4th and 2 Decision: Maximizing likelihood of victory?

Last night there was a very interesting and controversial coaching decision in the Patriots v. Colts game. Here is the setup: there are 2 minutes left in the game, the Patriots have a 4th and 2 on their own 28 yard line, and they are winning 34-28. Conventional football wisdom says to punt it. And by conventional wisdom I mean that's what coaches have always done in the past. But that does not mean it is right. Belichick ended up going for it on 4th down, not getting it, and the Colts proceeded to get a touchdown and win the game.

Making the decision is actually really easy after you come up with some subjective probabilities. Basically if you estimate the probability of converting the 4th down, the probability of the Colts scoring a touchdown if they don't make the 4th down, and the probability of the Colts scoring a touchdown if the Patriots punted, then you can say what the best decision is to maximize the Patriots chance of winning. A calculator was created to make this analysis http://belichick-decision.heroku.com/. (The check box is by far my favorite part of this calculator) By my estimations (60%, 70%, 33%) going for it gives you a 72% chance of winning and punting gives a 67% chance of winning. So going for it maximizes the Patriots chance of winning. Any reasonable percentages make the call a close one, although it seems to me going for it is probably better.

Since the decision to go for it on fourth down backfired, as would be expected, the high-paid commentators on ESPN, etc. are almost unanimously saying this was a horrible coaching decision without any cogent analysis. I spent much of this afternoon being frustrated by the commentary on ESPN on what seemed to me to be a pretty simple decision under uncertainty problem. This episode did allow me to discover http://www.footballoutsiders.com/, a nice website which had spot on analysis of the Belichick decision.

Of course all of the above analysis assumes the coach is trying to maximize the likelihood of winning. Given the agency problem and the nature of the NFL, this is probably not true. Fact is, Belichick is probably the only coach in the league that would have gone for it on 4th and 2. He also happens to be the coach in the NFL with the most secure job. Other coaches probably would have been trying to minimize regret/ex post criticism. If the Patriots had punted and lost, no one would have blamed the loss on that decision. As we have seen, losing after going for it, an unconventional move, is blamed on the decision. A coach worried about losing his job would be most interested in making decisions that would not be subject to ex post criticism and hence would have punted even if it did not maximize the likelihood of winning.