Noise about Noise: The Good Coach/Bad Coach Fallacy

It is 4th and inches from the 50 yard line. The defense lines up with nine in the box, with a cornerback covering the loan wide receiver and the safety playing a bit closer than usual. The quarterback snaps the ball. The safety breaks in to blitz. The running back executes a play fake. The quarterback bombs it to his wide receiver, who has the safety beat. Touchdown.

“What a great call!” exclaims the color commentator.

Your first reaction might be to agree. After all, the play worked. The safety blitzed, leaving the wide receiver with one-on-one coverage. The quarterback came through, delivering a well-placed ball for a quick score. Credit the offensive coach for the play, and discredit the opposing coach for choosing to blitz.

Well, maybe not.

Let’s investigate how perfect coaches would play this situation. To simplify the situation greatly, suppose the offense can choose whether to call a run or a pass. The defense can choose whether to defend the run or the pass. The defense wants to match, while the offense wants to mismatch. To further simplify things, suppose the defensive benefits for matching are the same whether it is pass/pass or run/run. Likewise, the offensive advantages for mis-matching are the same whether it is pass/run or run/pass.

(These are strong assumptions, but the claims I will make hold for environments with richer play calling and differing benefits for guessing correctly/incorrectly.)

If all that holds, then the game is identical to matching pennies:

In equilibrium, both players flip their coins. Note that as long as the opponent is flipping his coin, the other player earns a fixed amount (zero in this case) regardless of which strategy he selects.

This is a necessary condition to reach equilibrium. If one strategy was even slightly better in expectation given the opposing strategy, then the player would always want to play the superior strategy. For example, if running was even slightly better than passing given the offensive’s expectations about the defense, then the offense must choose to run. But then the defensive coach’s strategy is exploitable. He could switch to defending the run and expect to do better. But the defensive coach is supposed to be superhuman, so he would never do something so foolish.

As it turns out, the only strategies that don’t leave open the possibility of exploitation are the equilibrium strategies. Thus, the superhuman coaches should play according to equilibrium expectations.

Now consider how this situation looks to the observer. We only see outcome of one play. But note that all outcomes occur with positive probability in equilibrium! Sometimes the offense does well. Sometimes the defense does well. But any given outcome is essentially chosen at random.

This makes it impossible to pass judgment in favor or against any coach. Certainly all real world coaches are not perfect. But on any given play, one superhuman coach looks foolish while the other superhuman coach looks great. Consequently, on any given real world play, we cannot tell whether the result was a consequence of terrific coaching on one side (and bad coaching on the other) or just pure randomness.

Thus, we have the good coach/bad coach fallacy. Commentators are quick to praise the genius of the fortunate and lambast the idiocy of the unfortunate, but there simply is no way of knowing what is truly gone on given the information. On-air silence might be awkward, but it beats noise about…noise.

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