I finally took the time to add in GS/GA/G% to the main page of
the GRaNT sheets. I think it makes it easier to visualize why teams are ranked where they are, since Wins and Losses don't factor into anything and records are really only being shown for informational purposes. You'll notice between similarly ranked teams that if one has a better G%, it'll have a corresponding weaker SOS, and vice versa:
You'll also notice that now that we've got a much more substantial sample size, the SOS's make way, way more sense. To wit, WCHA teams occupy 7 of the top 8 positions, with only Cornell breaking up the clean sweep by sneaking into 8th (lol. Alol, even).
Also I'm realizing I never answered your last post!
Say two teams start their season by playing each other to a 5-5 draw. You say that each also has a mythical 1-1 draw against BYE on their record. How does GRaNT consider this: that each team has scored 6 of 12 total goals, or that it scored 50% of the goals once, and then 50% of the goals a second time? From your description, I'm assuming the former.
Well -- it's actually more the latter, but in a way it's both. I'll explain.
This is calculated the same way KRACH is, except we use goals instead of wins. The goal of the calculations (both for KRACH and GRaNT -- but I'm going to describe it for GRaNT here) is to get the goals scored vs. goals allowed that actually happened (expressed as goal percentage, "G%", calculated as GS/[GS+GA] ) to match up exactly with the G% that you would EXPECT to happen based on the playing of each individual game.
Your "Expected" G%" is:
[ Expected G% in game 1, plus
Expected G% in game 2, plus
Expected G% in game 3, plus
... , plus
Expected G% in game N ] ,
divided by N games.
"Expected G%" in a given game is calculated by [Team 1 Rating ] / [Team 1 Rating + Team 2 Rating ]. Before you start the iterative calculations, you assume everyone has the same rating. After you run the calculation, everyone has a new "rating", so you throw it back into the mixer and recalculation it dozens of times until everyone's "ratings" stop changing by more than 0.00001 or so on each successive run.
The "actual" G% which you are trying to match is your season-long GS/[GS+GA]. So, yes, season-long GS and GA are factors in the calculation. But that's exactly how KRACH is calculated as well: you take a game-by-game "expected number of wins" regardless of the actual result (for example 0.95 wins for a Stonehill vs. Minnesota game, etc. etc. from each individual game divided by number of games) and assign ratings such that that sum exactly equals the number of wins you actually had in a season -- the latter of which just takes the overall season-long final number of wins, and doesn't care who those wins were against.
Put another way:
In GRaNT,
- Your Expected G% is based on a hypothetical game between those two teams, but your Actual G% is based on what already happened.
- Your Expected G% is calculated based on each individual opponent, but your Actual % is calculated as one season-long number.
Just as,
In KRACH,
- Your Expected # of wins is based on a hypothetical game between those two teams, but your Actual # of wins is based on what already happened.
- Your Expected # of wins is calculated based on each individual opponent, but your Actual # of wins is calculated as one season-long number.
That would at least get away from having to add in the imagined 1-1 tie.
I don't
really need to add in the imagined 1-1 tie. As soon as everyone has scored 1 goal and allowed 1 goal in a given season, it's not needed anymore. But in the interest of preventing the possibility, and since adding it to everyone's set of results means it doesn't affect the rankings, I like having it in there. KRACH does the same with a "phantom tie," which is no longer needed once everyone has at least 0.5 wins and 0.5 losses, but it's kept in there regardless.
Why should a 1-0 win be more decisive than a 5-1 win...
The thing is, in the long run, it's not. An individual shutout breaks the math, but over the course of a season (or by including that phantom 1-1 tie), the math resolves itself. If you have a 1-0 win and a 1-1 tie, and another team against the same opponent has a 5-1 win and a 1-1 tie, then Team 1's G% is 0.667 and Team 2's is 0.750. That 5-1 win is indeed a better win than 1-0.
...and why is a 2-1 win better than a 7-5 win?
I think, again, over the long run, the math supports this. One 2-1 win is too small of a sample size to say anything definitively. But if you are winning 2-1 in every game, you're doubling up your opponents every time you hit the ice. If you're winning 7-5, you're allowing nearly as many goals as you're giving up.
It becomes a lot clearer in the extremes: If a basketball team wins 100-97, but a hockey team wins 4-1, one of those results is far more decisive than the other even though they are both 3 point margins. A 7-5 result is a more "basketball-like," high scoring, high variance result than 2-1 is over the long run.
I'm not really questioning your implementation so much as the basic premise, that a higher ratio of goals scored provides insight into a team's strength, more so than so other metric, like margin of victory/defeat.
I think of it as being just another tool in the toolbox. If you've got two teams that are undefeated but one is winning games 2-1 and one is winning games 5-1, I think that tells you something of value ahead of a potential game between those two teams that can't be captured by KRACH. A lot of times what I'll do if I want to compare two close teams is I'll look at them in KRACH, and if they're pretty close, I'll pull up GRaNT to see if they have more separation there to provide a little clarity.
For what it's worth, I still to this day consider KRACH to be the gold-standard of all the rankings. I consider it to be as close to mathematical beauty as you'll ever get due the fact that (1) you're directly measuring a team's ability to earn wins, which is what each team is ultimately trying to do, not score goals, (2) the calculation boils down to "how many wins should you have at this rating vs. how many do you actually have," and (3) its beauty is in its simplicity and its lack of needing to do anything arbitrarily (i.e. there's no decision that needs to be made on "how much weight do we put on SOS vs. how much do we put in a win" like a traditional calculation like RPI/NPI needs to).
Thanks again for all of the effort that you put into providing us with calculators!
And cheers to you for caring enough to dive into the numbers with me!
