How do they tie break?

Nice, complete and accurate analysis by filiusdextris! Here's a question....assuming Nakamura and Caruana qualify for the Candidates tournament by GP points (which is the most likely case), then which 2 players will qualify by rating points? Topalov and Kramnik? Are there any scenarios where So gets in?

with the frequency with which these games enter into time trouble, Carlsen is going to have a blast playing any of these opponents. I wish them all the best

The statement that for Jakovenko "Anything less than that will leave him out of the competition" is not true. He can qualify with a draw as well, provided Caruana loses and other results go the right way for him. All told I have Jakovenko at an estimated 19.5% chance of qualifying (and the not-quite-dead-yet Tomashevsky with about a 0.3% chance).

There's also a ridiculously unlikely scenario where it's technically possible for both Jakovenko AND Tomashevsky to advance (it will happen roughly 1/5500 times).

Gelfand

If Gelfand defeats Karjakin, and if Giri defeats Caruana, and if Nakamura draws Jakovenko; and if Dominguez draws Grischuk, Gelfand would tie 1st-3rd with Nakamura and Jakovenko, winning 140 GP points, having 310 overall. Caruana would tie for 4th-6th with Giri and Dominguez, winning 80 GP points, having 310 overall. In that scenario, Tomashevsky would, at best, tie for 7th-8th with Grischuk, winning 55 GP points and 307 overall.

Nakamura would finish first in the GP standings with 347 GP points, Gelfand and Jakovenko and Caruana would tie for 2nd with 310 points. Under the first tiebreak, cumulative game result points, Caruana has 18.5; Jakovenko has 17.5; Gelfand has 16.5. Caruana would advance as 2nd place in the GP series, with Jakovenko as first alternate, and Gelfand as second alternate.

Instead of scenarios where Gelfand finishes tied for 2nd, finishing 4th on tie-breaks, it might be better to finish clear 3rd. Gelfand cannot pass Caruana on the final GP standings, even if the latter loses. This requires a decisive Nakamura-Jakovenko result (either way), and of course a Gelfand win vs. Karjakin. Gelfand would win at least 100 GP points with a 3rd-4th place tie with Dominguez, a 110 if Dominguez does not win, and a 140 if he also passes Caruana, for respective possible totals of 270, 280, or 310. If Nakamura wins in this scenario, Gelfand always finishes ahead of Jakovenko. But if the decisive game favors Jakovenko, Nakamura can finish 3rd in the overall GP standings behind Caruana/Jakovenko, ahead of Gelfand in some cases. Tomashevsky can also catch him in some scenarios. Gelfand does better when Dominguez doesn’t catch him, and when he also passes Caruana in the current event. If both of these happen, his clear-2nd finish would only lose to Tomashevsky if the latter finishes clear 7th, possible if Tomashevsky wins and Jobava defeats or draws Svidler.

Karjakin

Karjakin can best finish in a five-way tie at 6 points for 2nd-6th place with Giri, Caruana, the Nakamura-Jakovenko loser, and the higher of Grischuk-Dominguez. This would give him 98 GP points and a total of 255. He would still finish behind Tomashevsky, Caruana, Nakamura, Jakovenko for 5th place overall in the GP standings no matter what happened. This is still better than anyone else, even Mamedyarov, who also will finish at least behind either Karjakin or Gelfand.

Caruana

If Caruana wins, he wins 155-170 GP points, the lesser number if Nakamura-Jakovenko is decisive. Since we know Nakamura can win and advance (see above analysis for Nakamura), the same is true for Caruana who has the same current score, but more GP points coming in. There is a highly unlikely scenario where Nakamura won’t advance with either 1st or 2nd with a draw (see same analysis) because Tomashevsky can catch him, but Caruana has an additional 23-point buffer to eliminate this possibility.

If Caruana loses to Giri, and Nakamura draws with Jakovenko, Caruana will still finish ahead of Jakovenko if either Gelfand or Dominguez wins or both (see above analysis for Jakovenko), which dilutes Jakovenko’s point total too much. He defeats Jakovenko only on tie-break in the case of one extra winner (see same), but that’s good enough. He even defeats Jakovenko and Gelfand on tie-break at 310 if that winner is Gelfand, but again, that’s good enough (Caruana’s 6.5+6.0 +6.5 vs. Gelfand’s 6.5+3.5+6.0). At worst case in that scenario, Caruana finishes tied for 5th-6th with Giri, behind Nakamura, Jakovenko, Dominguez, and Gelfand. He would gain 75 GP points, putting him at 305 total, ahead of Jakovenko at 297. However, in this case, Tomashevsky could finish clear 7th by defeating Vachier-LaGrave which would give him 60 GP points and 312 total, narrowly advancing over Caruana. This requires Svidler to also draw or lose against Jobava, thereby not catching Tomashevsky and thus diluting his standing.

Tomashevsky

Based on the above analysis, Tomashevsky can finish 2nd overall in the GP series, thus qualifying for the Candidates, but only if both Gelfand and Dominguez win, which would dilute second-place ties too much in various scenarios. In the case of Caruana winning and a Nakamura-Jakovenko tie, he only needs to win his own game to pass Nakamura. In the case of Caruana losing to Giri, he also needs Jobava to defeat or draw Svidler in order to pass Caruana.

Nakamura

If Nakamura wins, he wins 155-170 GP points for a total of 362-377. Tomashevsky cannot catch him, even if he finishes clear 6th, his highest possible finish, with 70 GP points, and a total of 322. He would finish either first or second in the final GP standings depending on how Caruana does.

If Nakamura draws with Jakovenko, he finishes ahead of Jakovenko in the GP series since he had more points coming in and they finish on the same score here. If Nakamura and Caruana both draw, he finishes behind Caruana in the GP series since he had less points coming in and they finish on the same score here. If Nakamura draws, and Caruana loses, Nakamura needs to make up the difference of 23 points in the current GP standings. If Nakamura draws with Caruana losing, the worst he can do is a four-way tie for first with Jakovenko, Gelfand, and Dominguez. Caruana would tie with Giri in that scenario for 5th-6th and 75 GP points, but Nakamura would gain at worst 127 points, easily erasing the current 23 points deficit.

So, if Nakamura draws, and Caruana draws or wins, the only person who could then push Nakamura out of 1st or 2nd and a ticket to the Candidates is Tomashevsky. In a five-way tie for 1st place, Nakamura nets 118 points. Added to his current total, his final GP point total would be 325. Tomashevsky could finish at best with a tie with Giri for 6th-7th and 65 points. Added to his current total, his final GP point total would be 317, still behind Nakamura. However, if Caruana defeats Giri, and Nakamura draws into a four-way tie for 2nd-5th, Nakamura wins 105 points, going to 312 points. In this scenario, Tomashevsky can finish clear 6th by defeating Vachier-Lagrave and gain 70 points, pushing his total to 322. Even a shared 6th-7th tie with Svidler (if the latter beats Jobava), still surpasses Nakamura.

Thus, if Nakamura draws, in order for him not to finish at least 2nd in the final GP standings with a ticket to the Candidates, all of the following must occur: Caruana must defeat Giri (with Black), Gelfand must defeat Karjakin (with Black), Dominguez must defeat Grischuk, and Tomashevsky must defeat Vachier-Lagrave.

Jakovenko

If Jakovenko wins, he wins 155-170 GP points for a total of 325-340. Tomashevsky cannot catch him, even if he finishes clear 6th, his highest possible finish, with 70 GP points, and a total of 322. He would finish either first or second in the final GP standings depending on how Caruana does.

If Jakovenko draws, he would finish behind Nakamura. To finish in 2nd place overall, Caruana must lose. Jakovenko would gain between 127-155 points, depending on whether Dominguez and Gelfand can win their games, taking a tied share of first place. In a four-way tie for first place among Jakovenko, Nakamura, Gelfand, and Dominguez, Jakovenko only gains 127 points, bringing his total to 297. In that scenario, Caruana ties for 5th-6th winning 75 GP points, and surpassing Jakovenko with 305. In a three-way tie for first place among Jakovenko, Nakamura, and only one of Gelfand, and Dominguez, they each gain 140 points, bringing Jakovenko’s total to 310. In that scenario, Caruana finishes tied 4th-6th with Giri and one other among the group of Gelfand, Dominguez, Karjakin or Grischuk. Caruana would gain 80 points, bringing his total to 310, the same as Jakovenko. Caruana defeats Jakovenko on the first tie-break as Caruana’s cumulative score across the GPs (6.5 + 6.0 + 6.0) is higher than Jakovenko’s (4.5 + 6.5 + 6.5).
Therefore, Jakovenko can only advance to a 2nd place final GP standing, and a ticket to the candidates, with a draw if Caruana loses and both Gelfand and Dominguez draw or lose. Note that in this scenario, Jakovenko gains 155 points for a first-place tie for a total of 325. Tomashevsky cannot catch him, since his highest finish in this hypothetical, a clear 7th, adds 60 GP points to his current total of 252 and a final tally of 312, short of Jakovenko.

The likelihood of this happening is very, very small. Tomashevsky has easier ways in but they involve Caruana Losing, not winning.

It's worth noting that Tomashevsky is not mathematically eliminated from qualifying yet! All 6 games will need specific results, but in the unlikely situation where Caruana defeats Giri, Nakamura and Jakovenko draw, Gelfand defeats Karjakin, Dominguez defeats Grischuk, Svidler does not defeat Jobava and Tomashevsky defeats MVL, then Tomashevsky will qualify for the candidates (along with Caruana)! ;-) Nakamura will still likely qualify for the Candidates tournament on rating points.