Football And Game Theory

One more post on football for now. In the previous post, I noted that while you could gather statistics on whether blitzing was a more effective form of defense than not blitzing, such statistics were not worth much for practical application. I want to take a moment now to show a more valid way of analyzing the game.

The crux of the matter, as I noted before, is that play-calling is a guessing game between the offense and the defense. If the offense knows whether the defense will blitz or not, they can call an appropriate play to take advantage of that. Let’s set up an example with real numbers. We’ll call the case where the defense plans for a blitz ‘DB’, and the defense plans for a conventional defense ‘DN’. Likewise, ‘OB’ and ‘ON’ for whether the offense calls a play designed to handle a blitz or not. I’ll just make up some numbers for how the outcomes play out, but mindful of TMQ’s statistics, I’ll make blitzing versus a prepared offense a costly mistake:
· ON vs DN (conventional play) – 5 yard gain average.
· ON vs DB (defense blitzes against unprepared offense) – 4 yard average gain.
· OB vs DN (offense holds blockers back in anticiption of blitz that never comes) – 4 yard average gain.
· OB vs DB (offense is expecting the blitz) – 10 yard average gain.

This can be represented as a matrix:

	ON	OB
DN	5	4
DB	4	10

There is a whole branch of mathematics called game theory devoted to understanding scenarios like this. This example in fact can be solved easily. Before I state what the solution is, you can make a reasonable prediction just by looking at the numbers. Basically, the defense doesn’t want to blitz very much, because if they do and the offense catches them at it, the defense loses big. But at the same time, it won’t be optimal for the defense to never blitz, because then the offense will always choose ON and pick up the five yards each time. So a good defense will blitz some, but not most of the time. Likewise, the offense will know that the defense will blitz infrequently, and so will plan accordingly, calling ON most of the time, but OB occasionally.

Now for the solution. Assuming perfectly rational opponents who can’t be predicted, the optimal strategy for the defense in this example is to blitz one seventh of the time, chosen randomly, and run a conventional defense the other six-sevenths. The optimal strategy for the offense is also to run a play for the blitz one seventh of the time, and run a normal play the other six-sevenths. This works out to an average of 4.85 yards per play. (All that’s assuming I did the math correctly.)

One interesting property of this result is that in this optimal case, the average yards per play is the same whether the defense blitzes or not. In fact, one way to look at solving the problem is to find that crossover point. If you gave up more yards blitzing, then that should tell you that your strategy is not optimal and you should blitz less. Likewise if you gave up less yards on your blitzes, you should blitz more. Coming full circle, if Easterbrook’s statistics (i.e. that blitzes perform statistically worse than non-blitzes) are valid and can be applied to real football, they do show that teams should blitz somewhat less than they do.

So TMQ’s methodology could be given firmer mathematical grounding. It would require a lot of data – for best results you would have to break things out per team, and find out how each defense and offense with and without the blitz. You would want to use probability distributions instead of simple averages, as above. Once you have the data, you could apply game-theoretic techniques to figure an optimal strategy for a pair of teams. Then, and only then, can you criticize a team for blitzing too much (or not enough), because then you have a take on what would have been better.

I should mention that I take a dim view of the chance of success of such a programme. I doubt you can really break out the statistics like that, or that they would be that reliable, and I also doubt you could factor in all the non-numeric issues, such as beating up on the quarterback. But such an effort would be more grounded and noble than the cheap shots over at TMQ.

My key point here is that once you buy into this game-theoretic approach, the whole TMQ idea of pointing out individual cases when blitzes fail, and mocking the teams involved, becomes quite silly. Of course calling a blitz is bad when the offense is ready for it. But you still need to do it; getting burned at times is the price you pay for applying an optimal strategy.