FOOTBALLNOMICS: An Economic Approach to Roster Management, Part 1
Let's go back to ECON 101 and formalize some useful ideas for college football teams.
College football coaches in today’s game are faced with a difficult optimization problem: How do I balance playing time to maximize wins, develop talent for the future, and keep everyone happy? An oft-used economic model may provide some insight.
College Football Teams as Firms
College football teams are firms. Their inputs are labor (players) and capital (snaps those players play at different levels of quality), and they use technology (facilities, strategy, strength programs, analytics, coaching) to produce wins with those inputs. That’s a little reductive, and frankly a little icky to reduce the entirety of college football to the sum of widgets, but in a general sense, provides us a foundation upon which to develop some useful theories.
These firms - going forward, I’ll stick to teams - seek to maximize wins using players, subject to some constraints. Some of those constraints are more concrete than others: you can only have 11 men on the field, you can only field so many players on a roster, you can only play 12 games, so you’re capped at regular season wins. Others are more abstract: you can only devote so much time to installing plays and developing players, and you’re subject to things like departure from the transfer portal, the entire behemoth that is recruiting and talent acquisition, and strength of competition that can vary wildly across teams and from year to year. All of those are interesting features in a model, but for now, suffice it to say that our working understanding of a college football team as a firm is simple: Teams employ players via snaps to produce wins, and they have to strategize in a way that considers not just this season, but future seasons.
College Football Players as Households
Players, on the other hand, are households. They consume playing time and develop for the future, with the goal of performing as well as possible while helping the team win. (SIDE NOTE: it’s entirely plausible there are cases where the incentives surrounding player performance and team success do not align - bowl opt outs stand out as a glaring example - but as we’re building up the basics, we can generally assume the incentives align, most of the time). Players come in as prospects, they develop, they contribute, they graduate, and new players come in. Player contribution is increasing with experience - holding talent constant, more experienced players will contribute more to their teams - and quality - holding experience constant, more talented players will contribute more to their teams.
Instead of the traditional idea that households work to receive wages, and with those wages, they consume and save, we’ll instead say that players work to receive time in the program, and with that time they can play on the field or develop for the future.
Model Based Thinking
What is a model? In data science and statistics, a model is a data-generating process, some assumptions, and an estimator. In economics, a model is something different. It’s a structured system defined mathematically to help identify important facts and features of the world. It’s a simplification of reality that helps us isolate interesting dynamics. In the world of economics, generally, models are going to be concerned with marginalism: how does utility/output change when decisions are made or variables change or things go wrong?
The rest of this post will be an outline of a model for college football roster management, specifically focused on the question of how to manage playing time versus development for prospects in order to maximize the sum of your discounted future wins.
It may be a little mathy, and I’ll do my best to sum up each section in a way that lets you skip the boring stuff, but formalizing the structure of ideas is useful, and by the end of this post, I’ll provide a single equation that serves as a foundation for understanding the trade-off between playing prospects young and developing them for the future.
The Model
The Solow Model for growth is a well-known framework for understanding how firm decisions and household preferences affect an economy’s gross domestic product. It features a “fundamental equation” that relates savings today to output tomorrow, thus linking production across periods and consumer preference to firm output. One of many limitations of this model is that savings - the decision about what resources to dedicate to the future - are exogenous, determined outside the model.
Given that the savings decision is of interest in our quest to understand how much playing time you should give prospects, I’ll build out a college football team’s economy using an extension of the Solow Model - an overlapping generations model. The OLG model, developed by a few folks but most notably and first by Paul Samuelson in 1958, provides two important features: different kinds of consumers, as opposed to one representative consumer, and an endogenous (defined by and responsive to changes in the model) savings function.
I’ll spare more detail here, and apologize in advance to any economists or my doctoral advisor reading this - I’m going to take some liberties with the framework, but if you stick with me until the end, I think we’ll all land somewhere tolerable.
Without further ado, let’s get economical.
College Football Programs with Overlapping Generations
We’ll consider consecutive time periods as t, t+1, t+2…. This could correspond to seasons, or more simply could correspond to development windows for teams. At any given period, there are N total players. Players can either be prospects or starters. The total number of players in a given period is the number of prospects in this period and the number of prospects from last period (N_t and N_{t+1}). Each player “lives” for two periods: they enter as a prospect, the matriculate to a starter, and then they die graduate. There is potential for growth, as it’s possible to have more prospects than starters, but I’m going to tiptoe around that for now, as it requires opening the Pandora’s box of attrition.
The “economy” is populated by a single team. As mentioned above, the team uses the factors of production to produce output. For our example today, the output they produce is “wins”, although it could be NFL Draft Picks, Conference Championships, Playoff Appearances, or what have you.
Players
A player enters college with no playing time as a baseline. The key decision for a prospect is whether to give him playing time or develop him. The idea of development being in contrast to playing time is different for positions and for players of different quality - don’t worry, that’s one of the first assumptions we’ll abandon in future posts. For now, players are all of a similar quality, and they can either play in a given period or choose to improve that quality through development.
We’ll refer to development as s_t. Development turns into productive snaps in time period t+1. The player uses the rest of his time playing snaps and producing wins.
Starters do not develop, they only play snaps and produce wins.
There is an inherent budget constraint on players, young and old.
For prospects, their total time is divided into playing time and development.
For starters, their time is
A starter’s productive capital (read: snaps) is a function of the choices of his development as a prospect, and the team’s ability to improve players throughout their career (R_{t+1}).
More simply, coaches choose to play players early, or they choose to develop them for the future, and a player’s development can be improved by the team’s efforts to develop prospects.
For young players, they want to pick development time as to maximize the discounted sum of their utility.
To find the maximum, you just take the first derivative. I’ll skip some math, especially with no functional forms here, and highlight that we want to equate the utility a prospect gets from playing now to the discounted value of playing time as a starter.
That gives us the following equation, which will look familiar if you know who Euler is.
If a coach chooses to play a prospect less (decreases playing time in period t and increases development time in period t), he decreases the player’s lifetime utility by the left-hand side of the equation. We’ll call that the marginal cost of developing a prospect.
What’s the benefit to developing a prospect? Quality of playing time in period t+1 increases by the right hand side. That’s the marginal benefit of developing a prospect.
We know from ECON 101, our general goal is to have marginal cost equal marginal benefit. In the equation above, we’ve satisfied that, and have a framework to evaluate they effects on a player, and even to solve for an optimal level of development to maximize utility.
Are you still with me? We got a little formal there, but in high-level terms, we’ve put some equations on the relationship between a coach’s choice to develop a player or play him as a prospect and winning. The problem is two-fold. 1, what is utility? We’d need a functional form on the equations above, and that can be pretty influential on our ultimate result. 2, as college football teams, while we do certainly care about the well-being of our players, we care more strongly about our performance as a program. So instead of developing this further, we’ll take this structure and add the most important ingredient: how teams convert players into wins.
A Theory of Teams
A team produces wins using capital (snaps played) and labor (players), enhanced (or diminished) by team-specific factors like coaching, analytics, strategy, etc (we’ll call that A, our technology).
Simply enough, wins today are a function of how good we are at converting player-snaps into wins (A), and which players (N) we play, and what their level of development is (K).
Teams choose which players to play and what opportunities they give young players to maximize wins.
From a standard profit maximization sense, firms will earn 0 profits when they are maximizing their production function. In the example of college football teams, we can think of O wins as the “ceiling” of a team, and any deviations from optimal will result in a reduction of wins.
Maximizing the production function gives us two important results:
The team-specific ability to convert player-snaps into wins is improved by R, which depends on our technology and our wins today.
The opportunities (time) we give prospects is determined by our technology and our wins today. We’ll consider those opportunities fixed as we land the plan here, but effectively, this could be a way to devote program hours and coaching time across prospects. We’ll also explore that more.
Ok, So What?
We have an equation documenting the trade-offs between playing time and development for a prospect. We have an equation linking team success to player inputs. What we need now is an equilibrium.
In a typical overlapping generations model, equilibrium features about nine equations that characterize the economy. We can substitute some plain text for letters, though, and skip straight to the good stuff, arriving at an equation for understanding the optimal level of prospect development to maximize the sum of discounted future wins, based on past wins, our team-specific ability to convert snaps to wins, our technology, and current level of development.
The key variable in the equation is player snaps next period, characterized by K_{t+1}. Why? Well, because player snaps this period is already decided. The total quality of snaps we can put on the field this season is a function of how many players we' have developed and how much we have developed them.
In equilibrium, prospects divide their time between playing time and development.
Starters contribute quality snaps to wins, but they do not develop any further.
So the fundamental equation that gets us to equilibrium in this economy is
Playing time for prospects today plus playing time for starters today plus the quality snaps we will play next year is equal to the number of players we play today, how well we develop them, and our ability to convert player snaps into wins.
It’s more complicated than it sounds. But, you don’t have to trust me, you can just logically get to this equation. That’s all economics is, really, is applying a formalized structure for thinking to complex problems in order to examine interesting features. And we can examine interesting features here.
Skipping over some unnecessary equations, the most pressing question for teams is “In equilibrium, how much should I develop my prospects relative to how much I play them?”. A well-founded strategy here can help coaches determine the costs and risks associated with going all-in in a single year, understand the trade-offs in playing prospects and developing them, and identify where returns to strategy, strength and conditioning, and other off-field improvements can affect their process.
The Key Equation, or “Making it all make sense”
If you’ve skipped to the bottom, I applaud you. You don’t really need much from the above sections beyond the following:
Prospects can develop or play snaps.
Starters play snaps, and their quality is based on how much they developed.
Teams use snaps to win games.
Teams have to make a decision about how much to develop prospects and how much to play them.
Now for the result:
Fans of the chain rule will recognize this structure. Without the annoying algebra that got us here, we can say that the key insight from this entire post is this:
The quality of snaps a team can put on the field next season is a function of how much they develop their prospects. The optimal level of development (relative to playing time) for prospects depends on how well the team converted player snaps to wins this season, how many more wins they would contribute with higher-quality snaps, and their strategy and coaching.
You might be sitting at home, having stumbled through this entire post, saying “well, of course!” To an economist, having someone read through a model and say that it’s obvious might be considered a compliment. In this post, I’ve build the foundations to link player development and team performance across years, helping us to identify dynamics and how different strategies for player development might affect wins.
This fundamental equation, the linkage of wins today to wins tomorrow, gives us a key direction to begin to peel back layers. From that equation alone, we have several empirical questions we can go to the data and try to answer for teams:
How good are different teams at converting player snaps to wins?
How good are teams at developing players? (Returns to development time)
What is the marginal effect of adding player snaps at different positions?
What is the marginal effect of adding development at different positions?
How do time preferences for winning align with development cycles, and what can that tell us about talent acquisition strategies (prospects v portal)?
And more. By making a simplified version of reality, we’ve isolated a framework to estimate the trade-offs between developing prospects and playing prospects, a key decision for coaches in college football. Next time, we’ll start to add positional importance into the model to better account for differences in development and differences in contributions to winning.