Game Theory in NFL Playmaking
Game Theory in NFL Playmaking
http://www.advancednflstats.com/2008/06/game-theory-and-runpass-balance.html
Football is often characterized as a physical sport, one where the reality of a team’s athletic abilities outweighs any potential intellectual shortcomings. However, it is much more than that. Game theory is more prominent in football than any (perhaps most) other sports, ranging from choosing to run left or right to calling the opening flip. This article is written by an aerospace engineer who has developed a site that looks into the more economical and mathematical side of football and looks at deeper statistical trends than traditional football stats like touchdowns and yards. Brian Burke wanted to give a general idea of how offensive and defensive coordinators proportion their plays and why they do it.
The article specifically delves into one of the seemingly simple, yet overwhelmingly evasive decisions that occur each play: whether to run or pass and how often to do it. Of course, a tremendous amount of complexity and variety accompanies each run and pass play, so of course there are a nearly infinite amount of additional variables and conditions to account for, but the following idea developed by the author of this article attempts to simplify The following charts explains based on past research what the generalized gain or loss of the offense choosing a strategy and what the defense chooses:
|
Defense |
||||
|
Run D |
Pass D |
Blitz |
||
|
Offense |
Run |
-3 |
4 |
6 |
| Pass |
9 |
-3 |
-5 |
|
Keep in mind that the numbers do not represent yardage, but benefits or detriments based on the other results. For instance, the -3 is likely more representative of simply a 1-2 yard gain, which is not exactly considered good in most cases. In addition, there are many more detailed strategies than this, but it is simplified to illustrate the point.
As demonstrated in class, there is no pure strategy Nash equilibrium for this game, but there is in fact a mixed strategy Nash equilibrium, as in the best strategy for offense is to utilize an unpredictable mix of pass and run. With this mixed strategy, there exists a Nash equilibrium for both offense and defense that is found through potting the relative proportions on a graph and finding the proper percentage of each. For the offense, an approximate 63% run and 37% pass provides the optimal mix in an unpredictable way that will result in the highest “score” possible for the offense. Using these percentages for the defense, we realize that the optimal mix for them is somewhere that involves both run defense and pass defense, but no blitz is the optimal mix. This is because the blitz is not optimal given the offensive choices, and would only be viable if the offense passed 50% of the time or more, which is not the case. Thus, if the defense chose to run a blitz at the offense’s best strategy, it would result in the offense being able to reach a higher equilibrium, thus ruining the Nash equilibrium. It’s also interesting to note that there are no pure strategies in this case. Overall, this article deeply connected with what we recently discussed in class about game theory and Nash equilibriums and provided a direct application into sports to explain it.