Game Theory and Great Running Backs

Should a team that adds a superstar running back run more often? It seems like a dumb question, but the answer isn't so simple.

In 2006 the Minnesota Vikings ran the ball 388 times for 1820 yards, a 4.1 yard per attempt average. In 2007 the Vikings added rookie sensation Adrian Peterson who helped the Vikings run 440 times for 2634, a 5.3 yard per attempt average. That's an increase of 52 runs from '06 to '07, a 13% increase. (Rush attempt figures do not include QB runs or kneel-downs).

Part of the increase might have been due to the Vikings ' two more wins in '07 than in '06. Since running is a safer option late in games with leads, we could expect a few additional run attempts, but certainly not 52. With the addition of a star RB like Peterson, should a team like the Vikings have run even more? Or did they run too much?

Conventional wisdom might say of course. Why wouldn't a team use its biggest advantage more frequently? But think about it from the point of view of an opposing defense. They know how big a threat Peterson is, and would likely orient their strategy toward stopping the run. Knowing this, what should the offense do? There are 3 options:

1. Run more often than before because every run will bring an increased payoff.
2. Pass more often because the defense will be stacking the line with run defenders.
3. Run the same proportion as before. Considerations 1 and 2 balance out, and payoffs will increase for both passing and running.

From my recent article of game theory and strategy, recall this simple payoff matrix (below). An offensive run play against a run defense will usually result in a setback. But against a pass defense it will be more successful. Similarly, a pass against a run defense is likely to be more successful. The payoffs don't necessarily represent yards or points, but rather a proportional preference for the various expected outcomes.

Before star RB		Defense
		Run D	Pass D	Blitz
Offense	Run	-3	4	6
	Pass	9	-3	-5

Now let's add a star RB to our team. Let's say our expected run outcomes increase by 1 full unit against each defensive strategy. Our passing outcomes remain unchanged. The new payoff matrix looks like this:

With star RB		Defense
		Run D	Pass D	Blitz
Offense	Run	-2	5	7
	Pass	9	-3	-5

How would this change affect the optimum run/pass balance for the offense? The original payoff matrix is illustrated in the chart below. The x-axis represents the run/pass mix of the offense. The left side of the chart indicates the payoffs vs. the defensive strategies if the offense runs 100% of the time, and the right side of the chart indicates the payoffs if the offense passes 100% of the time. In between are the payoffs given various proportions of running and passing.
The optimum play mix, where the payoff to the offense is maximized, is the intersection of the red and blue lines at point b. In the example (before adding the star RB), the offense should run 63% of the time and pass 37%. As long as the offense optimizes, the blitz (orange) strategy is not part of the defense's optimum strategy mix. We can ignore it for now.

Graphically, when an offense's run performance increases, the payoffs on the left side of the chart move upward. The payoffs on the right side of the chart (the passing payoffs) stay fixed. In effect, we get a hinged upward swing of all the payoff functions. As run performance increases, notice what happens to the optimum play mix (the intersections).

The optimum ratio of running and passing remains constant. Running yields bigger payoffs, but defenses would have to adjust, allowing the passing game to face run defenses more often. In the example where we added 1 unit of payoff to every run play, the intersection is where y_{RUN D}= y_{PASS D}. Therefore:

-2 + 11x = 5 -8x
19x = 7
x = 7/19 = 0.37

The optimum balance remains 37% pass and 63% run. Theoretically, no matter how much the running game improves (or worsens), and no matter what the original payoffs were, the optimum balance should remain the same.

Did the Vikings run too often? Who knows. In theory, all things being equal, the addition of a superstar RB shouldn't alter an offense's play selection. The passing game should open up, and the offense will improve overall. In fact, that's what happened to the Vikings between '06 and '07. With a very shaky Tavaris Jackson under center, they still improved from 5.4 net yards per pass attempt to 5.8 after adding Peterson.

So if the relative strengths of the offense doesn't affect the theoretical optimum run/pass balance, what does? Increasing pass performance would just lead to the same balance. The graph would be hinged on the left instead of the right, but the optimum proportion would remain constant.

The result is the same when the defense improves or worsens. Oddly, the only way to achieve a shift of the optimum run/pass balance is for either or both offensive strategies to improve against one of the opponent strategies but not the other. For example, the optimum balance would shift toward the pass if an offense improved both their run and pass performance against a run defense without improving against the pass defense. Confused? Perhaps you should be--such a scenario would be hard to imagine. How would a team's running or passing improve against a run defense but not a pass defense?

Admittedly, this is a highly theoretical analysis. In reality play selection has much more to do with game situations than anything else. But the analysis does explain why teams with apparently vastly different passing and running abilities all tend to share similar run/pass proportions. With rare exceptions, teams stick very closely to the NFL average 45% run proportion regardless of how their running game compares to their passing game.