When making models, it is important to qualify why you use the assumptions that you do. Which is why I chose to write down the methodology when publishing Magnus Carlsen’s chances of keeping the World Chess Championship throne.
However; there has been some questions as to why the model was not adjusted for color, as well as the fact that this is a championship match which makes everything different. Both of which are valid concerns, but lead to the question: How do we adjust for this?
On the question of championship matches being different, I can agree. And so does Garry Kasparov, who really ought to know. But just how different are they, and how to adjust the model for it?
What I chose to do was to analyze the 15 championship matches since 1985 (ignoring tournaments), noting each player’s initial rating, the winner, if there was a tie-break as well as how many games were won with either color and how many were drawn. If we assume that rating has no influence on the outcome, then the wins should be fairly evenly distributed. They are not. The stronger player won 8 of the 15 matches, with 3 needing a tie-break, and the lower rated player winning 4. (For clarification, in the Kramnik vs Topalov match, I ignored the forfeited game 5 and have the lower-rated Kramnik winning outright instead of after tie break.)
Based on this, it seems that rating does influence the outcome. Further, the rating difference is particularly large this year (71 points), so I chose to look at the matches where the rating difference was larger than 50 points to see how that would influence the result. And of those 8 matches, the highest rated player won 5, 1 was decided by tie-break, and 2 won by the lowest rated player (Kramnik over Topalov being one of them). A low sample size, yet good information to have when deciding on the model to use.
In addition, I looked at all the 254 games played (including yesterday’s draw) to see the win distribution based on color. 59 were won by white, 27 by black, while 168 ended in a draw (23.2%, 10.6%, and 66.2% respectively). So this gives us quite a bit of background when trying to adjust for color.
Do for the alternate model, I chose to place 50% weight on my original model, and 50% weight on color. Which gives us the following win chances per game:
|Anand white||Carlsen white|
This is different from the initial model and more heavily skewed towards draws and wins by Anand, so using this alternate model should reduce Carlsen’s chances somewhat.
I then ran 1,000 simulations of the games and got that according to this model, Carlsen’s original chances of winning were 63.3%, Anand’s 22.4%, with a 14.3% chance of the need for a tie break. This is very much on par with the historical data of matches between players with a rating difference larger than 50 points, so this might be an even better model than the original.
Following the draw in game 1, Carlsen’s win chances increased to 64.7%, Anand’s was reduced to 20.0%, with the chance of a tie-break increasing to 15.3%.
This seems intuitively to be more correct, and we will start publishing the results from both models following each game.
Which leaves us with these probabilities after game 1:
|Rating model||Adjusted model|
Thank you for the comments and input to my models. Now we look forward to game 2.