Carlsen Will DESTROY! Anand - World Chess Championship 2013.

[gerold]

In a world chess championship. Anything can happen. It all depends on how well the players have prepared for the match and also how well one player may play vs. another single player. How well anyone may play at the time of the matches. So many variables to consider with top rated players on any given day.

If Anand wins will that make the rating system invalid in some cases?

[Laskos]

Hello:

I took a look to this recent article and I decided to do my own simulations under the same assumptions (I did not take into account colour influences; a simulation is finished when there is a 6-6 result or one of the players has more than 6 points).

The first one:

Anand–Carlsen 2013

A 12 game match is far shorter than my recommended 26 games, so should we be worried that the weaker player might win by chance? Normally this would be a concern, but here there is one factor working in our favor: a relatively wide gap in the ratings. On the September rating list, Carlsen is rated 2862 and Anand is at 2775. According to the ratings formula, Carlsen is expected to score 0.62 points per game on average in the match. Since Anand is so much lower rated than Carlsen, it is unlikely that he can score an upset even in such a brief match.

So what will happen? Let’s run some simulations of the match. In their classical games, 20 out of 29 were drawn, so the 66% draw rate is still a reasonable assumption. Of the remaining nine games, Anand actually has the edge: six wins, three losses. However, several of his victories occurred before Carlsen reached his full strength; Carlsen won both of the last two decisive games. The ratings system forecasts that Carlsen will average 0.62 points per game. If we combine this with the 66% draw rate assumption, then in each game Carlsen has a 29% chance of winning and 5% chance of losing. Here are the results from running 40,000 simulations of the match:

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Probability that Carlsen wins 90.6%
Probability that Carlsen wins by a statistically significant margin (7-3 or better) 26.0%
Probability that Anand wins 3.0%
Probability of a drawn match 6.4%

I took a draw ratio of 66% and a win ratio (from Carlsen's POV) of ~ 29.625% (in other words, an Elo difference of 2862 - 2775 = 87 Elo). I got the following results after 100,000 simulations:

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100000/100000
Carlsen wins:  91068  Anand wins:   2756  Drawn matches:   6176

------------------------

The second one:

But before we conclude that Anand is doomed, there are a few caveats. Drawn matches are resolved by four rapid tiebreak games, and Anand is well known to be an excellent rapid player. More importantly, these simulations put complete confidence in the Elo rating system, though other statisticians have devised more accurate ways to calculate ratings. It would be interesting to hear their thoughts on the upcoming match. Thirdly, it seems difficult to accept that Anand has just a 5% chance of winning in each game, even though this is the only possible result if we assume that the draw rate is 66% and that the players perform as predicted by the rating system. With these assumptions, Anand has less than a 50% chance of winning a single game in the match!

Also, one observer pointed out that the draw rate of 66% probably isn’t valid for matches in which the rating gap is large. Since Anand and Carlsen were not always so far apart, their draw rate in their previous games might not be a good indicator of how often they will draw in the world championship. To get a better estimate of the draw rate and Anand’s winning chances, I searched my database looking for games that met these requirements:

•Both players 2700+
•Rating gap of 75 – 100 Elo
•Not a rapid or blitz game
In these games, the higher rated player wins 30.7% of the time, draws 60.7%, and loses 8.6%. Based on these results, Carlsen’s expected score would be 0.61 points per game, which is very similar to what his rating would forecast. However, the draw rate has declined and Anand’s winning chances have increased. Using these assumptions, I ran 40,000 simulations:

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Probability that Carlsen wins 85.7%
Probability that Carlsen wins by a statistically significant margin (7-3 or better) 24.8%
Probability that Anand wins 6.1%
Probability of a drawn match 8.3%

Anand’s chances have certainly improved, but he is still very much the underdog unless there is something badly wrong with the rating system. However, both players’ ratings have been very stable lately. Anand has been between 2770 and 2790 for more than a year, and Carlsen has seen even less volatility since January 2013 (min. rating: 2861, max rating 2872). So assuming that the ratings are accurate is not unreasonable.

I used the same probability values (30.7%, 60.7% and 8.6%) than Wilson. I got the following results after 100,000 simulations:

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100000/100000
Carlsen wins:  85542  Anand wins:   5979  Drawn matches:   8479

------------------------

The third one:

However, in their previous games Anand has a tendency to slightly outperform his rating when he is playing Carlsen. After comparing his average rating to his performance rating, it appears that Anand gains 33 points in strength when Carlsen is on the other side of the chessboard. This pattern is present in both their early games (pre-2010) and recent games (2010-2013). But it is not statistically significant, so it could be just luck. Nevertheless, there remains a possibility that the pattern is real, so how would it affect the forecast? I subtracted 33 points from the rating gap to form an adjusted rating gap. Now that the gap is not so large, it makes sense to return to the 66% draw rate. I plugged these assumptions into the silicon oracle:

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Probability that Carlsen wins 77.2%
Probability that Carlsen wins by a statistically significant margin (7-3 or better) 14.6%
Probability that Anand wins 10.3%
Probability of a drawn match 12.5%

My parameters for this assumption are the same as in the first case, except that now Elo difference = 87 - 33 = 54 Elo. I got the following results after 100,000 simulations:

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100000/100000
Carlsen wins:  75679  Anand wins:  11225  Drawn matches:  13096

------------------------

The last one:

The brevity of the match mitigates Anand’s disadvantage: in a 24 game match with a 66% draw rate, he loses in 98% of the match simulations and wins only 0.7% of the matches. But in a twelve game match, maybe he’ll just get lucky.

Getting back to 87 Elo difference and new stopping rules (12-12 result or a player with more than 12 points), this is what I got after 100,000 simulations:

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100000/100000
Carlsen wins:  98278  Anand wins:    584  Drawn matches:   1138

I think that my results are not very different from Wilson's ones, but one thing is sure: I do not dare to give a forecast.

Regards from Spain.

Ajedrecista.

Interesting, Jesus. The folks who are betting in UK give 2 to Anand and 4/11 to Carlsen, in percentages, 69% to Carlsen and 31% to Anand.

I think that the simulation with 54 Elo points gap (adjusted by the particular gameplay between these two players) and 66% draw rate is the best. That gives

Carlsen wins: 75.7%
Anand wins: 11.2%
Drawn match: 13.1%

It seems to approach to what the betters think.

[Laskos]

Interesting, Jesus. The folks who are betting in UK give 2 to Anand and 4/11 to Carlsen, in percentages, 69% to Carlsen and 31% to Anand.

I think that the simulation with 54 Elo points gap (adjusted by the particular gameplay between these two players) and 66% draw rate is the best. That gives

Carlsen wins: 75.7%
Anand wins: 11.2%
Drawn match: 13.1%

It seems to approach to what the betters think.

Also, If Anand snatches _one_ extra victory, say from his opening preparation, and the number of games is 11 (+ 1 for Anand as a plus), then the percentages change pretty dramatically.

Draws 66%, Wins 24.7%, Losses 9.3%, Elo gap 54 points, for the remaining 11 games.


12 games result:

Carlsen wins: 54.1%
Anand wins: 25.7%
Drawn match: 20.2%

So, the preparation, especially opening preparation, is pretty vital here.

[Ajedrecista]

Hello Kai:

Interesting, Jesus. The folks who are betting in UK give 2 to Anand and 4/11 to Carlsen, in percentages, 69% to Carlsen and 31% to Anand.

I think that the simulation with 54 Elo points gap (adjusted by the particular gameplay between these two players) and 66% draw rate is the best. That gives

Carlsen wins: 75.7%
Anand wins: 11.2%
Drawn match: 13.1%

It seems to approach to what the betters think.

Interesting to know it! Just for comparison, Wilson's results and my results for this scenario are:

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                            Carlsen wins   Anand wins   Drawn match
Wilson (40,000 simulations)    77.2%         10.3%         12.5%
   Me (100,000 simulations)    75.7%         11.2%         13.1%

Surely lots of people do not know how to translate bets into probabilities. Here is a little help:

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UK system of bets: X means that you earn X with each monetary unit bet. That is: for example you bet 1 british pound and there are two possibilities:
- You win: the betting house pays to you X + 1 british pounds (you get a benefit of X british pounds).
- You lose: you have lost 1 british pound.

Of course you are not forced to bet 1 monetary unit: if you bet n british pounds, then:
- You win: the betting house pays to you n*(X + 1) british pounds (you get a benefit of n*X british pounds).
- You lose: you have lost n british pounds.

I do the math with the European system: instead of 2 and 4/11, I use 2 + 1 = 3 and 4/11 + 1 = 15/11 respectively. Then:

1/3 + 1/(15/11) = 16/15; 16/15 - 1 = 1/15 ~ 6.67% of expected benefit for the betting house.

Prob.(Carlsen wins the WCC 2013) = 1/[(16/15)*(15/11)] = 11/16 = 68.75%.
Prob.(Anand wins the WCC 2013) = 1/[(16/15)*3] = 5/16 = 31.25%.

Of course: 11/16 + 5/16 = 1 (100%).

I hope to be right.

Regards from Spain.

Ajedrecista.

[Laskos]

Hello Kai:

Interesting, Jesus. The folks who are betting in UK give 2 to Anand and 4/11 to Carlsen, in percentages, 69% to Carlsen and 31% to Anand.

I think that the simulation with 54 Elo points gap (adjusted by the particular gameplay between these two players) and 66% draw rate is the best. That gives

Carlsen wins: 75.7%
Anand wins: 11.2%
Drawn match: 13.1%

It seems to approach to what the betters think.

Interesting to know it! Just for comparison, Wilson's results and my results for this scenario are:

Your numbers are better than Wilson's, I added mine:

Code: Select all

                            Carlsen wins   Anand wins   Drawn match
Wilson (40,000 simulations)    77.2%         10.3%         12.5%
Jesus (100,000 simulations)    75.7%         11.2%         13.1%
Kai  (10,000,000 simulations)  75.76%        11.13%        13.11%

Surely lots of people do not know how to translate bets into probabilities. Here is a little help:

Code: Select all

UK system of bets: X means that you earn X with each monetary unit bet. That is: for example you bet 1 british pound and there are two possibilities:
- You win: the betting house pays to you X + 1 british pounds (you get a benefit of X british pounds).
- You lose: you have lost 1 british pound.

Of course you are not forced to bet 1 monetary unit: if you bet n british pounds, then:
- You win: the betting house pays to you n*(X + 1) british pounds (you get a benefit of n*X british pounds).
- You lose: you have lost n british pounds.

I do the math with the European system: instead of 2 and 4/11, I use 2 + 1 = 3 and 4/11 + 1 = 15/11 respectively. Then:

1/3 + 1/(15/11) = 16/15; 16/15 - 1 = 1/15 ~ 6.67% of expected benefit for the betting house.

Prob.(Carlsen wins the WCC 2013) = 1/[(16/15)*(15/11)] = 11/16 = 68.75%.
Prob.(Anand wins the WCC 2013) = 1/[(16/15)*3] = 5/16 = 31.25%.

Of course: 11/16 + 5/16 = 1 (100%).

I hope to be right.

Good translation of UK bets into probabilities.

Regards from Spain.

Ajedrecista.

[Graham Banks]

Carlsen Will DESTROY! Anand - World Chess Championship 2013..........
Anthony.R.Brown

You an Astrologer or what ?...........

Far from it. He lives under a bridge.

[Ajedrecista]

Hello Kai:

Your numbers are better than Wilson's, I added mine:

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                            Carlsen wins   Anand wins   Drawn match 
Wilson (40,000 simulations)    77.2%         10.3%         12.5% 
Jesus (100,000 simulations)    75.7%         11.2%         13.1% 
Kai  (10,000,000 simulations)  75.76%        11.13%        13.11%

1e+7 simulations! Well done. I tried to manage a faster simulator by omit printing intermediate results... indeed I made it a lot faster! Some examples of the case of D = 66% and Elo difference = 54 (please note that my simulator is deterministic, so I get the same results for 100,000 simulations than previously):

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    1000/    1000 (one thousand).
Carlsen wins:      750  Anand wins:      113  Drawn matches:      137

Approximated elapsed time:    0.3 seconds.

---------------------------------------------------------------------

   10000/   10000 (ten thousand).
Carlsen wins:     7567  Anand wins:     1110  Drawn matches:     1323

Approximated elapsed time:    0.6 seconds.

---------------------------------------------------------------------

  100000/  100000 (one hundred thousand).
Carlsen wins:    75679  Anand wins:    11225  Drawn matches:    13096

Approximated elapsed time:    3.6 seconds.

---------------------------------------------------------------------

 1000000/ 1000000 (one million).
Carlsen wins:   758826  Anand wins:   110846  Drawn matches:   130328

Approximated elapsed time:   34.5 seconds.

---------------------------------------------------------------------

10000000/10000000 (ten million).
Carlsen wins:  7579279  Anand wins:  1110827  Drawn matches:  1309894

Approximated elapsed time:  685.5 seconds.

My numbers are very similar to yours. After 10,000,000 simulations:

Code: Select all

                               Carlsen wins   Anand wins   Drawn match 
Kai   (10,000,000 simulations)    75.76%        11.13%       13.11%
Jesús (10,000,000 simulations)    75.79%        11.11%       13.10%

Regards from Spain.

Ajedrecista.

[Ajedrecista]

Hello again:

I improved A LOT the speed of my simulator. For some unknown reason, I do not get anymore the results that I got previously but I get very similar results. Here is the code, just to probe I am not making up the results:

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program WCC_2013_simulator
implicit none
integer(1),parameter::games_per_simulation=12
integer(4),parameter::simulations=1000000000 ! 1e+9 simulations.
integer(1)::n,half_points_Carlsen,half_points_Anand,finished
integer::i,Carlsen_wins,Anand_wins,drawn_matches
real(2)::D,L,random_number,Elo_difference,t0,t1
t0=cpu_clock@()
Elo_difference=5.4d1
D=6.6d-1
L=1d0/(1d0+1d1**(2.5d-3*Elo_difference))-5d-1*D ! From Carlsen's POV.
Carlsen_wins=0;Anand_wins=0;drawn_matches=0
do i=1,simulations
  half_points_Carlsen=0;half_points_Anand=0
  finished=0
  do n=1,games_per_simulation
    random_number=random@()
    if (random_number<L) then
      half_points_Anand=half_points_Anand+2
    else if (random_number<L+D) then
      half_points_Carlsen=half_points_Carlsen+1
      half_points_Anand=half_points_Anand+1
    else
      half_points_Carlsen=half_points_Carlsen+2
    end if
    if (finished<1) then
      if (half_points_Carlsen>12) then
        Carlsen_wins=Carlsen_wins+1
        finished=1
        continue
      else if ((half_points_Carlsen>11) .and. (half_points_Anand>11)) then
        drawn_matches=drawn_matches+1
        finished=1
        continue
      else if (half_points_Anand>12) then
        Anand_wins=Anand_wins+1
        finished=1
        continue
      end if
    end if
  end do
end do
write(*,'(I19,A)') simulations,' simulations:'
write(*,*)
write(*,'(A,I19)') 'Carlsen wins:  ',Carlsen_wins
write(*,'(A,I19)') 'Anand wins:    ',Anand_wins
write(*,'(A,I19)') 'Drawn matches: ',drawn_matches
write(*,*)
t1=cpu_clock@()
write(*,'(A,F7.2,A)') 'Approximated elapsed time: ',(t1-t0)/2.999d9,' seconds.'!2.999 GHz in my PC.
end program WCC_2013_simulator

It is not a perfect simulator but at least it is fast from my POV. Here are some results (I have changed slightly the output):

Code: Select all

1e+5 simulations:

Carlsen wins:                75570 (75.57%)
Anand wins:                  11234 (11.23%)
Drawn matches:               13196 (13.20%)

Approximated elapsed time:    0.13 seconds.

-------------------------------------------

1e+6 simulations:

Carlsen wins:               758437 (75.84%)
Anand wins:                 110594 (11.06%)
Drawn matches:              130969 (13.10%)

Approximated elapsed time:    1.36 seconds.

-------------------------------------------

1e+7 simulations:

Carlsen wins:              7578786 (75.79%)
Anand wins:                1109760 (11.10%)
Drawn matches:             1311454 (13.11%)

Approximated elapsed time:   13.38 seconds.

-------------------------------------------

1e+8 simulations:

Carlsen wins:             75803427 (75.80%)
Anand wins:               11092305 (11.09%)
Drawn matches:            13104268 (13.10%)

Approximated elapsed time:  133.98 seconds.

-------------------------------------------

1e+9 simulations:

Carlsen wins:            758061630 (75.81%)
Anand wins:              110894166 (11.09%)
Drawn matches:           131044204 (13.10%)

Approximated elapsed time: 1239.24 seconds.

Summarizing:

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                          Carlsen wins   Anand wins   Drawn match 
Wilson (4e+4 simulations)    77.2%         10.3%         12.5% 
Kai    (1e+7 simulations)    75.76%        11.13%        13.11%
Jesús  (1e+9 simulations)    75.81%        11.09%        13.10%

I will not do more simulations: one billion is enough.

Getting back to the world of bets, and assuming my probabilities, bets could be more less:

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               European system    UK system
Carlsen wins:       1.25            0.25
Anand wins:         8.54            7.54
Drawn match:        7.23            6.23

Expected benefit for the betting house: 5.54%.

(Of course 'drawn match' referes to the 12 classical games only).

Regards from Spain.

Ajedrecista.

[Ajedrecista]

Hello:

Carlsen raised his rating 8 Elo in the recently finished Sinquefield Cup. Doing a billion of simulations with Elo difference = 54 + 8 = 62 Elo = 2870 - 2775 - 33, and draw ratio = 66%:

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1e+9 simulations:

Carlsen wins:            802360043 (80.24%)
Anand wins:               83813921  (8.38%)
Drawn matches:           113826036 (11.38%)

Approximated elapsed time: 1282.14 seconds.

Formerly we had (Elo difference = 2862 - 2775 - 33 = 54 Elo, and draw ratio = 66%):

Code: Select all

1e+9 simulations: 

Carlsen wins:            758061630 (75.81%) 
Anand wins:              110894166 (11.09%) 
Drawn matches:           131044204 (13.10%) 

Approximated elapsed time: 1239.24 seconds.

Maybe a draw ratio of 66% is too high. Who knows?

Regards from Spain.

Ajedrecista.

[Laskos]

Hello:

Carlsen raised his rating 8 Elo in the recently finished Sinquefield Cup. Doing a billion of simulations with Elo difference = 54 + 8 = 62 Elo = 2870 - 2775 - 33, and draw ratio = 66%:

Code: Select all

1e+9 simulations:

Carlsen wins:            802360043 (80.24%)
Anand wins:               83813921  (8.38%)
Drawn matches:           113826036 (11.38%)

Approximated elapsed time: 1282.14 seconds.

Formerly we had (Elo difference = 2862 - 2775 - 33 = 54 Elo, and draw ratio = 66%):

Code: Select all

1e+9 simulations: 

Carlsen wins:            758061630 (75.81%) 
Anand wins:              110894166 (11.09%) 
Drawn matches:           131044204 (13.10%) 

Approximated elapsed time: 1239.24 seconds.

Maybe a draw ratio of 66% is too high. Who knows?

Regards from Spain.

Ajedrecista.

All our results contradict a bit the betting ods of major betting houses. They give 69% to Carlsen and 31% to Anand, with the odds to go into tiebreak at 30%.