
Math Meets Football: The Transitive Property Is Real - subnaught
http://www.theplayerstribune.com/math-meets-football-the-transitive-property-is-real/
======
patmcguire
This past NFL season there was a perfect cycle where every team was both
transitively better and worse than every other team:
[http://i.imgur.com/GRC6lT1.png](http://i.imgur.com/GRC6lT1.png)

~~~
madcaptenor
It's also happened in 2013 ([http://www.washingtonpost.com/blogs/early-
lead/wp/2013/11/26...](http://www.washingtonpost.com/blogs/early-
lead/wp/2013/11/26/nfls-circle-of-parity-in-one-neat-graphic-2013-edition/))
and 2012
([http://www.reddit.com/r/nfl/comments/131qfn/parity_2012_the_...](http://www.reddit.com/r/nfl/comments/131qfn/parity_2012_the_circle_of_life_in_the_nfl/))
and 2011 ([http://www.sportsgrid.com/nfl/nfl-parity-
chart-2011/](http://www.sportsgrid.com/nfl/nfl-parity-chart-2011/)) and 2010
([http://www.sportsgrid.com/nfl/nfl-parity-
photo/](http://www.sportsgrid.com/nfl/nfl-parity-photo/)) and 2009
([http://mathfridge.blogspot.com/2010/11/nfl-2009-parity-
wheel...](http://mathfridge.blogspot.com/2010/11/nfl-2009-parity-wheel.html)).
In 2007 the Patriots were undefeated, and in 2008 the Lions were winless; I
can't find any instances of this concept in 2006 or before. But I suspect that
this perfect cycle has a very good chance of happening if there is no
undefeated or winless team.

~~~
patmcguire
I think it's more common now, with the switch from five-team divisions to
four-team ones. More distributed schedules, something like that, although I
don't have the mathematical knowledge to prove that.

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tunesmith
I had a website about this for several years called beatpaths.com . Some of
the graphs for it are still up, and another person took the idea and continued
auto generating similar graphs over at beatgraphs.com.

It was fun, but not really revolutionary. You'd often end up with cycles, and
I had a system to discard the "beat loops" (smallest loops first) until you'd
end up with a DAG.

You could make predictions by saying Team A would probably beat Team B if it
had a path to team B in the graph (the basic transitive property), but still,
the performance wasn't better than Vegas.

I went as far as coming up with some custom tsort algorithms; various
tiebreakers, to come up with a full power ranking that fully respected the
implied orderings in the graph.

In pro football, there's a dummy algorithm you can use to test your homemade
custom snazzy algorithm. The dummy algorithm is: pick the team with the better
record, and if they tie, pick the home team. If your algorithm doesn't match
it, then your algorithm isn't worth a whole lot. My tsort algorithms didn't
regularly beat it. If I had altered the dummy algorithm to just say: 1. Pick
the team with the better record, 2. pick the team that has a beatpath to the
other, 3. pick the home team - then I probably would have beat the dummy
algorithm by a very little bit. But still, not close to Vegas performance.

In college football, trumpeting 88% is dumb because college football regularly
has crap matchups. I had some full beatgraphs to college football, too - they
were huge and loosely connected, and its performance wasn't surprisingly good
or anything.

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ugexe
This should save everyone a few minutes: _Okay, let’s get back to the big
question: Does the transitive property work in college football?_ _Yes._ _..._
_But sometimes it might take a little imagination_

I wish I could go into detail on why everything about this article is wrong,
but I can't. Luckily the author discredits his own premise, but unfortunately
not until you get to the end.

~~~
epmatsw
Exactly. "It works, except when it doesn't".

Considering he even says that it only holds 88% of the time, you would expect
1 or 2 games out of every teams 12-game season to go the "wrong" way. That is
more than enough to screw up any ranking you want to make based on the
transitive property when you take into account that the top 25 teams all range
between 0 and 3 losses.

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wetmore
The author is also a guard for the Baltimore Ravens.

~~~
madcaptenor
A month ago he and some coauthors published a paper "A Cascadic Multigrid
Algorithm for Computing the Fiedler Vector of Graph Laplacians" in a math
journal ([http://arxiv.org/abs/1412.0565;](http://arxiv.org/abs/1412.0565;)
note that the arXiv date is earlier than the publication date, since arXiv is
a preprint server) and the media had a field day with the "NFL player
publishes math paper" story.

