The Two Point Conversion When Eight Down

[MyTurn2Raise]

David,
I haven't read the entire thread yet, but want to give my reaction having read the initial post and argument. I'll go back and read later.

First, a 50-50 OT proposition is a huge mistake IMO. Given that one team has scored two touchdowns in a row to tie the game, I feel they are much stronger than 50/50 to win the game.

Second, you cannot view 2-point plays in an independent vacuum. They are a part of a whole game and whole season. Teams develop and practice only so many key plays to get 3 yards when needed throughout the year. By going for 2, teams are showing their tendencies and given away their strategy needlessly IMO. Football is much more complex than you are giving it credit for based on the problem you present in the OP.

[Magic_Man]

[ QUOTE ]
First, a 50-50 OT proposition is a huge mistake IMO. Given that one team has scored two touchdowns in a row to tie the game, I feel they are much stronger than 50/50 to win the game.

[/ QUOTE ]

A lot of people are saying this, so I thought I'd crunch some numbers. At first, I will graciously allow that extra points are made at 100%:

2 points:
Hit, kick = .42 * 1.00 = 0.42 to win
Miss, hit = .58 * .42 * OT = .2436*OT to win

Kick:
kick, kick = 1.0 * 1.0 * OT = OT to win

Thus, for kicking to be better, we need:
OT > .2436*OT + .42
OT > 0.555 = 55.5%


If extra points are only X, we need:
2 points:
Hit, kick = .42 * X = .42*X to win
Hit, miss = .42 * (1-X) * OT = .42*OT - .42*X*OT to win
miss, hit = .58 * .42 * OT = .2436 * OT to win

Kick:
Kick, kick = X*X*OT to win = X^2*OT to win

So, for kicking to be better, we need:

X^2 * OT > .42*X + .42*OT - .42*X*OT + .2436*OT

Trying some values, for kicking to be better:
X = 1.00, OT > 55.5%
X = 0.99, OT > 56.8%
X = 0.98, OT > 58.1%
X = 0.97, OT > 59.5%

I would also like to point out that if a team is 56.8% to win in OT, the chances of making an 2pt go up as well. Letting conversion success be C:

2pt:
hit, kick = C*X = C*X to win
hit, miss = C*(1-X)*OT = C*OT - C*X*OT to win
miss, hit = (1-C)*C*OT = C*OT - C^2*OT to win

kick:
kick, kick = X^2*OT to win

X^2*OT > C*X + 2*C*OT - C*X*OT - C^2*OT
X^2*OT - C*X - 2*C*OT + C*X*OT + C^2*OT = F(C,X,OT) > 0

Taking partial derivatives, the sensitivities to the different parameters are:
dF/dX = 2*X*OT - C + C*OT
dF/dC = -X - 2*OT + X*OT + 2*C*OT
dF/dOT = X^2 - 2C + C*X + C^2

for typical values X = .99, C = .42, OT = .50, the sensitivies are:

X: 0.78
C: -1.075
OT: 0.7323

Thus, if we assume these are fairly close to the real values, the most sensitive parameter is how often we make two point conversions. It is almost 50% more important than how often we win in OT.

~MagicMan

[MyTurn2Raise]

[ QUOTE ]
Trying some values, for kicking to be better:
X = 1.00, OT > 55.5%
X = 0.99, OT > 56.8%
X = 0.98, OT > 58.1%
X = 0.97, OT > 59.5%

[/ QUOTE ]
I would think OT would be that high for a team that overcomes a 14 point deficit late in the game


[ QUOTE ]
I would also like to point out that if a team is 56.8% to win in OT, the chances of making an 2pt go up as well. Letting conversion success be C:

[/ QUOTE ]

a valid point and interesting externality, which does indeed show this question is much more complicated when moved to a real world situation as opposed to the assumptions in the OP that makes the decision obvious.

[Magic_Man]

[ QUOTE ]
I would also like to point out that if a team is 56.8% to win in OT, the chances of making a 2pt go up as well. Letting conversion success be C:

2pt:
hit, kick = C*X = C*X to win
hit, miss = C*(1-X)*OT = C*OT - C*X*OT to win
miss, hit = (1-C)*C*OT = C*OT - C^2*OT to win

kick:
kick, kick = X^2*OT to win

X^2*OT > C*X + 2*C*OT - C*X*OT - C^2*OT

[/ QUOTE ]

Just in case I lost anyone on the partial derivatives part, let's try some numbers. I will assume X = 0.99, as this is fairly well established.

For kicking to be better:
C = .420, OT > .568
C = .425, OT > .575
C = .430, OT > .583
C = .435, OT > .590
C = .440, OT > .597
C = .445, OT > .605
C = .450, OT > .612
C = .455, OT > .619
C = .460, OT > .626
C = .465, OT > .634
C = .470, OT > .641

This shows the idea of the sensitiviy of C; if we are off by about 10% in our estimate of C, (.42 --> .46), then we have to also be off by 25% (.5 --> .626) in our estimate of OT for kicking to be correct.

~MagicMan

[Kristian]

It's basically a problem of risk. No, not risk of losing, risk of making a bad decision.

If the coach knew with 100% certainty that his teams converts 2-pointers with exactly 42% likelihood, I am sure you could convince him, or his GM that the 2-pointer is the best way to go.

His problem is that he doesn't know this, there is some chance that his team converts with a smaller probability, and thus that he is actually making a bad decision going for 2.

You might argue that on a league wide scale over the last X years, everyone converts with 42% frequency, so 'everyone' would benefit from going for it every time. But this is not a valid argument since a specific coach should not care how every other team does, only about his own team. And sufficient statistics are not available for his own team.

In other words he is facing a decision with a large unknown factor. This is an uncomfortable situation for most people, so naturally, he takes the conservative approach.


On a completely different level, the fact that he is down by 14 late might be an indicator that his guys are having problems gaining 2 tough yards when they need them. They probably just scored throwing deep [img]/images/graemlins/smile.gif[/img].

[stinkypete]

[ QUOTE ]
Another thing I find interesting is that apparently all teams everywhere seem to think their 2 point conversion success rate is below 50%. Maybe this is true in the NFL, but surely there are lopsided college match-ups where a team would expect its success rate to be above 50%. If this were the case they should be going for 2 after almost every touchdown.

[/ QUOTE ]

if the matchup is that lopsided they might be better off just reducing variance and going for 1.

on the other hand, if a team is significantly worse than the opponent otherwise, but decent at 2 point conversions, they might be better off going for two even if their 2 point conversion rate is significantly less than half of the 1 point rate.

[David Sklansky]

You guys are all just wrong. I am almost sure that the great majority of the coaches have no idea that they will win more often given the 42% and the 50% probabilities. They probably think that the two pointer has to be 50% to make it worth it. The various excuses you are giving them are not their reason, though they might use them if confronted with the truth.

[CanadaLowball]

yeah.. but those 42% and 50% figures arent always "givens".

[MyTurn2Raise]

[ QUOTE ]
You guys are all just wrong. I am almost sure that the great majority of the coaches have no idea that they will win more often given the 42% and the 50% probabilities. They probably think that the two pointer has to be 50% to make it worth it. The various excuses you are giving them are not their reason, though they might use them if confronted with the truth.

[/ QUOTE ]

Have you ever heard the song "Uncle Walter" by Ben Folds Five?

[Kristian]

[ QUOTE ]
You guys are all just wrong. I am almost sure that the great majority of the coaches have no idea that they will win more often given the 42% and the 50% probabilities. They probably think that the two pointer has to be 50% to make it worth it. The various excuses you are giving them are not their reason, though they might use them if confronted with the truth.

[/ QUOTE ]

You are certainly entitled to that opinion, but I hope you will agree that you have not mathematically proven that any coach should always make the two-point conversion attempt in the situation mentioned?

I will grant you that it is odd we NEVER see it, as simple math shows it to be 'profitable' at least in some situations.