FOOTBALLNOMICS: The Half-Life of Talent
"This is a process. It's a process. It's a process."
The first question on a fan’s mind when a new college coach steps to the podium is inevitably, “How long until we’re competitive?”
In an age of chaotic rosters, unlimited transfers, and more money than we all know what to do with, understanding the nature of roster turnover challenges some preconceived notions of the sport. For example, consider the Year Zero - the idea that a coach walks in to a situation so bad, nothing that occurs on the field that year can be held against him while he gets his footing and sets things in the right direction.The era of Year Zero is over. Coaches now are expected to come in hot, bringing talent with them, and getting results on the field.
The canonical mixture model from physics describes how the composition of a substance in a container changes with inflows and outflows. A tank begins with an initial composition, and as new water flows in and old water flows out, over time, the concentration changes predictably. A college football roster works the same way.
The Roster Mixture Model
Consider, if you will, a college football roster as a “tank” containing players with an average talent level T(t) at time t. Every year, players leave - to the portal, to graduation, to the NFL draft - and new players arrive - high school recruits, transfers in. As those players move, the composition of the roster - the overall talent level - changes. The Roster Mixture Model can tell us how long it will take for the tank of talent to reach a sufficient level of composition, and in doing so, can provide strategic insights about roster management for college football teams.
Let’s define:
T(0) as the inherited state of the roster
T* as the target level of talent the coach deems acceptable for competition (A Blue Chip Ratio > 50%, perhaps?)
r as the roster turnover rate, how many players to gain and lose annually
TI as the average talent of incoming players
A note about TI: in this model, we’ll assume the new coach coming in has a fixed recruiting ability, presumably such that TI >= T(0), meaning the new coach should be at least no worse at talent acquisition than his predecessor. I won’t get into this, but allowing that parameter to change endogenously with the model complicates things in a way that isn’t worth our time for the key insights here. Consider this problem more generally as quantifying the notion of “how quickly can a coach get his guys in here?”
Then the rate of talent change on a roster follows this equation:
I sat through differential equations (and passed, thank you) so you didn’t have to, so I’ll skip the boring parts and present the familiar solution to this problem:
The talent level of a roster approaches the incoming talent level with a time constraint of T = 1/r, meaning the speed of your roster composition efforts are limited by the turnover rate of your roster. Which is admittedly a clunky way of saying “you can only bring in so many guys every year”.
Think of TI as the “ceiling” of what your roster talent can be. Then, in that case, what we want to know is “how quickly can we expect our coach to get his roster to the level he needs?” The answer depends on turnover and the difference between our coach’s recruiting ability and our current roster.
There are a couple of key parameters here you’d need your coaching staff to determine
TI, the talent ceiling or goal
Turnover rate (r): how many players can you afford to shuffle in and out ever year
With those in hand, you can apply this mixture model to your roster to estimate how quickly you can get your roster up to a competitive standard. Additionally, should you choose to employ an army of nerds, you could work through the relationship between talent and winning to give you a competitive expectation for each season. You could also start to think about recruiting wins and losses in terms of timelines. If we miss out on a 5 star player and have to replace him with a 3 star, what does that do to our incoming talent and how much does that put us back in the timeline? I’m being a little fast and loose with the math there, but as I mentioned in earlier installments, the point of these kinds of models is not decimal point precision, but rather structured and strategic thinking about the decisions we make.
A wrinkle we must address especially in football is that not all positions are created equal. A five-star kicker might increase your talent composition, but he may not move the needle on wins or competition relative to a five-star quarterback. Conversely, that five star quarterback is much harder to acquire and is more prone to transferring, affecting both sides of the equation. To address the differences in impact and scarcity across positions, we have to break the mixture model down into a position-weighted mixture model.
The Position-Weighted Mixture Model
Accounting for the relative values of positional importance, we can express the state of the roster like this:
In plain words, roster composition is the sum of how important that position group is to winning times the total talent composition of that player group. w_i represents the positional importance, and that means that each position will evolve according to its own mixture equation. This isn’t complicated, it just means we have more containers of water to manage.
So we can solve our mixture model as above for each position group and come to a similar conclusion: the rate of roster improvement depends on the difference in our new coach’s ability to recruit that position and the rate at which we can turn over a position. The key wrinkle here is that turnover rates vary dramatically by position due to market dynamics.
Let’s assume due to scarcity, you can only turnover 20% of your offensive line every year. That means, to get it to your standard requires a 5-year horizon (1/.2). But a more liquid position like running back might have an r of .5, meaning you could get it to the desired standard in just 2 cycles. This would mean you should be planning for offensive line 2-3 recruiting cycles ahead, whereas running backs could be more of playing the market year to year.
That scarcity and liquidity of positions introduces the idea of how a program should allocate it’s resources in a rebuild. Let’s say you just got hired at State and you need to figure out where you should start first on the rebuild. With this framework, your coaching staff could determine the positional importance weights above (as I’ve mentioned before, using the franchise tag in the NFL isn’t a terrible starting point) and assess the market for players to determine turnover rates, defining your best resource strategy something like this:
In plain language, we have an empirical approach to understanding which positions are high-impact and low liquidity. Holding market liquidity constant, we should prioritize higher-impact positions. Holding positional impact constant, we should prioritize lower-liquidity positions. And the fun part of this is that those positional weights and turnover rates are not universal constants! Each coach has a well defined sense of what they need on their roster to compete, and codifying those preferences can allow team identity to guide recruiting strategy.
From this, a coach can employ a “portfolio rebalancing strategy” of sorts, with clear benchmarks about how long it will take to turn over a roster, as well as prioritize according to the talent gap. Simply multiplying the above investment equation by the gap between the inherited talent and the goal talent gives you a priority score for a position group i:
Some other potential extensions to this position-specific model:
Supply constraints - talent is not uniformly distributed within and across positions
Weak and strong link systems - in which phases of the game will a single talent move the needle? In which phases do we need to “raise the floor” to help us succeed?
Positional interdependencies: If the defensive secondary needs an elite pass rush to succeed, we can include that in our mixture equation to help shape our strategy:
\(\frac{dT_{CB}}{dt} = r_{CB}(TI_{cb} - T_{CB}) * \alpha*T_{DE}\)Now, our CB room can only adapt to our sufficient talent level if the pass rush adapts as well.
Progress monitoring: Clear goal setting can help coaches evaluate areas of need and dedicate more personnel and resources (or try new strategies) to stay on schedule and close the gaps.
Conclusion
As is my style, the above is just an exercise in placing some mathematical structure on things we already know.
Roster-building is an art, but that doesn’t mean that science can’t help us. The mixture model approach can help programs assess their needs, allocate their resources, and set clear expectations for the timeline of a rebuild. Including positional value, market scarcity, positional interdependence, and more can provide insight into where a team should spend its resources - both monetarily and in terms of man house - to improve the most, and this flexible framework can be tailored to a coach’s specific preferences about what positions he needs on his roster to reach his goals. In addition, many of the above concepts provide launching points for analytical projects that could sharpen a program’s edge in thinking about building their roster. Thanks for reading.
Great read and analytical approach to team building when, like you said, there’s a ton of factors at play regarding position units alone