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03/18/2010 05:12 am ET Updated May 25, 2011

# Belichick's Decision: Flip a Coin, or Trust the Team?

If you write, talk, or scream about football for a living, you probably had something to say about Bill Belichick's decision on Sunday night. In short, the Patriots had the ball with just over 2 minutes to go on their own 28. It was 4th and 2, and instead of electing to punt, Belichick decided to go for it. The pass play came up just short, and the Colts gained possession and went on to score a touchdown for a 35-34 victory.

This post is not really about the correctness of that call, though. As I see it, it's been handled quite convincingly. Bill Barnwell shows here (as well as Brian Burke here, and Wayne Winston here) that Belichick likely gave his team a better chance to win the game by going for it on 4th and 2. Instead, it's about the consequences of making a choice that is mathematically correct when it's going to make everyone upset.

It's hard to say precisely how much better the Patriots were off going for it. But let's say the win probability difference is not greater than 10%. Is that 10% worth the inevitable firestorm that would result if he gambled and lost? This might not be true for Belichick, but almost any other coach would have to think about the effect it would have on his job security and on his relationship with the players.

Let's say you're an Olympian. You're about to enter a final race, one that you have a 49% chance of winning. A corrupt French judge offers to flip a coin to determine the outcome of the race. Heads, you win, tails, you don't. After making it that far, would you really take the offer? I have a feeling that very few would. Even fewer would like it if their coach took the offer for them.

When a coach makes a decision like Belichick's, he's doing something very close to that. If the Patriots had converted on 4th down, they would have had roughly a 100% chance of winning the game. If they didn't, they probably had somewhere between a 15-50% chance of winning the game (depending on how the Colts offense and Pats defense are evaluated). That means that Belichick singlehandedly based 50-85% of the outcome of the game on just one play. Granted, a play is not a coin flip -- it is based on the players' performance. But the outcome of one play is a lot more random than the outcome of one drive.

My guess is that the average Patriot doesn't understand the win probabilities behind the decision evaluation (nor does he have to). But I do bet that he has a surprisingly keen understanding of which decision is more random. I bet the defense got the following message from that decision: "I'd rather base the outcome of the game on one play run by the offense than on you guys making a stop over the course of an entire series." Is that an unfair characterization of the complex decision that Belichick had to make? Certainly. But it's what inevitably results, especially when you have Rodney Harrison on your team.

This sort of decision happens all the time, in all sports -- do you shoot a 2 at the buzzer to send a game into overtime or a 3 to try and win it in regulation? Do you play small ball to get 1 tying run in the bottom of the 9th, or swing away to get 2 or more to win? In both of these cases, the mathematically correct decision is probably to go for the win in the moment. But the reason for that involves the team's relatively lower chances of winning the game itself through more skill.

So what does all this mean? I'm not necessarily saying that any of this is as important as an increase in win probability. Maybe you think that NFL players ought to realize that it's worth sacrificing an opportunity to cause a win for any greater chance to win. They're professionals who don't have to be babied when they don't get the ball. If you believe that all this crap about psychology and confidence is overblown (and I think we can agree that at least some part of it is), then the above means absolutely nothing.

But it's a lot murkier if you think that these aspects of sports matter a lot. I'd love to say that this stuff doesn't matter, but I think it does. My feeling is that as much as professional athletes should let their coach do the right thing, a lot of them won't like it to the point where it affects their play.

So Belichick not only has to crunch the numbers, but he also has to ask himself, is that extra 10% in win probability worth the potential problems in the locker room afterward? Does it cost the Pats more than 10% in win probability over the course of the regular season? And what about the problems that would result from a loss that was the defense's fault?

I have no clue. But a good coach should.

This post originally appeared on the Harvard Sports Analysis Collective's blog. Check it out!