A model-based approach to football strategy.
|December 27, 2005|
The Week 16 game between Atlanta and Tampa Bay went deep into overtime. With 1:08 left in the extra session, Atlanta faced 4th-and-2 at their own 24-yard line, and had to decide whether to go for it or punt. Atlanta coach Jim Mora decided to punt.
The Falcons were 8–6 going into their Saturday game against Tampa Bay. By the time the overtime started, Dallas had already won to go to 9–6, and Washington was about to do the same. Another playoff contender, Minnesota (8–6), would not play its Week 16 game until Sunday.
The Falcons understood that they would be eliminated from the playoffs if they lost to the Buccaneers. Even if Atlanta defeated Tampa Bay, the Falcons would have to beat the Panthers in the finale to make the playoffs. So for purposes of calculating Atlanta's relative likelihood of making the playoffs with a win or tie versus Tampa Bay, we can assume that Atlanta defeats Carolina in Week 17.
Barring an improbable loss at home against New Orleans in Week 17, even a tie against Atlanta would earn Tampa Bay a playoff berth. Therefore, if the Buccaneers gain possession, they will not want to risk a loss. This implies that if the Falcons punt with 1:08 left, their probability of winning will be very small.
We will assign the following probabilities: 0.4 for Minnesota defeating Baltimore on the road (Week 16), 0.85 for Dallas defeating St. Louis at home (Week 17), 0.75 for Washington defeating Philadelphia on the road (Week 17), 0.5 for Minnesota defeating Chicago at home (Week 17), and 0.87 for Tampa Bay defeating New Orleans at home (Week 17). From these probabilities it follows that Minnesota has probability 0.2 of finishing at 10–6. As we will see, our conclusions would be unlikely to change as a result of reasonable alterations to these input probabilities.
If Atlanta and Tampa Bay play to a tie (but Atlanta goes on to defeat Carolina), Atlanta finishes 9–6–1. Since Atlanta would be 0–1–1 versus Tampa Bay, they would finish third in their division. The Falcons would then get a Wild Card only if Minnesota, Washington, and Dallas all fail to reach 10–6. This has probability
|(1−0.2)(1−0.75)(1−0.85) = 0.03.|
If the Falcons defeat the Bucs (and Panthers), they finish 10–6. The Falcons win their division if Tampa Bay loses at home to New Orleans in Week 17 (probability 0.13). Otherwise, the Falcons finish second in their division. Atlanta's Wild Card possibilities, governed by the NFL Tiebreaking Procedures, become complicated, in some cases coming down to "strength of victory." We won't list all the scenarios in which 10–6 earns Atlanta a playoff berth. However, at a minimum, Atlanta certainly gets a Wild Card if at most one of Minnesota, Washington, and Dallas reaches 10–6. Therefore Atlanta's probability of earning a Wild Card if they finish 10–6 but don't win their division is more than
|0.2(1−0.75)(1−0.85) + (1−0.2)(1−0.75)0.85 + (1−0.2)0.75(1−0.85) + (1−0.2)(1−0.75)(1−0.85) = 0.298.|
It follows that at 10–6, Atlanta's probability of making the playoffs either as a division winner or a Wild Card is more than
|0.13 + (1−0.13) 0.298 = 0.39,|
which is thirteen times as much as their probability of making the playoffs if they finish 9–6–1.
The explanation for the disparity is that at 10–6, Atlanta either wins the division, or at worst finishes second. At 9–6–1, Atlanta finishes third. Under the tiebreaking procedures, this makes a huge difference. The third-place team in a division is precluded from the #5 playoff seed, and can vie for the #6 seed only if the second-place team in that division has already secured the #5 seed.
The rest of the analysis is straightforward. Since a win is thirteen times as valuable as a tie, Atlanta's objective is to maximize
|pw + pt /13 ,|
where pw is the probability of a win and pt is the probability of a tie. Now, Atlanta's probability of picking up the first down, if they go for it, is about 0.55. If at that point they choose to go all out for a win, then (conditional on picking up the first down) our model for the
says that their probability of winning the game is about 0.23. Therefore, Atlanta's probability of winning the game if they decide to go for the first down, and then go all-out to score, is
Copyright © 2005 by William S. Krasker