UPDATE: Congratulations to the Saints and the city of New Orleans! Although it was mostly about the talent on the field, this game could do a lot of good for smarter NFL strategery. Expect a lot more onside kicks next season!
Let's take a quick look at the Saints' decision to go for the touchdown on fourth down and about a yard-and-a-half with 1:55 to go in the first half. Although Pierre Thomas got stuffed, it was probably the best play by a wide margin.
How come? Because even if you miss, the Colts are pinned inside their own 2 yard line. This makes a lot of difference: having the ball on your own 1 or 2 is worth about -1.5 points, according to David Romer's seminal paper (that is to say, the value of possession is outweighed in this instance by your incredibly poor field position). Conversely, having the ball on the 20 or thereabouts after a made field goal and a kick-off is worth +0.5 points -- your opponent has a positive scoring expectation. (There's also the case of a missed field goal, in which the Colts would have the ball at about their own 9; that's worth -0.5 points.)
Overall, considering both the Saints would score on the play and what they'd do to the Colts' field position, their decision matrix looks something like this:
We say a touchdown is worth 6.5 points, for example, which is the value of the 7 points you score less the 0.5 points in expectation you lose by giving the ball back to your opponent on their 20. The other cells in the matrix are calculated in the same fashion.
Assume that you have a 98 percent chance to make the field goal from the 1-and-a-half. Using the matrix above, that means your expectation from kicking is a net of +2.46 points.
What if you go for it instead? How often would you have to score the touchdown in order to make going for it the right play? Only about 20 percent of the time -- the main reason being, again, that failing to score but sticking your opponent on their own 1 yard line is almost as good as converting the FG but giving the ball to the opponent on the 20.
Certainly, a consideration of the scoring margin (the Saints trailed by 4), the short clock, and the opponent (the Colts can move the ball in a hurry) also deserve consideration here. But those considerations cut both ways and would probably not outweigh the ample mathematical justification for going for the touchdown. Sean Payton made the right call.
UPDATE: How about that onside kick to start the second half?
Once again using the estimates from Romer's paper, giving yourself possession on your own 45 (if you recover the kick) is worth +2.0 points; giving your opponent possession on your 45 is worth about -2.3 points. Compare this with giving the opponent their ball on their 20 after a regular kickoff, which is worth -0.5 points ... although often they'll return the kick and wind up with slightly better field position than that, so let's call it -0.7.
Given those estimates, the onside kick breaks even if you convert it about 37 percent of the time. And according to Football Outsiders, "surprise" onside kicks are in fact recovered in the neighborhood of 55 percent of the time or so. So this, too, looks like a good call -- and probably something that teams should be doing a little bit more often than they do.