Shouldn't teams be going for two more often?

One of the central assumptions which is made all the time in (Baseball) Sabrmetrics is that teams always try to maximize their runs on offense and minimize the runs they allow on defense (and with pitching). Everything is cast in terms of expected value. I actually think this causes significant issues in baseball as well, though it's reasonable, but that will wait for another time.

The question I want to look at here concerns the game of football. Basically, teams now have 2 point conversion rates of significantly over 50% (actually, I'm not sure that this is true right now; but a few years back, it had crept up to about 60%). And if that's true, you expect to get more points out of going for two than in just kicking the regular ole PAT. So why don't they do that? Are they dumb, set in their old-fashioned ways, or is there some deeper reason?

The answer is... well, it's complicated. But it's certainly not such a cut-and-dried "Go for two!" as the above might lead you to believe. And what it comes down to is game theory.
What you want to use here, insomuch as it's possible (by the way, don't you think words like 'insomuch' are just fantastic?), is backward induction. Basically, if you are at the end of the game (the clock is showing zeroes), and you've just scored a touchdown, you are either a) in the lead, in which case it doesn't matter; b) trailing by 3 or more - again, it doesn't matter; c) tied - obviously kick the extra point (assuming you can do this mroe realiably than scoring two of course); d) down by 2 - obviously go for two, otherwise you lose for sure; and in this case, if you make it, you win with .5 times the probability that you make the two, assuming you have a 50% chance to win in OT, which is likely not totally correct, but ought to be quite close; or most interestingly e) you're down by one. In this case, you can take the extra point and go to OT (actually, there's a non-zero chance you miss the extra point and lose anyway...), winning .5 of the time, or you can go for 2 and have everything ride on the one play. If you do, then you win with probability p2, where p2 is the chance of making a 2-point attempt, and lose with probability q2, which is defined as (1-p2), the probability of missing a 2-point attempt. So as soon as this goes over 50%, you really ought to be going for it in these end-of game situations. And coaches by-and-large DON'T do this in the NFL, despite good 2-point conversion rates. Now, ok, the chance you win in overtime isn't exactly 50%, and if you believe you have the better team (on the day, in the given conditions), then this should factor in somewhat, BUT you also know that you can't be THAT much better, if you've tied them after 4 quarters, and the rules are set up in such a way to make it a little bit of a crapshoot. Also factor in that kicking the PAT isn't 100%, and you really should be going for it in these end-of-game scenarios.

On the other hand - let's say you have scored a touchdown to put you up 6 pending the PAT with oh, 1:30 left. In this case, there are a few things going on. First off, there's basically just going to be one possession left (adjust the 1:30 as necessary to make this true). Given that, if your opponent doesn't score a touchdown, whether you go for 2 or not doesn't matter. So we only consider the cases where they do. If you go for 2, you make with a probability of p2, and if you do, they will go for 2 as well. They will also make with a probability of p2, meaning there's a p2^2 chance of going into OT from this. They'll miss with a probability of q2, meaning that you get an outright win p2*q2 of the time. Of course, you miss with a probability of q2, and if you do, and this is the key point here, they can take the 'worse play' from the expected points point-of-view just to take the lead, which is of course what really matters, by just kicking the PAT. They will make that with probability p1, which is somwewhere very close to 1. Your alternative is to go for one, and as we've just seen, if you do, they are going to win p2 of the time (assuming they score, of course), meaning you win q2 of the time. Now, if we assume that kicking the PAT is an automatic one point, then we see that, in this situation, the expected win% when going for 1 is q2, and the expected win% when going for 2 is (p2^2)/2+p2*q2. And, setting them equal, this means that you should go for 2 when the chance of making it is at least 2 - SQRt(2) ~= .586. So actually we see that you ought to go for it in these situations as well, if the chance is really 60%.

You can keep going back further and further. Next, it's that you each get one more possession, so you've got a chance to score again, then they get more possessions, on and on and on. And pretty soon, there's just too many possibilities, you really can't say that things are independent of each other any more, and moreover, lots of other options get thrown into things. For instance, if you look at the situation where you have just scored to go down 1, and you have to choose whether to 'safely' tie it up or go for a lead, with some small amount of time left for the opponent to score back on you, there's lots of things to consider. If you go for it and make, any score by the opponents will still win, so you lose whenever that happens. If you take the one though, the same is still true. But of course, up one, if they don't score, you just win, whereas if you were tied, it goes to OT, so that's in favor of going for it. On the other hand, if you miss, then you are handing them the game, right? Well, you can always onside kick. But the better chance you get of that, the more you should just go for doing that anyway, in all situations. And on and on - the later decisions and possibilities affect what you are doing now.
Furthermore, you have to take field goals into account as well - 7 gets you ahead of two field goals, but not three, just as 8 does, but 6 only puts you level with teh team that makes a pair of field goals. And it is not very likely that the 8+7 from a 2 point and a one equalling 5 FG, or the 8+8>5*3 is going to be important. So the higher the chances of field goals, basically, the more attractive going for one rather than 2 is.

Nevertheless, and most importantly in close games, which is where the choice is most likely to matter, consistency is indeed valuable - taking a chance at a big lead is not so valuable as assuring SOME lead, and so even when expectation values favor going for 2, you should still situationally go for 1 rather often. And in fact, most of the determinations are made on a situational basis.
And as I look up the actual rates.... they're really hard to find. But they're apparently in the ballpark of 48-49% over long periods of time, and 98.5%ish on 'regular' PATs. So that is actually a relatively important deviation from the presumred 100%. And you have to look, of course, at the actual 2 teams playing. LEAGUE percentages may be at X, due to the overall balance of offense and defense in the league (although the chances that we see in practice are probably also somewhat correlated to waht kind of teams GO for 2 more often, which is probably not just an every-team-does-it-the-same-amount kind of thing) whereas in a particualr game, it might be a defensive slugfest, where the percentages are much lower, or an offensive shootout, where they are much higher. Of course, on one hand, because we KNOW that there is significant spread in the talent levels of offenses and defenses, it really OUGHT to be the case that SOMETIMES the 2 point percentage is significantly higher, which means we really SHOULD be seeing more conversion attempts. And realistically, while you shouldn't go for it EVERY time, you should see it more often than we do. (Incidentally, if you want ot find what the true rate should be, you want to try to take out things botched field goal attempts, or raelly, to increase your sampling, you should have a lower bound at whatever the 4th down conversion percentage is from the 2 (it's a lower bound because on 4th down there is some additional aversion to throwing an interception or having a fumble which can get returned or even losing a big chunk of yards, none of which you have on a 2pt attempt, where every failure to get in the end zone is equivalent), or probably 3rd down and goal conversions from the 2 are a good approximation to what your rate is - but I don't know how to find these numbers super handy either!).

Finally, we can look at a very clear-cut case where... the coaches get it wrong by siding too much on the side of caution (surprise surprise). In the case where you are very very late in the game and down by 14 (actually... this probably extends to basically ANY time you're down by 14, come to think of it, because you also get the 8+6 bonus from making and then scoring two field goals tying it for you...) and you score a touchdown. Go for 2! If you miss, you can always go for 2 again on your next score, and if you make that, you're no worse than if you'd made 2 singleton PATs; if you make, on the other hand, you can safely kick a PAT on your next score to take the lead! And if you miss both, a field goal STILL puts you in the lead. Of course, most of the time it won't matter, since you most likely either won't make up the whole 2 TD deficit by the end of the game or will MORE than make up for it, but when it does... even if we discount the possibility that you get TD + 2PT + FG + FG to tie... you only need a conversion rate of (3-SQRT(5))/2 or ~.382 to make it worth it. And that's assuming that kciking an extra point is a 100% thing. Which means that given the real situation, it's more like a 3/8 chance needed - maybe even less. And if the real rate is closer to what you have in reality, say 50%, you would expect to come out ahead 50% of the time it makes a difference, and behind only 25% - a 2-to-1 proposition which is going unexploited.