A model-based approach to football strategy.

January 4, 2006


Goal-Line Gambles

Two of the most widely discussed coaching decisions of the 2005 season involved goal-line gambles. With 0:05 left in the Week 9 game between Oakland and Kansas City, Chiefs' coach Dick Vermeil elected to go for the win from inside the Raiders' 1-yard line, rather than kick a field goal to send the game to overtime. A week later, with less than a minute left in the game between Washington and Tampa Bay, Buccaneers' coach Jon Gruden chose to attempt to take the lead with a two-point conversion rather than kick the extra point to tie the score.

Vermeil's decision was very simple. Kansas City was right to go for the touchdown provided the probability of scoring was more than 0.5. Notwithstanding the estimation biases discussed in the next section, Kansas City's actual probability of success exceeded that threshold by a large margin.

Commentators have said that Tampa Bay was also right to go for it provided their probability of success exceeded 0.5, but that is not quite true. The Buccaneers require a higher chance of success. As we show in an appendix, Washington's opportunity to score following Tampa Bay's try raises the hurdle for Tampa Bay to go for two. The full analysis of Gruden's decision is complicated, and our conclusion that the Buccaneers were right to attempt the two-point conversion isn't nearly as compelling as the corresponding result for Kansas City. In fact, the results suggest that if John Madden had been coaching the Redskins, it might have been right for the Buccaneers to kick the extra point.

Biases in Estimates of Success Probabilities

When teams elect to go for it on 4th-and-goal, with the ball officially at the 1-yard line, the observed success rate is around 65%. Since that figure is important for our analysis, we will begin with some issues that affect its interpretation.

In the official NFL play-by-play records (PBP), the spot of the ball is always recorded as an integer number of yards from a goal line. According to the NFL's Guide For Statisticians1, if the ball lies entirely between two yard stripes, the rounding is in the direction of the defender's goal. This implies that whenever the nose of the ball is less than about 5 feet from the defender's goal, the PBP will say that the ball was at the 1-yard line. (The only exceptions derive from the requirement that the spot of the ball and the necessary line for a first down be distinct positive integers. So, if the ball is two inches from the defender's goal with one inch to go for a first down, the PBP will say that the ball was at the 2-yard line with 1 yard to go for a first down.)

Because of the rule for rounding, balls that are officially at the defender's 1-yard line will, on average, be slightly less than a yard from the goal. Offenses do not always go for the touchdown when the ball is officially at the 1-yard line, but even if they did, the observed success rate would be an upward-biased estimate of the probability of success when the ball is really one yard from the goal. However, this bias due to rounding is small.

The more important bias is the "selection bias." The likelihood that a team will go for the touchdown depends systematically on the actual position of the ball. For example, teams are more likely to go for it when the ball is an inch from the goal than when the ball is nearly five feet from the goal. Consequently, among cases in which the ball was officially at the 1-yard line and the offense chose to go for it, we expect that the average distance from the goal was significantly less than a yard.

A similar bias presumably affects the comparison between running and passing when it is 4th-and-goal and the ball is officially at the 1-yard line. In that situation, as Aaron Schatz of Football Outsiders has noted, the success rate is higher for running than for passing. However, one would assume that teams are more likely to run when the actual spot of the ball is closer to the goal. This would explain at least part of the disparity in the success rates. Notice, though, that there is no way to verify this or the previous hypothesis without knowing the true position of the ball.

Analysis of the Decisions

The biases discussed in the previous section have no effect on the conclusion for Vermeil's decision. Kansas City was less than half a yard from the goal line—probably closer than the average even for teams that go for it when it's officially 4th-and-goal at the 1-yard line. This suggests a win probability around 0.65 if they go for it, compared to about 0.5 if they kick the field goal and go to overtime. The difference of 0.15 is very large for a controversial decision. Had Vermeil chosen instead to kick, the choice would have been a leading candidate for worst strategic decision of the year.

Gruden's decision isn't so easy to analyze. Let's review the situation. With 0:58 remaining in the 4th quarter, Tampa Bay scored a touchdown to close to within a point. Washington was called for delay of game prior to the extra point, and (invoking their option under Rule 11-3-3-c) the Buccaneers elected to have a 5-yard penalty enforced on the subsequent kickoff. The extra point was then blocked, but Washington was called for offsides, and the enforcement of the distance penalty under Rule 11-3-3-d moved the ball half the distance to the goal. The Buccaneers then decided to attempt a two-point conversion. Washington had two timeouts, Tampa Bay none.

Taking into account the biases mentioned earlier, we estimate that Tampa Bay's probability of success from 1 yard out is 0.6. If the try succeeds, then following the kickoff from the 35-yard line, Washington will go all-out to score. Our model for the two-minute drill says that Washington's probability of scoring is about 0.24.

If instead the two-point try fails, Tampa Bay will attempt an onside kick. Over the last four seasons, 13% of anticipated onside kicks were recovered by the kicking team. The success rate has fallen each year, to just 4 successes in 47 attempts so far this season. The decline is most likely routine sampling variation. Still, it's possible that teams are getting better at defending against onside kicks. With that possibility in mind, we will use 0.12 for the probability of a successful onside kick. If the Buccaneers recover the onside kick, their probability of winning is 0.41, according to the model for the two-minute drill. It follows that if the Buccaneers go for two, their probability of winning the game is about

0.6(1−0.24) + (1−0.6)0.12(0.41) = 0.476.

The alternative for the Buccaneers is to kick the extra point to tie the game. Since the ensuing kickoff is from the 35-yard line, a touchback is likely. (Actually, according to the model, the Redskins would have to return the ball to about the 23-yard line just to compensate for the time that would be taken off the clock by a runback.) The question is, can kicking the extra point give Tampa Bay a higher win probability?

The answer depends on what strategy Washington will adopt when they gain possession. To state the obvious, if Washington simply runs out the clock, Tampa Bay's win probability is 0.5, which exceeds their win probability of 0.476 if they go for two. Remarks by John Madden suggest that he would run out the clock in that situation. We believe that most coaches would try to score, though not too aggressively. Certainly, the Redskins would not go all-out to score, as they would when trailing. According to the model, going all-out from the 20-yard line with 0:58 and two timeouts gives Washington a 0.23 probability of scoring. However, it probably gives them an even greater chance of losing in regulation. From 2002 through 2004, in late-game drives when the offense trailed by one score and punting was not an option, 23% of the drives (coincidentally) resulted in a score, but 33% ended in a turnover, and another 8% ended in a four-and-out. That's not a problem when the offense trails, because a score by the opponents is no worse than having time expire; but it's a disaster if the score is tied.

If we modify the model so that Washington punts on fourth down when they are out of field-goal range, but otherwise goes all-out to score, their probability of scoring drops to 0.19. This is the upper limit for Washington's likelihood of scoring, and probably still introduces too much risk of a turnover. By being a little less aggressive, perhaps Washington could achieve a 0.16 probability of scoring, with just a 0.02 probability that Tampa Bay scores. A strategy by Washington with those characteristics would make Tampa Bay's win probability

0.02 + (1−0.02−0.16)0.5 = 0.43

if Tampa Bay kicks the extra point to tie the game, compared to 0.476 if the Buccaneers go for the two-point conversion. So, although we recognize the imprecision of some of the numbers, we estimate that going for the two-point conversion raised Tampa Bay's win probability by roughly 0.046.

1 This information was provided by Chris Hoeltge of

Copyright © 2006 by William S. Krasker