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.

Thanks for your great posts about the Elo ratings, I found them really helpful!

Is there any way to calculate the likelihood of a draw as well? Theoretically win/loss/draw probability should be 33% if both have the same Elo, right? I’m asking because I want to adapt this for football matches.

I believe the elo system itself does not answer your question. Mr. Elo did not need to make the assumption that if Expected Value of the game is .7, how often the underdog will get a draw and how often he will win. That is, your assumption of 33/33/33 is not correct: In high level games (GM) the distribution is probably 15/70/15 while in my games (elo=2000) the distribution is more like 40/20/40.

So I believe you have to model this yourself.

I see you invoked the Boltzmann function, but let’s say you have values of the cdf, for example with mean 1500 and std deviation for the sum of two cdf as 200*sqrt(2), how would you calculate the probability of a 1700 player beating a 1500 player with cdf values 0.7 and 0.5, respectively?

I cannot find anywhere on internet where this comes about (almost like asking how the Elo chart figures were calculated but in more general terms and NOT looking at rating differences of the two players but instead making the probability based off cdf values for each of the players).