# Sports: Elo-Rating and Win Probability / Carlsen vs. Anand

The Elo rating system is a commonly applied tool to judge the performance of players in sports. Most associate it with chess, but it is in use in other sports as well, for example American Major League Baseball and Basketball. A nice thing about the rating system is that you can easily estimate the win probabilities given the difference in Elo rating of the opponents.

So if the difference is zero (opponents of equal strength), the win probability is obviously 0.5 = 50 %. If one player has a 200 Elo points advantage, his win probability is about 76 %. In case you prefer a table over a graph, you can find the corresponding values here. And for those interested in formulas, I found that the Boltzmann function models the values extremely well (r stands for Elo rating, p for the win probability):

p = 1 – 1 / ( 1 + exp ( 0.00583 * r  – 0.0505) )

For example, plugging in r = 200 leads to p = 0.75 = 75 %, close enough. For a very rough linear approximation this formula will do:

p = 0.5 + 0.001 * r

By the way, what sports event is happening right now in India? Right, the World Chess Championship. At the moment Anand and Carlsen are locked in a bitter battle to death (ok, a little too dramatic, but still very interesting) and I just finished watching their brilliant third match, which was yet another draw. Here’s the current top ten for chess:

 Rank Name Title Country Rating Games B-Year 1 Carlsen, Magnus g NOR 2870 0 1990 2 Aronian, Levon g ARM 2801 6 1982 3 Kramnik, Vladimir g RUS 2793 9 1975 4 Nakamura, Hikaru g USA 2786 17 1987 5 Grischuk, Alexander g RUS 2785 17 1983 6 Caruana, Fabiano g ITA 2782 25 1992 7 Gelfand, Boris g ISR 2777 11 1968 8 Anand, Viswanathan g IND 2775 0 1969 9 Topalov, Veselin g BUL 2774 6 1975 10 Mamedyarov, Shakhriyar g AZE 2757 11 1985

As you can see, Carlsen is currently the leading chess player with an Elo rating of 2870 and Anand is on rank eight with a rating of 2775. The difference is 2870 – 2775 = 95 points, which translates into a win probability of about 63 % for Carlsen and 37 % for Anand. If you have to bet, bet on Carlsen, but it’s going to be a close one. Both have shown a great performance so far, though it seemed that in the third match Anand did miss some opportunities. I’m looking forward to tomorrow’s match.