A model-based approach to football strategy.
|September 29, 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 2:00 remaining in the 2nd quarter, and leading 7-3, Kansas City faced 4th and 2 at the Houston 6 yard line. In a controversial decision, Chiefs coach Dick Vermeil chose to go for the first down. However, Priest Holmes was stopped short; and after taking over on downs, the Texans were able to move down the field and score a field goal before halftime.
We can simplify the analysis by supposing that it's 4th and goal from the 2 yard line. This situation is obviously more favorable for going for it than is the actual situation. So, if it turns out to be better to kick the FG on 4th and goal from the 2 yard line, it's certainly better to kick on 4th and 2 from the 6.
According to the
if Kansas City attempts a FG, their probability of winning the game is either 0.765 or 0.678, depending on whether the kick succeeds or fails. Assuming it succeeds with probability 0.985, Kansas City's probability of winning if they kick is
If the Chiefs go for it, their probability of winning the game is either 0.864 or 0.682, depending on whether they succeed or fail. The probability of success is the same as the probability of a successful two-point conversion, which we take to be 0.4. So, if the Chiefs go for it, their probability of winning the game is
There are three factors that lead to this result. First, the probability of picking up the required yardage is only about 0.4. Second, Kansas City leads in the game, which tends to make going for it less desirable. Finally, with under two minutes left in the half, the effect of field position is smaller than usual.
With 0:12 left in the first half, Washington faced 4th and goal at the 2-foot line, and had to decide whether to go for the TD or kick a FG. They trailed 7-0 at the time.
With the half about to end, field position is not an issue. We need to know only Washington's probability of winning the game as a function of the score at halftime.
According to the
Washington's probability of winning the game will be either 0.229, 0.321, or 0.475 at the start of the second half, depending on whether they trail by 7, or by 4, or the game is tied. (Washington's chances are less than 0.5 if the game is tied at halftime because they kick off to start the third quarter.) Assuming that Washington has about a 0.6 probability of scoring from 2 feet, and assuming for simplicity that either a FG or an extra point would be a sure thing, then Washington's probability of winning the game is 0.321 if they kick the FG, versus
Dallas coach Bill Parcells made an puzzling decision to challenge the spot of the ball with 3:42 left in the 3rd quarter, leading 14-3. If the ruling is upheld, Washington has 1st and 10 at the Dallas 16 yard line, and according to the Model , Washington's probability of winning the game is 0.193. If the ruling is reversed, Washington presumably faces 4th and about a foot, still at about the Dallas 16 yard line. In that situation it turns out that Washington should go for it rather than attempt a 34-yard FG. Assuming the Redskins have a 0.75 probability of making the first down, their probability of winning the game if they go for it is 0.166. Thus, even if the ruling on the field is reversed, Washington's probability of winning the game is reduced by just 0.027. Since Parcells couldn't have thought there was more than about a 10% chance of reversing the call, the expected benefit of challenging is a reduction of only about 0.0027 in Washington's probability of winning the game. This small benefit comes at the likely cost of a timeout and Dallas's final challenge. Of course, Parcells might have thought it unlikely that a better opportunity to challenge would arise prior to 2:00 left in the game, and that leading by 11 points, an extra timeout was unlikely to be valuable.
Much has been written about Washington's use of two timeouts to prevent delay-of-game penalties in the third quarter. However, from reading the critiques, one might conclude that if the Redskins had gotten the plays off without having to call timeout, everything would have been fine. To us the more interesting question is, when you're behind by 11 points late in the 3rd quarter, why are you using the whole play clock?
More generally, there were twelve plays during that possession for which the game clock was running at the time of the snap (or would have been if Washington hadn't called timeout). Examination of the video tape reveals that for those twelve plays, an average of 36 seconds elapsed from the end of the previous play to the snap (or timeout). On four of those occasions, more than 40 seconds elapsed. (We're talking about real time, not game time; the latter can be smaller because the game clock stops temporarily after a sack or if the ball carrier goes out of bounds, but the former is the true measure of a team's effort to conserve time.) This is a tempo more appropriate for a team that leads by 11 points than for a team that trails by 11. By simply reducing the average time to 32 seconds, which would have required no change in strategy other than to notice that you shouldn't waste time when you're behind, Washington could have saved 0:48 of game clock on that possession alone — the equivalent of more than a full timeout.
With 12:54 left in overtime, the Rams went for the first down on 4th and 1 at their own 41 yard line. Although the Rams converted, the decision was called "somewhat ill-advised" in the nfl.com Recap .
The footballcommentary.com Dynamic Programming Model is not applicable to overtime. However, we can analyze this decision using a model we built for studying the effect of wind in overtime, as described in a previous article . Obviously there is no wind indoors, but we can use that model to get estimates for a team's probability of winning, if they gain possession early in overtime at a particular yard line.
According to that model, a team that has 1st and 10 at their own 42 yard line early in overtime has about a 0.73 probability of winning. The corresponding probabilities for their own 18 yard line (where the Saints would be expected to start if the Rams punt) and the opponent's 42 yard line (where the Saints take over if the Rams go for the first down and fail) are 0.59 and 0.84, respectively.
So, if the Rams punt, their probability of winning the game is about
Baltimore chose to kick a FG on 4th and goal from the Cincinnati 2 yard line, with 9:47 left in the 1st quarter and no score. According to the
if the FG is good Baltimore's probability of winning the game is 0.583, compared to 0.509 is the FG misses. Assuming it succeeds with probability 0.985, Baltimore's probability of winning the game if they attempt the FG is
On the other hand, if the Ravens go for the TD, their probability of winning the game will be either 0.693 or 0.536, according to whether or not they score. The probability of scoring is presumably the same 0.4 that applies to two-point conversions. Therefore, if the Ravens go for it, their probability of winning the game is
After scoring a TD to go ahead 23-9 with 8:45 left in the game, Baltimore attempted a two-point conversion. However, as can be seen from the Chart , it's clearly better to kick the extra point if you lead by 14, regardless of how much time is left, unless you think your probability of success on the two-point conversion is much higher than the league average. Of course, the Ravens had a very high probability of winning the game at that point, regardless of whether they kicked of went for two.
The Colts led 38-31 with 8:29 left in the game, and had 4th and 1 at their own 46 yard line. They decided to punt. According to the
the Colts's probability of winning is 0.882 if they punt, 0.92 if they go for the first down and make it, and 0.818 if they go for it and fail. Assuming a 0.7 probability of making it, Indianapolis's probability of winning the game if they go for it is
Interesting issues arose in the endgame. After Green Bay used its final timeout at 1:55, Edgerrin James scored on 3rd and goal from the 1 yard line. The Packers now needed two touchdowns in the final 1:49 to tie. The question is, would the Colts have been better off if they had deliberately not scored on 3rd down, and kicked a FG on 4th down? The Packers would then have been able to tie with a TD and a FG, and win outright with two TDs — but would have had only about 1:05 and no timeouts to work with, which is getting close to a physical impossibility.
Actually, the Packers might have been better off letting James score on 1st and goal from the 2 yard line, with 2:54 left. Doing so abandons the hope of trailing by just 10 points, but leaves them with about 2:50 on the clock and one timeout. Of course, if that strategy were actually optimal, the Colts could frustrate it by refusing to score.
With 10:26 remaining in the 3rd quarter, Miami trailed 3-0, and faced 4th and about two inches at their own 47 yard line. Miami coach Dave Wannstedt chose to go for it. However, the Dolphins were stopped, and Pittsburgh took advantage of the excellent field position by scoring a field goal.
According to the
Miami's probability of winning the game is 0.443 if they pick up the first down, and 0.321 if they fail. Assuming that the probability of picking up two inches is about 0.8, Miami's probability of winning the game if they choose to go for it is
Copyright © 2004 by William S. Krasker