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Friday, December 26, 2008

Football 2 Point Conversion Trivia

In this post on no third solution, David Z refers to my post where I give an analysis of "go for a first down vs. punt" when behind at the end of a football game. My point was that, due to pro-State brainwashing, the coach will take the "less risky" action of punting, when going for the first down maximizes the chance of winning the game. The reason is that, if you go for the first down and miss, the coach is blamed for losing the game. If you punt and later lose, then the decision to punt is not usually denounced as wrong.

David Z makes an incorrect analogy about 2 point conversions. Suppose you are down by 7 points at the end of the game and score a touchdown. Do you go for 2, or kick the extra point? Statistically, your chances in overtime are 50%, the extra point is near-automatic, and less than 50% of 2 point conversions are made. Under those assumptions, you should kick the extra point.

A coach may be reluctant to go for 2 in the "Behind by a 7 points and score a touchdown" scenario, but statistically, a 2 point conversion is made less than 50% of the time. If you believe that you are less than 50% to win in overtime (your players are tired or injured), or if you have a good play to call, then it makes sense to go for the 2 point conversion.

Statistically, the NFL concluded that "winning the coinflip in overtime" and "who wins the game" are uncorrelated. Over several seasons, the winner in overtime only wins the coinflip nearly 50% of the time, although there was one season when the coinflip winner won very often.

The problem is that if you win the coinflip, receive the kickoff, and then gain 2-3 first downs, you can kick a 30 yard field goal and win. This is perceived as unfair for the team that kicked off, as the other team scored a field goal and they never got the ball. There are several solutions:

  1. Make the goalposts narrower and higher, making it harder to score a field goal from the 30 yard line.
  2. If the team winning the coinflip scores a field goal on their first possession, then the other team gets 1 possession to score a touchdown and win (they cannot kick a field goal to tie).
  3. Adopt a system similar to college, where each team is guaranteed an equal number of possessions in overtime. The drawback of this system is that the overtime can drag out longer, leading to player injuries.
My main point was that, due to pro-State brainwashing, coaches are reluctant to make a decision that maximizes their chance of winning, but they immediately lose and look like a fool if they're wrong. The problem is that, due to pro-State brainwashing, people judge by outcomes and not a priori odds. If going for the first down maximized the chance of winning the game, then the coach should not be denounced if he goes for a first down and fails to make it. For example, computer simulations could be used to determine an approximation of the odds backing each choice.



There's one interesting bit of 2 point trivia that is almost never discussed. (I actually saw this problem in Bridge World magazine initially, in the context of matchpoint strategy when you follow an anti-percentage play but maximize your matchpoints. For example, with 2 suits missing 9 and the queen, statistically playing for the drop is best, but finessing in the first suit is like going for a 2 point conversion in the below scenario. Finessing is only slightly unfavorable compared to playing for the drop.)

Here are the assumptions of the problem.
  1. An extra point try is automatic, made 100% of the time.
  2. A 2 point conversion is made 45% of the time.
  3. In overtime, each team has a 50% chance of winning the game.
  4. You are behind by 14 points. There is enough time for you to score two touchdowns, but not enough time for you to catch up if the other team scores again.
  5. Assume that you are going to score two touchdowns, you prevent the other team from scoring, and no other scoring will occur.
  6. After the first touchdown, the coach goes for a 2 point conversion.
What are your chances of winning the game, given the assumption that you do score two touchdowns and prevent the other team from scoring?

If the coach naively kicked two extra points, he would have a 50% chance of winning in overtime. What are the odds of winning if he goes for 2 after the first touchdown? Hint: The odds are better than you think!

I'll give you a chance to figure it out.

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If the coach makes the first two point conversion, then he will kick the extra point after the second touchdown. This is a 45% chance of winning the game.

If the coach fails the first two point conversion, then he will go for another two point conversion after the second. If the second succeeds, then the coach has a 50% chance of winning in overtime. This is a 0.55*0.45*0.50 = 12.375% chance of winning the game.

If both two point conversion tries fail, then the game is lost in regulation. The media decry the coach as an idiot (even though he made the statistically correct choice).

Notice that there is a 45% + 12.375% = 57.375% chance of winning. This is greater than the 50% chance from naively kicking the extra point both times.

The key point is that, if the first 2 point conversion try succeeds, then the coach can conservatively kick the extra point after the second touchdown. If the first 2 point conversion try fails, then the coach gets a chance to get back to par with a second 2 point conversion try.

The same analogy occurs in the Contract Bridge matchpoint example. Suppose you finesse in the "9 never" situation. If you're right, you play the 2nd suit normally, winning an extra trick than everyone else. If you're wrong, you can finesse in the second "9 never" situation and catch back up.

In a real game, if the coach went for 2 and succeeded early, then the other coach would change his strategy, minimizing the gain. Assuming an equal number of touchdowns scored by each team, then the 2 point strategy I outlined gains.

That's an interesting non-intuitive calculation that is rarely quoted.

5 comments:

Anonymous said...

I could never understand why people care about sports. I understand playing them, I enjoy that too.

But what do you care if someone else lost or won, who, how, where? Especially when it is a business sport, where everything is a setup to make the most money, where all the judges are paid and everything is supersafe with a state license?

Wish I could understand this sport thing...Then again, may-be I don't.

citizen stefish said...

why do some people care about stamps, or coins? or bird watching? or watching porn? it interests them, and entertains them.

i'd rather watch a good game than most other stuff. i could sit here and read another post on LRC about how evil the fed is, but then again i already know that. i don't know, however, who is going to win the game.

that's why i still read this blog, because it actually offers some new info. i am done with actually paying attention to anything political, what politicians are doing, what the thieves over at the federal reserve are doing, etc. i have stopped worrying about these people and have decided that since there is really very little that i can do to stop them, i am going to just let them keep stealing and murdering and kidnapping people and bringing The End a little closer. i have turned my thoughts toward agorism and trying to come up with something workable. but i am not going to spend my days being miserable anymore. sometimes i like to waste time by watching a decent athletic competition.

Anonymous said...

Good point, citizen stefish. But then, this could be easily replaced with another binary questions, such as:

- Will there be rain tomorrow, and if yes then at what time and how much?

- Will my cat go to the left or to the right of me, if I stand in it's way?

My point is that first, there isn't much to know. There will be a draw or one of the teams will win with some count.

Secondly, it has already been decided who and with what count, and so, it is not really "happening" in front of your eyes. It is sort of like watching a prerecorded event on VHS, and although you already know the outcome, still losing your cool rooting for a team.

That being said, you right, may-be the problem is with me, as I could never understand collectors either!

I mean I do understand a pleasure of having a rare coin in your possession, only if it had some history to it, otherwise any old (not rare)coin is the same pleasure to hold and to think of. I could never understand collectors of pokemon cards, postal stamps, coins in those cases where they collect items specifically manufactured for collectors!

Anyway, FSK will probably say this is off the point and be right at that.

FSK said...

You can say "I enjoy watching a football game." without being a pro-State troll. As long as you're aware of the direct and hidden pro-State advertisements, then watching a football game is harmless.

You don't have to always work towards a free market agorist revolution. You can take breaks and do other things.

The Presidential election was in many ways fixed so Obama won. The Super Bowl winner probably hasn't already been determined, and will be decided based on the merits of the players and the luck/circumstances of the game.

In many ways, coverage of sports is actually honest. It's one of the few things the mainstream media covers as honestly as possible. Nearly completely honest coverage of sports helps provide an illusion of legitimacy to coverage of other "news". The designated hitter rule in baseball is debated more honestly than any mainstream criticism you'll see of the Federal Reserve.

Of course, professional sports leagues have a State-endorsed monopoly. That doesn't mean it's immoral to watch the game.

citizen stefish said...

yea i have learned to ignore that stuff. if a game starts at "3", i won't turn it on until 3:10 so i can skip the flag-waving, national anthem, the salute to "our heroes", etc. it sickens me.

but if the state collapsed tomorrow, football would still be played. we can't erase "football" from our collective memories. as long as humans exist in the land currently called the USA, this game will be played. if the nfl collapsed with the state, the guys would eventually get together and play the game anyway if they truly loved it. eventually salaries would be worked out with the best players making more. someone would broadcast it on the airwaves, and people would watch by the millions.

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