Why don't coaches understand fundamental math?

[aggie]

Situation: it's 4th and goal from the 4 yard line with a little over 7 minutes on the clock and we're down by 19.

Our team kicks the field goal. The argument of course is you take the sure points and it puts you within 2 scores of tieing the game. If you go for it and miss, game over.

This is all true but it misses the entire point which is to MAXIMIZE OUR OVERALL CHANCES OF WINNING THE GAME. How could a high level coach who is paid millions of dollars get this wrong?

Scenario 1, Kick the field goal:
1. Successfully kick field goal
2. Score touchdown
3. 2 point conversion
4. Another touchdown
5. Another 2 point conversion
6. win in overtime

Scenario 2, go for it!
1. TD on 4th and 4
2. Score a touchdown
3. xtra point
4. Another TD
5. Another Xtra pnt.
6. We Win!

Points 2&4 cancel eachother out in both scenarios so let's look at the approximate probablilities of points 1,3,5,(6) in each scenario and figure our odds of success:

Scenario 1:

Kick field goal (~95%) * 2point conversion (~45%) * 2nd 2 point conversion (~45%) * win in OT (~50%) = .096

Scenario 2:

Go fo it (~35%) * xtra point (~95%) * xtra point (~95%) = .31

So basic math proves going for the field goal decreases our chances of winning by a factor of more than 3. It's not even close. (please let me know if i'm missing something here)

It consistently amazes me how often coaches in all sports ignore math and instead use prevailing wisdom/tradition in making game changing decisions.

Please discuss...

[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.

[aggie]

I guess baseball is getting closer but it's not there. Here is a facinating read on the subject:

Moneyball: the art of winning an unfair game

I agree with everything else you said....It's simply amazing with the amounts of money involved in professional/collegiate sports taht sporting franchises don't seem to take this stuff seriously

[DrVanNostrin]

[ QUOTE ]
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.

[/ QUOTE ]
Good post. David Sklansky started a thread on this exact subject sometime last year.

[Lottery Larry]

[ QUOTE ]
Scenario 2:

Go fo it (~35%) * xtra point (~95%) * xtra point (~95%) = .31

[/ QUOTE ]

How do we factor in the chance of being able to kick the ending field goal for the tie, rather than having to score a TD, if things don't work out?

If you assume 3 scores are needed to tie or win, won't you go for 2 on the first touchdown? if you miss it, then 2 more TDs + XP gives you the win.

If you make it, you can go for 2 on the second TD as well. If you make it, then you DON'T have to score a TD to tie/win. If you don't make it, you score a TD + 1 to tie.

Should the calculations reflect this? Since you're not including the % difficulty of scoring 2 TD against a preventative defense, maybe that's not part of the analysis.

Below, I'm just trying to calculate the chance of going for 1 and leaving a chance for a game-typing FG.
This may not be the correct calculation, since I can't see it being only a 0.6% chance of tying the game and then winning in OT (?)-

<font color="blue">Scenario 1:

Kick field goal (~95%) * 25% score TD * 2point conversion (~45%) * 25% score TD * 2nd 2 point conversion (~45%) * win in OT (~50%) = 0.006372421875 </font>


<font color="green">Scenario 2:

Go fo it (~35%) * xtra point (~95%)* 25% score TD * 95% XP * {0.30 *[25% score TD * xtra point (~95%)] +0.70*[70% long FG * 50% win in OT]} = 0.0225328125</font>

Wow- that would mean you're 4x likely to win going for it now rather than kicking the FG now.

[rufus]

Contrasting this, in 2001 the Pats used an intentional safety, that may have helped them win a game. It's especially noteworthy that that's a trade of points for field position, which is much more difficulty to evaluate and more questionable.

Considering that more than 14 points down, and less than 5 minutes to go also translates to needing the defense to basically pull a 3 and out anyway, the down team should be willing to play a whole lot more 4th downs than they typically do.

[pzhon]

See http://www.pigskinrevolution.com for a tool that ought to be used, but is not. Any team that starts to use that tool will gain a significant advantage, perhaps enough to win an extra game per year. Any rational assessment would be that an extra game is worth millions of dollars to a professional team. They could choose to win more, or they could save millions of dollars in player salaries and win the same amount.

Football teams don't just blow the decisions in the endgame. A common mistake is punting on 4th down. "Yet on the 1,100 fourth downs where Romer found it would be best to go for it, teams kicked 992 times." It's very commonly correct to go for it when it is 4th and 3, or even 4th and 4, and it can be right when it is 4th and 8 even with plenty of time left. No team goes for it on 4th down when it is marginally correct.

Decisions in other sports are not nearly as bad as the ones in football, although there are many individual decisions which are pretty bad.

[rufus]

There are limitations on the tech allowed on the sidelines, so you'd have to have someone trained to make those calls. (Not that the NFL really has any shortage of budget there.) Even so, it would probably be a good idea to try to condense things into rules of thumb of some kind.

[pzhon]

In live competitive backgammon (from which I know Frank Frigo and Chuck Bower, who are behind PigSkinRevolution.com), you aren't even allowed to use pencil and paper. Nevertheless, all top players, and most serious ones, train with computer programs. It's simply a huge mistake for the football teams not to use the available tools.

Of course, the first coach who starts to make unusual decisions will not necessarily be rewarded for it. However, a mediocre coach could become a great one if he did.

[rufus]

[ QUOTE ]
Of course, the first coach who starts to make unusual decisions will not necessarily be rewarded for it. However, a mediocre coach could become a great one if he did.

[/ QUOTE ]

Honestly, it seems like coaches tend to be over-conservative, so more aggressive play calling should be popular with the crowds, and maybe with Vegas as well.

I'd be curious to see what the average log of the effect of coaching decisions on the result is, rather than just the flat averages. It tilts the tables a bit, but I think it would draw a bit more attention if the coach rankings aligned better with the league standings.