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Math Meets Football: The Transitive Property Is Real

23 points by subnaught 11y ago

ugexe 11y ago
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 11y ago
  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.
- gohrt 11y ago
  That's the joke.
patmcguire 11y ago
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 11y ago
  It's also happened in 2013 (http://www.washingtonpost.com/blogs/early-lead/wp/2013/11/26/nfls-circle-of-parity-in-one-neat-graphic-2013-edition/ http://www.washingtonpost.com/blogs/early-lead/wp/2013/11/26...) and 2012 (http://www.reddit.com/r/nfl/comments/131qfn/parity_2012_the_circle_of_life_in_the_nfl/ http://www.reddit.com/r/nfl/comments/131qfn/parity_2012_the_...) 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.html http://mathfridge.blogspot.com/2010/11/nfl-2009-parity-wheel...). 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 11y ago
    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.
wetmore 11y ago
The author is also a guard for the Baltimore Ravens.
- madcaptenor 11y ago
  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.
tunesmith 11y ago
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.

23 points by subnaught 11y ago

ugexe 11y ago

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 11y ago
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.
gohrt 11y ago
That's the joke.

patmcguire 11y ago

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 11y ago
It's also happened in 2013 (http://www.washingtonpost.com/blogs/early-lead/wp/2013/11/26/nfls-circle-of-parity-in-one-neat-graphic-2013-edition/ http://www.washingtonpost.com/blogs/early-lead/wp/2013/11/26...) and 2012 (http://www.reddit.com/r/nfl/comments/131qfn/parity_2012_the_circle_of_life_in_the_nfl/ http://www.reddit.com/r/nfl/comments/131qfn/parity_2012_the_...) 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.html http://mathfridge.blogspot.com/2010/11/nfl-2009-parity-wheel...). 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 11y ago
  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.

wetmore 11y ago

The author is also a guard for the Baltimore Ravens.

madcaptenor 11y ago
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.

tunesmith 11y ago

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.