KRACH ratings work on an odds scale. Consider, for example, Brown, rated
at 219, and Yale, rated at 113; Brown is reckoned to have odds of 219/113
= 1.94 to 1 of defeating Yale on neutral ice. If you prefer
probabilities, this comes out to 219/(219+113) = 0.66 of Brown winning.
 
The ratings are calculated using only game results: win, loss or tie. I do
not use the score, or shootout results. I also only use games between the 44
teams, ignoring non-div I or Canadian opponents. (This might explain any
discrepancy between the W-L records I give, and others you might see.) The
ratings are scaled to give an "average" of 100 -- I could equally well have
used any other figure without affecting the odds, but 100 seemed like a
nice number.
 
Let me show you KRACH at work on a tiny example. Suppose three teams, A, B
and C, play a round-robin on neutral ice, with the following results:
 
A defeats B
A ties with C
B defeats C.
 
Thus A is 1-0-1 with 3 points, B is 1-1 with 2, and C is 0-1-1 with 1 point.
Counting a tie as half a win, A has 1.5 wins, B 1 and C 0.5.
 
Now, I claim that the KRACH ratings are A: 213, B: 100, C: 47. If this is
so, then:
 
Prob. A def. B is 213/(213+100) = 0.68; prob. B def. A = 1-0.68 = 0.32;
Prob. A def. C is 213/(213+47)  = 0.82; prob. C def. A = 1-0.82 = 0.18;
Prob. B def. C is 100/(100+47)  = 0.68; prob. C def. B = 1-0.68 = 0.32.
 
(We're ignoring the possibility of a tie for the moment.)
We can calculate an "expected number of wins" for each team, by adding up
the probabilities for that team's games.
A's win probs are 0.68 and 0.82, so their expected number of wins is
0.68+0.82=1.50;
B's expected number of wins = 0.32 + 0.68 = 1.00;
C's expected number of wins = 0.18 + 0.32 = 0.50.
 
Based on counting ties as half a win (and otherwise ignoring them), you see
that each team's expected number of wins is equal to the number of wins that
team actually had. This is what determines what the KRACH ratings are: *the
ratings for which each team's observed and expected number of wins are
equal*.
 
All right, so I picked those ratings out of the air, like any good math text
(!) To actually calculate them, there's no simple approach that leads
directly to the answer; all you can do is a kind of successive
approximation, continuing to adjust the ratings until the observed and
expected wins are "close enough". There are several different ways to do
this, but they all follow a plan like this one:
 
1. Set all the ratings equal to 100.
2. Calculate the observed and expected wins for each team.
3. For those teams with more observed than expected wins, move their ratings
up a bit. For those teams with fewer observed than expected wins, move their
ratings down a bit.
(This is deliberately vague! The different methods differ in this step.)
4. If the observed and expected wins for each team are "close enough", stop.
Otherwise, go back to step 2 and repeat.
 
After a few repetitions of steps 2 and 3, things settle down, and we have
our final ratings.
 
In practice, I add a couple of refinements to this. I include a home ice
advantage, which is estimated by trying to match the observed and expected
number of home-ice wins; this has the desirable effect of making a road win
against a team more valuable than a home win when estimating a rating.
 
Also, especially early in the season, teams that have won nearly all (or lost
nearly all) of their games tend to have unreasonably high (or low) ratings
-- possibly even infinite ones -- so I include a "damping factor" that pulls
all the ratings towards 100 slightly, hopefully yielding more reasonable
estimates of the probabilities of one team defeating another.
 
Well, that was supposed to be a nice, concise explanation. It wasn't.
Apologies to those who didn't want to wade through it but had to anyway....
 
Feel free to e-mail me if there's more you want to know about this. Though,
given that I'm going away soon, you might have to wait a while for an answer
:-)
 
Cheers,
Ken.
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Ken Butler                          |     at    [log in to unmask]
Dept. of Mathematics and Statistics |     or    [log in to unmask]
Simon Fraser University             |---------------------------------
Burnaby, B.C., Canada V5A 1S6       |     Ora asht nje pikerisht.
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