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Schottenheimer's DecisionOn December 18, 2005 with 2:09 remaining in the San Diego/Indianapolis game, Michael Turner's 83-yard run increased the Chargers' lead to eight points (25-17) with a conversion attempt forthcoming. Many coaches have two-point conversion "cheat sheets" -- usually based on score differential -- that they use to determine whether or not to go for two. Up eight, most suggest going for two and a potential 10-point lead. However, San Diego coach Marty Schottenheimer (known for his conservative style) sent out the kicking unit. The extra point would conclude the scoring, as the Chargers held on to win 26-17, ending the Colts' winning streak at 13 and handing them their first loss of the season. But was it the right call? AnalysisUnlike some of our other strategy essays, the complexity of this problem precludes constructing a concise yet representative decision tree. So we'll answer this problem numerically using the SportsQuant football simulator. Specifically, we'll estimate the probability of the Chargers winning the game if Schottenheimer calls for a point-after-touchdown (PAT or extra-point) or a two-point conversion. For each strategy 10,000 simulated game conclusions were examined. The results suggest two things:
In Schottenheimer's particular situation, the percentage play was to kick the extra point. In this case, the difference between an eight and a nine point lead is far greater than between a nine and a ten point lead. This late in the game, a nine point lead means that the Colts need two scores, which is unlikely given how little time remains. On the other hand, a 10 point lead means that a touchdown and a field goal (after a successful onside kick) results in overtime rather than a loss. While no one will argue that a 10 point lead is better than a nine point lead, the risk of an unsuccessful attempt resulting in an eight point lead (which Indianapolis could have overcome with a touchdown and a two-point conversion) was too great. It is interesting to ask whether the optimal decision would have changed if there was more time remaining. In other words, if Indianapolis was likely to have at least two more possessions, would playing for the 10 point lead improve San Diego's chances of winning? Again, we'll turn to the SportsQuant football simulator. The following figure gives a plot of the probability that the Chargers win if they attempt the extra point (black) or the two point conversion (red) for various amounts of time remaining in the fourth quarter. Estimated probabilities are based on 10,000 simulated game conclusions for each value of time remaining (ranging from 0:00 to 15:00 in 15 second increments). The plot shows that at any point in the fourth quarter, Schottenheimer would be best served to attempt the extra point, rather than go for two.
But the extra point isn't an unconditionally dominant strategy. If there is enough time remaining, the two-point attempt becomes more appealing. The next graph is a comparison of smoothed win probabilities throughout the second half. As the previous plot noted, attempting the extra point (black) is the dominant strategy at all points in the fourth quarter (i.e. 15 minutes or less remaining). However, the two-point attempt results in a higher probability of winning as long as there is more than 2:36 remaining in the third quarter. This change in strategy occurs with 17:36 remaining (vertical dashed line) where the PAT and two-point lines cross.
DiscussionIn light of the preceding analysis, it is tempting to conclude that leading by eight points, a coach should go for two if there is more than 17:36 remaining and kick the PAT otherwise. However tempting, this conclusion is incorrect. At the end of the game, the value of an additional point or two depends greatly on the score differential. For example, there's a huge difference between leading by a point and trailing by a point with a few seconds left. However, when there is a great deal of time remaining, the probability of winning is approximately linear in point differential (i.e. the value of a point is roughly constant). Therefore, early in the game, the coach should act to maximize the expected number of points his team scores. (In late-game situations, maximizing expected points and probability of winning do not necessarily lead to the same decision, but early in the game, the two agree. For an explanation, see our discussion on the marginal value of an additional point scored.) Maximizing expected points leads a coach to attempt the PAT if there is enough time remaining. People often complain that even when they ask a statistician a simple question, the answer involves lots of qualifiers and caveats. However, this problem illustrates why such caveats are necessary. The question, which sounds straightforward, is "when leading by eight, should the coach try for one or two?" The answer, which is anything but straightforward, is "it depends." The dependence is due to the time remaining (which influences how many possessions remain and how likely certain combinations of scoring events are). However, despite the caveats, the two-point strategy for a coach leading by eight can be summarized succinctly: try for two if the time remaining is between 33:30 (3:30 remaining in the second quarter) and 17:46 (2:46 remaining in the third quarter); kick the extra point at all other times. Back to Top |
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