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Magnus Carlsen Is More Than An Odds-On Favorite To Win The World Chess Championship

The World Chess Championship Match begins Saturday in Sochi, Russia. Defending champion Magnus Carlsen, the Norwegian wunderkind, will take on Viswanathan Anand, the former champion from India.

The match comprises 12 games. Players score one point for a win and half a point for a draw. First to six and a half points becomes champion. If the 12 games end 6-6, there will be tie-break games. The match could last until Thanksgiving.

Carlsen, the 23-year-old world No. 1, is favored to beat Anand — but just how big a favorite is a complicated question. And the betting market may be underestimating Carlsen’s chances.

Anand is the world No. 6 and qualified by winning the Candidates Tournament in March. He has been a grandmaster since 1988 — two years before Carlsen was born. But Carlsen, a former child chess prodigy, has since achieved the game’s highest rating ever. His official FIDE Elo rating eclipsed Anand’s for the first time in 2008, when Carlsen was just 17.

Around here, we use Elo ratings to analyze to a lot of things — the NFL, the Women’s World Cup, Scrabble. I’m going to take the analysis back to its source: chess.

As I write, Carlsen’s Elo rating is 2863. Anand’s is 2792. As such, Carlsen is expected to take slightly more than 0.6 points from each game, in the form of wins and draws.$$E_C=\frac{1}{1+10^{(R_A-R_C)/400}}$$ where $$E_C$$ is Carlsen’s expected points, $$R_C$$ is Carlsen’s rating and $$R_A$$ is Anand’s rating.

Note, I’m not adjusting for a possible first-move advantage (playing with the white pieces). I also do not simulate Elo rating updates during the match itself. In other words, the players’ current Elo ratings remain their Elo ratings throughout the match.

">1 (Wins are worth one point, draws a half-point.) But the Elo system does not address the specific win-draw ratio that can be expected. For example, an expectation of 0.6 points per game could mean an expectation of 0.6 wins and zero draws, or 0.5 wins and 0.2 draws, or 0.4 wins and 0.4 draws, and so on. So, to calibrate a forecast of the match, we’ll have to make an important assumption. To inform this assumption, let’s dig a bit deeper into the prevalence of draws.

Top chess players draw a lot. Research shows that great players like Carlsen and Anand draw far more often than rank amateurs — about half of all top players’ games end in draws. Grandmasters rarely make obvious errors, so decisive, winning breakthroughs are hard to come by. Carlsen and Anand, against one another, may draw even more. They’ve met before, including in last year’s World Championship. Excluding speed games and exhibition games, their record is tied: six wins apiece and 28 draws. It’s a smallish sample, but that’s 70 percent draws.

I ran 100,000 simulations each of the championship match, assuming a number of values for draw prevalence.2 The simulations are essentially draws from a multinomial distribution over Carlsen’s wins, draws and losses, adjusting the probabilities to maintain Carlsen’s 0.6-points-per-game expectation while accounting for levels of draw prevalence. Here is Carlsen’s probability of beating Anand in the 12-game match, as a function of the prevalence of draws:

The more likely draws are, the better Carlsen’s chances to defend his crown. If draws are expected a quarter of the time, say, Carlsen has about an 80 percent chance to win the match. If they’re expected half the time: an 85 percent chance. Three-quarters of the time: nearly 95 percent.

The intuition is straightforward. As draws become more common, Carlsen must be expected to win a higher proportion of individual decisive games — games where someone wins — to account for his higher Elo rating. That increased edge in decisive games would give him an increased edge to win the match. For example, in the absence of draws, Carlsen is expected to win 60 percent of decisive games. If draws are expected half the time, Carlsen is expected to win 70 percent of decisive games. If draws are expected three-quarters of the time, it’s 90 percent.

Given this analysis, bookmakers may be underestimating the defending champ’s chances. At Ladbrokes, Carlsen is a 1-4 favorite, and Anand is an 11-4 dog. Factoring out the vig, this implies a 75 percent chance of a Carlsen championship — that’s the chance I estimate if draws are never expected to occur. At a 50 percent draw clip, though, his chances are more like 85 percent.elsewhere Carlsen is a slightly stronger 2-9 favorite, and Anand is a 41-13 dog. Still, factoring out the vig, this puts Carlsen’s chances at just 77 percent.

">3

None of this is to say that Carlsen would be better off playing with the intention of drawing more often, necessarily. Rather, it speaks to the natural occurrence of draws expected when these two players get together. Carlsen playing specifically for draws may be a weaker player overall.

While Anand won the right to be the challenger, a few other top players would have a better chance against Carlsen, according to the Elo ratings. I ran the same simulations of the championship match, this time pitting Carlsen against each of the other top 10 players in the world.they are, in order, Fabiano Caruana, Alexander Grischuk, Veselin Topalov, Levon Aronian, Viswanathan Anand, Hikaru Nakamura, Sergey Karjakin, Anish Giri, and Shakhriyar Mamedyarov.

">4 (For this chart, I assume draws are expected in half the games.) With the exception of Fabiano Caruana, the 22-year-old Italian, Carlsen would have a better than 80 percent chance against anyone in the world:

Of course, all of this takes the players’ Elo ratings, and the Elo system, at face value. An Elo rating is necessarily only an estimate of a player’s true average skill, based on actual game outcomes, and the system itself makes statistical assumptions. And there may be “intangibles” at play, too: how the contestants’ styles of play clash, Carlsen’s discomfort with the match arrangements, or a bitter taste left in Anand’s mouth by last year’s match, for example. My simulations are agnostic about those.

## Footnotes

1. This is derived from the Elo system formulae, specifically the following equation:

$$E_C=\frac{1}{1+10^{(R_A-R_C)/400}}$$ where $$E_C$$ is Carlsen’s expected points, $$R_C$$ is Carlsen’s rating and $$R_A$$ is Anand’s rating.

Note, I’m not adjusting for a possible first-move advantage (playing with the white pieces). I also do not simulate Elo rating updates during the match itself. In other words, the players’ current Elo ratings remain their Elo ratings throughout the match.

2. If the match is simulated to go to tie-break games, I assume the outcome after that is a coin flip, since those games would be played at a much shorter time control, and the variance is likely higher.

3. As I write, elsewhere Carlsen is a slightly stronger 2-9 favorite, and Anand is a 41-13 dog. Still, factoring out the vig, this puts Carlsen’s chances at just 77 percent.

4. As I write they are, in order, Fabiano Caruana, Alexander Grischuk, Veselin Topalov, Levon Aronian, Viswanathan Anand, Hikaru Nakamura, Sergey Karjakin, Anish Giri, and Shakhriyar Mamedyarov.

Oliver Roeder was a senior writer for FiveThirtyEight. He holds a Ph.D. in economics from the University of Texas at Austin, where he studied game theory and political competition.