## Monday, June 20, 2011

Today I'm going to look at adding MOV to Bradley-Terry style ratings.

Recall that in Bradley-Terry style ratings, we're trying to select a rating R for every team such that the odds of Team I winning a matchup with Team J turn out to be equal to:

Oddsi = Ri / (Ri + Rj)
So how do we determine R?  The trick is to look at our set of historical outcomes and find the values for R that maximize the likelihood that what actually happened would have happened.  (This is called maximum likelihood estimation, or MLE.)  Various mathematically gifted folks figured out that you could iteratively determine the proper values for R using this equation:
Ri =Wi * 1 / (Ri + Rj)]-1
where Wi is the number of wins by Team I, and the sum is over all the games played by Team I.

Those of you who have been following along carefully from home will recognize this as the update equation from the KRACH rating, and indeed, KRACH is just Bradley-Terry for (originally) hockey teams.

Spartan Dan over at "The Only Colors" blog suggests in this posting a rating system that awards a variable amount for each win based upon the MOV:
W(mov) = 1 / (1 + e-mov/k)
In this equation, "k" is a scaling constant.  The effect of this equation is to award a 1/2 win for an MOV of zero, about 3/4 of a win for an MOV of "k", and then asymptotically to 1 win for greater MOVs.  The idea here is to limit the impact of "blowouts" on the rating system.

It's fairly easy to plug this equation for W into our implementation of KRACH.  With k=5 (as suggested by Spartan Dan), it gives this performance:

Predictor    % Correct    MOV Error
Govan (best)73.5%10.80
KRACH (original) 71.5%11.50
KRACH + MOV (k=5)  72.0%11.36
KRACH + MOV (k=10)  72.2%11.34

As you can see, it provides a small improvement over KRACH without MOV.  Some experiments show that performance is maximized for k=10, but not by a large amount over k=5.

As I noted in the original posting on KRACH, since the Bradley-Terry method is aimed at producing odds, it isn't particularly suited for predicting MOV.  (Although it's reasonable to think that a team with greater odds of victory is likely to win by more points.)  Even so, it's a little disappointing that adding MOV doesn't provide more improvement in the Bradley-Terry model.