Kicking from the 50--what's the right play?

I can't have been the only one cringing towards the end of halftime last week in expectation of a personal foul penalty about to be wasted by blasting the upcoming kickoff through the endzone.

Can't just complain about it based on intuition, though--what do the numbers say?

There are basically three choices when kicking off from the 50 after a personal foul:

1) kick it through the endzone for the easy touchback

2) kick onside

3) kick short of the goal line, either high or squibbed.

I'll compare (1) and (2)--I have no data on kicks in play from the 50 so I can't analyze (3). Mathlete? Anyone?

The essential and oft-copied graph:

Case #1 is easy to analyze: the receiving team gets the ball at the 25 That's somewhere between 0.5 and 0.6 expected points for the receiving team. Since I go into this convinced kicking onside is the right choice I'm going to give the benefit of the doubt to the other side everywhere, so we'll call it 0.5

For case #2 I'm going to assume that the ball ends up at the 40--the average is probably a little over ten yards on an onside kick but again I want to err on the conservative side. There's also a tiny possiblity of the kicking team actually returning the kick instead of falling on it; if anyone has any data that indicates this materially impacts the calculation please let us know in the comments.

Let the probability of the kicking team recovering the kick be x. If the receiving team recovers they get the ball at their own 40, worth 1.4 expected points. If the kicking team recovers they get the ball at the opponents' 40, worth 2.5 expected points.

Expected points for the receiving team, then, are (1.4) (1-x) - (2.5) x, or

1.4 - (3.9)x

Breakeven is when this equals 0.5:

1.4 - (3.9)x = 0.5

x = (0.9)/(3.9) = .23

give or take 2 or 3 percentage points from any errors I made eyeballing the graph.

That's pretty close to the actual rate of recovery on onside kicks, but remember that most onside kicks are taken in desperation mode at the end of games when the receiving team fully expects them. According to advancednflstats.com the rate of recovery on a surprise onside kick is 55-65%. (That, of course, is for NFL players and NFL rules; I'm not aware of any NCAA data on this.)

If the receiving team doesn't defend the onside kick like they would in an end-of-game situation, kicking onside from midfield is a little over one point better than the touchback.

If the receiving team does send the hands team on and defend the onside kick, it's just about a push, with perhaps a very slight advantage to the touchback.

My proposal is:

1. kick onside, always, from the 50 until teams start defending it.

2. if the receiving team sends out their hands team to defend the kickoff, kick short of the goal line and cover the kick, making use of the short field and the absence of the usual blockers.

--------

Under the pre-2012 rules, with the touchback only coming out to the 20 and the onside kick going to the opponents' 45 instead of the 40, the break even point was somewhere around 35%. It probably still made sense to kick onside, but the situation wasn't as clear-cut as it seems to me to be now.