I saw a claim floating around Twitter that for Nakamura to qualify for the Candidates Tournament (finish in the top two of the final Grand Prix standings), he *HAD* to outscore Tomashevsky by at least 1.5 points over the final 5 rounds, which would be an extremely difficult task. I doubted this could be true if if our numbers were right (estimating Naka at 30% to qualify) so I ran a few simulations until I managed to find an exception. The following is a set of results in all remaining games that allows Nakamura to finish second in the final Grand Prix standings despite “only” outscoring Tomashvesky by 1 point (3/5 to 2/5). As a bonus, this scenario even involves Naka losing a game, and still getting in!:
Round 7:
| Caruana 1/2 – 1/2 | Gelfand |
| Jakovenko 1/2 – 1/2 | Grischuk |
| Karjakin 1/2 – 1/2 | Jobava |
| Nakamura 1/2 – 1/2 | Vachier-Lagrave |
| Giri 1/2 – 1/2 | Tomashevsky |
| Dominguez 1/2 – 1/2 | Svidler |
Round 8:
| Gelfand 1/2 – 1/2 | Svidler |
| Tomashevsky 0 – 1 | Dominguez |
| Vachier-Lagrave 1/2 – 1/2 | Giri |
| Jobava 0 – 1 | Nakamura |
| Grischuk 1/2 – 1/2 | Karjakin |
| Caruana 1 – 0 | Jakovenko |
Round 9:
| Jakovenko 1/2 – 1/2 | Gelfand |
| Karjakin 0 – 1 | Caruana |
| Nakamura 1 – 0 | Grischuk |
| Giri 1/2 – 1/2 | Jobava |
| Dominguez 1/2 – 1/2 | Vachier-Lagrave |
| Svidler 0 – 1 | Tomashevsky |
Round 10:
| Gelfand 1/2 – 1/2 | Tomashevsky |
| Vachier-Lagrave 1 – 0 | Svidler |
| Jobava 1/2 – 1/2 | Dominguez |
| Grischuk 1/2 – 1/2 | Giri |
| Caruana 1 – 0 | Nakamura |
| Jakovenko 1/2 – 1/2 | Karjakin |
Round 11:
| Karjakin 0 – 1 | Gelfand |
| Nakamura 1/2 – 1/2 | Jakovenko |
| Giri 1/2 – 1/2 | Caruana |
| Dominguez 0 – 1 | Grischuk |
| Svidler 1/2 – 1/2 | Jobava |
| Tomashevsky 0 – 1 | Vachier-Lagrave |
Final Khanty-Mansiysk Standings:
| Name | Score | GP Pts |
| Caruana | 8.5 | 170 |
| Nakamura | 6 | 113 |
| Dominguez | 6 | 113 |
| Gelfand | 6 | 113 |
| Karjakin | 5.5 | 75 |
| Svidler | 5.5 | 75 |
| Vachier-Lagrave | 5 | 50 |
| Grischuk | 5 | 50 |
| Jakovenko | 5 | 50 |
| Jobava | 4.5 | 20 |
| Giri | 4.5 | 20 |
| Tomashevsky | 4.5 | 20 |
Resulting Final Grand Prix Standings:
| Player | Baku | Tashkent | Tbilisi | Khanty-Mansiysk | TOTAL |
| Fabiano Caruana (ITA) | 155 | 75 | 170 | 400 | |
| Hikaru Nakamura (USA) | 82 | 125 | 113 | 320 | |
| Boris Gelfand (ISR) | 155 | 15 | 113 | 283 | |
| Evgeny Tomashevsky (RUS) | 82 | 170 | 20 | 272 | |
| Shakhriyar Mamedyarov (AZE) | 35 | 125 | 75 | 235 | |
| Sergey Karjakin (RUS) | 82 | 75 | 75 | 232 | |
| Dmitry Jakovenko (RUS) | 30 | 140 | 50 | 220 | |
| Teimour Radjabov (AZE) | 50 | 50 | 110 | 210 | |
| Dmitry Andreikin (RUS) | 20 | 170 | 10 | 200 | |
| Leinier Dominguez (CUB) | 10 | 75 | 113 | 198 | |
| Peter Svidler (RUS) | 82 | 20 | 75 | 177 | |
| Alexander Grischuk (RUS) | 82 | 40 | 50 | 172 | |
| Maxime Vachier-Lagrave (FRA) | 75 | 40 | 50 | 165 | |
| Baadur Jobava (GEO) | 75 | 40 | 20 | 135 | |
| Anish Giri (NED) | 40 | 75 | 20 | 135 | |
| Rustam Kasimdzhanov (UZB) | 35 | 15 | 75 | 125 |
The exact scenario is of course extraordinarily unlikely (as is any one specific set of results), but it demonstrates a few key things that have to work in Nakamura’s favor in order for him to pull off second place without outscoring Tomashevsky by more than one point.
First, Caruana runs away with the event while other players at the top of the standings stumble, allowing Nakamura to catch up with (and tie for) second place. This is critical because the top three places earn (or split in the case of a tie) bonus Grand Prix points.
Second, the players at the bottom of the field bounce back in the final rounds, and catch up with Tomashevsky, so that he earns fewer points than he “normally” would with a 2/5 finish.
Please double check my math, but I’m pretty sure that this scenario is a valid case in which Nakamura could qualify (somewhat convincingly) for the Candidates Tournament with a 2nd place finish in the Grand Prix, while NOT outscoring Tomashevsky by 1.5 (or more) points over the final five rounds at Khanty-Mansiysk. If I’m wrong, let me know, so that I can identify the flaw in my model ASAP!
Update: How unlikely is this exact scenario, by the way? Extremely, but that’s true of all specific cases. Obviously, since a draw is the most likely of the three possible results in any given game, the single most likely scenario would be for every game in the final five rounds to happen. Even that, which again is the MOST LIKELY scenario, would happen less than 1 in 70 million times! With three possible results in each game (white win, draw, or black win) there are 3^30 possible permutations for how the event could finish up!
Chess by the Numbers 
The stronger GMs are showing the way in Khanty-Mansiysk (Caruana, Karjakin …and Naka).
Caruana is treating the GP leg as just ‘another’ tournament, as if to say that this is how naturally good he is!
Karjakin probably feels that as 2nd placer in the last Candidates Tournament, it’s humiliating if he cannot even just qualify for it again. Being thrown out from the Grand Chess Tour is Sergey’s (and all the other Russian players feel the same) strongest motive to tell Kasparov and the GCT organizers that they are wrong … just look at MVL!
Naka to outscore Toma by 1.5 pts? don’t bother! Naka is playing cool, as Karjakin has observed in the 1st round (something like “…you prepared well for the game”). Naka has so far kept his cards close to his chest even if he needs a win to booast his GP standing. Like Toma, he feels he is atop the pedestal with his every chance to make it to the Candidates, i.e., via ratings or by organizer’s (Sinquefield/USCF) nomination, althu he said at the US Championship that Khanty-Mansiysk is his “most important tournament”.
Conversely, the other strong player, Grischuk, is not motivated but is having fun. He is in Khanty-Mansiysk to fulfill his obligation, just being present and play. But watch him in World Cup, it will be different.
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Your final score table lists Nakamura at 6 and Tomashevsky at 4.5, a 1.5 point difference.
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Yes, but only a one point difference *during the final five games* since he was already half a point ahead through round 6. That was the point of contention. Whether he had to outscore by Tomashevsky by 1.5 in just those games (finishing 2+ points ahead).
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Oh. That wasn’t clear to me as well. I understood it that Naka had to outscore Toma by 1.5 in the whole tournament, not in the last games.
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Got it, thanks for clarifying. I’m enjoying your Grand Prix coverage!
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At last, Naka gets his first win(!) over MVL, the favourite whipping boy. Naka only needs to stay within half point behind Karjakin (which he is now) to counter Karjakin’s charge.
IMO Naka just need 1-pt over Toma (but has to finish amongst Top 3) to dislodge Toma from slot #2 in GP series.
Recall that Naka’s huge 50 GP points lead over Karjakin is because of only ½-pt difference between their scores at Tashkent (Naka 6.5 vs. Karjakin 6). He can repeat the same here at Khanty-Mansiysk against Toma!
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an interesting point is that it may be in Grischuk’s interest to lose against Nakamura in round 9, although this lowers Grischuk’s rating
so if Grischuk loses, he may be accused of cheating (or however you call this), even if this wasn’t his intention
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Grischuk trying to get in by rating is dangerous, given how close Kramnik (and So and Aronian) are. I wouldn’t bet on him losing on purpose, dropping his rating further.
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I’ve been mulling over this idea for a couple days. It’s really hard to tell whether losing that game would benefit Grischuk or not mathematically. Morally it’s much simpler of course.
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Not that dangerous. It actually makes sense if he could bring himself to do it. If he gifts Nakamura a win and helps him qualify it opens up another rating slot for an alite player. It is unlikely Karjakin can make it by rating and it is impossible for Tomashevsky to do it at this point in the cycle so Grischuk would be slamming the door shut on one of his countrymen so another top 10 player could fight for rating qualification since Nakamura, who is likely to be in the final chase for rating qualification would be “off the market” for a slot. In other words he is closing the door on an option he doesn’t have so he can open a window only he and a few others can reach.
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One could do way better (/worse, depending on how you look at it). If Nakamura finishes a clear second and gets 140 points, then Tomashevsky needs a third place, and it cannot be split in more than two. A three-way split third will not do it for Tomashevsky. Even if Nakamura ends up in a two-way split second on 125 points, then a four-way split fourth is not enough for Tomashevsky. A three-way split fourth leaves it to the tiebreaks.
It was – and I think stil is – possible to see Nakamura end at 6 points and a whole bunch of guys have 5.5: let MVL have the full point over Domínguez and let Karjakin draw against Gelfand but lose against Grischuk. Domínguez and Karjakin and Gelfand and Svidler and Grischuk and MVL are all 5.5. That means you can allow Tomashevsky another full point.
Looks like it still is possible even after today’s rounds. Give Caruana four points, and adjust Nakamura to end up at 6. It seems possible that nobody else gets more than the field average of 5.5 (in which case there will necessarily be more than two).
Give Tomashevsky 3 points (losing against MVL); give MVL four points (unlikely, given his shape but theoretically possible). Work it down from there: Karjakin 1/2 against Gelfand and 1/2 against Grischuk. Domínguez as above except losing to Tomashevsky.
Gut feeling says the puzzle would easily match up. It should be theoretically possible for Nakamura to score 2 in the remaining games while Tomashevsky scores 3, and Nakamura still goes to the candidates.
It wouldn’t surprise me if it is still possible after round 8. Nobody but Caruana will hit the 6 points mark by then. Domínguez could beat Tomashevsky, but Tomashevsky could still score 3/3. Karjakin could win but lose the rest. Gelfand could win tomorrow and against Karjakin but lose the rest. But then I think Jakovenko has to lose against Caruana, or draw and someone work out the four-way split fourth spot.
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With yesterday Naka’s win, Toma’s loss and Karjakin’s draw, the fight for GP slots has changed favourably for Hikaru. However, the situation is still liquid and unless Naka finished at least solo Top 3 (not a 3-4 split), he can be dislodged such as in the following scenario:
Rank Pts Name/s Calculation Khanty GP
—————————————————————————————————
1-2 7 Karjakin / Caruana (140+170) / 2 = 155
3-4 6.5 Naka / Dominguez (90+110) / 2 = 100
6-7-8 5.5 Toma / Gelfand / Giri (50+60+70) / 3 = 60
Here, Karjakin gain 55 points whilst Toma is only 40 points behind vis-a-vis Naka. Then GP slot #2 shall go to Toma and #3 go to Karjakin.
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