THE BLOG
03/18/2010 05:12 am ET Updated May 25, 2011

# Belichick: The Long and Short of It

My colleague Jeff Sagarin of USA TODAY fame has come up with an elegant simplification of whether or not Belichick should had gone for it on 4th down in Sunday night’s Colts-Patriots game. The decision hinges on your estimates of the following three parameters:

MAKE= chance you make first down

LONG = Chance Colts score TD if Pats punt

SHORT = Chance Colts score if Patriots fail to make 1st down after going for it.

Patriots should go for it if

MAKE + (1- MAKE)*(1-SHORT)  > 1 - LONG

MAKE + 1 - SHORT - MAKE + MAKE*SHORT  > 1 - LONG

SHORT(MAKE - 1) >  -LONG

MAKE - 1  >   -LONG/SHORT

MAKE      >  1 - LONG/SHORT.

Thus we are indifferent between punting and going for it if MAKE = 1-LONG/SHORT.

The following table gives some estimates of parameter values for which the decision is a “toss up.” Note if LONG exceeds value in table then Patriots should have gone for it.

 Chance of Making 1st Down LONG SHORT 0.45 0.44 .80 0.50 0.40 .80 0.55 0.36 .80 0.60 0.32 .80

 Chance of Making 1st Down LONG SHORT 0.45 0.385 .70 0.50 0.35 .70 0.55 0.315 .70 0.60 0.28 .70

I think these cases show that at worst, the decision was a tossup. I think most people thought LONG was at least 0.40 and SHORT could not exceed 0.80. My personal estimate was LONG = 0.5 and SHORT = 0.8 so Bill should have gone for it if chance of success was at least 37.5%. Put in your own estimates and  see what decision you would have made.  I think Jeff’s elegant solution gives an easy to use rule of thumb that coaches can use to inform future decision making. The main lesson to be learned from this controversy is that differences in opinion are usually due to differences in estimates of relevant  parameters, not differences in mathematical methodology.