Rating changes and probability calculations for chess world championship

Question

I have some interesting questions that have to do with the rating changes and calculations for the Anand-Carlsen world championship. Most of this has to do with solving a system of linear equations.

First, some preliminary information:

Carlsen's live rating: 2870, Anand's live rating: 2775

If Carlsen wins a game, he gains 3.7 points; draw -1.3 points, loss -6.3 points.

If Anand wins a game, he gains 6.3 points; draw 1.3 points, loss -3.7 points.

The match will be 12 games, with WIN=1 point, DRAW=0.5 points, LOSS=0 points with the winner being the first to 6 points. Therefore, the number of standard rated games in the match can vary based on all the ordered possibilities of wins, draws, and losses.

My questions are:

1. What are the possibilities for Carlsen to cross 2900 in live rating?

2. What are the possibilities for Anand to cross 2800 in live rating?

3. What are the upper and lower bounds for Carlsen's rating in the match, given the stipulations?

4. What are the upper and lower bounds for Anand's rating in the match, given the stipulations?

Thanks!

Answer 1

Hints:

What is the maximum number of points that Carlsen can gain? That makes 1 easy

For 2, you need to list the cases. Order does not matter, just the number of wins, losses, and draws. Note that a player can score 6.5 if he has 5.5 and wins the last game.

For 3 upside, only one result increases Carlsen's rating. How many games can he win? For the downside, note that two draws give Anand a point, as does a loss, but the loss hurts more.

4 is like 3, upside down.

Answer 2

The answer for $1$ is NEVER, because even if Carlsen has the theoretically best possible tournament with $7-0$, his rating would go to $2895.9$ only.

The answer for $2$ is if:

Anand wins the match $7-0$, making his rating $2819.1$

OR

Anand wins the match $6.5-5.5$ (order of game results does not matter) by:

6 wins, 1 draw in a 7-game match: new rating = 2814.1
5 wins, 3 draws in a 8-game match: new rating = 2810.4
4 wins, 5 draws in a 9-game match: new rating = 2806.7
3 wins, 7 draws in a 10-game match. new rating = 2803.0

The answers for $3$ and $4$ are:

Carlsen's maximum rating, $2895.9$, is attained if he wins $7-0$, and thus Anand's minimum rating, $2749.1$. Anand's maximum rating, $2819.1$, is attained if he wins $7-0$, and thus Carlsen's minimum rating $2825.9$.

Answer 3

You've sorted 1, 3 and 4. Here's part 2, which I've just trawled through:

Assuming that the match stops when one players gets 6.5 or more match points, the possibilities which take Anand's rating above 2800 are as follows:

0 D and 7 W (new rating of 2819.1)

1 D and 6 W (new rating of 2814.1)

2 D and 6 W (new rating of 2815.4)

3 D and 5 W

4 D and 5 W

5 D and 4 W

6 D and 4 W

7 D and 3 W

8 D and 3 W

10 D and 2 W