re: 4th and 2 on the their own 30 - Discuss it here (Merged 9X)
I haven't had time to look at the statistics but what if the ball was on the 9yrd line, with 2yards for the 1st down.. was probability for a completion still on our side?
We SHOULD HAVE PUNTED!!!
Here's a post by our own Kasmir (written after the Atlanta game) which illustrates the intellectual problem many of you are having quite succintly. The math he applies is not quite correct, but the basic idea is there and you can read Romer's article for the actual equations. I encourage you to read this and Romer if you have time:
Good old Bayesian "paradox", i.e. fans are understandably applying Bayesian logic, which in this case happens to strongly disagree with classical probability analysis
Aside from end of game and end of half effects, field positions have consistent point value, taking *everything* into account. Romer purports to have created such a function, from data compiled from 1998-2000 play analysis. At its simplest: from his tables, 1st and 10 on your own 25 yard line is worth about half a point. Your opponent having 1st and 10 on your 25 yard line is worth about -3 points. So the break even on the decision to go for it on 4th and 1 is about 86%. That is, if Belichick thought the probability of getting a 1st down was greater than 86%, it was a mathematically justifiable decision according to Romer.
You can reasonably question the particular tables Romer used, and question whatever offsets Belichick may have or should have applied. But you can't reasonably question the tradeoff itself. It apparently makes sense makes sense to go for it on 4th and 1 in from your own 25 surprisingly more than you would think based on what coaches actually do.
That's where the Bayesian part comes in. We all *know* that NFL coaches *don't* go for it in those circumstances, so our Bayesian filter violently disagrees with Romer's recommendation. How could so many coaches be so wrong for so long? Bayesian logic suggests Romer's analysis must be flawed.
But we're left with uncomfortable counter datum that Bill Belichick appears to agree with Romer, so maybe Romer is onto something, lest we conclude Belichick is a fool sometimes.
So how *could* so many coaches possibly be so wrong? Romer considers the obvious, that they're risk averse, i.e that they'd rather punt the ball and slightly increase their probability of losing the game according to decision theory rather than to go for it and slightly increase the probability that they'll lose their job to the Bayesian mob. Romer dismisses this possibility because he believes losing games it what actually loses jobs for coaches.
But I think he underestimates the magnitude of the Bayesian driven risk to coaches. Just in reading this thread, you can see how powerfully the disbelief is in the Romer recommendation amongst clearly intelligent fans assessing the decision of a coach whom they believe to be one of the greatest of all time. Even *Belichick* is pilloried by his most fervent fans *after he won*; how then can a lesser coach, perpetually worried about job security, possibly be expected to put his reputation at risk for a small and controversial increased expectation of winning? The answer is clear: coaches fear ridicule more than they fear losing. The fear perpetuates a questionable practice, which reinforces the Bayesian disbelief, which perpetuates the fear. This status quo is quite resistant to change.
