Infinite Depth in Advanced College Football Ratings

Advanced systems automatically consider strength of opposition to an inifinite depth.

> In the explanation of "ADVanced" ratings you say
> "Advanced systems automatically consider strength of opposition to an
> inifinite depth. "
>  
> Technically that can't be true unless the program that produces the rating
> is still running and has never produced a report. I think what you meant was
> that "advanced systems automatically consider strength of opposition until
> all teams are related by a well-defined minimum 'distance' metric."
>  
> For football, all teams are related by no worse than an opponents'
> opponents' opponents' opponent relationship, so while the number of
> connections is large, it is not infinite. What we need is a word for "as far
> as possible."
>  
> I would also argue that the statement is not correct anyway.
>  
> The definition of advanced is:
> Ratings systems are "advanced" if they do any of the following: 
> 
> *       Repeatedly recalculate the ratings until they stabalize 
> *       Use a least-squares fit 
> *       Use simultaenous equations 
> *       Use a neural network 
> 
> A rating may meet one of these criteria and NOT include SOS to an "infinite
> depth." The requirement to include SOS to its "ultimate" depth is different
> than the requirement to use a least squares fit (to what?).
>  
> Anyway, it would take me a minute to find an "advanced" method that doesn't
> get past Opponents' opponents and therefore violates the "infinite depth in
> SOS" comment. And technically, there's no such thing.
Let's consider the simplest advanced system:
    team_rating = average_opponent_rating + fraction_of_games_won
The key point is each rating's SOS (average_opponent_rating) depends not on the opponents' results but on their ratings. A team's rating depends on its opponents' ratings each of which in turn depends on its opponents' ratings each of which in turn depends on its opponents' ratings each of which in turn depends on its opponents' ratings each of which in turn depends on its opponents' ratings each of which in turn depends on its opponents' ratings each of which in...and so on forever.

Multiplying both sides by games played and subtracting opponent ratings from both side, we get:

    games_played*team_rating - opponent_ratings = games_won
This is not enough to get unique answer since if you had a set of ratings that worked, you could add some number to all the ratings and get another set of ratings that worked. Thus, we add the provision:
    sum_all_ratings = 0
For a specific example, we'll use the record of an isolated conference--the New England Small College Athletic Conference (NESCAC) as of 10/23/99. Here are the game results. "Win" or "Loss" refers to how the team at the left did. Each game appears in the table twice.

AmherstBatesBowdoinColbyHamiltonMiddleburyTrinityTuftsWesleyanWilliams
AmherstLossWinWin Loss  Win 
BatesWin   Loss LossLossLoss
BowdoinLoss  Win LossLoss Loss
ColbyLoss  WinLossLoss Win 
Hamilton  LossLoss LossLossLoss 
MiddleburyWinWin Win   LossLoss
Trinity  WinWinWin Win Loss
Tufts WinWin Win Loss  Loss
WesleyanLossWin LossWinWin    
Williams WinWin  WinWinWin 

This gives us the equations:
+5*Amherst-Bates-Bowdoin-Colby-Middlebury-Wesleyan=3
-Amherst+5*Bates-Middlebury-Tufts-Wesleyan-Williams=1
-Amherst+5*Bowdoin-Hamilton-Trinity-Tufts-Williams=1
-Amherst+5*Colby-Hamilton-Middlebury-Trinity-Wesleyan=2
-Bowdoin-Colby+5*Hamilton-Trinity-Tufts-Wesleyan=0
-Amherst-Bates-Colby+5*Middlebury-Wesleyan-Williams=3
-Bowdoin-Colby-Hamilton+5*Trinity-Tufts-Williams=4
-Bates-Bowdoin-Hamilton-Trinity+5*Tufts-Williams=3
-Amherst-Bates-Colby-Hamilton-Middlebury+5*Wesleyan=3
-Bates-Bowdoin-Middlebury-Trinity-Tufts+5*Williams=5
+Amherst+Bates+Bowdoin+Colby+Hamilton+Middlebury+Trinity+Tufts+Wesleyan+Williams=0

The first ten equations are redundant. Any one of them can be dropped so we get 10 equations in 10 unknowns. These can then be solved in a finite amount of time.

> But the example isn't "infinite depth." By the time you get to opponents'
> opponents' opponents' opponents you've connected all of the teams. 
Being connected is not good enough. Say, for example, you tried to solve the set of equations iterative by starting all teams with a rating of zero. At each iteration you use ratings for opponents from the previous iteration so the results are not order dependent. After 5 iterations you've gotten to use the opponents' opponents' opponents' opponents ratings but the ratings still have not stabalized. You need to go further until you have what is effectively infinite depth.
> I understand that completely. But now we're talking about the algorithm, not
> about the "depth of SOS".
> 
> That's the clarification I suggested. My algorithm needs 28 iterations to
> get to its "ultimate" consideration of SOS, yours might need 33. But if
> we're using the same set of games, the "SOS depth" is the same and is not
> "infinite."
> 
> Like I said, I knew exactly what you meant, and agree everybody else
> qualified to have an opinion does, too. I was just suggesting more precise
> phrasing. "Infinitely-variable depth" would work as well. But since "depth"
> isn't defined outside of the context of an algorithm, I thought "infinite
> depth" could be improved upon.
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David L. Wilson / dwilson@engr.wisc.edu