ELO Ratings

Baseball Prospectus today devotes a column to ELO ratings for baseball teams ($ reg.), after one of its writers stumbled upon ELO ratings for national soccer clubs. Since I had never stumbled on ELO ratings either for baseball or soccer, I found this very interesting.

The ELO rating system was developed as an alternative rating system for ranking chess players and has since been adapted as a method for ranking competitors in all sorts of situations. The system assigns a ranking to each player, which together produce the probability of a win by each player. Given ratings of Ra and Rb for competitors a and b, the probability of a win by player a is 1/(1+10^((Rb-Ra)/400)) and the probability of a win by player b is 1/(1+10^((Ra-Rb)/400)). Thus, if the two competitors have the same ratings, the probability of a win is 50% for each side.

The ratings are updated after each match, so that player a's new rating, Ra', is based on the outcome of the match and the expected outcome of the match:

Ra' = Ra + K(Actual Outcome - Expected Outcome)

For the soccer ratings referenced above, Actual Outcome = 1 for a win, .5 for a draw, and 0 for a loss and the expected outcome is the probability of winning based on the ex ante ratings. The "updating factor" K is varied based on the importance of the match in question and the margin of victory in the match (follow the link for details). The system also adjusts the win probability for home-field advantage. Nate Silver's baseball ELO system recalibrates the system for baseball teams.

The advantage of such a system is that it is easy to use to compare teams and accounts for strengths of schedule and more heavily weights recent performance, so it should give an accurate measure of the team/player's current abilities. This is especially useful for team sports, where the make-up of teams changes over time.

So, with all the background out of the way, what do the ELO ratings reveal? First, the baseball ratings, updated through Monday 6/26:

• The White Sox are baseball's best team (1573), closely followed by the Red Sox (1552), Mets (1549), and Tigers (1548).
• The Royals (1419) and Pirates (1437) bring up the rear.
• The Marlins are totally average (1500). [By convention, a ranking of 1500 is taken to be exactly average.]
• The best team at any point in the 2000s: the October 11, 2001 Oakland A's (1624)
• The worst team at any point in the 2000s: the September 23, 2003 Detroit Tigers (1335)
  
• The best "year-end" team in the 2000s: the 2004 Red Sox (1609), surely helped out by winning 8 straight to end the post-season.

What about soccer? Well, the quarterfinals are set, so here's what the odds should be, based on current ELO rankings:

• Argentina (1975) v. Germany (1946): Argentina 54.16% (giving Germany the 100 point home field advantage boost gives the edge to Germany at a 60.08% win probabilitity)
  
• Italy (1974) v. Ukraine (1788): Italy 74.47%
  
• England (1970) v. Portugal (1980): Portugal 51.44%
  
• Brasil (2075) v. France (1973): Brasil 64.27%

For comparison's sake, here's the current tradesports.com win probabilities, based on the assumption that the share price equals the expected win probabilities:

• Argentina (1975) v. Germany (1946): Germany 52.6%
  
• Italy (1974) v. Ukraine (1788): Italy 78.5%
  
• England (1970) v. Portugal (1980): England 59.5%
  
• Brasil (2075) v. France (1973): Brasil 69.7%

The tradesports difference is largest for England and Portugal, likely because tradesports knows that Portugal is down 2 players due to red cards received against Holland, while the ELO rankings does not know this.