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24 Oct 2004

*Guest Column by William Krasker*

Each week at * footballcommentary.com** *I analyze some notable coaching decisions. In this article I re-examine a few of those decisions. In some cases I have expanded on the analysis I presented originally, and in other cases I have modified my original conclusions as a result of comments from readers.

Trailing 7-3 with 12:10 left in the 2nd quarter, the Vikings had 4th-and-goal from inside the Philadelphia 1-yard line. Minnesota coach Mike Tice decided to settle for a field goal.

My analysis of this decision is an application of the *footballcommentary.com* Dynamic Programming Model, which provides a framework for estimating a team's probability of winning the game in various situations. Here, for simplicity, I will assume that an extra point or a 19-yard field goal is a sure thing. This has no meaningful effect on the results.

If the Vikings kick the field goal, their probability of winning the game is 0.42. If they go for it, their probability of winning is either 0.551 or 0.364, depending on whether or not they make it. Assuming that the probability of scoring from inside the 1-yard line is 0.6, we find that Minnesota's probability of winning if they go for it is

My preferred measure of the impact of an in-game coaching decision is the amount by which it changes the probability of winning the game. Here, Mike Tice lowered his team's probability of winning by 0.056. This is a very large amount for a coaching decision, and is in fact the same order of magnitude as some serious blunders by players. For example, a fumble at midfield early in a game lowers the team's probability of winning by between 0.05 and 0.1.

(The biggest coaching blunder I can dream up is kicking a field goal from the opponent's one-inch line on the game's final play, when trailing by four points. This absurd decision lowers the team's probability of winning by at least 0.6. I have no idea what is the biggest blunder a coach might actually make.)

Part of the reason why going for it is better than kicking in this situation is field position. If Minnesota kicks the field goal, or if they go for the touchdown and make it, the Eagles will receive a kickoff. However, if the Vikings go for it and come up short, Philadelphia takes over near their own 1-yard line. It's interesting to find out how much of the advantage that going for it has over kicking is due to field position. So suppose that the Vikings go for the touchdown and fail. If the Eagles could then receive a kickoff rather than start near their 1-yard line, Minnesota's probability of winning would be 0.327 rather than 0.364. The difference between these numbers comes into play only when the Vikings fail to score, which happens with probability 0.4. It follows that of the 0.056 by which going for it improves Minnesota's probability of winning, the portion that is due to field position is

Following Week 2, I examined a decision Denver coach Mike Shanahan made near the end of Denver's game at Jacksonville. Trailing 7-6 in the final minute, with no timeouts, Denver was driving toward the Jacksonville goal line. On 2nd-and-10 from the Jacksonville 24-yard line, Denver ran the ball for a 1-yard gain, and Jacksonville used its second timeout to stop the clock with 0:37 left. In one of the more controversial decisions of the week, Shanahan decided to run one more play before bringing in the field goal unit. Unfortunately for Denver, Quentin Griffin fumbled, and the game was over.

My original analysis focused on the benefit of picking up a few more yards prior to the field goal attempt. We all have a tendency to think of a team as being either "in" field goal range or "out of" field goal range, when in fact there is nothing magic about any particular-yard line. NFL place-kickers make about 76% of their field goals from 41 yards, but the chance of success rises about 1% per yard as the distance decreases. Therefore, to a first approximation, a play that (expectationally) gains 3 yards increases the probability of making the FG from 0.76 to 0.79.

As an offset to this benefit I considered the possibility of a lost fumble. If we let *p*_{f} denote the probability of losing a fumble, we find that Denver's probability of taking the lead is *p*_{f} )0.79

Readers pointed out one factor that I glossed over, and one that I omitted entirely. In my original analysis I ignored the possibility of a penalty. Even if both teams have the same probability of a penalty, it hurts Denver on net, because the benefit of being five yards closer is somewhat less than the harm from being five yards farther out. In addition, the chance of a penalty on the offense -- particularly the dreaded false start -- is presumably larger than the chance of a penalty on the defense. Nevertheless, the possibility of a penalty doesn't reduce Denver's chances of taking the lead by very much. Even if we ignore the possibility of a defensive penalty, assume a 5% chance of a penalty on the offense, and use 0.66 for the success probability on a 46-yard field goal, the possibility of a penalty reduces Denver's probability of taking the lead by at most

Finally, in my original analysis I casually stated that if Denver runs another play, it forces Jacksonville to use its final timeout. I cited this as an additional benefit of running another play. My assumption was that Jacksonville would want to preserve time for a potential scoring drive of their own. I didn't consider the possibility that Jacksonville might let the clock run and hope that Denver can't get the field goal off before time expires. Essentially, if the Jaguars call timeout, they are betting that they can score in the final half minute, whereas if they don't call timeout, they're betting that Denver won't have time to kick the field goal. It's not clear which of these is a better bet; neither one is very good. At any rate, if it's optimal for Jacksonville to call timeout, then forcing them to do so is indeed an additional benefit of running another play. But if it's optimal for Jacksonville to let the clock run, the clock-management benefits of running another play are smaller, or possibly negative.

In summary, running another play gains a few yards on average, which is of significant benefit. However, this benefit is substantially offset by the possibility of a fumble or penalty. I now feel this decision was a very close call.

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.

In my original analysis of this decision I chose to analyze a simpler situation, whose analysis would be less dependent on the assumptions of the Model. However, as a reader pointed out, the thought process that allowed me to draw conclusions about the actual situation contained a flaw. So here I'll simply apply the Model directly.

According to the Model, 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 try to pick up the first down, their probability of winning the game is either 0.848 or 0.681, depending on whether they succeed or fail. We will assume that the probability of picking up the required 2 yards when near the opponent's goal 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

The result would be different if there were more time remaining in the half. According to the Model, the crossover point occurs with about four minutes left. With more time than that, it's best for the Chiefs to go for it, because Jacksonville's poor field position if the Chiefs fail to pick up the first down becomes an important factor. The closer it is to halftime, the less important field position becomes. Indeed, very close to halftime, field position becomes irrelevant; all that matters are Kansas City's probabilities of scoring before halftime if they go for it or kick.

Actually, even with a lot of time remaining, going for it never becomes very much better than kicking in this particular case. The reason is that a 4-point lead is favorable for kicking a field goal. When you're ahead by 4 points, a field goal gives a disproportionate benefit relative to a touchdown, compared to what one might expect by comparing their point values.

San Francisco's long opening drive reached a decision point with 9:51 left in the 1st quarter, when the 49ers faced 4th-and-14 at the St. Louis 30-yard line. San Francisco chose to punt, but the kick went into the end zone for a touchback. The 49ers netted 10 yards on the kick.

Obviously, when they decided to punt, the 49ers were hoping for better. However, it's not realistic to expect a pooch kick to pin the opponents back near their own goal line. That happens sometimes, but it seems like touchbacks occur about as often. I generally assume that following a pooch kick, the opponent's expected starting field position is their 10-yard line. If anything, this assumption is generous to the kicking team.

So, by punting the 49ers should have been expecting to net about 20 yards. It seems that coaches systematically overvalue 20 yards at that point on the field. We will use the Model to compare the decision to punt to the alternatives of attempting a field goal, or going for the first down.

According to the Model, if San Francisco punts, their probability of winning the game is 0.485.

If instead they attempt a FG, their probability of winning the game is either 0.545 or 0.45, depending on whether or not the kick is good. One can check that any success probability in excess of 0.37 is sufficient to make the FG attempt superior to the punt. (By this I mean that

Alternatively, San Francisco could have tried to pick up the first down. For the sake of argument, suppose that if the 49ers go for the first down and make it, their expected gain on the play is 16 yards. Then the Model says that if the 49ers go for it, their probability of winning the game is 0.59 if they succeed and 0.46 if they fail. One can check that San Francisco needs at least a 0.2 probability of success to make going for the first down superior to punting, and hence the choice between punting and going for it is a close call. But instead of punting, the 49ers should have attempted a field goal. The expected gain from punting is so small that it's worth trying the FG even if it's chances of success are relatively small.

In Week 5, after falling behind Seattle 27-10 with 8:47 left in regulation, St. Louis completed an astonishing comeback that culminated in an overtime victory. Much of the post-game commentary focused on the the way the comeback was facilitated by Seattle's poor clock management. Leading 27-24 with 2:40 left in regulation, on 1st-and-10 at their own 36-yard line, Seattle threw an incomplete pass which stopped the clock at 2:35. Seattle's motive for passing is clear: Since St. Louis has no timeouts, one more Seattle first down settles the matter. However, as Russell Levine noted in an article at * Football Outsiders, * if Seattle simply runs the ball three times, being careful to use the play clock each time, their worst case is that the Rams get the ball back with just under 0:30 remaining. A St. Louis score is then very unlikely, though not impossible.

I found the argument that Seattle should have run on first down sufficiently convincing that it doesn't require further analysis. But what should Seattle do after throwing the incomplete pass on first down? To try to analyze this question, I built an * ad hoc * dynamic programming model that operates at the level of individual plays. A model of that sort would not normally be computationally feasible, but it is in this case because the Rams have no timeouts, and a Seahawk first down ends the game. Here are the main features of the model: On fourth down, Seattle punts, and St. Louis's probability of winning is proportional to the time on the clock when they get the ball. On first, second, or third down, Seattle chooses whether to run or pass. There is a probability distribution for the yards gained on a run. I assume a pass is thrown far enough to pick up the first down, and that its probability of completion is a decreasing function of the number of yards to go. The model includes the possibility of a sack. The number of seconds consumed on a play is modeled in detail, including its interaction with the two-minute warning.

Using plausible parameter values, the model says that the correct play on 2nd-and-10 at the 36-yard line, with 2:35 left, is to run the ball and let the clock wind down to the two-minute warning. The strategy then becomes more interesting. On 3rd down, the model says to pass if there are between 5 and 10 yards to go for a first down, and run otherwise. This unexpected result actually makes some sense. If it's 3rd-and-short, Seattle has a good chance to pick up the first down even with a run, which has the additional virtue of keeping the clock moving. If it's 3rd-and-long, neither a run nor a pass has a high probability of gaining the required yardage, so once again it's better to guarantee that the clock runs. But in the intermediate range, the chances of picking up the first down by passing are good enough to risk stopping the clock.

As a reader pointed out, when the model says to run, you have to assume your opponent knows you are going to run. Similarly for passing. So one should probably assume that the expected gain on a running play, and the probability of a completion on a pass play, are smaller than they would be if the opponents were unsure of what's coming. For this reason, is would be better if the model allowed for * randomized * strategies, in which the offense runs with probability *q* and passes with probability *q*.

Finally, one issue that often comes up in the discussions of these strategy analyses on Football Outsiders is why I almost always write that the correct decision would be to go for the first down on 4th-and-1.

I addressed those questions in my May 29 article "Some Common Strategy Errors" as well as in the "Go For It" Tables on footballcommentary.com. As the article says, "teams should always go for it on 4th-and-1 if they trail, are not in field goal range, and are not too deep in their own end. Closer to either goal line, or when leading, or with more than a yard to go for a first down, it is still often correct to go for it, but these cases must be considered on an individual basis."

Given the position on the field, the score, and the time remaining, the Tables tell you what your probability of picking up the first down has to be to justify going for it. As an example, if you're on your own 40-yard line, trail by 7, and there is 27:00 left in the second half, then you should go for it if your probability of picking up the first down exceeds 0.47. For most teams, that would correspond to about 3 yards to go. Of course, the gain from going for it isn't significant unless your probability of picking up the first down significantly exceeds 0.47. On 4th-and-2, even I am not going to make an issue of it if the coach decides to punt. On 4th-and-1, though, I feel punting is a mistake.

If we still assume that there is 27:00 left and we are at out own 40, but now suppose we LEAD by 7, then we need a 0.56 probability of making it to justify going for it. In this case, I'm not going to make a fuss even if the coach punts on 4th-and-1.

I sympathize with those who wonder if the Model always says to go for it. It must seem that way, but of course it isn't. For example, if your opening drive stalls at your 20-yard line and it's 4th-and-5, the Model says to punt. Coaches make that decision correctly all the time, but it isn't interesting enough to appear in a weekly strategy analysis. Since coaches often punt when they should go for it, and almost never go for it when they should punt, it's not surprising that I mainly discuss the punts.

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