Why don\'t coaches understand fundamental math?
 

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...

Re: Why don\'t coaches understand fundamental math?
 

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.

Re: Why don\'t coaches understand fundamental math?
 

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

Re: Why don\'t coaches understand fundamental math?
 

[ 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.

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Good post. David Sklansky started a thread on this exact subject sometime last year.

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
Scenario 2:

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

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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.

Re: Why don\'t coaches understand fundamental math?
 

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.

Re: Why don\'t coaches understand fundamental math?
 

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.

Re: Why don\'t coaches understand fundamental math?
 

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.

Re: Why don\'t coaches understand fundamental math?
 

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.

Re: Why don\'t coaches understand fundamental math?
 

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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.

Re: Why don\'t coaches understand fundamental math?
 

Not only are the teams losing out, so are the announcers. I think the fans would value comments like, "They should go for the touchdown. Going for it won't win more points here, but failure means turning over the ball on the goal line. That is worth 1 point more than kicking the ball off after a successful field goal," or, "They need to realize they are in a desperate situation since losing by 3 points is the same as losing by 30. They need to play for there to be 3 or 4 more touchdowns in this game, even if that means they get those touchdowns only 1 time in 3. They have to go for it here."

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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.

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What do you mean? If a mistake drops a team's winning chances from 10% to 5%, that is both cutting their chances in half, and increasing their opponent's chances by less than 6%.

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
[ QUOTE ]

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.

[/ QUOTE ]
What do you mean? If a mistake drops a team's winning chances from 10% to 5%, that is both cutting their chances in half, and increasing their opponent's chances by less than 6%.

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Right, but there's only one side making each decision. The issue is that if you're looking at probability to win then the log of the decision makes more sense -- a coach on a strong team is usually going to be doing less damage with poor decisions than a weak coach will. Of course, a strong team is less likely to have tough decisions anyway.

Re: Why don\'t coaches understand fundamental math?
 

Ok, I still don't think the logarithmic measure is good at all. It would make sense if being undefeated for the season were all that mattered. Do you really want to say dropping from a 0.01% chance to win to a 0.001% chance to win (and this type of error may be common) is more serious than dropping from 80% to 20%?

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
Ok, I still don't think the logarithmic measure is good at all. It would make sense if being undefeated for the season were all that mattered. Do you really want to say dropping from a 0.01% chance to win to a 0.001% chance to win (and this type of error may be common) is more serious than dropping from 80% to 20%?

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Yeah. OTOH you must admit that the difference between 70 and 80 percent is less of an issue than the difference between 10 and 0. There's probably some sort of happy medium.

Considering real-time analysis is feasible, it might be interesting to provide the commentators with ZEUS - even if it's just for 4th down, conversion, and free kick situations.

Re: Why don\'t coaches understand fundamental math?
 

aggie i understand and agree, but i will respond "hello risk aversion."

Re: Why don\'t coaches understand fundamental math?
 

If you'd like another example. Consider a basketball game in the bonus with 30 seconds left on the clock. The team with possession of the ball is trailing by 4. Does anyone else on this forum think the best shot to take (given a team of standard skill distribution) is a 3-pointer?

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
aggie i understand and agree, but i will respond "hello risk aversion."

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If the coach goes against CW and is wrong, he risks getting fired.

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
aggie i understand and agree, but i will respond "hello risk aversion."

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I really don't think that's an accurate description. Looking at the analysis there are plenty of examples where there is basically no downside to the better choice because there is very little difference between 1 and 2 or 4 and 5 point leads in the last minutes of the game.

It's intellectual laziness or ignorance.

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
If you'd like another example. Consider a basketball game in the bonus with 30 seconds left on the clock. The team with possession of the ball is trailing by 4. Does anyone else on this forum think the best shot to take (given a team of standard skill distribution) is a 3-pointer?

[/ QUOTE ]

It's too early to risk a 3 point shot with 30 seconds to go . It's much better to take the best shot available during your possession and hope that you can make it within 2 points . Sometimes , the best shot available is a 3 point shot and you should go ahead and take the risk .

Re: Why don\'t coaches understand fundamental math?
 

Those who argue that you should take a 3 tend to flatly look at 3-point percentages and argue that the chances of making them so good that you should do that instead.
They fail to realize that a FORCED 3-pointer has a far lesser chance of success. It's apples and oranges. The guy that makes 40% of his 3-pointers all season long did so because he took them when the shot was open.

If you set things up in the situation with 30 seconds left to "get the ball to Miller because he's 40% to make this 3-pointer" then you are missing the point because he's almost definitely not going to be 40% in that forced situation.


The stats-geeks have been invading baseball and have been changing strategies the past few years.
I believe the same will happen in football with such scoring-decision situations in the years to come.
There are some coaches who are less afraid to take chances and may have some possiblity of actually getting this stuff.
As I've said before in other threads about this...some teams actually have some pretty smart players.
MIT has a football team I believe. So do Harvard and Yale and Princeton and Stanford among other schools. Surely there have to be some math majors on some of these teams who could understand something like this and maybe even one or two of them could become a coach in the future.

As I recall, an assistant coach at James Madison who is in a wheelchair has posted on these forums and is a poker player. I forget his name but he linked to an article about himself that was pretty impressive.
So he understands the general idea of EV and poker concepts, etc and is an assitant coach on a Div 1-AA team.

Getting the picture? Not ALL of these guys are total jock idiots as the perception goes.
And some of them are much bigger risk-takers then you might think. It's not all conservative play-calling either.

If there's an edge to be had and they can actually learn that it is real then they will indeed go for it. It's only a matter of time.

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
If you'd like another example. Consider a basketball game in the bonus with 30 seconds left on the clock. The team with possession of the ball is trailing by 4. Does anyone else on this forum think the best shot to take (given a team of standard skill distribution) is a 3-pointer?

[/ QUOTE ]

edit: pps is points per shot

there are players whose pps is high enough that they should be electing to take the best shot available, rather than shoot a 3.

Re: Why don\'t coaches understand fundamental math?
 

In one day cricket, the team to bat first makes their total in 50 overs, and the other team tries to reach it without losing 10 wickets (one innings each). The team batting second has a clear information advantage in knowing what sort of score they are aiming for (much like having positional advantage in poker).

The first team opens the batting behind a veil of ignorance (of how well they'll do), and thus must take a moderate approach in terms of aggression. Say they get off to a good start by losing only 1 wicket in the first 25 overs, they can then change their strategy and try and post a big total by taking more risks. However, a team batting second has a better chance of making a big total if they know the target from the start.

However, not once have I heard a commentator, captain, or coach speak of this - all that is discussed in considering whether the captain who wins the toss should bat first or second is the condition of the pitch, the weather, and the intimidation factor of a big total! And incidentally, teams will choose to bat first the majority of the time.

...

Similar information advantage applies to the follow-on in test matches (rejecting the follow-on defies logic IMO).

And one last thing that annoys me is the persistent talk of batsmen down the order having higher batting averages because they get a lot of not-outs.

Re: Why don\'t coaches understand fundamental math?
 

What coach made the decision to kick a FG when down by 19 with 7 minutes to go? That's such an unbelievably retarded decision that there is no reason he should employed.

I can understand that coaches have the habit of playing it safe, and frequently err on the side of caution by taking the "sure thing". But when you're down by 19 late in the 4th, you simply cannot do that. Did this actually happen, or is it just a hypothetical?

That coach is trading a ~5% chance to win, for a .000000001% chance to win. It's so stupid that I tried for 3 minutes to think of something to compare it to, but I couldn't think of anything that stupid.

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
What coach made the decision to kick a FG when down by 19 with 7 minutes to go? That's such an unbelievably retarded decision that there is no reason he should employed.

I can understand that coaches have the habit of playing it safe, and frequently err on the side of caution by taking the "sure thing". But when you're down by 19 late in the 4th, you simply cannot do that. Did this actually happen, or is it just a hypothetical?

That coach is trading a ~5% chance to win, for a .000000001% chance to win. It's so stupid that I tried for 3 minutes to think of something to compare it to, but I couldn't think of anything that stupid.

[/ QUOTE ]

FWIW this did happen...It was coach Dennis Francione of Texas A&amp;M and he will almost definitely be fired after the season (this decision will have had nothing to do with it).

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
In one day cricket, the team to bat first makes their total in 50 overs, and the other team tries to reach it without losing 10 wickets (one innings each). The team batting second has a clear information advantage in knowing what sort of score they are aiming for (much like having positional advantage in poker).

The first team opens the batting behind a veil of ignorance (of how well they'll do), and thus must take a moderate approach in terms of aggression. Say they get off to a good start by losing only 1 wicket in the first 25 overs, they can then change their strategy and try and post a big total by taking more risks. However, a team batting second has a better chance of making a big total if they know the target from the start.

However, not once have I heard a commentator, captain, or coach speak of this - all that is discussed in considering whether the captain who wins the toss should bat first or second is the condition of the pitch, the weather, and the intimidation factor of a big total! And incidentally, teams will choose to bat first the majority of the time.

...

Similar information advantage applies to the follow-on in test matches (rejecting the follow-on defies logic IMO).

And one last thing that annoys me is the persistent talk of batsmen down the order having higher batting averages because they get a lot of not-outs.

[/ QUOTE ]

I don't know if I agree with the premise. It was conventional wisdom 20 years ago that batting second was better in 50-over games because the batting team could accurately pace their scoring. It is only recently that there has been a trend towards batting first, with the Sri Lankans of the early 90s, and now the Aussies apparently attempting to exert maximum pressure on their opponents from the start of the match.

It could also just as reasonably be argued that the team bowling second might have an advantage, allowing the fielding captain to schedule his bowling changes with a pretty good idea of the path that the batting side have to take to get to the target.

Either way I'm not aware of any real statistical analysis on the subject.

PS Don't get me started on night watchmen.

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
Similar information advantage applies to the follow-on in test matches (rejecting the follow-on defies logic IMO).


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Not true. Depending on what day you are on, it is usually better to get in and 'blast'. While you are doing this you are also deteriorating the pitch and fatigueing your fielding opponents.

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And one last thing that annoys me is the persistent talk of batsmen down the order having higher batting averages because they get a lot of not-outs.

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I can see your point here. But to take it to the next level one could again argue that fatigue is not such a factor for lower order batsmen especially 8-11 because they will always be at their peak fitness during their innings (as they are out faster or its all over) i.e their averages will never be directly in line relative to the rest of teh team.

JT

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
[ QUOTE ]
Similar information advantage applies to the follow-on in test matches (rejecting the follow-on defies logic IMO).


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Not true. Depending on what day you are on, it is usually better to get in and 'blast'. While you are doing this you are also deteriorating the pitch and fatigueing your fielding opponents.


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You completely disregarded informational advantage (my point). If rain had closed in on the final day of the Tassie test like it was threatening to do, Ponting could have ended up with a draw, which would most likely not have happened if he sent Sri Lanka in again, since we could have made whatever 4th innings total at an appropriate pace. And yes, obviously there are other considerations.

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And one last thing that annoys me is the persistent talk of batsmen down the order having higher batting averages because they get a lot of not-outs.

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I can see your point here. But to take it to the next level one could again argue that fatigue is not such a factor for lower order batsmen especially 8-11 because they will always be at their peak fitness during their innings (as they are out faster or its all over) i.e their averages will never be directly in line relative to the rest of teh team.



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Semantics. In fact, lower order batsmen more often have to bat in a more extreme fashion (aggressively or defensively) which would certainly have an average lowering effect.

Re: Why don\'t coaches understand fundamental math?
 

[ QUOTE ]
I don't know if I agree with the premise. It was conventional wisdom 20 years ago that batting second was better in 50-over games because the batting team could accurately pace their scoring.

[/ QUOTE ]

Interesting. I was not aware of this.

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It is only recently that there has been a trend towards batting first, with the Sri Lankans of the early 90s, and now the Aussies apparently attempting to exert maximum pressure on their opponents from the start of the match.

It could also just as reasonably be argued that the team bowling second might have an advantage, allowing the fielding captain to schedule his bowling changes with a pretty good idea of the path that the batting side have to take to get to the target.


[/ QUOTE ]

Fair point, but I think that the batting team has much more control in this regard.

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PS Don't get me started on night watchmen.

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Lol, I have a feeling I'll agree with you on this one.