Carlsen-Anand: A Mathematical Analysis

by admin on November 11, 2014

Today a seismic shift happened in the world championship match between Magnus Carlsen and Viswanathan Anand. Vishy finally won a game! Last year, you might remember, Carlsen defeated Anand without even losing a single game. Even though Anand went into the match as the world champion, he didn’t even look as if he was in Carlsen’s league.

Many people, including me, were not too excited about this rematch because it seemed unlikely that a year could have made very much difference. And it seemed as if our expectations were confirmed when Carlsen demolished Anand in the second game of the match, going up 1½-½. Given Anand’s inability to seriously threaten Carlsen last year, that lead appeared as safe as when the San Francisco Giants have a 1-0 lead after the second inning with Madison Bumgarner pitching.

However, there were a few things I failed to take into account. Anand has shown before that he can come back from a crushing defeat with a crushing victory. He did it against Boris Gelfand in 2012: after Gelfand drew first blood with a win in game seven, Anand turned right around and embarrassed Gelfand with a 17-move win in game eight. Likewise against Veselin Topalov in 2010. Topalov won the first game of the match, but Anand turned the tables on him in the very next game. In fact, it would appear that Anand is never more dangerous than when he is behind 1-0!

Does this mean Anand will win the match? No. But it does mean that Carlsen will be tested this time. Last year, Anand had all the pressure on him because he was the defending champion against the wunderkind, Carlsen. This time Anand has no pressure on him. Nobody was expecting him to win this match. He can just play chess and let Carlsen struggle with the weight of expectations.

Nevertheless, objectively Carlsen is still the heavy favorite. This morning I read an interesting pre-match article on (I love it when chess gets noticed in the regular media!) explaining why Carlsen was a heavier favorite than the betting lines (4-1) suggested. The reason has to do with the prevalence of draws in high-level chess. If you expected 50 percent of the games in the match to be drawn (probably an underestimate) then Carlsen should be an 85 percent favorite. If you expected 75 percent of the games to be drawn (probably a slight overestimate) then Carlsen should be a prohibitive, 97 percent favorite!

Oliver Roeder, the author of the article, based his analysis on Elo ratings. To maintain a 70-point rating difference over Anand, Carlsen has to score on average 0.6 points in every game. If you hypothesize that they are going to draw 3/4 of their games, then Carlsen will get only 3/8 of a point (0.375) out of those games. That means he is expected to score 0.225 points out of the remaining 0.25 games that are decisive. In other words, he is expected to win 9 out of every 10 decisive games against Anand! This percentage goes down if the frequency of draws decreases. For instance, if draws happen in 2/3 of the games, which I think is more realistic, then Carlsen only needs to win 4/5 of the decisive games to maintain his 70-rating-point advantage.

Now let’s re-do Roeder’s analysis for the current match situation, after three game have been played. If Carlsen wins 4/5 of their decisive games, and if you know that there will be three more decisive games in the match (consistent with the assumption of 67 percent draws), then the chance that Anand wins two or more of those three games is 0.104. In other words, Anand has a 10.4 percent chance of winning the match.

There is one flaw with this calculation. The fact that we expect 2/3 of the games to be draws doesn’t mean that there actually will be 6 draws. It’s simply the most likely of several possible outcomes. To be more careful we should analyze each of the possible outcomes and their probabilities. Also we run into the complication of a tie match if the number of decisive games is even. But it turns out that all these details scarcely affect the probability of Anand winning. With a pencil and paper calculation (as opposed to Roeder’s 100,000 computer simulations) I get 10.0 percent.

However, there is some very important caveats that affect both Roeder’s analysis and mine. First, they do not take into account any psychological factors, which as I’ve mentioned could be quite big. Also, the analysis is predicated on the belief that the Elo system is an accurate forecaster of the result of head-to-head matches. This assumption, in my opinion, is incorrect. I do not think that Carlsen would maintain a 70-point rating lead over Anand if Anand was the only person he ever played. The 70-point lead reflects, at least in part, Carlsen’s greater ability to destroy 2600-level and 2700-level players. But their different strengths against lower-rated players, which is quite relevant to tournament play, has no relevance at all to match play.

Part of being a mathematician is being aware of the limitations of mathematics. The Elo system is a beautiful mathematical construct, and it does give us a way of objectively analyzing the likelihood of various outcomes. But when you take it outside of the kind of situation it is designed for, you have to take its predictions with a grain of salt. Therefore, I’m going to overrule the mathematics and say that, in my opinion, Anand now has a 20 percent chance of winning the match.

What do you think, esteemed readers? Am I being too optimistic for Anand? Am I overestimating the psychological effect of Anand’s finally breaking through and winning a game? Let me know!

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{ 3 comments… read them below or add one }

Simon November 11, 2014 at 4:29 pm

One idea: It is customary to assume that the results of the games are independent of each other when performing these analyses. Based on my tournament results, this is not true: when I win a game, I’m likely to win another one, and when I lose a game I’m likely to win another game. So, I can imagine the results as being a Markov chain, whose stationary distribution is what Roeder claims the distribution is. I don’t know how to estimate the transition matrix, but based on what I said above, I think Anand’s chances should be somewhat better than they are based on the independence assumption, given that he won the last game. (Although, there’s also the fact that Carlsen has an extra white now, which ought to matter a bit as well.)


Edward November 11, 2014 at 6:39 pm

Dana, I agree with your analysis. Do you think Carlsen has a tendency to get bored because he usually wins so easily? Perhaps this loss will bring out Carlsen’s best and he won’t lose again.


admin November 11, 2014 at 9:06 pm

Edward, I do think it’s a hazard of becoming world champion. For so many years you’ve worked toward and obsessed over that goal. Then you’ve got it, and what else is there to motivate you? Fischer lost his motivation. Karpov didn’t lose it for a long time, but that’s partly because in the early years he had to prove he was legitimate, and then in the later years he had such intense rivalries with Korchnoi and then Kasparov. Carlsen is more in the category of Fischer. He’s climbed his mountain, he’s way above everyone else. How much does it mean to him to stay on top? That’s what we’re going to see.


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