As the games progress, I am placing xxx in front of the possibilities that are eliminated.
DISCLAIMER: The following results are generated manually by me, mostly in my head. Errors are possible and maybe likely!
Following are all the possibilities for whether Magnus Carlson or Maxime Vachier-Lagrave will win the 2017 Grand Chess Tour. Magnus leads MVL by three GCT points going into the tournament, and more points will be awarded depending on their respective finishes in this tournament: 13 points for clear first, 12 points for a playoff winner, 10 for second, 8 for third, 7 for fourth, 6 for fifth, 5 for sixth, 4 for 7th, 3 for 8th, 2 for 9th, 1 for 10th. Points are divided equally in case of ties.
In the tournament, MVL leads Magnus by a half point going into the final round, and has a chance to win the tournament (and hence the GCT). But the possibilities if MVL doesn't win the tournament are extremely complicated. All five games have a possible influence on the GCT outcome.
Going into the final round, we have Ian Nepomniachtchi in clear first with 5.5, Fabiano Caruana in clear second with 5, and MVL in clear third with 4.5. Magnus is in a pack of 4 players at 4, and the standings are rounded out with three players in last place, with 3 points.
The first and third place players, Ian and MVL, play each other, so that MVL can catch Ian by beating him. Fabi plays a bottom player Adams. Carlsen plays another 4-point person, Aronian, while the other two 4-pointers, Nakamura and So, play 3-pointers.
If MVL wins, he is guaranteed to stay ahead of the entire 4-point pack. If MVL draws, he can be tied by any of the 4-point pack who wins, or tied by a losing Fabi, dropping down from above. If he loses, he can be tied by drawing 4-pointers, or passed by winning 4-pointers.
If Magnus draws, he would be tied by his opponent Aronian. He would be passed by a winning Nakamura or So, or tied by a drawing Nakamura or So. If Magnus loses, he could be tied with a losing Nakamura or So, as well as the 3-point players that beat Nakamura, So, or Fabi.
Nakamura and So's respective games are equivalent when it comes to affecting the relative GCT standings of Magnus and MVL. So for example, when we consider the case in which one of them draws and the other wins, it doesn't make any difference whether Naka draws and So wins, or vice versa. This condition is indicated by the entry "Naka/So DW" in the charts below.
In general, Magnus benefits if Fabi does well. If MVL wins, Magnus benefits if Nakamura and So don't win. If MVL loses, Magnus benefits if Nakamura and So do well.
Lets now look at the specific cases.
If MVL wins and Fabi wins, Fabi wins the tournament, and MVL and Ian tie for second. MVL gets 9 GCT points.
If MVL wins and Fabi loses, MVL and Ian tie for first and have a playoff. MVL gets 12 points if he wins, and 10 points if he loses.
If MVL wins and Fabi draws, I'm not sure of the playoff situation. It's a three-way tie and Ian wins the first tiebreaker with more wins. Now you apply the second tiebreaker (games against each other) to determine the number two player for the playoff. The question is, do you apply that tiebreaker to just the two players left, or do you apply it to all three players? If the former (which is the way the US National Football League does it, but they state that explicitly), then MVL and Fabi are tied, having drawn each other. In that case, there is no playoff; but do you declare Ian the winner or does it fall back to a three-way tie? MVL gets either 10 or 9 GCT points, depending on the answer.
If you apply the second tiebreaker to all three players, then MVL beats out Fabi with his win against Ian, and Ian and MVL do the playoff. MVL gets either 12 or 9 GCT points.
If Fabi loses, then MVL scores at least 10 points. In that case, Magnus needs 7 points to tie MVL, 7.5 to win. Magnus could join Fabi in a tie for third place by winning. The problem for Magnus is, the two Americans Nakamura and So could join them, too, by winning their games. If neither win, Magnus gets 7.5 points and wins the GCT by a half point. If exactly one of the two Americans wins, Magnus get 7 points and there is a GCT playoff. If both Americans win, Magnus drops to 6.5 points, and MVL wins the playoff.
If Fabi loses and Magnus doesn't win, he can do no better than a tie for 4th with Aronian, for 6.5 points, and MVL will win the GCT with his 10 points.
If Fabi doesn't lose, then the bar for Magnus winning the GCT is lowered by the possibility of MVL scoring 9 GCT points. If Magnus wins his game, he gets either clear 4th (7 points), or a tie for 4th with a winning Nakamura and/or So (6.5 points if one wins; 6 points if both win). If Magnus draws, then he might still tie for fourth with Aronian, if neither Nakamura nor So wins. In that case, Magnus get 6.5 points if neither Nakamura nor So draw; 6 points if one draws, 5.5 points if both draw. Magnus needs at least a tie for 4th: if Magnus draws and either Nakamura or So wins, Magnus loses the GCT in all cases.
Note that there are several scenarios above in which MVL loses the tournament playoff, and then has to play a GCT playoff against Magnus. That would be a long day!
Thanks for looking, Jen!
In this case, MVL is tied with Aronian, either for second place with Fabi if Fabi loses, or for third place if Fabi draws or wins. The tie can also be joined by a winning Nakamura and/or So. Each Naka/So win dilutes MVL's tie, but also pushes Magnus down the standings. Each Naka/So draw also pushes Magnus down the standings, but does not dilute MVL's tie. Each Naka/So loss promotes Karjakin/Anand into the tie with Magnus a 4 points, diluting his GCT point total. And a Fabi loss promotes Adams into a tie with Magnus.
Magnus has chances only if Fabi wins or draws, and Nakamura and/or So wins. Here's the chart:
If Fabi Loses:
and then MVL is with and Magnus is with GCT result:
Naka/So DW T-2--5 7.75 pts, T-7--8 3.5 pts. MVL wins by 1.25
xxx Naka/So LL T-2--4 8.33 pts, T-5--10 3.5 pts. MVL wins by 1.83
xxx Naka/So LD T-2--4 8.33 pts, T-6--9 3.5 pts. MVL wins by 1.83
xxx Naka/So LW T-2--5 7.75 pts, T-6--9 3.5 pts. MVL wins by 1.25
xxx Naka/So DD T-2--4 8.33 pts, T-7--8 3.5 pts. MVL wins by 1.83
xxx Naka/So WW T-2--6 7.20 pts, T-7--8 3.5 pts. MVL wins by 0.70
If Fabi wins or draws:
and then MVL is with and Magnus is with GCT result:
Naka/So DW T-3--5 7.0 pts, 7th 4.0 pts. tied, playoff
xxx Naka/So LL T-3--4 7.5 pts, T-5--9 4.0 pts. MVL wins by 1/2
xxx Naka/So LD T-3--4 7.5 pts, T-6--8 4.0 pts. MVL wins by 1/2
xxx Naka/So LW T-3--5 7.0 pts, T-6--8 4.0 pts. tied, playoff
xxx Naka/So DD T-3--4 7.5 pts, 7th 4.0 pts. MVL wins by 1/2
xxx Naka/So WW T-3--6 6.5 pts, 7th 4.0 pts. Magnus wins by 1/2
This time MVL is ahead of Aronian. But, once again, MVL is tied with Fabi for second if Fabi loses, or in third place otherwise. Nakamura and/or So can once again tie MVL by winning. Magnus is tied with Aronian, and could be ahead of, tied with, or behind Nakamura and So. Magnus is immune to being tied from the 3-pointers below.
MVL has chances only if Fabi loses. Here is the chart:
If Fabi Loses:
and then MVL is with and Magnus is with GCT result:
Naka/So DW T-2--4 8.33 pts T-5--7 5.0 pts. MVL wins by 1/3
xxx Naka/So LL T-2--3 9.00 pts T-4--5 6.5 pts. Magnus wins by 1/2
xxx Naka/So LD T-2--3 9.00 pts T-4--6 6.0 pts. tied, playoff
xxx Naka/So LW T-2--4 8.33 pts T-5--6 5.5 pts. Magnus wins by 0.83
xxx Naka/So DD T-2--3 9.00 pts T-4--7 5.5 pts. MVL wins by 1/2
xxx Naka/So WW T-2--5 7.75 pts T-6--7 4.5 pts. MVL wins by 1/4
If Fabi wins or draws:
and then MVL is with and Magnus is with GCT result:
Naka/So DW T-3--4 7.50 pts T-5--7 5.0 pts. Magnus wins by 1/2
xxx Naka/So LL 3rd 8.00 pts T-4--5 6.5 pts. Magnus wins by 1.5
xxx Naka/So LD 3rd 8.00 pts T-4--6 6.0 pts. Magnus wins by 1
xxx Naka/So LW T-3--4 7.50 pts T-5--6 5.5 pts. Magnus wins by 1
xxx Naka/So DD 3rd 8.00 pts T-4--7 5.5 pts. Magnus wins by 1/2
xxx Naka/So WW T-3--5 7.00 pts T-6--7 4.5 pts. Magnus wins by 1/2
MVL is behind Ian, Fabi, and Aronian. MVL is also passed by Nakamura and/or So if they win, or tied with them if they draw. Magnus is passed by Nakamura and/or So if they win or draw, and is tied by one of the 3-pointers if they lose, or if Fabi loses.
MVL has a chance only if Nakamura and So do poorly. Here is the chart:
If Fabi Loses: and then MVL is with and Magnus is with GCT result: Naka/So LL 4th 7.0 pts, T-5--10 3.5 pts. MVL wins by 1/2 Naka/So LD T-4--5 6.5 pts, T-6--9 3.5 pts. tied, playoff Naka/So LW 5th 6.0 pts, T-6--9 3.5 pts. Magnus wins by 1/2 Naka/So DD T-4--6 6.0 pts, T-7--8 3.5 pts. Magnus wins by 1/2 Naka/So DW T-5--6 5.5 pts, T-7--8 3.5 pts. Magnus wins by 1 Naka/So WW 6th 5.0 pts, T-7--8 3.5 pts. Magnus wins by 1.5 If Fabi wins or draws: and then MVL is with and Magnus is with GCT result: Naka/So LL 4th 7.0 pts, T-5--9 4.0 pts. tied, playoff Naka/So LD T-4--5 6.5 pts, T-6--8 4.0 pts. Magnus wins by 1/2 Naka/So LW 5th 6.0 pts, T-6--8 4.0 pts. Magnus wins by 1 Naka/So DD T-4--6 6.0 pts, 7th 4.0 pts. Magnus wins by 1 Naka/So DW T-5--6 5.5 pts, 7th 4.0 pts. Magnus wins by 1.5 Naka/So WW 6th 5.0 pts, 7th 4.0 pts. Magnus wins by 2