Game Theory in Professional American Football
There are a myriad of offensive philosophies in American Football including the Option, the Spread, and the West Coast style offense. These philosophies use a range of unbalanced strategies from pass-heavy to run-heavy and are common at all amateur levels of the game due to high deviations in talent among different positions; however, when it comes to the professional level, where players are on higher, more even footing, teams tend to use a more balanced approach (Pro style) in order to succeed.
Modelling the game of American Football as a two-person zero-sum game between the Offense and the Defense results in the following chart. The strategies of the offense include running or passing the ball and the strategies of the defense include defending against the run or defending against the pass.
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It is clear that the best strategy for each side is a mixed strategy where the probability of both Run and Pass being 50%. This game accounts for most situations in the NFL, but what if a team has a star player that gives that specific team a slight edge in one category. For example, a team with a star quarterback would result in a game where the Offensive Pass payoff would be a ++ when the Defense played the Run and the same in every other regard. How much would this affect each team’s strategy? It would be intuitive to think that the Offense should pass more; however, the Defense would be prepared for that. Thus, it would be better for the Offense to run the ball more often than passing in that case. This type of thinking applies to any other type of situation in the game and shows, through a Nash Equilibrium, that staying balanced remains the best strategy for both sides.
Source: http://www.fieldgulls.com/seahawks-analysis/2015/1/9/7517551/understanding-effective-play-calling-with-game-theory