A model-based approach to football strategy.
|October 20, 2004|
In this article we discuss some notable coaching decisions from selected games. Many of the analyses use the footballcommentary.com Dynamic Programming Model .
With 1:49 left in the 2nd quarter, the Rams faced 4th and goal with a foot to go for a touchdown. They trailed 14-7 at the time. Commentator John Madden said that he would always take the field goal, which we generously interpret as a statement about himself rather than a strategy prescription. St. Louis coach Mike Martz chose to go for it.
says that if the Rams go for it, their probability of winning the game is 0.472 if they make it and 0.233 if they are stopped. (The first number is less than 0.5 partly because extra points aren't a sure thing, but more importantly because St. Louis kicks off to start the second half.) If we assume that there is a 0.6 probability of gaining the foot, then the Rams' probability of winning the game if they go for it is
One can check that going for it remains the better decision as long as the probability of scoring exceeds 0.36. This required probability is smaller than the 3/7 one might expect using an expected-points calculation, because down by 7 points, a touchdown is disproportionately better than a field goal. (We discussed this issue in more detail in a previous article .)
With 9:54 left in the 3rd quarter, and the score 14-14, the Rams faced 4th and about a half yard at their own 35 yard line. "That's one you have to punt," said Madden, and in this case the Rams did indeed punt. But the
says that if the Rams punt, their probability of winning is 0.477. If they go for it their probability of winning is 0.54 or 0.406, depending on whether or not they make it. Assuming a 0.7 probability of gaining the half yard, we obtain
Philadelphia coach Andy Reid, who occasionally calls for onside kicks in unusual situations , elected to try an onside kick to start the second half. The Eagles led 13-0 at the time.
According to the Model , if Philadelphia kicks deep, their probability of winning the game is 0.877. If they kick onside, their probability of winning is 0.912 if they recover the kick but 0.854 if Carolina gains possession. One can check that for the onside kick to make sense, Philadelphia's probability of recovering it has to exceed 0.39.
It's interesting to contrast this with our finding for the Week 4 game between Tennessee and San Diego . With 12:26 left in that game, Tennessee coach Jeff Fisher called for an onside kick, and we found that he needed only a 0.29 chance of success to justify it. The difference is that the Titans trailed by 14 points. Leading by 13, the onside kick requires better odds.
There is actually some reason to suspect that Philadelphia's probability of recovering the kick was around 0.39. In a Nash equilibrium (a concept we discussed in a previous article ), the kicking team's strategy is usually a randomized strategy, in which they kick onside with some probability that is neither zero nor one. (There are obvious exceptions, such as when they trail with under 2:00 left and have no timeouts.) However, the kicking team is willing to randomize only if kicking deep and kicking onside yield the same probability of winning the game. So in equilibrium, the receiving team chooses its players and formation to make kicking deep and kicking onside equally attractive. We may discuss this equilibrium in more detail in a later article.
With 13:13 left in the game, New Orleans trailed 31-21, and faced 4th and 1 at the Minnesota 27 yard line. The Saints elected to attempt a field goal. According to the
their probability of winning is either 0.138 or 0.057, depending on whether or not the kick is good. The average NFL place-kicker makes about 68% from 45 yards, and if we use this value, it follows that the Saints' probability of winning if they choose to kick is
If instead New Orleans goes for the first down, their probability of winning is 0.161 if they make it and 0.061 if they fail. Assuming a 0.7 success probability, it follows that the Saints' probability of winning if they go for it is
During Minnesota's possession that began at 13:08 in the 4th quarter and culminated at 7:04 with a touchdown that gave them a 38-24 lead, one of the commentators complimented them for eating up so much of the clock. Actually, during that drive the Vikings behaved as though they were unaware of the desirability of running time off the clock. There were nine plays during that drive when the game clock was running at the time of the snap (or when Minnesota called timeout). On those plays, the Vikings left a total of 75 seconds on the play clock.
Finally, during Minnesota running back Moe Williams's 49-yard run with 1:56 left in the game, neither he nor Fred Thomas, the New Orleans defender who tackled him, seemed aware that the only way the Vikings could lose was for Williams to score. If he scores, the Saints can hope for a quick TD and a recovered onside kick, but if Williams is tackled, the Vikings take a knee twice and the game is over. Once he has picked up the first down, Williams should down himself. Of course, pigs will fly before a ball carrier deliberately passes up a touchdown.
On the opening drive of the game, the Jets reached 4th and 1 at the San Francisco 42 yard line, and elected to punt. According to the
if the Jets go for it, their probability of winning the game will be 0.534 if they make it and 0.447 if they fail. Assuming a 0.7 probability of making it, the Jets' probability of winning if they go for the first down is
Twice during the game the Jets attempted a two-point conversion, first with 3:15 left in the 3rd quarter, trailing 14-9 ("Too early in the game for me," said the television commentator) and again with 11:41 left in the 4th, leading 15-14 ("Chasing the points here"). However, as can be seen from the Chart , coach Herman Edwards was correct in both cases, assuming the probability of a successful two-point conversion is 0.4. In the first case, the decision to go for it raised the Jets' probability of winning from 0.272 to 0.282, and in the second case from 0.59 to 0.6. Actually, in order for it to be correct to go for two, the Jet's probability of success only has to exceed about 0.28 in both cases.
With 0:25 left in the game, the 49ers took over at their own 20 yard line, trailing 22-14 with one timeout remaining. Tim Rattay was sacked and fumbled, but San Francisco recovered. The 49ers could have called timeout with as much as 0:14 on the clock. Had they done so, there would have been time for a pass down the sideline, possibly followed by a Hail Mary. However, San Francisco didn't call timeout until 0:02, and could run only one play.
With 3:52 left in the game, Jacksonville trailed 16-14, and faced 4th and 1 at their own 33 yard line. Jacksonville coach Jack del Rio decided to go for it.
If the Jaguars punt, their main hope for winning the game is to force the Chiefs to go three-and-out. In that case, after the exchange of punts, Jacksonville can expect to get the ball back somewhere near where it is now. So whether they punt or go for it, Jacksonville's aim is to get 1st and 10 at around their own 33 yard line.
If the Jaguars go for it, they have about a 70% chance of picking up the first down. Their chances of forcing the Chiefs to go three and out are much smaller. (So far this year Kansas City has gone three-and-out on only 33% of their possessions; the corresponding figure for 2003 was 29%.) So Jacksonville's chances of getting to 1st and 10 are better if they go for it than if they punt. Moreover, if they go for the first down and make it, they will have much more time than if they get the ball back after a Kansas City possession and punt. It seems clear, therefore, that del Rio's decision go for it was correct.
Leading 17-13 with 8:45 left in the 4th quarter, the Bills chose to settle for a field goal on 4th and goal at the Miami 2 yard line.
According to the
if they make the FG, their probability of winning the game is 0.855, whereas if they miss and Miami takes over at the 20 yard line, Buffalo's probability of winning is 0.767. If the probability of making the FG is 0.985, then Buffalo's probability of winning if they attempt the FG is
If instead the Bills go for the TD, their probability of winning is 0.963 if they score and 0.818 if they are stopped short. The probability of scoring is presumably the same 0.4 applicable to two-point conversions. Therefore Buffalo's probability of winning if they go for the TD is
An interesting situation would have arisen if Miami had scored a late TD to send the game to overtime: The wind was so strong (Rian Lindell's 43-yard attempt barely made it to the goal line) that it would have been correct for the coin-toss winner to take the wind rather than the ball. Of course, there is no chance that either coach would have actually done so.
The Seahawks make the right decision by going for two after they scored a touchdown with 11:05 left in the game, to close the deficit to 20-15 prior to the try. According to the Model , their probability of winning the game is 0.232 if they go for two, and 0.215 if they kick. This calculation assumes a 0.4 probability of a successful two-point conversion, but going for two remains optimal as long as the probability of success exceeds 0.2.
With the Patriots leading 30-20 and 0:04 left in the game, Seattle threw an incomplete pass. However, New England was called for pass interference in the end zone. Although the clock read 0:00, the game can't end on a defensive penalty, and Seattle was allowed another play from the New England 1 yard line. The Patriots then stopped Mack Strong for no gain, and the game was over.
New England should have let the Seahawks score on the final play. There wasn't even a theoretical possibility that Seattle could win the game, and points scored or allowed are so far down the list of playoff tiebreakers that New England shouldn't have taken any risk of injury in order to preserve the margin.
Copyright © 2004 by William S. Krasker