# One more thought on going for two

Submitted by El Jeffe on October 6th, 2009 at 10:11 PM
Sorry for being a little late to the party, but here's a couple of other thoughts on the going for two question. If you're sick of the whole bloody mess, you should probably stop reading about now.

First off, I largely agree with ikestoys's diary (http://mgoblog.com/diaries/down-14-and-going-2). I have often thought that football is a game that rewards aggressive play calling, like going for two and on fourth down more often, and fake punts from your own 20... Eh...

Anyway, I disagree with a couple of points ikestoys made, both explicitly and implicitly, and I thought I'd chuck 'em out here.

Trials are not independent
This point was made by a commenter in the original diary, but the basic idea of treating the different sorts of trials (going for 2, going for 1, overtime) as independent events (and therefore as amenable to the application of the mathematics of garden-variety probability theory) is flawed.

In football the outcome of one trial affects the probability of another trial even occurring, and not in predictable ways. Let's say UM had made the first two-point conversion. Would State have played their next drive differently than they did? Maybe, maybe not. Perhaps they would have come out throwing and scored a field goal to go up by nine. We have no way of knowing how things would have unfolded in that alternative universe.

Relative frequencies are not probabilities
Second, and another point made by a commenter, is that ikestoys treats relative frequencies (the proportion of successful two-point conversions) as the same thing as probabilities of success. They are not. That's like saying that because 1% of adults die of lung cancer, you have a 1 in 100 chance of dying of lung cancer. Do you smoke? If so, then your probability is surely higher. If not, it's lower. The point here is that the probability of success of a two point conversion depends on many factors, as various people have noted.

Nevertheless...

Because relative frequencies =/= probabilities, I thought it would be interesting to see how the probabilities of winning fared if you didn't assume the probability of a successful two-point conversion was 0.44. So, two graphs for your viewing pleasure. The y-axis is the probability of winning the game after all events have unfolded (post-touchdown try after TD 1, TD 2, and possibly overtime). The x-axis is the probability of success of the two-point conversion (I limited the range of this probability to between 20 and 80%).

Graph the first
In the first graph, I have plotted the cumulative probabilities of winning for two strategies: going for 2 after scoring a TD to be down by 8 (iketoys's strategy--the black line), and going for 1 (RichRod's strategy--the blue line). The only thing I have allowed to vary is the probability of success of a two-point conversion (on the x-axis).

Couple things:
• Note that I have reproduced the probability ikestoys does, where the dashed red line intersects with the black curve at about 57% when Pr(success) for a two-point conversion is 44%.
• Note also that despite ikestoys's implicit claim that going for two is always the better move, if the probability of success falls below 35.5%, it is better to go for 1, as RichRod did. I'm not suggesting that this is what the probability would have been, though people's comments about a dog-tired Tate, a driving rain, etc., make this idea not too farfetched).
Graph the second
There are two other variables in the process: the probability of a successful PAT (which I held constant at 0.95), and the probability of winning in OT. The latter probability doesn't change the black curve below much, so I left it at 50/50, as did ikestoys.

In the graph below, the three non-black curves represent three different probabilities of winning in overtime: 40% (orange), 50% (blue), and 60% (green).

The only thing to take away here is that if you believe your probability of winning in overtime is high (based on your style of play, being at home, etc.) and if you believe your probability of a successful 2-point conversion is less than 44%(ish), then you should adopt RichRod's strategy. If you believe that your chances of winning in OT is 50/50, and you believe your chances of scoring on a two-pointer are > 35%, then you should follow ikestoys's strategy.

In conclusion (I know, finally)
Of course, coaches don't think this way in the heat of a game. Again, I basically agree with ikestoys, but the story is a bit more complex.

I like the analysis by both you and ikestoys until we get to your last point--that if you're better in OT than going for two, force OT, and if you're better at going for it, then get it over with now--isn't that more or less common sense? "In the heat of the game," doesn't a coach just go with his team's strength?

Agreed, though you have to wonder if that's precisely what Chris Petersen was thinking against OU:

"Man, we're only in this position because of a crazy-ass unlikely turn of events. We probably don't have the horses to win in OT."

+

"Dude, I have this AWESOME two point Statue of Liberty play!"

= eschew OT and try to win in regulation.

You're on El Jeffe! Will it be wiffle ball bats? passive aggressive barbs? lurking variables? Choose your weapons and bring a second. We meet at dawn!

MGV,

It is "common sense" or just plain logical (usually but not necessarily the same thing).

But there is a lot more involved in a coaches decision that the parameters listed here. How have his players been performing that day (flat, hyped up, intense? Does he think the other coach will be caught off guard? Are your players or the crowd urging you to go for it? Does your "gut" say "take a chance?" Are you playing to win or are you playing "not to lose" (both have their place).

This analysis is quite interesting, but it also quite academic and has only limited application to a coaches decision making process on the field.

Does Les Miles sit on the side lines generating graphs off of MS Excel? Hell no, he's playing by the seat of his pants.

Another possibility to consider is the case where UM goes for 2 both times, does not make it either, but still has some time left to possibly score again, perhaps after recovering an onside kick. This would seem to be factor that makes going for 2 even more attractive.

What the heck? I would have went for 2 after the second touchdown simply because of the way the team had come back to score twice at the end of the game. Put the dagger completely through the heart and end the dang thing! Was Tate tired? Was the whole team tired? YES. The little bit of rest before the start of overtime really didn't work out, did it? Go by the seat of your darn pants, score the 2 point conversion and feel the spirit of thousands of Sparty fans just leave the stadium!

I like faking the XP and going for 2 on the 1st touchdown as I think it would be totally unexpected so that would increase your probability of success. I do agree with the initial premise that every action causes a differnt reaction, a successful 2 pt conversion on the 1st TD changes the coaches thinking and the emotions of both teams significantly making it difficult to predict percentages from there.

I absolutely agree with KBLOW....these diaries are so complex. I totally understand what is going on but I don't think this kind of stuff is what does it for me. Maybe this is why I hate my grad level research class right now.......

Trials are not Independent

True, but then the math is incredibly complex. Obviously, this isn't the best argument for the probability argument, it is merely a "back of the envelope calculation" that gets you to a place where you can begin a discussion about proper strategy. Before Saturday, Michigan had won all of the overtime games that they had played (if I recall correctly...someone can correct me if I am wrong). So previously, playing for overtime was a strategy that worked at Michigan. Remember, "Statistics never lie, but liars use statistics."

Relative frequencies are not probabilities

True, but with more trials, and proper use, they become probabilities (as N ->\infty). You want to make the proper application of the statistics. If you are a male, non-smoker, you should only look at the statistics for male non-smokers when determining your risk for lung cancer. Because of this reason (how many times has Michigan gone for 2 vs. MSU) there may be more difficulty in generating the proper statistics.

However, I do like the graphs you make--though it is not clear what is on the y-axis. I believe that you call it probability for winning. Did you calculate this by:

(Probability 2pt make)(Probability 1pt make)
+(Probability 2pt make)(Prob 1pt fail)(P-OT win)
+(Probability 2pt fail)(Prob 2pt make)(P-OT win)

It doesn't look like it as you only have one curve for the "two point strategy" on the second graph. It seems like you only compared kicking XPs with the probability to win in regulation using the 2-point strategy. Also, you would have to calculate the probability for the "kick the XP" strategy as:

(Probability 1pt make)(Probability 1pt make)(P-OT win)
+(Probability 1pt fail)(Probability 2pt make)(P-OT win)

I do think that it would be interesting to consider what happens with the probability on the 2nd two point try relative to the first as the defense has already stopped one 2 point try.

Conclusion

Of course, this mathematical mumbo-jumbo is not how things are decided on the field.

How many of you (like me) were thinking about the Braylon game when Michigan scored that last TD? How many State fans were thinking the same thing...or at least "How did we blow another one to Michigan?" I was quite confident going into OT.

Maybe the strategy that is not mentioned [EDIT: scwolverine mentions it] is best, kick the XP on the 1st TD and go for 2 at the end. This way the game is over now--on one play--and I've got a hot QB/offense that is high on adrenaline, and their defense is defeated and back on their heels. If we do this now, we might just catch them blowing another coverage or thinking pass when we've got some MINOR RAGE in store for them. Strike while the iron is hot. Emotion and instinct are things that we cannot model mathematically. (And my wife cheers when I say this. What can I say, I'm a nerd and she's a free spirit.) Yet they may be some of the more important factors to consider in this problem.

Thanks for the comments. Some replies:

Trials are not Independent

True, but then the math is incredibly complex. Obviously, this isn't the best argument for the probability argument, it is merely a "back of the envelope calculation" that gets you to a place where you can begin a discussion about proper strategy.

Agreed. After noting that, I went ahead and used the same method that ikestoys did. I just wanted to be pedantic. Also, though, it's why people (and maybe chimps) value present resources more than the potential for future resources.

Relative frequencies are not probabilities

True, but with more trials, and proper use, they become probabilities (as N ->\infty). You want to make the proper application of the statistics. If you are a male, non-smoker, you should only look at the statistics for male non-smokers when determining your risk for lung cancer. Because of this reason (how many times has Michigan gone for 2 vs. MSU) there may be more difficulty in generating the proper statistics.

Somewhat agreed. Relative frequencies are only probabilities if the process is truly stochastic. There is lots of non-randomness in 2-point conversions (though there probably is some randomness too). Agreed that it would be silly to try to assess all the causes of success on the fly, which is probably why coaches fall back on much less random sources of points (PATs) unless obviously compelled to.

As for the calculations,

1. They were all done using ikestoys's formula, as you show.

2. For graph the second, it turns out that varying the probabilities of winning in OT doesn't affect the two-point strategy all that much. It obviously affects the OT strategy because, if Pr(sucessful PAT) = 0.95, then Pr(OT) = 0.95(0.95), or 0.9025.

On the note that the non-independence (or the interdependence, I suppose) of trials makes the math "incredibly complex"--I'm not aware of a branch of mathematics that can begin to predict the impact one failed 2-pt conversion will have on the team's next attempt. Since trials DO affect each other and we can't model them, we can do no better than RR on the sidelines in making the right call, and as another commenter on this thread pointed out, at least he knows which players are focused/fatigued/looked good in practice/might get too nervous/etc.
Your point about Michigan never losing in OT and the quote--was that sarcastic? With all the talk of statistical significance of different trends, we'd have to recognize that a team's performance in OT the previous year has little/no bearing on their expected performance this year, let alone the historical program record, right?
Lastly, I prefer the version that goes, "Figures never lie, but liars figure."

It can be tough to buck the conventional wisdom. Here's an example:

When Barry Switzer was coach of the Dallas Cowboys, there was a game where he faced 4th & 1 deep in his own territory, went for it, and failed. He was widely ridiculed for the decision, and his image in Dallas never recovered.

Now, we all know that statistically 4th & 1 heavily favors the offense. If the math were all that counted, teams should go for it almost every time. But they don't, and they won't. Even Mesko's botched fake punt was statistically the right thing to do, and yet, it's hard to find anyone who approves of it.

Going for two is a closer call, and I don't think Rodriguez would have faced the same ridicule for that as he did for Mesko's failed punt fake. But if Michigan had gone for two and missed both, thereby losing by 2 points in regulation, there certainly would have been plenty of second-guessers who said, "Our chances in overtime were better." Hardly anyone objects when you play for OT in that situation. The perception is that that's smarter football.

There are so many situational issues that come into play here, particularly for the potential 2 point conversion at the end of the game, that it's really hard to apply any statistics to it.

The offense (or at least Forcier) is clearly tired. Is the defense equally (or more tired) at the end of that series?

Is it still raining giving a potential advantage to Michigan WR's (they know where they're going, defenders do not)?

Are Michigan's WR's beating Sparty's CB's on a consistent basis? Can Greg Mathews beat his man on the same play that beat ND?

Is Brandon Minor healthy enough to pound the ball in on a cutback that has worked so well all season but hadn't been executed all game?

Conventional wisdom tells you to go for two on the road as an underdog (like CMU did). Personally, I don't think of Michigan in the same terms as I think of CMU. I would have taken my chances in OT just like RR did. That's more of a pride thing than anything else.