Math Meets Football: The Transitive Property Is Real

With the College Football Playoff National Championship fast approaching, I’d like to give some predictions for the big game …

But before I get to that, I want to tackle (pun intended) a topic very close to my heart. No, I am not exposing a major college football scandal this week (in case you missed it, check out my first piece on academic majors in college football here), but I would like to look into one of my favorite concepts, both in mathematics and in sports: the transitive property.

Ring any bells?

Remember third grade math class, with Mrs. Such-n-such? She told you if a > b and b > c, then a > c. Transitive property!

Remember talking football with Uncle So-n-so? In 2008, he told you that because the New England Patriots beat the Dallas Cowboys 48-27, and the Dallas Cowboys beat the New York Giants (twice) during the regular season, that the then-undefeated Patriots were a lock to beat the Giants in the Super Bowl and become the first team to go undefeated since the ’72 Miami Dolphins. The Pats beat the Boys and the Boys beat the Giants — so it was obvious that the Pats were going to beat the Giants. Transitive property!

And it makes sense. That’s what we all learned in school, right?

Well, it didn’t quite turn out like that. Eli Manning gets the ball down four with 2:39 left in the game, makes a big third down conversion with the help of David Tyree’s helmet, hits Plaxico Burress on a slant-and-go for a 13-yard touchdown, and the Giants win the game.

If this is not ringing any bells, please, please, please send me your address. I will send you a copy of Super Bowl XLII. Consider it a belated Christmas present from me to you. Why? I’m just that type of guy.

From an emotional perspective, the transitive property flies in the face of the notion of “any given Sunday,” that unwritten rule that no underdog is ever truly outmatched. This hallowed belief, in conjunction with the absurdity of comparing football divisions to a “totally ordered set,” results in the transitive property getting a pretty bad rap amongst football fans.

You may be thinking, “What’s a totally ordered set?”

I’m glad you ask!

A totally ordered set is a group of elements paired with a binary relation that is antisymmetric (a≤b and b≤a implies a=b), transitive (a≤b and b≤c implies a≤c), and total (either a≤b or b≤a).

In football terms, a given division is a “totally ordered set” if all the teams in the division play each other, and the transitive property works 100 percent of time.

That’s more than enough set theory to last you a lifetime. For those of you who lack a taste for numbers, feel free to pack it in. This is a judgment-free zone.

Still here?

Okay, let’s get back to the big question: Does the transitive property work in college football?

Yes*.

To examine this phenomena, let’s look at one of the most competitive and dominant divisions in college football. Yes, we’re talking the SEC West.

Since 2007, the SEC West has had 28 AP Preseason Top 25 teams (19 in the top 15, and nine in the top five), 27** AP Postseason Top 25 Teams (22 in the top 15, and 10 in the top five), 12 BCS bowl game appearances (five wins), seven BCS National Championship appearances, and five BCS National Championships.

To put it plainly for non-college football fans, the SEC West is traditionally very, very good.

What’s even more impressive is that since 2007, the transitive property has held a stunning 88.78 percent of the time. Meaning that if you pick any three teams in the SEC West, where Team A beats Team B and Team B beats Team C, then nearly 9 out of 10 times Team A will beat Team C.

Between 2007-2014, the SEC West was 14 games and a total of 69 points away from being a totally ordered set. That means that each year, on average, the SEC West is less than a combined nine-point swing (in roughly two games) away from being totally ordered, and therefore fulfilling the transitive property completely.

Want to see for yourself? Below, you can choose any year from 2007 to present, and look at the point differentials of all the in-division SEC West games. Each entry shows the number of points the team in the given row beat (or if negative, lost to) the team in the given column. For example, in 2007 there is a +36 in the top-right entry, indicating that Arkansas beat Ole Miss by 36 points. In red are the games that must be adjusted for the division to reach its nearest totally ordered set. Note that the SEC West final rankings do not always match up with the closest perfectly ordered set.

Below, you can see a distribution of all the point differentials that the transitive property fails by (number of points that any given three teams are away from fulfilling the transitive property). Of the 23 times where the transitive property fails, 15 of them were decided by a field goal or less, six by one score, and only two by two scores. Note that although the SEC West was a total of 14 games away from being a totally ordered set, this does not imply that the transitive property fails 14 times.

All of this is well and good, but I did promise national championship predictions, so here I go. We’re talking Ohio State vs. Oregon.

I’m calling it: Ohio State is going to win.

Why, you ask? It’s quite elementary … you see, Ohio State beat Penn State 31-24, who beat Boston College 31-30 (go State!), who beat USC 37-31, who beat Arizona 28-26, who, of course, beat Oregon 31-24.

That’s right, I called it. The Big Ten is making a comeback. The transitive property says so.

Actually, let me re-evaluate that prediction … I mean, Oregon did beat Florida State 59-20, who beat Wake Forest 43-3, who beat Virginia Tech 6-3, who beat Ohio State 35-21.

So rest easy Oregon fans, the Ducks are going all the way.

But while Oregon may win the National Championship, I know who’s really the best team in college football: the Massachusetts Institute of Technology!

Yeah, I said it! Don’t look so shocked … don’t you know?

MIT beat Salve Regina 48-26, who beat Norwich 48-21, who beat St. Lawrence 10-7, who beat Morrisville St 31-14, who beat Utica 52-41, who beat Buffalo St 31-21, who beat Manchester 60-32, who beat Trine 29-28, who beat Millikin 28-14, who beat Elmhurst 21-14, who beat Augustana IL 17-10, who beat Loras 34-17, who beat Simpson IA 27-24, who beat WI River Falls 22-6, who beat S Dakota Tech 43-28, who beat Wm Jewell 62-44, who beat Missouri S&T 34-27, who beat St. Joseph’s IN 14-13, who beat Valparaiso 31-10, who beat Butler 17-3, who beat Stetson 49-41, who beat Marist 22-14, who beat Jacksonville 17-16, who beat San Diego 35-18, who beat W New Mexico 23-17, who beat Ft Lewis 24-17, who beat CSU-Pueblo 23-22, who beat Sam Houston St 47-21, who beat McNeese St 38-22, who beat Northwestern LA 35-28, who beat Louisiana Tech 30-27, who beat Illinois 35-18, who beat Penn State 16-14, who beat Boston College 31-30, who beat USC 37-31, who beat Arizona 28-26, who beat Oregon 31-24!!!!!

There you have it. And it only took 37 applications of the transitive property to prove it. Go Engineers! Want to know how many it takes to prove your team would whoop the Oregon Ducks? Give the good people at MyTeamIsBetterThanYourTeam.com a visit.

It’s been real. I’ll be here, playing football, doing math, and, on occasion, talking about both for your reading pleasure. Until next time …

*But sometimes it might take a little imagination

**Week 16 rankings used instead for ’14 season