The transitive property. You might remember learning about it in a high school algebra class. You’ve also certainly heard it applied as an argument for college football preeminence - Team A beat Team B, and Team B beat Team C, therefore Team A must be better than Team C.
Such arguments are simplistic and often lead to contradictions (what if a fourth team, Team D, has beaten Team A, but lost to Team C?). But it’s still fun to chain together transitive arguments for your team’s superiority over another, Kevin Bacon style. This post serves as your interactive guide to creating such arguments as we near the midpoint of the college football season.
Most of what follows will be familiar to you if you’ve read my work in previous seasons. I’ll be explaining everything along the way, though, so don’t be shy if you’re a newcomer! The first thing to keep in mind is that I only consider games between two FBS opponents. If you’re interested in similar analyses that include lower-division teams, check out this site.
A final note before we dive in - the interactive bits in this post are best experienced on a desktop/laptop or tablet. If you found us on your phone, I’d suggest coming back on something with a larger screen if you want to play around.
Want an overview of the strength of your team’s transitive property arguments? Then you’ll want to take a look at the transitive property reports below. What do you find here? As an example, go ahead and choose Kansas State from the drop-down menu. The report tells you that K-State has 103 transitive wins, meaning that of the other 127 teams in FBS, you can make a transitive argument for Kansas State beating 103 of them.
Who are these teams? To find out, scroll down to the “Wins detail” table. The row in which a team is found signifies the length of the transitive argument necessary to reach them. The first row of the table identifies K-State’s direct wins, i.e. the teams they have beaten on the field, Florida Atlantic and Texas Tech. The next row, the “Level 2 wins”, are the teams that have lost to either FAU or TTU. Level 3 teams are the teams who have lost to a level 2 team, and so on down the table.
Directly underneath the “Transitive wins” line is the “Avg Path Length”. This is the average number of games necessary for K-State’s 103 transitive wins. The more teams you have higher up in the “Wins detail” table, the lower this number will be. So the best way to shorten your average path length is to win more games, and beat better teams.
Of course, we can play this same game in the reverse direction, counting the number of teams the Wildcats have lost to by transitive arguments. Scrolling down to the “Losses detail” table will tell you who these teams are, and the average path length is also displayed.
That leaves the “Win rank”, “Loss rank”, and “Combined rank” lines as the last items to explain, which we will get to in the next section.
In this section, we use information from the reports to generate a ranking of all FBS teams. We do this in three steps - first we rank the teams based on the quality of their transitive wins. Then we create another ranking based on their losses, and finally we combine the two for our final ranking.
How should we judge the quality of a team’s transitive wins? Clearly, beating more teams is more impressive than beating fewer, so our first step is to sort all teams based on raw number of transitive wins. Ties are broken by average path length, with shorter paths being preferred over longer ones.
Losses are judged in an analogous way, with fewer losses and longer paths being preferred. The final, combined ranking is determined by averaging each team’s wins and losses rankings, then sorting in ascending order.
The results? Clemson is our number one, just barely outpacing Texas A&M in average win length. You’ll see some head-scratchers here, but a few of them can be explained away by the ranking’s ignorance of FBS vs. FCS games (I’m looking at you, Wazzu and Iowa).
Interactive Path Finder
Alright, so now you know all of your team’s transitive property victims, but what are the actual arguments? You can find them using our interactive path finder. For example, if you choose Kansas State in the first drop-down menu, then Texas in the second, you’ll uncover the 9-game argument proving why the Wildcats should beat the Longhorns this Saturday. You can try to find a shorter argument if you want, but you won’t find one. All the paths shown here are as short as possible (though alternate paths of the same length may exist). Do you want to see the reverse argument? Then change the third drop-down from “wins” to “losses”, and you’ll find an 8-game argument for Texas beating K-State.
The paths are given in plain text form and also plotted out on a map of the US. Seeing the paths on a map, though, may play with your perception of what paths are “short.” For example, there’s a nice, clean argument for Washington beating New Mexico, due to Washington’s season-opening victory over Rutgers and the Scarlet Knights subsequent win over New Mexico. Plot it on a map, though, and that near coast-to-coast-to-coast trip looks anything but short. To take into account the point-to-point distances between schools, flip the fourth drop-down menu from “games” to “miles” to reveal a 3-game path that never ventures east of the Mountain time zone.
That’s it! Click around and have fun! For those who are curious, the most ridiculous argument available is one of several 17-game paths. You can find them by selecting the winning team as either Georgia State or Louisiana-Monroe, and ending at one of Akron, Maryland, Memphis, Minnesota, or Wyoming.