I've written about this before and I'm sure I am not the only one who has thought of it. But I felt it was important to bring it up with this group because so many people have a hard time fully grasping that serious decision making errors are made by mathematical illiterate "experts" regardless of the amount of experience they have. This most often occurs when the situation doesn't arise often enough for mere observation to be a reliable guide. Sometimes you just have to do the math. (I can hear some of you saying that if the situation rarely comes up, who cares. But what about if many different types of situations requiring math come up?)

The particular example I use here occurs when your football team scores a touchdown in the final minutes of the game when down by 14 points. Aside from extreme circumstances they should now go for two. (And then two again if they miss and score later.)

The math that proves this is amazingly easy and I will get back to that in a minute. But first I want to discuss the often stated objection that a coach might use. Namely "you haven't taken into account the psychological deflation a failed attempt will cause". My reply is twofold. First I do not for a minute believe this to be true. Players know the overall strategy which includes a miss now and a success later. To think that a miss psychologically deflates players to the point where it makes up for a fairly significant increase in the theoretical chances of winning doesn't seem right. I can't prove that though.

But the bigger problem with the coaches statement is that it was a RATIONALIZATION after the fact. If it was said by someone who understood the math and legitimately felt it should be rejected, fine. But this kind of thing is usually said by coaches AND ALL SORTS OF SUPPOSED EXEPERTS IN MYRIAD FIELDS to defend their stupidity. They didn't have any idea that theoretically the two point conversion or any of the myriad of mathematically based conclusions experts in other fields are unaware of, is the right decision. So they scramble to justify their ignorance. (Another weak excuse is that "my owner would fire me because he doesn't know the math". So spend three minutes explaining it to him.)

I changed my mind about doing the calculation. Anybody on this forum who can't do it should be ashamed of themselves.
Assume a two point attempt is 42%, a one point attempt is 98%, overtime is a 50-50 proposition, and there is time for one more score at best.

David,

"I changed my mind about doing the calculation. Anybody on this forum who can't do it should be ashamed of themselves.
Assume a two point attempt is 42%, a one point attempt is 98%, overtime is a 50-50 proposition, and there is time for one more score at best."

One point way, you win 48% of the time.

Two point way, you outright win 18% of the time pre-OT.
you go to OT 48% and win 24% of the time
you lose outright in reg 34%

Which says go for one point.

OK, it's late, so for your amusement I'll post this here and come back in the morning to laugh with you all at whatever trivial math error I made and be ashamed as you've instructed.

David,

I don't think this applies for every team. What if they've been dominated up front all day and aren't controlling the line of scrimmage as they should be? If they can move the ball down the field and score at the 2:00 mark, they ought to just go ahead and take the FG - it's still a 1 possession game. Would your opinion change if the team had no timeouts and would be forced to attempt an onside kick? It would sure suck to drive the ball down the field again only to fail to complete another two pt conversion.

Even if the math justifies these slightly unorthodox playcallings, there is a danger in slipping too far into the "Mike Martz" syndrom where instead of pounding the ball up the middle on 4th & short you try some overly sophisticated double reverse and it ends up backfiring. There is more leeway for trickier play in defensive schemes imo, when the game is at a critical junction even if you have some minute mathematical edge you're better off not risking it and executing the simple & proven play.

Assuming you will make it into the endzone twice, going for it sends the wrong message to your team whom you are supposed to have faith in to either drive the ball in OT or make a big defensive stop, not to mention the heat that will undoubtedly be put on you from the GM, fans, and likely your players during the week should your brilliant idea spit in your face.

Btw on the subject of NFL, did you miss my question in the other ask me forum?

David,

Oh, nm, haha.

If they score 8 the first time, they don't go for 8 again, they go for 7.

I feel stupid for not having noticed this myself.

Are the numbers you gave somewhat accurate? If they are, I'm surprised that the differences in expectation of winning are that significant.

You could also argue that the fact that you're down 14 at this point is evidence that your chances of winning in overtime is more likely to be less than 50%.

I think what you are trying to teach here could be accomplished more effectively by showing those who find this difficult the methods you can use to simplify problems with multiple parts. Even if they figure this out using one of those methods they won't draw those connections. People aren't going to magically become smarter, but maybe they can be less stupid.

My impression is that nearly every coach would go by a card that factors in losing on the road 60% in the case where they truly believed they had exactly one more posession possible via on-side kick-- so your question is a few years old.

Huh?

You might want to choose an example where the result doesn't depend so closely on your assumed percentages.

To simplify the calculation, assume the extra point conversion is made 100% of the time. Let x be the fraction of the time that a two point conversion is made. If another touchdown is not scored the question is moot. So assuming another touchdown, if the two point conversion is tried on the first (and second if you miss) you win if you convert the first or convert the second and then win in overtime. ie.

x + (x)(1-x)(.5)

If you kick two conversions you win in overtime .5. So the question is, for what values of x is

x + x(1-x)(.5) > .5 ?

x + .5x - .5x^2 > .5

3x - x^2 > 1

x^2 -3x + 1 < 0

Solving for the zeros of the quadratic,

x = [3 (+or-) Sqrt(9-4)] /2

x = .38 or 2.6

Which means that "going for two" on the first try improves your 50% chance of winning in overtime - by kicking each time - when the chance of making the two point conversion is greater than 38%. The 42% you assumed satisfies this. But if the coach thinks the chance is less than 38% he should kick-it.

PairTheBoard

(Edited to fix the first expression)

Greensboro 15, Guilford 14
http://www.d3football.com/story.php?story=5351

"Down 14-6, Pride coach Marion Kirby surprised many in the crowd of 2,211 when he decided to forgo the traditional extra point and go for the two-point conversion.

"We went for two so we had a chance to win [if we scored another touchdown]," said Kirby. "We wanted to kick to win at 15-14. That's nothing new that I've done. We go for two then, if we make it, we kick to win it. If we not, we are going to go for two the next time to try to get it into overtime. It's just as simple as that.""

His explanation should sound fine to a non-mathematical fan -- I can just imagine hearing Al Michaels explaining it.

Great. Now let's start talking about the larger point.

The frequency that NFL coaches make awful decisions is embarrassing. They kick field goals when they should go for a TD/ first down, they punt when they should go for it, they kick the extra point when they should go for two... and they do it every week.

I had never noticed the situation you just pointed out, but yes, it's very clear. Going for two will win the game 5.32% more often (of the times when you score twice), given your numbers - and I'm almost certain the 2pt conversion rate is higher than that the past two years.

If you have a rational explanation for why every coach makes these errors, I'd like to know. Is it possible that nobody with access to NFL coaches/owners has ever done these calcs? Do you think coaches and owners aren't smart enough to understand it? I've heard the "coaches are afraid to make these unorthodox decisions and have them backfire, because they want to keep their job" argument, and I'm not convinced.

expected value for each instance (assuming all stats given are correct) [also assuming team only has time to score twice]
Two point conversion
.42*2+.58*0=.84
One point conversion
.98*1+.02*0=.98
By kicking the extra point you gain .14 points.
However, that's not the situation we're in.

down 14 go for 2
win .42 of time then .98 kick xp for win= .42*.98=.4116
lose .58 of time then win .42 to tie=.58*.42=.2436
(everything else is a loss)
exp value for that team then to win
.2436*.5=.1218
so win percentage is .1218+.4116=.5334

kick both to tie .98*.98=.9604
expected win
.9604*.5=.4802

Thus, it appears that by going for 2 you gain about 5% in win equity.

Even that being done; I believe there is a psychological value lost when a team is down 8 instead of 7 since they're now thinking "now we've got to do two things instead of one to tie the game". Also, I believe that (wish I had stats to either confirm or discredit this) the team blowing that lead is more likely to lose the game than the team with the momentum. If that combined is just a 6% net swing then obviously kicking is correct.

They always choose the "passive" play (e.g. not going for the two points). This type of mistake is hardly noticed (punting doesn't make a good highlight). "Going for it", on the other hand, is exciting and makes it to the highlight real -- so if it backfires it gets a lot of attention.

So coaches make the mistakes that don't get attention, and then can blame the game on whatever didn't go great, e.g. turnovers. In a way, they may be maximizing their own $EV. A highlight-real backfire loses them a lot more (in terms of long term compensation) than a mistake nobody notices.

Often this is also the other way around. Some coaches know but realize the heat they would stand if it didn't work. Most just aren't smart enough to know any better. It's that way in every major sport. It's the classic battle of the traditional view vs the educated view of how things should go. The analysts with the most TV air time are the previous version so that's why that's still currently prevailing despite how many times it's been proven that they are incorrect.

Id like to see a stat about how many times a team converts a two point conversion on their second attempt in the same game.

Most teams only have a few plays that they plan on running in a 2pt situation, though i guess if they were always following this strategy they would practice it more often, but im guessing that the more 2pt plays u run in a game the lower the success rate becomes.

I don't buy that every coach in the league actively makes decisions that he knows gives his team a smaller likelihood of winning, simply because he is afraid of a highlight reel. Some coaches might, but several NFL coaches strike me as the type who aren't afraid of taking risks and failing.

In this particular case, the problem for the image concious coach is that he will look bad a large part of the time. More than half the time they will miss the two pointer. And then they will usually not score again. Instead of a simple loss caused by not scoring again after a normal 1 point conversion, people will talk about the unconventional decision to go for the missed Two. Going for Two only makes the coach look good when they make it, 42% of the time, and then score again which is unlikely. It makes him look bad at least 58% of the time when they likely don't score again.

PairTheBoard

I agree they wouldn't actively decide something like this. Though I do think the idea has some effect on them, maybe subconsciously.

Speaking of coaches in general, I think this is probably true. However, that doesn't explain situations like the one David posted about, where it is very clear what the correct move is, and yet every singe coach gets it wrong every single time. Surely somebody has explained this to one coach or owner, right?!?!?!

You do realize that going for 2 will be criticized on TV every day (less so if it works) heavily by every TV "expert" right? Then the fan base gets angered and wants a new coach. Then coach gets fired. Coaches aren't going to do anything that might get them fired. (outside of the crappy on field coaching) It's correct mathematically will not work as an explanation to 90% of the population.

I was thinking that too. The basic math involved has been around way longer than the game. I do not believe that every owner and coach choose to ignore it or are too stupid to understand it, especially when there is so much to gain by winning more often. Someone would try it out eventually, it would work, and the practice would spread.

This leads me to think that maybe there is, in fact, more to the game situation than it appears, or that coaches do avoid highly visible and risky plays for personal reasons.

This makes me think of Mutual Fund portfolio managers who are not shy about padding their image at the expense of EV when they add hot stocks to the Fund's portfolio just before reporting time just to make it look better, regardless of whether they think the stocks will continue to outperform.

PairTheBoard

Whatever. "I have confidence in my team," will. "We had momentum going our way, and I wanted to give our boys a chance to win it in regulation. And, of course, if we missed, we'd get another shot."