Fourth and 2

Last night at the end of the Patriots-Colts game, leading by 6 with a little more than 2 minutes to play Bill Belichick decided to go for a first down at their own 34 yard line. Conversion fails, Colts Score, Sports commentating mayhem ensues.

Let us at least look at the basic problem being proposed in this case. The simplest way to look at this is the following equation:  You go for it if  x + (1-x)*y >  z

x: probability of Patriots converting the 4th down.

y: probability of Colts failing to score a TD from 34 yards.

z: probability of Colds failing to score a TD from the 70-80 yards.

If the Patriot’s chances of converting combined with their chances of stopping the colts when they fail to is greater than the chances of the colts scoring from further then they should go for it. For various values of x,y,z that are very realistic going for it proves to be a good decision.

We can complicate it by asking what x,y,z really represent. What’s we’re implicitly weighing is the efficacy of the (Patriots Offense vs Colts Defense)  against the (Colts Offense vs Patriots Defense)  The better both teams are performing offensively the better the result for going for it versus punting. Another factor that arises in this question of going for it is the the value/penalty of having to go an extra 40-50 yards.  y is essentially the Touchdown function at 34 yards. Whereas z is the Touchdown function at 70 yards. (y= f(34) , y=f(70) ) Yards to end-zone have a diminishing return. Going from the 5 yard line to the 10 yard line is a much greater penalty than going from the 45-50. For better offensive teams the penalty for each incremental yard is less than it would be for poor offensive teams. (In the extreme scenario the Colts + Raiders both have a HIGH likelihood of scoring from 1 yard out, but the Colts have a significantly greater likelihood from scoring 50 yards out than the Raiders would)

I don’t propose to have an answer as to which is actually the correct decision but simply ask that people analyze and consider the different factors and solutions in this scenario. It drives me crazy that  most of the sporting world refuses to do any analysis but instead exclaim “Play the percentages and punt.” Exactly what percentages are people referring to here. Surely there must be a study that the coaches, sports writers and radio talk show hosts have all studied. My guess is that they simply relied on conventional practices and based on history believe them to be best.