Who Should be Calling Timeouts? A Simplified Bayes Theorem problem - PIT @ BAL, Thursday November 28, 2013

In an individual game, football is a zero-sum game.  Whatever is good for me is bad for my opponent, and whatever is bad for me is good for my opponent.  So, if I should call a timeout, my opponent definitely shouldn't, right?  What do the announcers think?

Down by 8 points with under 4 minutes to go, the Pittsburgh Steelers need to score a Touchdown and convert a 2 point conversion (2PAT) just to tie it up and potentially send the game to overtime.

The Situation

Pittsburgh is behind Baltimore 14-22 with under 2 minutes left to go in the game.  They need a touchdown and a 2PAT just to tie and potentially send the game to overtime.  Steelers QB Ben Roethlisberger threw an apparent touchdown pass to Heath Miller at 1:52 left, but the replay official rules he was down at the 1/2 yard line.  Announcer Cris Collinsworth remarks that it's a good thing the Steelers haven't scored yet since it allows them to burn some more time off the clock, presumably so if they score, the Ravens can't come back and win the game.  When he said that, the message board I was following at the time blew up and said how wrong he was in case the Steelers don't convert the 2PAT.

After a run for no gain, Baltimore takes its last timeout with 1:32 left.  Should they have taken that timeout?  If not, should the Steelers have done so with 2 timeouts remaining?

This leaves the Steelers with potentially conflicting goals.  If they do tie the game, they want to take as much time off the clock.  However, if they fail to do so, they want as much time on the clock as possible if they recover an onside kick to try to win.

The Factors

The most important, and obvious, factor is whether Pittsburgh can score or not.  However, this point is entirely moot.  At the 1 yard line, even with only 1:32 left, the clock is not the enemy.  Also, if the Steelers can't score in 3 more downs, it doesn't matter whether what's left on the clock since the Ravens will be able to run out the clock.

Another factor is the likelihood the Steelers will recover an obvious onside kick.  This number is less than 20% (to be generous).  If the Steelers don't recover the kick, then the clock doesn't matter.

The other factor that's in play is the likelihood of actually converting the 2PAT.  If the Steelers are likely to convert the 2PAT, then the Ravens want more time on the clock to mount a comeback.  If they're less likely to convert, the Steelers should be the one calling the timeouts to give themselves more time to make an onside kick to try one last gasp at a winning field goal.  

The final factor of course is the clock itself, and how that affects the winning chances of the Ravens in regulation play.  

Though these factors are important (to varying degrees), the only actual decision that needs to be made is how much time the Steelers want on the clock.  This requires us to look at this question:

Which is more likely?

1. The Steelers will convert the 2PAT (so they want less time on the clock)
2. The Steelers will fail the 2PAT and recover the onside kick (so they want more time on the clock)

The Calculations

Which is more likely?

The first order of business is to figure out which is more likely to happen, the Steelers convert the 2PAT, or they fail then recover an onside kick.  There's one other possibility, and that's if the Steelers do NOT recover the onside kick.  In that case, it doesn't matter at all how much time is left on the clock, so we need to compare the likelihood of the cases which matter.

Converting the 2PAT

Assuming the Steelers score of course, how likely are they to convert a 2 PAT?  The average team converts an expected 2PAT at a rate of 47.9%.  The other number which can be theoretically looked at is the possibility of converting a 4th and 2 at the goal line.  The 4th down calculator at Advanced NFL Stats has this at 55% chance of success.  Why the discrepancy?  The 4th down calculator uses actual data from NFL games.  This data will probably be biased towards teams who think they have a shot at converting (vs. kicking a field goal) on "normal" 4th downs.  On a 2PAT, teams who go for it, typically have no choice in the matter based on the score of the game. 

This leads us into a game theory question as to whether to try a pass or a run, which is a topic for another time.  From the above links, you can see the success rate of running the ball is much higher than that of a pass.  In the actual game, Steelers RB Le'Veon Bell got hurt on the 2nd down try making it less likely the Steelers would convert with a run, so this number might be pushed down a little further.  The Steelers chose to pass the ball in this situation, but they couldn't have known about the injury when they had to make the decision to run the clock down or not.

For ease of calculation, let's split the difference and call it a 50% chance of converting.

Recovering the onside kick after a failed 2PAT.

Looking at average onside kick data might help determine the likelihood of recovering an onside kick, however those numbers include both obvious and surprise onside kicks.  The Steelers will obviously try an onside kick here, which has a much lower chance of success than a surprise onside kick.  The success rate for an obvious onside kick has been estimated anywhere between 15 and 25%, but data from those attempts are mostly from before a 2009 rule change which limits where players on the kicking team might line up.  For ease of calculation, let's call this 20%.

But we're not done here.  The real comparison is how likely the Steelers are to convert the 2PAT with how likely they are to miss the 2PAT and recover the onside kick.  These two probabilities are independent of one another.  That is, the chance of converting a 2PAT has nothing to do with the likelihood of recovering the kick.  (The fact that they do not have to try an onside kick if they do convert the 2PAT doesn't affect the chances of recovering the kick itself).

Since they are independent probabilities, the probability of both of these events occurring will be the probability of one multiplied by the probability of the other.  In math speak, P(A U B) = P(A)*P(B). The U symbol on the left side stands for Union.  This is asking us for the probability of both of these things occurring.  Our actual expression is:

0.50 * 0.20

0.10 

or a 10% chance of missing the 2PAT and recovering the onside kick.

But how much does it matter?

We can't end here.  The probability of converting the 2PAT is 0.50, the probability of missing the 2PAT but recovering the onside kick is 0.10, and the remaining probability is where the Steelers don't recover the onside kick.  The true probability we need to look at is WHEN IT MATTERS, what's the situation we're probably in.

What on earth does this mean?  Well, recall the Steelers still have a 40% chance of missing the 2PAT and not recovering the onside kick.  If this happens, the amount of time left on the clock doesn't matter at all.  In order to figure out what the probability of each situation is WHEN IT MATTERS, we need to use a simplified version of Bayes Theorem.  Don't worry Bayes Theorem lovers, I'm sure we'll revisit it later.  The simplified version is to take the chance of one event, divided by the chance of all events that matter, in this case, the 2 events we're worried about.

1) The probability, given we're in a situation where it matters, to have converted the 2PAT is 0.50/0.60 = .83333

2) The probability, given we're in a situation where it matters, to have missed the 2PAT and recovered the onside kick is 0.10/0.60 = .16667

Conclusion

Let's see what the implications of this are: When the clock matters, there is an 83% chance we have converted the 2 point conversion and thus want to take as much time off the clock as possible.  Thus, the Steelers (and Cris Collinsworth) were right in wanting to take as much time off the clock as possible.

In the actual game, the Steelers scored a touchdown, but missed the 2PAT.  The onside kick attempt went a little short, putting it in the 40% where the clock didn't matter.

To do a fuller treatment of this question, we also have to ask ourselves, HOW MUCH does it really matter?  If it turns out that the Steelers will win the game 100% of the time it recovers an onside kick, then we need to treat the ~17% chance of recovering it with a lot more respect.