Questionably Empirical Decisions

On College Football

To many students, college football plays a large part in the quintessential university experience. Texas A&M, as a member of the Southeastern Conference, is certainly no exception. Every game day, tens of thousands of students and alumni flock to Kyle Field to watch the Aggies take on their latest opponent. There is an undeniable pageantry to it- from the early morning tailgates, to the spirited halftime performances and, above all, the long-held traditions that makes every university experience unique.

I am not here to wax poetic about college football, however. Rather, I am interested in a related topic. As those familiar with the sport know, a signficant portion of discourse revolves around hypotheticals: who will win each division, which quarterback is the Heisman frontrunner and, most importantly, which team will emerge victorious in this week's competition. It is these spirited debates, and their accompanying arguments, that has captured my interest.

Such conversations are fraught with peril though. In my experience, people exhibit a stronger attachment to collegiate sports as compared to their professional counterparts. I'd wager that, for alumni, there exists an inseparable bond between the university and their personal experiences. Conversations about the team brings back fond memories: stepping on campus for the first time as a freshman, impromptu late-night outings with close friends, and the profound feeling of accomplishment when walking across the stage at graduation. College is a formative experience, transforming young teenagers into adults fully prepared to take the next step in life. This is why hackles are raised when someone disparages your college team. It is not merely a criticism of the football program, but a vehement dismissal of the memories we hold so dear.

As such, I would like to avoid kicking this particular hornets' nest when talking about college football. My verdicts regarding determining team superiority must be empirical and devoid of any subjectivity. The million dollar question is: "What evidence will you use to substantiate your claims?"

Enter QEDSports- a service that provides the necessary arguments for why your team is the best in college football.

The Transitive Property

We are currently in the age of analytics in football. Gone are the days of just tracking wins, offensive yards, and turnovers. Instead, viewers are treated to an array of next-generation stats. These include average intended air yards, completion probability, and expected yards after catch, to name a few. While useful for the data nerds among us, these metrics can be somewhat hard to parse for the average fan.

QEDSports offers a refreshingly simple take on analytics- we only care about the final outcome of games. This makes our analysis crystal clear and prevents the user from having to Google "what is AYTS" every time the offense takes the field. To wit, the cornerstone of our arguments is the transitive property.

What's the transitive property? As explained by former Baltimore Ravens guard (and curent professor of mathematics at MIT) John Urschel:

"If team A defeats team B and team B defeats team C, then team A will beat team C"

Note that this idea is not limited to just a trio of teams. We can apply the property recursively, chaining together games to create a plausible path to victory for every matchup.

Constructing the Victory Graph

With our methodology in place, we now turn to the process of generating the so-called "path to victory". This requirement has a natural formulation as a graph problem. Formally, a directed graph $G$ is a collection of vertices $V$ and edges $E$, where each edge connects exactly two distinct vertices. In our college football example, we can think of every vertex as a team and each edge $(u, v)$ as a game in which team $u$ defeated team $v$. Graphs have a nice visual representation, as pictured below:

The benefit of this approach is that we can structure our question as a graph problem. Specifically, we're looking for the shortest path between two vertices. This is a classic problem in computer science, with Dijkstra's algorithm serving as the usual solution ¹ . In the above picture, our shortest path from $A$ to $E$ has length two, so team $A$ would defeat team $E$ through their common opponent $D$.

All that remains is to create a list of all games played in the 2025 season and generate the victory graph. The hard work is already done for us- CollegeFootballData has collected the relevant game data and exposes a REST API for developers. After a couple of requests, my graph, which includes over 16,000 games across all levels of college football, was created.

David and Goliath

Everything appears to work as expected. When asked why Texas A&M would defeat the reigning SEC champion Georgia, it replied:

1. Texas A&M def. Florida
2. Florida def. Florida State
3. Florida State def. Alabama
4. Alabama def. Georgia

which looks pretty reasonable. However, there is an obvious shortcoming with our methodology: what do we do with undefeated teams? A team with no losses has an indegree of zero in our victory graph, so the shortest path is undefined.

At present, only one undefeated team remains- the 13-0 B1G champion Indiana. With Indiana earning the #1 seed in this year's college football playoff, this presents quite an challenge for my Aggies. This obstacle is not insurmountable, though. We just need to exercise some creativity.

The Role of History

To take down Goliath, we introduce a concept called persistence. This is inspired by a forecasting technique of the same name, in which tomorrow's conditions are predicted to be identical to today's. You likely do this already when it comes to picking weather-appropriate attire. Yesterday was cold? Might want to bring a jacket. Rainy all week? Better pack an umbrella for today. This logic, I argue, can be seamlessly applied to college football.

As we all know, players and coaching staff tend to remain on the team that recruited them. Frequent transfers between institutions hamper academic progress, a high priority for every student-athlete. With all of their core pieces intact, the quality of a team ought to remain the same year-over-year. This aligns with our understanding of perennial powerhouses and underperformers. A team that only won three games, for instance, is destined to be a bottom-dweller in their division for years to come. Suggesting that said team could turn it around and compete for a national championship in only two years is utter nonsense.

The logical conclusion of this is that if Team A defeated Team B last year, persistence says that the result will be the same this year. With the persistence mechanic added, we arrive at the following convincing argument for why Texas A&M can best undefeated Indiana:

1. Texas A&M def. Notre Dame
2. Notre Dame def. Indiana (in 2024)

As for implementation details, it's as simple as gathering game data from previous years and adding the corresponding edges. There is one wrinkle though. We would like for persistence to be used only when it is impossible to establish a chain of victories within the current season. This can be achieved by adding a weight or cost to each edge. Should we assign a large enough weight ² to older games, the shortest path will prioritize current games instead.

A Step Away

Our ultimate goal for QEDSports is to provide a victory chain for any two teams. For our graph $G$, we require that there exists a path connecting any two vertices $u$ and $v$ (Should this hold, we say $G$ is strongly connected). I was using the networkx library, which provides a handy function for determining the number of connected components in a graph. After adding the persistence feature, I wrote a quick test script to check that there was indeed only one connected component:

cendres@MacBookPro qed-sports % python check_is_scc.py
connected components: 22

Oh.

After printing out the components, I was able to pinpoint the issue. The overwhelming majority of teams existed in a single strongly connected component. However, there existed isolated pockets, each containing only a handful of teams. These were exclusively teams in the lower divisions of the NCAA (i.e. DIII). This makes sense, since teams mainly take on opponents within the same division. The "cupcake" games are a notable exception to this rule, although these contests tend to be rather one-sided. This leads to a bubble enveloping these teams, preventing any outgoing paths from reaching the majority of the league. Without a win over a signature opponent, our algorithm will fail at generating a victory chain.

The LLM Shoehorn

My ideas for salvaging the situation were slim. I could try going back further in time, in hopes that persistence would give me games that connect the disparate components. Maybe I could contrive another reason for drawing an edge- perhaps margin of victory (or defeat) against common opponents? None of these options appealed to me, and I quickly grew discouraged. On the verge of giving up, I had an epiphany:

It's 2025. Just use generative AI.

By punting this problem to a language model, I can abdicate responsibility for inventing new metrics and rid myself of this mess. My weapon of choice is gemini-2.5-flash-lite, which appears to be the only "Big 3" model that I can get an API key for without providing my credit card information.

Integrating the LLM was trivial, as Google provides Python boilerplate for interacting with Gemini. The only remaining question was how to instruct the model to make its decisions. We could, of course, just rely on the training data to inform the model's response, but that's rather boring. Instead, I hit upon my final idea, one inspired by a particularly valiant bulldog:

"Generate explanations where the game is not played by the players, but by the mascots"

I wrote a quick system prompt for Gemini, instructing it to eschew the rules of conventional football in favor of my mascot contest. After gluing together the model service with the rest of the application, I asked how the Georgia Bulldogs could defeat the Tufts Jumbos, whose mascot happens to be an elephant. Here's what Gemini replied:

"The Georgia Bulldogs, led by their tenacious bulldog mascot, possess an undeniable advantage in raw power and tenacity. The bulldog embodies an aggressive, never-give-up spirit, translating to superior strength and a formidable presence in any challenge. Conversely, the Tufts Jumbos, despite their name suggesting size, are often depicted as more reserved or even bookish. While they might possess some strategic intelligence, they lack the sheer brute force and unwavering aggression that the bulldog brings to the field. In a mascot competition, the bulldog's unyielding drive and physical dominance would overwhelm the Jumbos, securing a decisive victory for Georgia."

Phenomenal.

Closing Thoughts

By harnessing the power of the transitive property, time travel, and generative AI, we are able to craft convincing explanations for any hypothetical college football game. Moreover, we use (un)Questionably Empirical Decisions for our arguments, which is where the name QEDSports comes from. This should hopefully prevent the emotional firestorm that usually accompanies debates about the superior college football team. Both parties can now rest easy, assured that their team is totally capable of beating any opponent.

As it happens, my Texas A&M Aggies have finished the regular season at 11-1, securing them a berth in the newly-expanded College Football Playoff. They face the #10 Miami Hurricanes, in what will be the first postseason game at Kyle Field in school history. I have no doubt the atmosphere, as always, will be electric.

Am I nervous? Maybe a bit. But I have an unwavering faith that my alma mater will pull through.

Why, you ask?

It's simple. Texas A&M clobbered LSU, who beat Clemson, who defeated Louisville, who upset Miami.

Q.E.D.