Adrian Peterson Should Rarely Touch the Ball


(I couldn't help but use an old football picture of me.)

Anyone who has seen A Beautiful Mind knows game theory can explain dating strategies, although the video is not a Nash Equilibrium.

The key is to choose the best outcome relative to what other people are doing.

Game theory can also explain something more important, football. Not what people in Spain call football. I am talking about the American gladiator fight known as football.

Football is a great laboratory for the game theorist. It has discrete plays. There is a short time to strategize between plays. This time could lead to more "rational" actions. There is an easy way divide plays (pass or run). The benefit of one team comes at the cost of the other. And the goal of any single play is simple, gain yards. These attributes make the model easier to work with and possibly more realistic.

As I've said before, economics can be counterintuitive. For football, game theory says that a good quarterback can lead an offense to pass less. How? 

A simple model of football involves two possible moves for each team. The offense chooses to run or pass and the defense picks a run or pass defense. If the defense guesses right, the offense loses yards. If they guess wrong, the offense gains yards.The expected gains/losses are below.


In this example, there is no dominant strategy for either team. Instead, each must split their actions between run and pass.

The offense will run the ball a proportion (q) of the time. q* is the best action for the offense and it is when the expected yardage for each run or pass is the same.  If they were different, the offense should choose the one with the higher expected gain. This wrong q would not be a Nash Equilibrium.

Two simple equations solve the problem-

  • if defense picks run, the expected gain is -3*q + 6*(1-q)
  • If defense picks pass, the expected gain is 3*q - 6*(1-q)
  • These are equal at q*: - 3*q + 6*(1-q) = 3*q - 6*(1-q) => q*= 2/3

If the offense passes the ball more the 1/3rd of the time, the defense will react by playing a pass defense more often. This will result in more sacks and loss of yards for the offense. The optimal strategy is 2/3rds run plays and 1/3 pass plays.

Now, the offense brings out a stud quarterback. They now are a passing threat. The gains/losses are again below.


The calculation is redone-

  • If defense picks run, the expected gain is -3*q + 15*(1-q)
  • If defense picks pass, the expected gain is 3*q - 6*(1-q)
  • These are equal at q*: - 3*q + 15*(1-q) = 3*q - 6*(1-q) => q*= 7/9

With a better quarterback, the offense will now run the ball MORE OFTEN. Since the defense knows about the quarterback, they will adjust their action. They will pick a pass defense more often.

We can generalize the problem to a general case-


Again, redo the calculation-

  • If defense picks run: Expected gain is -a*q + c*(1-q)
  • If defense picks pass: Expected gain is b*q - d*(1-q)
  • Setting these equal to each other: - a*q + c*(1-q) = b*q - d*(1-q) => q*=(c + d)/(a + b + c + d)

I do not expect Bill Belichick to change his strategy from this example. And, off course, the Vikings should not pass the ball more than they run.

Does this mean the model is a poor predictor? What assumptions need to be changed or added in this model? The simple model still teaches to think about both players in any situation.

(Note: This post is inspired by Mike Shor.)

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