Game Theory in American Football Play Calling
Burke, Brian. “Game Theory and Run/Pass Balance.” Advanced Football Analytics (Formerly Advanced NFL Stats), June 2008, http://archive.advancedfootballanalytics.com/2008/06/game-theory-and-runpass-balance.html.
In this article, Brian Burke analyzes how Game Theory can be applicable to choices made during the course of an American Football game. Specifically, the article breaks offensive choices down into whether to pass or to run. Ever since I was a kid watching the Patriots (I grew up in the Boston area), I’ve always wondered about the thought process behind the decisions made on each play call. When I was younger, I thought it was pretty much random; all I saw was that they would run sometimes and then pass sometimes. When I grew older, I realized that the decisions were far more structured and that there was a lot of thinking behind it. This website, Advanced Football Analytics, is incredibly intriguing and robust, as it has several articles analyzing topics such as player transactions, game theory in football decision making (like this article), and statistics on plays such as kicking a field goal, throwing a Hail Mary, and returning a kick off. As an in-depth website on football analytics, it provides for many interesting reads that show an intersection of what we learn in class with the real world, even in something that’s primarily a form of entertainment for most of us like football.
This article provides a hypothetical payoff chart that shows the potential payoff for the offenses given the choices of running and passing for the offense and the choices of pass defense, run defense, and blitz for the defense. The hypothetical values are represented in the following chart:
| Defense | ||||
| Run Defense | Pass Defense | Blitz | ||
| Offense | Run | -3 | 4 | 6 |
| Pass | 9 | -3 | -5 |
The units provided are that of “utility,” not necessarily a unit like yards or points. While the chart Burke provides only shows utility for the offense, the defense just has utility that’s the additive inverse of the offense’s. For example, the defense’s utility when playing pass defense against an offense’s pass would be 3, as that is the additive inverse of -3, the offense’s utility in that situation. These types of charts are of the same type as the ones we’ve seen in class when discussing Nash equilibrium and payoff matrices. Thus, we look to see if there exists a Nash equilibrium, which can come in the form of a dominant strategy. Burke summarizes the information in the graph and discusses the dilemmas the coaches face when making the decision of what play to call. For example, running could be considered safer for the offensive play caller because two of the results have positive utility, but passing has positive utility when against run defense, and in fact passing has the highest possible payoff of 9. Thus, there’s no dominant strategy for the offense. As we learned in class, just because there’s no dominant strategy doesn’t mean there isn’t a Nash equilibrium, which Burke addresses by stating that the real optimal path is through a mixed strategy in Nash equilibrium.
The most interesting section of the article follows, where Burke transfers this information from the table onto a graph in an attempt to find what the optimal course of action is for the teams. The graph has Run and Pass on sides of the horizontal axis, and then Run Defense, Pass Defense, and Blitz as 3 different lines with the values of utility on the vertical axis. Burke concludes that the optimal point for the offense lies at a point which is approximately 63% run and 37% pass. While this was a hypothetical situation, this shows how if we assign utility values to various situations such as run vs run defense, which coaches do based on experience, intuition, and statistics, we can hopefully find a distribution of run and pass that gives an optimal strategy. Sometimes, it could be a dominant strategy, but often times it will involve mixed strategies. I also like looking at this from the defense’s perspective, as knowing what the offense’s optimal point can affect how the defense calls their plays. For example, in the given case, defenses shouldn’t call blitzes because of the 63% run 37% pass optimal distribution of the offense. This is obviously different from the real world, where defenses use a myriad of strategies, including blitzes, and will try strategies even if they aren’t “optimal.” Burke acknowledges this discrepancy, as there is such a variety of plays and schemes that both the offense and defense can use that simplifying it like he did with the chart isn’t completely realistic, but the principles can still be applied to real life. While this article doesn’t give a method to completely “solve” the matter of football play calling, it does give valuable insight into how the decision making calculus that play callers go through would look like if we thought of these decisions as being a part of this created game. In the end, football is a game, and with any game, there is strategy involved that can be strengthened and better understood through the utilization of analytics and game theory.