Here are the stats on two-point conversions (all of them, admittedly, not just the ones at the end of games – though I’d bet that almost all of them are at the ends of games):
http://www.advancednflstats.com/2010/12/almost-always-go-for-2-point.html

As the writer points out, coaches are risk-averse (and kicks are seen as being far less risky than two-point attempts, even if the expected payoff might be slightly higher for the 2-point).

So – if you know that you’re going to need at least one of each, then you get the easy one *now,* which keeps you in *this* game, and worry about the hard one *if* you end up needing it (if the kicker somehow misses, you’re *still* at the point where you can ease up for your last possession).

If that is the case, and the kicker himself (as oppose to e.g. the coach) chooses which kick to make, he might just be trying to minimize the probability that he is the one who ruin the chances for the team.

I disagree with Rajiv’s theory that the motive is affecting the opponent’s behavior on offense. Our theory should apply in the simplest late-game situations where the team is down by 15 points, in which missing the two-point conversion at any point drives the win probability to zero. In such cases, it doesn’t matter what the opponent does when up by nine, and I expect the difference between the opponent’s offensive behavior when up by 8 or 7 is minimal.

Here are two possible explanations:

(1) The probability of a successful two-point conversion is greater after the second touchdown than after the first touchdown. Assume that an extreme amount of pressure has a negative effect on defensive performance. There’s much less pressure on the opponent’s defense when the conversion is attempted after the first touchdown, because there’s still a chance the opponent’s offense could still score or run out the clock without giving the ball back. Once the second touchdown has been scored, the defense knows that chance has passed, and the pressure is all on them to prevent blowing the 15-point lead. If you’re not convinced by the psychological story, here’s a physical story: defense requires more physical effort than offense. If the defense is relatively more tired than the offense after the second touchdown is scored, then the chance of success after the second attempt is greater.

(2) The information effect is always negative. The players and coaches know that they are entertainers, and that the longer they keep the game close, the longer people will watch, and the greater attention will be paid to their game on the postgame highlight shows. Going for two later creates much better drama. Having kept the game closer increases players’ and coaches’ expected future contract and endorsements, even if they lose. This benefit outweighs any negative effect of wasted effort.