Yet Another Revision

© Copyright 2008, Paul Kislanko

I had hardly published Weighted Retrodictive Ranking Violations 2.0 (and asked for comments) before I received suggestions for improvement from Kenneth Massey.

Dr. Massey suggested that the weighting factor in my formula would be more easily understood if the weight were at least 1 for all games (I'd required it be at least ½ and chose one that was at least 2/3.) The problem is that when I set the parameters in my formula to do that, ƒ(1,anyrank) becomes much too large.

(This is especially true when I try to use it to define another reasonable measure of schedule strength.. Using ƒ as defined in version 2.0, playing one game vs #1 would get you a top 10 schedule, even if you played Pop Warner teams in the other 11 games...)

It became obvious that if there is a function that applies to WRRV, Game Interest, and SOS, the ƒ(R1,R2) defined in WRRV 2.0 isn't it.

So instead of finding a function that looked useful for one purpose and trying to make it work for others, I refined the criteria. We want a function
ƒ: { 1,2,3,...,M } → [1, ω1]
with properties:
  1. a• ƒ(1) = ω1
    b• ƒ(M) = 1
  2. ƒ(i) > ƒ(i+1) > 1 for all i in { 1,2,3,...,M-1 }
    (This means ƒ′(i) < 0 for all such i)
  3. | ƒ′(i) | > | ƒ′(i+1) | for all i
    (Taken with 2 this means ƒ″(i) > 0 for all i)

We note:

so functions of the form
W(r) = Log ri Mj Mi+j
meet our criteria and for any ω1 we can find i and j so that W(1) = ω1. W(M) = 1 by the laws of logarithms, so all we have to do is choose the value we want for W(1). Some examples and their graphs are shown below.
ω1      W(r)
4      Log Mr3 M4
3      Log Mr2 M3
2      Log Mr M2
1+1/2      Log M2r M3
1+1/3      Log M3r M4
1+1/4      Log M4r M5
1+1/5      Log M5r M6

Graphs of W(r)

It turns out to be useful to have a whole family of functions. First, it may be that for the putative new schedule strength metric we may want a different ω1 than we do for characterizing a rating's ranking violations. And as Brien Aronov suggests, we may want more than one choice for the WRRV function itself:

Since the "rank" is a random variable with some variance, and (hopefully) this variance decreases as the season wears on.....why not make the weights less "equal" the farther down the season you go?

So it may be appropriate to weight the first rankings with W(1) = 1.2 or so and gradually work up to W(1) = 2.