Posted by Justin Kubatko on February 8, 2011
Today I would like to introduce a new feature that I think will be a lot of fun: the NBA Elo Player Rater.
The Elo rating system is a method for calculating the relative skill levels of players in two-player games. The creator of the system, Arpad Elo, was a professor of physics at Marquette University who wanted an improved chess rating system. Although the system has its roots in chess, today it is used in many other games.
We decided to take Elo's work and apply it to the ranking of basketball players. Our player pool consists of all NBA players who meet at least one of the following career criteria:
- 10,000 points
- 5,000 rebounds
- 2,500 assists
- 1,000 steals plus blocks
All players have an initial rating of 1500 points. These ratings are then updated by randomly selecting pairs of players and having them "play" each other.
We start by computing the win probabilities for each player (let's call them A and B):
P(A wins) = 1 / (1 + 10^((RB - RA) / 400)) P(B wins) = 1 / (1 + 10^((RA - RB) / 400)) where RA = rating for A RB = rating for B
After the winner has been determined, the ratings of the two players are adjusted. If A wins the match then the new ratings are:
RA_new = RA + K * P(B wins) RB_new = RB - K * P(B wins)
The "K" above stands for the "K-factor" and has a value of 24. (The K-factor was later changed to 10 --Ed.)
While if B wins the match then the new ratings are:
RA_new = RA - K * P(A wins) RB_new = RB + K * P(A wins)
P(MJ wins) = 1 / (1 + 10^((2450 - 2500) / 400)) = 0.571 P(WC wins) = 1 / (1 + 10^((2500 - 2450) / 400)) = 0.429
If MJ wins then the new ratings are:
MJ_new = 2500 + 24 * 0.429 = 2510 WC_new = 2450 - 24 * 0.429 = 2440
While if WC wins then the new ratings are:
MJ_new = 2500 - 24 * 0.571 = 2486 WC_new = 2450 + 24 * 0.571 = 2464
Here are a few more notes about the way we have chosen to implement the Elo rating system:
- This is a community-based project with the goal of rating the best players in NBA history. In each matchup, the user should choose the player who they believe was the better player. It is up to the user to determine how much weight to give to offense versus defense, peak value versus career value, regular season versus playoffs, etc.
- Before opening this up to the public we simulated 100,000 matchups in order to give the players more realistic starting ratings. These starting ratings do not necessarily represent the opinions of the owners of this site.
- Pairs are not chosen completely at random. The first player is randomly selected to begin the process. Following that, a second player with a rating within 250 points of the first player is randomly selected to complete the pair. We did this in order to prevent bizarre choices (e.g., Manute Bol over Bill Russell) from distorting the ratings.
As always, please send us some feedback if you have any comments or questions.