[AaronBrown]

I agree with your analysis, but there are even more obvious cases of this in football. It's one of the reasons I find it hard to take the game seriously. It's clear to any numerate person that the people are not trying to maximize chance of winning, which means (a) it isn't a sport, it's entertainment, and (b) all the violent nonsense that ruins people's lives isn't even justified by a serious attempt to win (not that winning a sporting event would be a good reason for violence, but at least it would be a reason).

In this situation, the coach expects to lose. If he goes for it on 4th and loses, he will be criticized as if that decision lost the game. If he kicks the field goal, the blame will be more generalized.

My particular favorite situation is a team down by 14, until it scores a touchdown with a minute to go. The only realistic chance of winning is to score a second touchdown, and hold the other team scoreless. So assume this happens, if it doesn't, nothing you do matters.

If you go for one point conversions both times, you have 0.95 x 0.95 x 0.5 = 0.45 chance of winning (assuming you get the second touchdown and hold the other team scoreless). If you go for a two point the first time, then a one point the second time if you make the two, and a two point the second time if you miss the two, you have 0.45 x 0.95 + 0.55 x 0.45 x 0.5 + 0.45 x 0.05 x 0.5 = 0.56 chance of winning. You can play with the numbers all you want, within reason it always comes out better to go for two. It's even more true in lower levels of football where the chance of the one-point is less than 0.95 and the chance of a two point is more than 0.45.

It's true this difference is less than your example, but it's a far more common situation, with a greater chance of winning (so a greater amount thrown away by the mistake).

There is quite a bit of literature showing teams punt too often, go for too many field goals and too few two-point conversions. They also manage the clock badly, if you're down a couple of touchdowns in the fourth quarter, saving a few seconds on the clock really matters (and if you're ahead, wasting a few seconds matters as well). But teams play at normal speed until much too late in the game.

NBA coaches tried far too few three-pointers when the rule was introduced, it took many years to get close to the right number (and they still don't take enough).

Only baseball was pretty close to proper management. People talk a lot of nonsense, but didn't do as much nonsense as other sports.