Play-by-play data and dynamic programming are used to estimate the average payoffs to kicking and trying for a first down under different circumstances. Examination of actual decisions shows systematic, clear-cut, and overwhelmingly statistically significant departures from the decisions that would maximize teams’ chances of winning.
The choice between kicking and going for it leads to an immediate payoff in terms of points (which may be zero) and to one team having a first down somewhere on the field. That first down leads to additional scoring (which again may be zero) and to another possession and first down. And so on. (Page 342.)
By describing the values of situations in terms of expected point differences, I am implicitly assuming that a team that wants to maximize its chances of winning should be risk-neutral over points scored. Although this is clearly not a good assumption late in a game, I show in Section IV that it is an excellent approximation for the early part. For that reason, I focus on the first quarter.
- Assume the Pats will win the game for sure if they go for it and make the first down. Let m be the probability that NE makes the first down if the Pats go for it. Also let q be the probability that the Pats win even when they go for it and don't get the first down (so q is the probability that NE either prevents an Indy TD or gives one up and then scores themselves).
- If New England punts, NE will either win the game (by not letting Indy score a TD or by letting them do so but subsequently scoring themselves), or NE won't. Let p be the probability that NE does win, if New England punts.
- If New England goes for it, the probability of winning is m + (1-m)q. That is, the Pats win when they make the first down or, having not made it, win anyway.
- If New England punts, the probability of winning is p.
- NE's win probability when it punts, p, is at least 50 percent here, but not more than 80 percent (yes, Indy's offense is great, but the Pats had intercepted Manning twice, and Indy did have only 28 points after 58 minutes, after all; plus, there's only 2 minutes left, a punt will put Indy somewhere between its own 20 and 30, and Indy needs a TD, not just a field goal).
- NE's win probability when it goes for it and doesn't get the first down, q, is at least 10 percent and not more than 40 percent. My thinking here is that going 30 yards is a LOT easier than going 70 yards, especially with so little time remaining.
- NE wins 50 percent of the time when it punts and 40 percent of the time when it goes for it and doesn't get the first down: p = 0.5, q = 0.4. This can be shown to be the most friendly-to-Belichick-and-Levitt case my bounds allow. It implies that Belichick should go for it if and only if m is greater than 1/6: NE has to have at least a 17 percent chance to convert on 4th and 2.
- NE wins 80 percent of the time when it punts and only 10 percent of the time when it goes for it and doesn't get the first down: p = 0.8, q = 0.1. This implies that Belichick should go for it if and only if m is greater than 7/9: NE has to have at least a 78 percent chance to convert on 4th and 2.