Written by Daniel Rosenkranz
Football is America’s most popular sport, with approximately 200 million people tuning in for last National Football League season’s Super Bowl, and for good reason (Deitsch, 2023). Football has history, is intense, and features a truly unique aspect of decision making. The judgment process that players and coaches go through for each play is quite reminiscent of the risk reward trade off present in economic theory. In football, each team has four tries on offense, called ‘downs’, to advance the ball 10 yards or the possession is forfeited to the opposing team. Recently, 4th down decisions have been a hot topic of debate. If a team attempts to convert and gain the necessary yardage, they risk scoring no points and giving the opposing team the ball. Depending on their position on the field, and the amount of yards needed to convert, the team may also elect to kick a field goal, which, if successful, scores 3 points. They can also punt, which would score no points but give the opposing team the ball deeper in their own territory. ESPN, the broadcaster of the NFL’s weekly Monday Night Football game, displays a percentage on 4th downs indicating the likelihood the offensive team will convert. The use of probabilities to analyze the coach’s decision is interesting, but it feels like a drastic simplification of the decision at hand. Much more is factored in than pure likelihood of conversion.
The decision of whether or not to go for it on 4th down can be modeled using an economic concept called game theory. Game theory weighs the strategies of different players and their potential outcomes. Players can formulate probability-based beliefs about other players actions, allowing them to calculate expected payoffs. In most games, players do not have the full information set (knowledge of the other players strategies and their payoffs), which is the case for football (Samuelson, 2016). The defensive coordinator does not know if the offense is going to elect a run or pass play but has to strategize in order to best assign coverages. By using the available information – which consists of game score/history, team strengths/weaknesses, location on the field, time remaining, and distance to convert – both coaches ‘play the game’ and receive a payoff based on the resulting equilibrium.
In order to illustrate the concept, a sample model of a play from the 2023 Michigan versus Ohio State game can be utilized. Before walking through the example, it is important to to understand the basics of a dynamic game. In the following game, there are two players. Player 1 is Michigan, and because they are on offense, they will choose first from two options: ‘kick’ or ‘go for it’. In real life, the defense, which in this case is Ohio State, can see if player 1 is electing to kick or go for it based on the formation they assume. This is known as perfect information. If player 1 elects to kick, the game ends and the payoffs are determined by whether or not the field goal is made. If player 1 elects to go for it, the players engage in a subgame. In this subgame, player 1 will elect to either run or pass, and player 2 will select a defense. For the sake of the model, it is simplified into four choices: run commit, blitz, man coverage, and zone coverage. Since the players do not know what the other will choose in the subgame, it is imperfect information, and is represented by the dashed line. The choices of players 1 and 2 are made simultaneously if player 1 has chosen to enter the subgame of ‘go for it’.
Michigan and Ohio State use the available options and implied probabilities to elect their strategy. In this instance, it was 4th down and 4 yards to go with 1:10 remaining in the 4th quarter, and Ohio State had no timeouts. Michigan head coach Jim Harbaugh believed that they were to receive the greatest utility by attempting a field goal in this specific situation. In real life, this decision is made by the coach, but in game theory this decision is made using expected value. Arbitrary values, based on typical expectations and situation-specific context, have been assigned as payoffs for each set of decisions in the model. The expected value is calculated by multiplying the payoff received by the probability that it occurs. While Coach Harbaugh could have elected to go for it, he believed that given the available information the greatest expected value would come from making the initial choice to kick rather than go for it.
The unique aspect of using game theory, is that it models situations that do not occur, yet factor into the situations that actually transpire. The decision to kick is not as simple as whether or not the ESPN renders the likelihood of conversion as too low, it is significantly more complex. With the ongoing controversy of when the right time is to go for it on 4th down, coaches like Harbaugh navigate a dynamic landscape of imperfect information, strategic calculations, and unexpected outcomes. Game theory provides a lens to understand the nuanced decision-making process, reminding us that football’s 4th and short is not just a play – it is economic theory unfolding on the gridiron.
References
Deitsch, Richard and Shea, Bill. The Athletic. “Super Bowl 57 sets TV ratings record.” The Athletic, https://theathletic.com/4478840/2023/05/02/super-bowl-57-tv-ratings-record/.
Samuelson, Larry. American Economic Association. “Game Theory in Economics and Beyond” American Economic Association, https://www.aeaweb.org/articles?id=10.1257/jep.30.4.107.