Trouble passing? Try a better running back.
We have examined a simplified version of American football through the lens of game theory. In this situation, the offense can choose to run or pass and the defense can choose to defend the run or defend the pass. Once we have the payoff matrix, it is straightforward to show that the Nash equilibrium occurs when the offense runs with a certain probability (say, p) and the defense defends the run with a certain probability (say, q). But how do we arrive at a payoff matrix that reflects the vast number of variables in real football? How does this affect the equilibrium probabilities and the payoffs? Specifically, how does the skill of a team’s running back (the player most responsible for the payoff in the “run” strategy) influence the game?
Suppose that a team with an average running back has the following payoff matrix.
|
Defense |
|||
| DR | DP | ||
| Offense | R | -3,3 | 4,-4 |
| P | 9,-9 | -3,3 | |
The equilibrium probabilities are then p = 0.636 and q = 0.37.
The payoff to the offense is then -3q + 4(1 – q) = -7(0.37) + 4 = 1.41.
If that same team acquires a star running back, should they run the ball more often (increase p)? Will their payoffs increase? Suppose that this star player allows the offense to increase all of its payoffs for the “run” strategy by 1. This also reduces the defense’s payoffs in these cases because we are dealing with a zero-sum game.
|
Defense |
|||
| DR | DP | ||
| Offense | R | -2,2 | 5,-5 |
| P | 9,-9 | -3,3 | |
We might expect that the defense will defend the run more often.
At equilibrium,-2q + 5(1 – q) = 9q – 3(1 – q) and q = 0.42, so the defense does defend the run more often.
Would it be a good idea for the offense to use their star player and run more? Or should they take advantage of the defense’s shift in strategy and pass more?
To find the offensive probability at equilibrium, 2p – 9(1 – p) = -5p + 3(1 – p) and p = 0.63.
Amazingly, it is not in the offense’s advantage to run or pass the ball more often than they used to.
The offense’s payoff is then -2q + 5(1 – q) = -7(0.42) + 5 = 2.06. So, by improving the running game, the offense can increase its payoff, but it will follow the same run-to-pass ratio as it did with an average running back.
The conclusions here are interesting. If a team improves their running ability, why would they not run more often? The reason lies in the payoff matrix: The better running skill adds 1 to the offensive payoff against both defensive strategies. This amounts to adding the same thing to both sides of an equation – the solution is unchanged.
In order for the offense to change their optimum probability, the payoff increases must be different against different strategies. In fact, this is exactly what happens to the defense. The defensive payoffs remain the same against the pass, but decrease against the run. Thus, the defense must accommodate the improved running threat by defending the run 42% of the time instead of the original 37%.
Keep in mind that while the offense uses the same probabilities of run and pass, their payoff certainly increases (from 1.41 to 2.07). Common football knowledge suggests that teams “run to set up the pass,” and this is precisely the reason. A better running game forces the defense to defend the run more often. Then the offense benefits when they pass against a run defense, which has the highest payoff of any strategy combination. For example, in 2007, the Minnesota Vikings drafted star running back Adrian Peterson and saw their quarterback, Tavaris Jackson (who was mediocre at best), improve his yards per passing attempt that season.
There is of course no single payoff matrix that applies to every play of a football game. There are a wide range of scenarios, variables, and unpredictable occurrences in the actual game of football. But if different payoff matrices could be accurately created for different situations (downs and distances, field positions, times of the game, whether winning or losing) and for more specific strategies (types of run and pass plays, different formations), the consequences and insights could be astounding.
http://www.advancednflstats.com/2008/06/game-theory-and-runpass-balance.html
http://www.advancednflstats.com/2008/06/game-theory-and-great-running-backs.html