyone wants to know that Wei Yi was 5751 days old when he broke the 2700 barrier, so I divide ages by 365.25 (to account for Leap Year) and present them as just “years old”, rounded to the appropriate number of decimal places for the particular purpose in play.

I hope you appreciate my precision.

*This isn’t technically true yet, but it appears that it will become true on March 1st, when the next FIDE rating list is published.