*[EDIT: There's been much contention about this. I post this in the comments, but it's worth posting here as well: If someone has a model to calculate win probability (other than the Advanced NFL Stats one, because it doesn't really work here), I'd love to see the analysis of win probably for the following (with 7:53 left on the clock and kicking the ball off):*

WP(extra point) vs. WP(2-pt make)*.4 + WP(2-pt miss)*.6

I think this is the crux of my argument. My assumption is that WP(extra point) is higher.]

WP(extra point) vs. WP(2-pt make)*.4 + WP(2-pt miss)*.6

I think this is the crux of my argument. My assumption is that WP(extra point) is higher.]

In the wake of Michigan's loss to Iowa, game theorists are criticizing Brady Hoke's decision to kick the extra point rather than go for two on Michigan's fourth-quarter scoring drive. Though I know it goes against traditional game theory to kick the extra point, I can't shake the feeling that it was the correct decision.

Hear me out. Michigan scored to make the game 15-24 with 7:53 remaining on the clock. Game theory states that Michigan should go for the two-point conversion because, if they miss, it will inform the coaches whether or not they need one or two scores with the remaining time, and can adjust their playcalling accordingly. But there's one play that this theory doesn't take into account: a successful, expected onside kick.

First, let's examine the path to victory that Michigan would have to follow for all of the possible scenarios (with the exception of missing the extra point because, well, if you do that twice in a game, shame on you):

Go for 1 | Go for 2 make | Go for 2 miss |

Defensive stop | Defensive stop | Defensive stop |

Touchdown drive | Touchdown drive | Touchdown drive |

2-point conversion | Extra point | Extra point |

Overtime | Overtime | Expected onside kick recovery |

Field goal drive |

Michigan, obviously, elected to kick the extra point. With that decision, the team was essentially playing for a coin flip (2-point conversions are successful about 40% of the time). Kicking off with 7:53 remaining, Michigan needed a defensive stop as well as a successful touchdown drive and two-point conversion. As the game played out, Michigan had ample time for that. Not only did the defense force a stop after 2:41, but they were able to fit in two more drives and a second defensive stop (offensive drives consisted of: 1:13 off the clock preceding a punt, and 2:15 off the clock ebfore stalling at the 3 yard line; the second defensive stop took 1:44). Had they been able to punch the ball into the endzone, they would've had a 40% chance of tying the game and sending it into overtime. From there, it's anyone's game.

If Michigan makes the two-point conversion, the same scenario unfolds as above, but instead of needing a two-point conversion after the possible tying touchdown they only need an extra point. There is ample time for this situation, and it is the best-case scenario.

The problem lies in missing the two-point conversion early in the quarter. If Michigan had gone for the two-point conversion and missed, it's a two-possession game (9 points) with 7:53 to go. In that instance, not only does Michigan have to force a defensive stop (which they do), but also receive the ball, march down for a score, successfully complete an onside kick (about which more later), and drive for another score. Not only does the successful completion of an onside kick present problems, but time then becomes a major factor.

Hypothetically, let's say that Michigan scores with 7:53 left in the game and misses its two-point conversion attempt. They're down nine, kicking off, and need two scores. If the defense produces the identical defensive stop, they will receive the ball on their own four yard line with 5:12 remaining. Michigan's fastest scoring drive of the day was its final one, which covered 57 yards and took 2:49 (0.337 yards/second). If Michigan continued that pace, a drive of 96 yards would take them 4:44. If we're generous and shave off a minute from that time, Michigan is left with 1:28 and probably zero timeouts. In that time, they have to successfully recover an onside kick and move the ball about 35-40 yards to have a makeable field goal for the win.

While this is doable, onside kicks present a huge gamble. According to Advanced NFL Stats (caveats about these being NFL numbers apply, but only barely),

Onside kicks in the NFL are successful 26% of the time. It’s true, but it’s also very misleading. Onside kick success rates are very dependent on whether the receiving team is expecting one...In this situation, Iowa would be expecting an onside kick, making the success rate somewhere at or near 20%. With a true freshman kicker who has never attempted a collegiate onside kick? Michigan's chances would be low.

When teams are expecting it, when WP is about 0.15 and below, the success rate is about 20%. But when teams aren’t expecting it, the success rate averages 60%.

The point is that going for two points earlier in the quarter, while it would pay off significantly if you convert (40%) basically puts the game out of reach if you fail (60%). Not only does clock management become a significant issue if you miss the two-point conversion early, but Hoke would also be asking his team to play nearly perfect football for close to 8 minutes. With Michigan's current team, that's asking too much.

This is why I feel like the "information" angle of game theory here is short-sighted. Would Michigan's coaches know whether or not they needed two scores? Sure, but getting two scores in that time-frame while playing errorless football is the equivalent of putting the game out of reach. By kicking the extra point, Michigan creates a potentially one-score game and forces Iowa to play errorless football (a fumble or interception changes the game significantly). Most importantly, it removes the variance of a successful, expect onside kick.

As it played out, if Michigan was able to score, that gives them a 40% chance of taking the game to overtime. But that same 40% chance of making the two-point conversion earlier in the quarter significantly increases the win variance and Michigan's ability to convert what is needed for an eventual victory. So while kicking the extra point may not make the game a guaranteed one-score game, it does indicate that you're playing for one more scoring drive. Because with the time remaining, playing for two is nearly impossible.

## 15 comments:

So missing the 2 pt conversion the first time would make you play a perfect game but would give you a shot. Missing the 2 pt conversion with 0:00 seconds left ends the game and you don't have any chance. Obviously its not an ideal situation but you are at least giving your team a chance if you miss the 2 pt conversion early instead of late.

Yeah, disagree completely. You argue that the onside kick means you probably won't win if you miss the two point with time on the clock. Well you definitely won't win if you miss it at the end of the game.

If you miss the two-point conversion early, you put the game out of reach for reasons that are largely out of your players'/coordinators' control. With a chance to send the game to overtime on a single play, you have significantly more autonomy. I'd rather go with this strategy.

Chris, consider the difference between the "Go for 2" columns: depending on if it's make or miss, the subsequent "defensive stop" and "touchdown drive" blocks are not the same. This is because you know that, if you miss the two-pointer, you adjust your tactics in the time remaining, and it will not be the same as your tactics supposing you make the two-pointer. This actually holds across all three columns as well: "defensive stop" and "TD drive" are generalized strategies, but they will be tactically different. And if you go for it earlier, you actually have more information about the outcome, meaning you have more control over your tactical approach.

I understand all of that in theory. What I'm saying is that practically speaking, if you miss the two-point conversion early in the quarter (in this instance specifically), the game is essentially over. What is required to win is so unlikely that you've more or less put it out of reach.

The point is, I think it's less likely you come away with a win/tie if you miss it early (largely because of the onside kick which has a 20% success rate) than you would if you played the game out and went for it at the end of the game (a 40% success rate). Sure, if you make the two-point conversion early on, it's a whole different ballgame, but that's a gamble that I'm not sure is worth taking.

But if you miss early you still have the chance to get an onside kick and score again. If you miss late you have a 0% chance of scoring again because time is out. Or if time isn't out then you still need a succesful onside kick.

Chris, thanks for your comment, but I'm going to have to agree with AVC here. The 2pt conversion is equally likely no matter when you go for it (that's an assumption). If you are supposing that you basically take yourself out of the game by missing early, then you have to assume that you are definitively taking yourself out of the game by missing late. The difference seems to be that: missing early gives you 20% chance of winning, and missing late gives you a 0% chance.

At any rate, I've enjoyed thinking about why game theory says what it says, so thanks for that.

If someone has a model to calculate win probability (other than the Advanced NFL Stats one, because it doesn't really work here), I'd love to see the analysis of win probably for the following (with 7:53 left on the clock and kicking the ball off):

WP(extra point) vs. WP(2-pt make)*.4 + WP(2-pt miss)*.6

I think this is the crux of my argument. My assumption is that WP(extra point) is higher.

I guess I am just failing to see how you would increase your WP by taking the extra point early. In this case then your WP would be 40% since a 2pt conversion is succesful 40% of the time. But if you went for the 2pt conversion after the first td your wp would be 40% + the chance that you win even after missign the 2pt conversion. All of this assuming that you score two TD's of course.

So my math I guess works out to

XP Early = 40% WP because you only have the one 2pt conversion attempt as time expires

2pt Conversion Early = 40% WP + The odds you score twice in the remaining time.

Even if the odds you score twice more aren't great you are adding to the 40% which is as high a WP as you can get by going for the Extra Point early. And by adding to the 40% that becomes the better option

Success rate and win probability are not equivalent. I don't think that math works out quite right.

That's certainly fair, definitely an oversight on my part.

I still stand by my opinion but acknowledge that my math is not the correct way to support my argument.

Either way, thanks for taking the time to respond. I read your blog daily and really enjoy the insights.

Good topic and subsequent debate.

Chris,

First off - good work here in general. I haven't been commenting but have been reading and enjoying the blog.

As for this post - I think you expended a lot of words but are missing the point entirely.

Point A: If you miss the 2 point conversion - at any time - you're kinda screwed. You're right that having a two-score deficit sucks (obviously), but it sucks a lot less if you have that information earlier. You can't include the ramifications of missing the 2 point conversion in one scenario but not the other (unless you think Michigan is somehow more likely to convert with 2 seconds left than otherwise.)

If you score with 2 seconds left and miss the 2 point conversion, your chances of winning are zero. If you score with 8 minutes left and miss the conversion, your chances are higher than zero. It's that simple. Well actually, it's not because....

Point B: The real drawback here is if two things happen: 1) you fail on the conversion AND 2) Iowa scores. That turns a 9 point deficit into 12 (instead of 11) for a FG and 16 (instead of 15) for a TD. Reality is that if Iowa scores you are probably losing either way because they'd have run down the clock - especially since they're not really a big-play offense.

The downside is very small. The upside is significant. This was an easy call to make.

Hoke's call was essentially 'ignorance is bliss'. Frankly, I find it kind of cowardly.

In order for this to be game theory you have to bring in the opponent, his reaction function and his beliefs about what your reaction function is. Absent such a discussion, this is dynamic optimization.

The idea that more information early is better ignores the fact that you're also giving Iowa more information. (Factoring that into the analysis makes this game theory.) Now, you might not think that Iowa's information gain is worth a lot because irrespective of the outcome, they're running three dive plays and punting. Even so, it's something you have to account for. Also, I think the argument over whether to go for 2 early or late really hinges on your belief as to whether it's easier to convert early or late and whether by getting the game to one score makes Iowa's defense better or worse. For instance, you could argue that by getting the game down to one score, it freaks out Iowa, makes a second TD easier and turns their 2-point conversion defense into butter. You could argue the opposite. I don't know, it's an empirical question.

The problem, as other people have pointed out, is that if you make the XP on the first TD, you are then planning the rest of the game around the fact that you will make a 40% conversion later. If you get the 60% instead and fail, the game is over. There's probably no time left for an onside kick. Going for the conversion on the first TD gives you more options and is the correct call. The conversion is always going to be 40% successful - if you try it first and fail, you know that you are now in a two-possession game. As Brian noted somewhere on mgoblog, an 8-point game is not one possession. It's one possession 40% of the time and 2 possessions 60% of the time. Our brains see it as one possession because we know that it's possible to get 8 points in one possession, but that is misleading. For the same reason, your table should have four columns - there needs to be one for "make XP first, miss conversion second".

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