Monday, 5 May 2014

You can beat Carlsen. Statistically.

The well-followed Shamkir chess tournament is over and it’s no surprise to see Carlsen walking out as the winner. However, he failed to collect the necessary points to break the ELO 2900 barrier and on the FIDE ratings list of May he gained just one extra point.

Looking at the Shamkir tournament results, I was surprised to see Radjabov’s achievement: Although the lowest rated player in the A-class, he performed best against Carlsen who is rated nearly 170 points higher than him. Yet Radjabov came out at fourth place.

This strong play of Radjabov made me wonder what the likelihoods are to win against a 2882 player and so I played around with the calculation of ELO rating numbers.

Unfortunately, the possibility of a draw in chess makes a statement of what your winning chances against your opponent are hard, if not impossible. Radjabov’s likelihood of winning a game against Carlsen were mathematically 28%. Or you could say he had a 0% chance of winning, 56% of drawing and 44% of losing. Or anything in between… A draw counts as half a win and half a loss in the system and that makes it tricky to give likelihoods for wins (if that’s possible at all). I leave those details to the more skilled mathematicians.

Many chess enthusiasts are eagerly looking forward for the re-match Carlsen vs. Anand. So what are Anand’s chances then to win (a single game) against Carlsen? With their current ratings (May 2014), the calculation predicts a 36% chance. Statistically that is and neglecting the possibility of a draw.

Since we are dealing with a mathematical model here and not the real world, we can of course ask what are YOUR chances to beat Carlsen in a game of chess? Again, we ignore draws here but you can easily extend this by remembering that a draw counts as half a win and half a loss. The following tables gives your chances of winning against a 2882 player (choose the closest match for your rating):

Your rating<12501300135014001450150015501600
Winning Chance<0.01%0.011%0.015%0.02%0.03%0.04%0.05%0.06%

Your rating16501700175018001850190019502000
Winning Chance0.08%0.11%0.15%0.20%0.26%0.35%0.5%0.6%

Your rating20502100215022002250230023502400
Winning Chance0.8%1.1%1.5%1.9%2.6%3.4%4.5%5.9%

Your rating24502500255026002650270027502800
Winning Chance7.7%10%13%16%21%26%32%38%

Because a picture tells more than a thousand numbers, I also plotted the results graphically. The coloured dots mark three selected players, and in brackets are their current ranks as per May 2014 FIDE list. The plot stops at Carlsen’s own rating of 2882 who has a 50/50 chance of winning against himself – should he ever step up to the challenge or being bored of everyone else…

ELO curve and winning chances against Carlsen

Next time you experience a series of defeats, don’t be disheartened: Mathematics tell you that you will eventually win against anyone, even an elite-player like Carlsen. Statistically. Perhaps one game in 20 years if you play a game a day… At least that’s the path to victory and a high self-esteem if you are a statistician (no offence intended, I respect this profession). The better advice for the real world is the old and well-known formula: training, study and practice. Lots of it. And statistically speaking, training gives you the highest likelihood of winning too. Enough number crunching for now, so let's go back to chess ;-)

Happy Chess,
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