Point(s) After Touchdown

We are two weeks into the era of the long extra point, and it has already produced some interesting results. An eighth of the way through the season we have seen more missed extra points than we did all of last year (a fact that was mentioned on just about every broadcast last week). But the most interesting part of all of this was what the Steelers did last weekend. On each of their first two touchdowns they chose to go for two rather than kick the longer extra point, converting both times and opening up an early lead that carried them into a blowout victory. No other team has shown this aggression so far, but that could change as the season goes along.

The math of this decision is interesting, if murky. For years now there have been debates about the logic behind the PAT decision, but it seemed mostly settled that kicking was the smarter option. NFL extra points have been converted at a rate of .988 since 1994 (as far back as Pro Football Reference’s play data goes). Two point conversions have been successful at a rate of .475 over that same time, resulting in .95 expected points, fewer than a standard extra point.

When the change was announced to move the extra point kick back thirteen yards, the immediate question was how this math would change. The short answer is, it’s not clear. Since the line for a two point conversion hasn’t moved, the expected points remain the same at .95. But the kick is now 33 yards in length, a distance that has been converted at a rate of .896 since 1994. By these numbers it would appear that the prudent move would be to go for two. But field goal kicking has taken an incredible leap in efficiency in recent years, and if you limit Pro Football Reference’s data to the past five years (a sample more representative of current kickers), the efficiency of 33 yard field goals jumps up to .946, a difference in expected points that can be explained by simple statistical variance.

Expected points is a useful metric early in games, when the added utility of each additional point is roughly equal . But late in games things get more complicated. Naturally when a team scores while down eight, the value of a successful two point conversion is worth more than twice that of a successful extra point. This is a simple, obvious deficit, one every football fan understands. Things get more interesting when a team is down by a number of points that isn't quite so neat.

Pittsburgh’s success this past weekend got me thinking of another instance involving the Steelers, late in their opening night matchup against the Patriots. Down fourteen points, Pittsburgh was driving for a touchdown that would have pulled the game within a single score. Normally in this situation a team would kick the extra point to cut the lead to seven (as the Steelers did when they scored with two seconds left on the clock). But with the way the math had changed, I started wondering if they would consider going for two, attempting to cut the lead to six points so an ensuing touchdown would give them the win rather than send the game to overtime.

It didn’t end up mattering, but my mind kept running through this after the game ended. So I constructed a very rudimentary win expectation model to try to figure out what the best decision is. To make this model, I had to make a couple of assumptions. The first is that when a game goes to overtime, each team has an equal probability of winning. This is a reasonable assumption backed up by observational evidence. So for a game that ends up tied, I’m considering that outcome as half a win.

The second assumption is that the decision being made does not affect the probability of other events in the game. Going for two doesn’t make ensuing touchdowns any more or less likely, and it doesn’t change whether the opposing team scores. This may not be quite true. If a team has a one point lead late in the game, the opposing team may be more aggressive trying to score in the remaining time than if the game was tied, slightly decreasing the .5 odds I mentioned above. This is a potential source of error, but I don’t think it is greatly significant.

With these assumptions, other probabilities end up dropping out. Regardless of whether a team goes for two or kicks the extra point on their first touchdown, their chances of scoring a second touchdown are the same. So the only numbers we have to take into consideration are the win probabilities made under the assumption that two touchdowns are scored and the opposing team does not score. In order to get these values (which I will refer to as “Scaled Win Probability”), I constructed outcome trees for each possible decision.

Using this chart, it’s fairly simple to figure out the win expectations. Assuming a probability of success on a kick of X, and the probability of success on the two point conversion of Y, we end up with the following values.

Extra Point Scaled Win Probability   =  .5*X² + .5*(1-X)*Y

Two Point Scaled Win Probability    =  Y*X + .5*Y*(1-X) + .5*(1-Y)*Y

Rather than just using the slightly unreliable probabilities I mentioned above, I created a table to show the difference between these two numbers for values of X between .85 and 1 and values of Y between .3 and .6. I subtracted the extra point value from the two point value to get the Scaled Win Probability Added, a measure of the amount a team's win probability would increase by going for two. I colored the table for ease of analysis, green cells indicating a team should go for two while red cells show when a team should just kick the extra point.

	0.85	0.86	0.87	0.88	0.89	0.9	0.91	0.92	0.93	0.94	0.95	0.96	0.97	0.98	0.99	1
0.3	-0	-0	-0	-0	-0	-0	-0	-0	-0	-0.1	-0.1	-0.1	-0.1	-0.1	-0.1	-0.1
0.31	0.01	0	-0	-0	-0	-0	-0	-0	-0	-0	-0	-0.1	-0.1	-0.1	-0.1	-0.1
0.32	0.02	0.01	0.01	0	-0	-0	-0	-0	-0	-0	-0	-0	-0.1	-0.1	-0.1	-0.1
0.33	0.03	0.02	0.02	0.01	0.01	0	-0	-0	-0	-0	-0	-0	-0	-0	-0.1	-0.1
0.34	0.04	0.03	0.03	0.02	0.02	0.01	0.01	0	-0	-0	-0	-0	-0	-0	-0	-0
0.35	0.05	0.04	0.04	0.03	0.03	0.02	0.02	0.01	0.01	0	-0	-0	-0	-0	-0	-0
0.36	0.06	0.06	0.05	0.04	0.04	0.03	0.03	0.02	0.02	0.01	0.01	0	-0	-0	-0	-0
0.37	0.07	0.06	0.06	0.05	0.05	0.04	0.04	0.03	0.03	0.02	0.02	0.01	0	-0	-0	-0
0.38	0.08	0.07	0.07	0.07	0.06	0.05	0.05	0.04	0.04	0.03	0.03	0.02	0.02	0.01	0	-0
0.39	0.09	0.08	0.08	0.07	0.07	0.06	0.06	0.05	0.05	0.04	0.04	0.03	0.03	0.02	0.02	0.01
0.4	0.1	0.09	0.09	0.08	0.08	0.07	0.07	0.06	0.06	0.05	0.05	0.04	0.04	0.03	0.03	0.02
0.41	0.11	0.1	0.1	0.09	0.09	0.08	0.08	0.07	0.07	0.06	0.06	0.05	0.05	0.04	0.04	0.03
0.42	0.12	0.11	0.11	0.1	0.1	0.09	0.09	0.09	0.08	0.07	0.07	0.06	0.06	0.05	0.05	0.04
0.43	0.13	0.12	0.12	0.11	0.11	0.1	0.1	0.09	0.09	0.08	0.08	0.07	0.07	0.06	0.06	0.05
0.44	0.14	0.13	0.13	0.12	0.12	0.11	0.11	0.1	0.1	0.1	0.09	0.08	0.08	0.07	0.07	0.06
0.45	0.15	0.14	0.14	0.13	0.13	0.12	0.12	0.11	0.11	0.1	0.1	0.09	0.09	0.08	0.08	0.07
0.46	0.15	0.15	0.15	0.14	0.14	0.13	0.13	0.12	0.12	0.11	0.11	0.11	0.1	0.09	0.09	0.08
0.47	0.16	0.16	0.16	0.15	0.15	0.14	0.14	0.13	0.13	0.12	0.12	0.11	0.11	0.1	0.1	0.09
0.48	0.17	0.17	0.16	0.16	0.16	0.15	0.15	0.14	0.14	0.13	0.13	0.12	0.12	0.12	0.11	0.1
0.49	0.18	0.18	0.17	0.17	0.17	0.16	0.16	0.15	0.15	0.14	0.14	0.13	0.13	0.12	0.12	0.11
0.5	0.19	0.19	0.18	0.18	0.17	0.17	0.17	0.16	0.16	0.15	0.15	0.14	0.14	0.13	0.13	0.13
0.51	0.2	0.19	0.19	0.19	0.18	0.18	0.18	0.17	0.17	0.16	0.16	0.15	0.15	0.14	0.14	0.13
0.52	0.21	0.2	0.2	0.2	0.19	0.19	0.18	0.18	0.18	0.17	0.17	0.16	0.16	0.15	0.15	0.14
0.53	0.21	0.21	0.21	0.2	0.2	0.2	0.19	0.19	0.19	0.18	0.18	0.17	0.17	0.16	0.16	0.15
0.54	0.22	0.22	0.22	0.21	0.21	0.21	0.2	0.2	0.19	0.19	0.19	0.18	0.18	0.17	0.17	0.16
0.55	0.23	0.23	0.22	0.22	0.22	0.21	0.21	0.21	0.2	0.2	0.2	0.19	0.19	0.18	0.18	0.17
0.56	0.24	0.24	0.23	0.23	0.23	0.22	0.22	0.22	0.21	0.21	0.2	0.2	0.2	0.19	0.19	0.18
0.57	0.25	0.24	0.24	0.24	0.23	0.23	0.23	0.22	0.22	0.22	0.21	0.21	0.21	0.2	0.2	0.19
0.58	0.25	0.25	0.25	0.25	0.24	0.24	0.24	0.23	0.23	0.23	0.22	0.22	0.21	0.21	0.21	0.2
0.59	0.26	0.26	0.26	0.25	0.25	0.25	0.24	0.24	0.24	0.23	0.23	0.23	0.22	0.22	0.22	0.21
0.6	0.27	0.27	0.26	0.26	0.26	0.26	0.25	0.25	0.25	0.24	0.24	0.24	0.23	0.23	0.22	0.22

Wow, that’s a lot of green. Yes, under most probability combinations, going for two gives a team a better chance of winning. If we use the probabilities I mentioned above for the new rules (.95 for the extra point, .47 for the two point conversion), we get a return of .12 Scaled Win Probability Added, well above the break even point.

So it now becomes very clear that the Steelers (and all teams trailing by 14 late in the game) should have gone for two. But the most interesting thing is, this isn't an effect of the rule change. Under the probabilities of the old rule (.99 for the extra point, .47 for the two point conversion), the Scaled Win Probability Added was still .1, significantly higher than the break even point. In fact, even if the extra point was a guarantee, a team would still have to have less than a 37% chance of converting on two to be better off just take the point.

This is just one example, the sort of situation that will come up a few times every season. But it is just one of many situations that is not nearly as clear cut as we have always perceived it. The math of the conversion may not have changed significantly, but the fact that it has changed at all gives us the opportunity to look at the strategy of football from a new and better perspective.