From: C.Thomas.Eyford.III@dartmouth.edu (C. Thomas Eyford III)
Newsgroups: rec.sport.football.pro
Subject: Re: potential new team ratings
Date: 15 Nov 92 20:48:29 GMT
 <1992Nov14.174907.7131@gn.ecn.purdue.edu>
Organization: Dartmouth College, Hanover, NH
X-Posted-From: InterNews1.0a5@newshost.dartmouth.edu

In article <1992Nov14.174907.7131@gn.ecn.purdue.edu>
nitin@gn.ecn.purdue.edu (Nitin Khanna) writes:

> Problem is that even if you square the winpct, your new
> ranking will be almost exatctly like the one above because
> the square of a # less than 1 will make it even smaller and
> have very little effect.  Obviously the biggest factor
>  in your equation is average margin of victory which ranges
> from something like 12 to 0.  The Giants have 2.88 and the
> Broncos something like .41.  So squaring the .667 of the 
> Broncos and the .556 of the Giants wont have any effect on
> the final rankings.  If you wanted to make winpct have a greater 
> effect you could do something like:
> 
> (winpct+2)*margin*strength
> 
> How about you make a ranking based on the above (in the event
> that you are bored senseless it will give you something to do).

Actually, adding a constant to the winning percentage devalues it. 
Using straight winning percentages, a team with a .800 winning
percentage is twice as good as a team with a .400 winning percentage. 
Adding a constant 2 means the former team has a 'weighted' winning
percentage of 2.800, compared to 2.400 for the latter.  All of a
sudden, the difference falls to 16%, rather than 100%.  Alternatively,
look at it this way: 

(winpct+2)*margin*strength 

can be converted using the distributive property to

winpct*margin*strength + 2*margin*strength

which, as you can see, plainly weights the other two categories.

To weight winning percentage, you should multiply it by a constant.
However, in an expression composed entirely of multiplied terms,
multiplying by a constant just serves as a scale factor; there is no
way to designate the constant as only multiplying the winning
percentage.  What could be done is to add a second term into the
equation.

winpct*margin*strength + (constant)*winpct

which will weight winning percentage, though at a relatively inexact
amount.  
I suggest a better way to weight the categories is to come up with a
better formula.  

Going on the assumption that we have three categories to assess,
winning percentage, margin of victory, and opponents' winning
percentage, we need to decide on a formula that will weight the correct
things, and not the wrong things.  For example, in the margin of
victory calculations, a team that scored exactly as much as their
opponent did would receive a 0, making their final rating 0.  Yet this
team would probably have somewhere close to a .500 record.  I think a
ratio PF/PA works much better, though it has its drawbacks as well.  A
team that who averages a 17-10 victory over its opponents gets the same
rating as a team that averages 34-20, or 51-30.  Or, as a more extreme
example, a team that averages a 7-3 victory would be rated evenly with
a team that outscores its opponents 49-21.

A ranking system does no better - giving teams a rating from 1 to 28,
as it does not reward truly outstanding performance.  A team that leads
the league with 400 yards total offense and a team that leads the
league with 800 yards total offense are both ranked number 1, though
clearly the latter team should be rated stronger than the former.  

I suggest a combination of the two methods.  The top team in each
category is rated at 1.000, and the bottom team is rated at 0.000. 
This then defines the spread.  In the case of winning percentage, the
top team would be Dallas at .889, and the bottom New England at 0.000. 
Thus Dallas would receive a 1.000 rating, and New England a 0.000
rating, with the spread being .889-.000, or .889.  All teams would then
have their winning percentage divided by the range (after subtracting
the low end of the scale to put the bottom team at 0.000) to yield
their weighted range.  San Francisco would be rated at 0.875, the
Broncos at 0.750, etc.

The same would be done for both the other categories, with San
Francisco and New England defining the spread in margin of victory with
13.2 and -14.3, respectively.  Subtracting the low end of the scale,
each team's margin of victory would be increased by 14.3 points, to
raise New England's to 0, and make sure everyone had a positive number.
 Then, you would divide each team's new margin of victory by 27.5 to
get their final ratings.  San Francisco would be rated 1.000, Dallas at
0.920, and New Orleans at 0.738.   The best and worse schedules belong
to Phoenix and Miami, respectively, with Phoenix's opponents winning at
a 0.629 clip, and Miami's at a .357 pace.  Phoenix would rate a 1.000,
Dallas a 0.364, San Francisco 0.184, and Miami 0.000.

Now the problem of weighting comes into play.  Obviously (in my
opinion) schedule strengths should not be weighted evenly with winning
percentage.  A more correct version of this formula would look like
this:

Power rating = [winning percentage ranking] + [margin of victory
ranking] +          
               [strength of schedule ranking]

with each ranking being multiplied by a scale factor.  This is
extremely important, as with small spreads such as opponents' winning
percentage, there might be a very small increment between first and
last, but it is still a full point difference (1.000 vs. 0.000).  

If anyone would like to hazard a guess at what these weights should be,
and maybe compute power ratings using this formula, I'd be interested
in seeing what sort of results you come up with.
  
-Tom

RL: Tom Eyford                        E-mail: eyford@mac.dartmouth.edu

"So, you'll put down your rock and I'll put down my sword, and we'll
try to kill each other like civilized people?"  
                                      -- Westley, _The Princess Bride_  
Newsgroups: rec.sport.football.pro