A model-based approach to football strategy.
|February 6, 2008|
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With 6:49 remaining in the 3rd quarter, the Patriots led 7-3, and faced 4th-and-13 at the Giants' 31-yard line. New England coach Bill Belichick made the remarkable decision to go for the first down rather than attempt a 49-yard field goal. We will examine that decision using the footballcommentary.com Dynamic Programming Model.
We begin by listing some of the outputs of the model. If the Patriots attempt a field goal, their win probability is 0.76 if the kick is good but 0.64 if it misses (and the Giants take over at their 39-yard line). The latter probability is also roughly New England's win probability in case of a sack. An incomplete pass gives New England a 0.65 win probability. A New England first down at the Giants' 13-yard line raises New England's win probability to 0.82. This could be regarded as an average outcome, conditional on the play resulting in a first down but not a touchdown. Finally, if the Patriots score a touchdown on the play, their win probability is 0.88.
The average NFL place-kicker makes a 49-yard field goal about 61% of the time. Using that estimate, New England's win probability if they attempt the field goal is approximately
|0.61(0.76) + (1−0.61)(0.64) = 0.713.|
A higher likelihood of making the kick might be warranted by the nearly ideal conditions at University of Phoenix Stadium, with the roof closed. Alternatively, Belichick might have lacked confidence in place-kicker Stephen Gostkowski. A reader who feel that Gostkowski's chances differed from 61% is invited to insert his or her own probability into equation (1).
Information provided by Aaron Schatz at Football Outsiders suggests that the likelihood of picking up a first down (including by defensive penalty) on 4th-and-13 is around 25%. Consistent with that figure, we will approximate the range of outcomes if New England goes for it as follows: 8% chance of a touchdown, 17% chance of 1st-and-10 at the 13-yard line, 69% chance of an incomplete pass, and 6% chance of a sack. With these inputs, New England's win probability if they go for the first down is
|0.08(0.88) + 0.17(0.82) + 0.69(0.65) +0.06(0.64) = 0.697,|
which is lower than the win probability obtained by attempting a field goal, but not enormously so. However, readers are welcome to supply their own inputs into equation (2).
Copyright © 2008 by William S. Krasker