A model-based approach to football strategy.
|January 25, 2005|
In this brief article we use the footballcommentary.com Dynamic Programming Model to analyze some coaching decisions from this weekend's games.
The most controversial coaching decision of the weekend occurred with 13:32 remaining in the 4th quarter of this game. Trailing 31-17, Pittsburgh faced 4th and goal at the New England 2-yard line. Steelers' coach Bill Cowher decided to kick a field goal.
According to the
if the Steelers kick, their probability of winning is 0.05. If they go for the touchdown, their probability of winning is 0.142 if they score but 0.03 if they are stopped. Using 0.4 for the probability of scoring from the 2-yard line (which is the same as a two-point conversion), we find that Pittsburgh's probability of winning if they go for it is
Since any model is subject to modeling errors, it's nice when you don't have to use one. It turns out that, in this case, we can determine the proper decision using a simpler analysis.
We begin with a naive expected-points calculation. The field goal gives 3 points. Going for the touchdown yields 7 points if Pittsburgh is successful and 0 otherwise. If p is the probability of success, then Pittsburgh's expected points if they go for it are 7p. This exceeds 3 provided
An expected-points calculation, supplemented by an adjustment for field position, yields a reasonable decision rule in the opening minutes of a close game. In this case the adjustment for field position obviously favors going for it: If the Steelers score, they kick off to the Patriots, but if they go for it and fail, the Patriots begin at about their own 2-yard line. So it would be right for the Steelers to go for it on 4th and goal from the 2-yard line even early in a close game. Since it's actually the 4th quarter and the Steelers trail by 14 points, it's all the more important for them to do so.
Near the end of the 1st quarter, Atlanta had the ball trailing 7-0, and was playing into a stiff wind on a very cold day. On 3rd and 1 at the Philadelphia 31-yard line, the Falcons ran for no gain. The Eagles then called timeout at 0:01, rather than let the quarter end, to prevent Atlanta from getting the wind for a field-goal attempt. The Falcons then had no choice but to go for it on 4th down.
Now, it only makes sense to prevent Atlanta from kicking a field goal if the field goal attempt is Atlanta's best strategy. We can investigate this question using the Model. If Atlanta tries the field goal, their probability of winning the game is 0.332 if it's good, but 0.242 if it misses and the Eagles take over at their 39-yard line. If Atlanta goes for the first down, their probability of winning is 0.347 if they make it but 0.25 if they fail. So making the first down is better than making the field goal, and going for the first down and failing is better than missing the field goal. Since the likelihood of picking up the first down exceeds the likelihood of making a 49-yard field goal (even with the wind), it's clear that Atlanta should go for it even if Philadelphia lets the quarter expire.
At best, then, Philadelphia wasted a timeout. At worst (if Atlanta might have kicked), Philadelphia wasted a timeout and forced Atlanta into a superior strategy.
Copyright © 2005 by William S. Krasker