Aaron Rodgers is the best quarterback in the NFL, and should be the highest-paid. We can all agree on that. But this guest column by Kevin Kolbe explains why salaries for other quarterbacks are all out of whack.

*Guest Column by William Krasker, reprinted from footballcommentary.com*

In this article we will analyze some notable coaching decisions from the 2003 NFL season. These are not necessarily the most momentous decisions, but they all lend themselves to discussion and analysis with more general application.

Three of the analyses use the *footballcommentary.com* Dynamic Programming Model. Those who are unfamiliar with the application of the Model can find more detailed examples in our analysis of Super Bowl XXXVIII.

Panthers vs. Cowboys

Colts vs. Buccanneers

Packers vs. Eagles

Lions vs. Vikings

Colts vs. Texans

Texans vs. Jaguars

In the wildcard round of the playoffs, Carolina led Dallas 13-3 with 0:04 remaining in the second quarter, and had 2nd and goal at the Dallas 1 yard line. Carolina coach John Fox decided to kick a field goal, settling for a 16-3 halftime lead.

What's interesting about this situation is that it requires no consideration of field position. Regardless of what decision Carolina makes, and regardless of the outcome, the game will resume with Dallas kicking off to start the second half. The only variable is the size of Carolina's lead. Because of this, the analysis of Carolina's decision forces us to focus on the value of additional points.

A naive analysis would base the decision on *expected* points. Let *p*_{TD} be the probability of a touchdown, and suppose for simplicity that the field goal is a sure thing. Then the expected points are 7*p*_{TD} if Carolina goes for the TD, and the naive analysis suggests that Carolina should go for the touchdown if *p*_{TD} > 3/7. *p*_{TD} is about 0.57 (see the data assembled by *Football Outsiders*). According to the naive analysis, then, John Fox made a large blunder.

The naive analysis implicitly assumes that the probability of winning the game is linear in the point differential. This assumption fails both locally and globally. It fails locally because certain leads are relatively unhelpful, compared to a lead that is a point larger or smaller. (That's why you go for two if you're ahead by one point late in the game.) It fails globally simply because the probability of winning the game must lie between zero and one: As our lead increases, diminishing returns must set in, and the incremental benefit of an additional point must get smaller.

The plot at left shows Carolina's probability of winning the game (on the vertical axis) as a function of their halftime lead (on the horizontal axis), as computed by the *footballcommentary.com* Dynamic Programming Model. Both kinds of departures from linearity are in evidence. First of all, we see that the 13-point lead John Fox settled for yields a relatively low probablity of winning (0.9005), compared to leads of 12 or 14 points (0.8878 and 0.9204 respectively). By itself, this would make Carolina's decision seem even worse than indicated by the expected-value analysis. However, the dominant feature of the graph is its overall concave shape, a consequence of diminishing returns. The value of a 13-point lead lies above the straight line connecting the values at leads of 10 and 17 points (0.8445 and 0.9537 respectively). Hence on balance, the results of the Model deviate from the expected-value analysis in a way that makes the field goal relatively more attractive.

Of course, this doesn't mean kicking the FG was actually *correct*. If Carolina goes for the touchdown, their probability of winning is

*p*_{TD} 0.9537 + (1-*p*_{TD}) 0.8445

Again assuming for simplicity that the field goal would be a sure thing, Carolina's probability of winning if they attempt the field goal is 0.9005. It follows that going for the TD is better if *p*_{TD} > 0.51.

In week 5, Indianapolis fell behind Tampa Bay 35-14 with 5:09 remaining in the 4th quarter, but staged one of the most improbable rallies in NFL history to win in overtime. We will analyze the Colts' extra-point strategy during their comeback.

More generally, suppose we trail by some number of touchdowns late in the game. There is just enough time for us to catch up, provided we have a lot of luck on offense and can hold the opponents scoreless. After each of our touchdowns, we need to know if we should kick an extra point or try for two.

The analysis is simplest for our final TD. Suppose we have just scored (presumably with time nearly expired), but have not yet attempted the extra point or points. If we are tied, we will certainly kick. If we trail by 2, we must attempt a 2-point conversion.

If we trail by 1, the decision depends on the probability *p*_{1} of successfully kicking an extra point, the probability *p*_{2} of a 2-point conversion, and the probability *p*_{OT} of winning in overtime. If we decide to kick the extra point, we win the game with probability *p*_{1}*p*_{OT}, whereas if we go for two we win with probability *p*_{2}. Since plausible values for *p*_{1}, *p*_{2}, and *p*_{OT} are 0.985, 0.43, and 0.5 respectively, the correct decision is to kick. None of this is a surprise.

Now consider the decision the Colts face after scoring the second TD of their comeback, but prior to the extra-point attempt. At this juncture they trail by 8 points, 35-27. If they decide to kick the extra point, there are two ways they can win. First, they can make the extra point, then make the extra point following their next TD, and then win in OT. Or, they can miss the kick, make a 2-point conversion following their next TD, and win in OT. So, if they decide to kick, their probability of winning the game (conditional on scoring another TD) is

*p*_{1} *p*_{1} *p*_{OT} + (1- *p*_{1}) *p*_{2} *p*_{OT}

If instead the Colts decide to go for two, there are three ways they can win. The first is that the 2-point conversion succeeds and then the extra point is good following their next TD. The second possibility is that the 2-point attempt fails, but they make a 2-point conversion following their next TD, and then they win in OT. The third possibility is that the 2-point try succeeds, the kick following their next TD is no good, but they win in OT. So, if the Colts decide to go for two, their probability of winning the game (conditional on scoring another TD) is

*p*_{2} *p*_{1} + (1- *p*_{2}) *p*_{2} *p*_{OT} + *p*_{2} (1- *p*_{1}) *p*_{OT}

Using the values 0.985, 0.43, and 0.5 for *p*_{1}, *p*_{2}, and *p*_{OT}, one can check that the Colts' probability of winning (conditional on scoring another TD) is 0.488 if they decide to kick, and 0.549 if they go for two. Thus, if you score a TD to close the gap to 8 points (prior to the extra-point attempt), and there is time for only one more score, you should go for two. This result is model independent, and holds as long as *p*_{2} > 0.374 *p*_{1} = 0.985 *p*_{OT} = 0.5

What should the Colts have done after the first TD of their comeback? More generally, suppose we score a TD to close the deficit to 15 points, and there is time for just two more scores. Should we kick the extra point or go for two? Interestingly enough, as we show in a mathematical appendix, it doesn't matter. Assuming plausible values for *p*_{1}, *p*_{2}, and *p*_{OT}, our probability of winning the game is the same whether we go for one or two.

In the divisional round of the playoffs, with 2:00 remaining in the first half, Green Bay led Philadelphia 14-7, and faced 4th and goal at the Philadelphia 1 yard line. Packer coach Mike Sherman made the controversial decision to go for it rather than kick the field goal, but Ahman Green was stopped for no gain, and the Eagles took over on downs.

This situation is quite similar to the one the Panthers faced in the wildcard round against the Cowboys, which we have already analyzed. Once again, a naive analysis based solely on expected points would say that Green Bay should go for the TD if the probability of success exceeds 3/7. A slightly more sophisticated analysis takes field position into account. Following a Packer score they must kick off, whereas if the Packers go for the TD and are stopped, the Eagles start near their own 1 yard line. So, an "enhanced" expected-value calculation that accounts for field position would suggest that Green Bay should go for the TD even if their probability of success is somewhat *less* than 3/7.

We will analyze Green Bay's decision using the *footballcommentary.com* Dynamic Programming Model. According to the Model, if the Packers attempt a FG, their probability of winning the game is 0.8389. If they go for the TD and make it, their probability of winning the game is 0.9165, but if they are stopped, their probability of winning is 0.7728. This implies that it's correct to go for the TD only if the probability of success exceeds 0.46 -- which is *more* than 3/7, not less. The intuition is that with Green Bay already ahead, diminishing returns come into play, so the additional points generated by a TD result in less than a proportionate increase in Green Bay's probability of winning. This effect is stronger than the effect of field position, which is weakened by the proximity to halftime.

Nevertheless, Green Bay's chances of scoring the TD from the 1 yard line were surely better than 46%, so Mike Sherman made the right call.

In week 3, Detroit trailed Minnesota 23-13 with 1:32 remaining in the game, and faced 4th and goal at the Minnesota 1 yard line. Lions coach Steve Mariucci chose to go for the TD rather than settle for the FG, but Joey Harrington's pass was incomplete, and the game was effectively over.

The decision Detroit faced is an example of a more general situation we wish to consider: It's 4th and goal to go, we trail by 10 points, and we have time for two scores provided we can stop the opponents quickly on their next possession (or recover an onside kick). Should we try for the TD or settle for the almost certain FG? If we attempt the FG first, we keep our hopes alive, but we make our subsequent task more difficult, and eliminate the possibility of winning in regulation.

Intuition suggests that if there is enough time for a reasonable shot at a TD on our subsequent possession, we should kick the field goal unless we're very close to the goal line. Conversely, if time is extremely short, we have no choice but to try for the TD even when the chance of success is small. In this section we will derive somewhat more precise guidelines.

If we decide to go for the touchdown, let *p*_{TD} denote the probability that the attempt is successful. Similarly, if we decide to begin with the field goal, let *p*_{FG} denote the probability that the attempt is good. Let *p*_{OT} denote the probability that we win the game if it goes to overtime (presumably 0.5). Finally, let *p*_{1} denote the probability of a successful kicked extra point.

If we kick the field goal and manage to get the ball back, a second field goal will do us no good. We will therefore treat the entire field as "four down territory," which increases our chances of scoring a touchdown. Let *q*_{TD} denote our probability of scoring a subsequent TD under these conditions.

It's easy to prove that sufficiently close to the opponent's goal line, we should attempt the touchdown for our initial score: If we opt for the FG first, our only winning scenario is that we make the FG, score a subsequent TD, make the extra point, and win in OT. The probability of this is *p*_{FG} *q*_{TD} *p*_{1} *p*_{OT}. *p*_{TD} *q*_{TD}. *p*_{TD} > *p*_{OT}

*p*_{FG} *q*_{TD} *p*_{1} *p*_{OT} < *p*_{TD} *q*_{TD}

In words, kicking the FG first is dominated by the strategy of going for the TD but *making believe* we kicked a FG. This shows that kicking the field goal first can't be optimal. To summarize: With 1.5 yards or less to go, the correct strategy is to try for the touchdown first.

This very clean result vindicates Steve Mariucci, but it doesn't tell us what to do when we're farther away from the opponent's goal line. The answer depends essentially on two factors. Not surprisingly, one of these is the probability that we score a TD on our next play if we decide to go for it. This is a decreasing function of how many yards we are from the goal line.

The second factor that the decision depends on is the relative likelihood of scoring a FG or a TD on our subsequent drive (assuming our initial possession resulted in a TD, so that either a subsequent TD or FG would be useful). Specifically, it's the probability of scoring a FG divided by the probability of scoring a TD. We will denote this ratio by *R*; it's a decreasing function of how much time is left when we get the ball back. To get a feel for reasonable values of *R*, note that with typical starting field position, and when time is not a constraint, about 20% of drives end in TDs, and about 12% end in FGs. So, under ordinary conditions, *R* is about

R |
P |
E |

0.6 | 0.43 | 2 |

1 | 0.38 | 2 |

2 | 0.30 | 1 |

3 | 0.24 | 1 |

5 | 0.17 | 1 |

10 | 0.10 | 1 |

In a mathematical appendix we show how these factors determine whether to kick or to go for it, and also whether we should attempt a 2-point conversion if we go for the TD and make it. The results are summarized in the Table at left. For these calculations we have assumed that the "four down" effect described earlier increases the probability of scoring a TD by a factor of 1.2. We also assume that 1- and 2-point conversions succeed with probability 0.985 and 0.43, respectively.

The first column of the Table -- the sole input -- is *R*. The last two columns are outputs. Column P is the *required* probability of scoring on the the initial TD attempt, to justify going for it. Column E tells whether we should attempt a 1- or 2-point conversion if we go for the initial TD and succeed.

To illustrate the use of the Table, suppose there is 1:55 remaining in the game, and we have no timeouts, so that if we score on our next play we will have to try an onside kick. If we recover it, we'll take possession with about 1:50 left, and we feel that with that amount of time remaining, a TD and a FG are equally likely. Thus, *R* = 1.*R*=1

With less time on the clock, we might feel (for example) that *R*=3

It may come as a surprise that with certain parameters, it's correct to go for two. This would never be right if we attempt the FG first; following the subsequent TD, we would kick the extra point and go to overtime. However, if we score the TD first, the situation is quite different. A successful 2-point conversion puts us in a position to win outright with just a FG, and even if it fails, we can still win with another TD. As long as the probability of the subsequent TD is large enough relative to the probability of a subsequent FG, it is correct to go for two.

We should add that a similar analysis can be applied if we are down by 11 points rather than 10, with time for just two scores. In particular, we can prove that if the probability of making the TD is more than half the probability of a successful 2-point conversion, then going for the FG first is dominated by the strategy of going for the TD but making believe we kicked the FG. Hence, if the probability of making the TD exceeds about 0.22, going for the FG can't be optimal. This simple result is enough to vindicate Arizona's week-17 decision to go for the TD, down 17-6 to Minnesota, facing 4th and goal at the Minnesota 2 yard line with 2:00 remaining in the game.

In week 17 versus Houston, Indianapolis lined up for an apparant 51-yard field goal attempt on 4th and 17, trailing 17-10 with 5:33 left in the game. However, the snap went to Vanderjagt, who punted. The ball was downed at the Houston 4 yard line.

Indianapolis coach Tony Dungy later explained his decision by pointing out that even with a field goal, the Colts would still need to score a touchdown to win. It's not clear if he took into account that after a field goal, a TD would win outright rather than merely get the Colts to overtime.

We can analyze this decision using the *footballcommentary.com* Dynamic Programming Model. Let's assume that on the punt, there is a 20% chance of a touchback, but if there is not a touchback, Houston's expected starting field position is their 5 yard line. Of course, after a missed field goal attempt, Houston takes over at their 41 yard line.

According to the Model, if the Colts attempt the field goal, their probability of winning the game will be 0.0584 or 0.1303, according to whether the attempt misses or is good. If they punt, their probability of winning the game is 0.0885. Let *p* be the probability that a 51-yard Vanderjagt field goal attempt would be good. We want to find how large *p* has to be to make

0.1303 *p* + 0.0584 (1 - *p*) > 0.0885

in other words, what does the success probability for the FG have to be to justify attempting it? The answer is, we need *p* > 0.42.

In week 4, Houston trailed Jacksonville 20-17, and had the ball at the Jacksonville six-inch line with 0:02 remaining in the game. Houston coach Dom Capers decided to go for the touchdown, and the Texans scored, winning the game 24-20.

Data from the 2003 season described in an article at *Football Outsiders* suggests that from the 1 yard line, the probability of making the TD is about 0.57. From the six-inch line, the probability is larger, and we don't think anyone has claimed that Houston's probability of scoring the TD was less than 50%. Nor has anyone argued that Houston's chances of winning the game in overtime would have been better than 50%. So Capers clearly increased his team's probability of winning the game by going for it.

One subtlety here is that in a regular-season game, winning and losing aren't the only possibilities. A game that goes into OT has somewhere around a 3% chance of ending in a tie. Therefore, a decision that increases a team's probability of winning doesn't necessarily decrease their probability of losing. In a close playoff race, a situation could arise in which a tie is as good as a win. A decision that precludes OT and slightly increases the probability of winning would then rationally be rejected, if going to OT reduces the probability of a loss.

On the other hand, probabilities sum to one. So, if going for the TD increases the probability of winning *by more than the probability that the game ends in a tie if we kick* (as is the case here), then it necessarily decreases the probability of losing. In this case we have what's called "stochastic dominance," and going for it would be preferred regardless of the standings. (Of course, none of this matters in a playoff game, which can't end in a tie.)

So no matter how carefully you look at it, it's clear that Dom Capers made the right decision. Nevertheless, we don't dispute that almost all coaches (99.999999%, joked Gregg Easterbrook) would have kicked. The only plausible explanation is fear of being second-guessed.

*William S. Krasker is a retired professor and the creator of footballcommentary.com, where this article first appeared. We thank him for allowing us to give our readers a chance to discuss it on our site.*

*If you are interested in writing a guest column, something that takes a new angle on the NFL, please email us your idea at info @ footballoutsiders.com*