Bills vs. Patriots: should Rex Ryan, Bill Belichick have onside-kicked?

In the first quarter of this past Sunday's Buffalo Bills versus New England Patriots game, with the Patriots finishing off a second consecutive touchdown drive to go up 14-7, Aaron Williams committed a taunting penalty, earning Stephen Gostkowski a kickoff from the 50-yard line. In the second quarter, a taunting penalty on Malcolm Butler after a missed extra point gave Jordan Gay the same kickoff position, from midfield, in a 21-13 game. In both instances, the kickers easily hit a touchback.

But weren't both coaches wasting prime field position? While kicking off from the 50 all but guarantees a touchback, it also affects one major game-changing tool: the onside kick.

Ordinarily, an onside kick is a very risky action, usually saved for high-leverage situations like late-game comebacks. That's because the cost of failure is very high: assuming you kick off from the 35, a failed onside kick awards the ball with less than half the field to go, and a success will likely leave the kicking team a significant portion of field remaining before they can score points. Onside kicks are among the most difficult plays in football. Per SportingCharts.com, only five kickers successfully onside-kicked in all of 2014, with an overall success rate of just 35 percent.

The taunting penalties, however, changed the situation in Week 2. By moving the ball forward 15 yards, the outcomes suddenly draw much more in favor of the kicking team. Let's assume an onside kick with an average depth of 13 yards (slightly more than the 10 yards necessary to avoid a penalty); if the kicking team recovers, they'll have the ball inside of their opponent's 40-yard line, putting them right on the edge of field goal range. If the receiving team comes down with the ball, they would start roughly 20 yards beyond where a touchback would place them - a significant leap, but one that still allows ample room for a defense to hold the offense.

Coaches often use decision matrices to make decisions on risky plays like onside kicks, two-point conversions, and fourth-down attempts. These matrices calculate the potential value of each outcome of a risky play given factors like the chance of success and the placement of the ball, giving a Go/No Go answer for the coach. We can build a similar model for onside kicks.

The chart below allows you to evaluate the risk of making an onside kick from the fifty. Using ESPN analyst Brian Burke's Win Probability Calculator, it determines the expected point value of a team receiving the ball at the 20, versus receiving it at, for our example's sake, the 37 or an opponent receiving the ball at that 37-yard line. By adjusting the probability of a successful recovery, we can calculate the overall expected value of an onside kick in this situation and find which is better. As you follow the X axis from left to right on the chart, the hypothetical probability of an onside kick increases from zero to 100 percent. The chart suggests that the mathematical break-even point for onside kicking from the 50 versus kicking a touchback exists if an onside kick has a roughly 24 percent chance of success. Any higher, it argues, and an onside kick is the optimal choice. If the probability is lower, than a touchback is the right call.

Let's put this calculation into context using Sunday's game. With the Patriots taking a 14-7 lead and capitalizing on short fields, an onside kick could lead to a 21-7 gap in the first quarter, turning the game into even more of a rout. They gave the ball to the Bills, who took a three-and-out and allowed the Patriots to score on their next possession regardless. Buffalo, meanwhile, could have recovered the ball at 21-13 following their touchdown and driven down the field for a potential equalizer. Instead, they allowed the Patriots to drive down the field and kick a field goal.

I'm not suggesting that onside kicking from the 50 is always the right call. While math might tell us it trumps the other options, there are factors like the individual kicker's performance, the performance of the hands unit, and the score differential that can make the decision more complicated. Math would suggest that a team should always attempt to convert a 4th-and-2 if the team averages three or more yards per play, but that doesn't work out in practice. Still, either team could have changed the flow of what ultimately became a one-score game, had either coach been willing to test a risky strategy when given favorable circumstances.