Holmgren's Dilemma
You're the head coach of a football team, and your team is protecting a one-point lead late in regulation. After scoring an insurance touchdown, you have two choices: kick the extra point for an 8-point lead, or go for two knowing that a successful conversion makes it a two-possession game and thus dramatically increases your chances of winning. If you play things by the book, you'd kick the PAT, comfortable in knowing your opponent needs a touchdown and two-point conversion just to force overtime. But is this really the best strategy?
Seattle Seahawks coach Mike Holmgren faced this decision in a game against the New York Giants on November 27, 2005. With his team leading 14-13 with just under 5:00 remaining, the Seahawks scored to push their lead to 20-13. Holmgren elected to kick the extra point to make it 21-13. The Giants scored a touchdown at the two-minute warning and converted the two-point attempt to tie the game. Seattle eventually won in overtime (24-21), but only after 3 missed field goals by New York kicker Jay Feely (one at the end of regulation and two more in overtime). How would things have turned out if Holmgren had called for the two-point conversion?
Mathematical Modeling
We'll assume that the probability of successfully converting an extra point is p1, the probability of successfully converting a two-point attempt is p2 and the probability of winning in overtime is one-half1. Furthermore, we'll assume that the probability of your opponent scoring a subsequent touchdown is q. Our last assumption is that your opponent will attempt a two-point conversion when trailing by 8, an extra point when trailing by 7 and will lose if trailing by 9. (The assumed conversion strategy is not only the optimal one for your opponent, but in agreement with conventional coaching behavior.) Now, let's examine your chance of winning for each strategy:
Go for two: There are four ways to win the game if you go for two:
Convert the two-point attempt.
Miss the two-point conversion and prevent the opponent from scoring a touchdown.
Miss the two-point conversion, allow a touchdown and prevent the opponent from converting the extra point.
Miss the two-point conversion, allow a touchdown and subsequent extra point but win in overtime.
These outcomes are illustrated in the decision tree below:
Assuming independence2 we can calculate the probability of each of these outcomes.
p2.
(1-p2) × (1-q).
(1-p2) × q × (1-p1).
(1-p2) × q × p1 × ½.
Similarly, there is only one possible way to lose under this scenario, and to do so, everything must go wrong. First you must fail on your own two-point conversion, then allow a game tying touchdown and extra point and then lose in overtime. The probability of all these events happening together is (1-p2) × q × p1 × ½.
Go for one: There are six ways to win the game if you kick the extra point and two ways to potentially lose.
Convert the extra point and prevent the opponent from scoring a touchdown.
Convert the extra point, allow a touchdown and prevent the opponent's two-point conversion.
Convert the extra point, allow a touchdown and subsequent two-point conversion but win in overtime.
Miss the extra point and prevent the opponent from scoring a touchdown.
Miss the extra point, allow a touchdown and prevent the opponent from converting the extra point.
Miss the extra point, allow a touchdown and subsequent extra point but win in overtime.
Assuming independence the probabilities of these outcomes are:
p1 × (1-q).
p1 × q × (1-p2).
p1 × q × p2 × ½.
(1-p1) × (1-q).
(1-p1) × q × (1-p1).
(1-p1) × q × p1 × ½.
Similarly, there are only two possible ways to lose:
Convert the extra point, allow a touchdown and subsequent two-point conversion and lose in overtime.
Miss the extra point, allow a touchdown and subsequent extra point and lose in overtime.
The associated probabilities are:
p1 × q × p2 × ½.
(1-p1) × q × p1 × ½.
Again, the potential outcomes can be organized using a decision tree:
1.
Since the teams must have been of comparable abilities to get to overtime, this assumption is not unreasonable. Another way of looking at this is that half the teams the end up in overtime win and half lose. (You'll notice that this implicitly neglects the slim possibility of a scoreless overtime resulting in a tie.)2. Essentially, independence states that the results of one event have no influence on the results of another.
Discussion
For either strategy, the potential ways to win (or lose) the game are mutually exclusive and collectively exhaustive3. Therefore, the probability of winning under each strategy can be determined by summing the probabilities of the various outcomes corresponding to a win. Therefore, the probability of winning is [1 - p1q(1-p2)/2] for the two-point strategy and [1 - p1q(1-p1+p2)/2] for the one-point strategy.
Based on the preceding arguments, the two-point strategy results in a higher probability of winning the game if and only if [1 - p1q(1-p2)/2] ≥ [1 - p1q(1-p1+p2)/2], or equivalently, when 2p2 ≥ p1. Assuming the probability of successfully converting an extra point is p1 = 0.96 and the probability of successfully converting a two-point attempt is p2 = 0.44, the optimal strategy is to attempt the extra point. It's clear, though, that even if the extra point was guaranteed (p1 = 1), the two-point strategy is preferable if p2 ≥ ½, since the two-point strategy results in a higher probability of winning when 2p2 ≥ p1.
Points vs. Winning
: The idea of maximizing expected points is different from maximizing probability of winning. For example, with one second left and trailing by 4 from their opponent's 25 yard-line, a field goal attempt maximizes a team's expected point total. However, a coach would attempt to score a touchdown because it gives his team a better chance to win. (Even a made field-goal results in a loss.) Conversely, with one second left in a tie game from the one-yard line, a coach would kick the field goal since it virtually assures a win, even though going for a touchdown would increase his team's expected points.Although this analysis considered only the probability of eventually winning, it is interesting that the probability of winning is directly related to the calculation of expected points, which, as noted in the previous paragraph, is not always the case. If our hypothetical coach was trying to maximize his team's expected point total, he'd go for two if (p2 × 2) ≥ (p1 × 1), or equivalently, if 2p2 ≥ p1. So in this case, the strategies which maximize expected points and probability of eventually winning coincide.
3. In probability nomenclature,
mutually exclusive (or disjoint) events are events which can not happen together. Collectively exhaustive events are those which encompass all possible outcomes. Therefore "mutually exclusive and collectively exhaustive" is a mathematically precise (and fancy-sounding) way to say "exactly one of these events will happen." Mutual exclusivity allows us to add up probabilities without worrying about "double-counting" outcomes that overlap one or more events. Collective exhaustion allows us to consider complementary events (notice how the probability of winning is exactly equal to one minus the probability of losing, which is much easier to compute).Further Analysis
Seattle's final touchdown was set up by Holmgren's decision to go for it on 4th and inches from the New York 3 yard-line. Commentators noted that Holmgren was trying to end the game decisively by going for it rather than kicking a chip shot field goal. While they praised him for his willingness to take chances, they were universal in ignoring the possibility that he try for two after the touchdown. If Holmgren had really been a risk-taker (and certainly if he had been interested in putting the game out of reach), he'd have gone for two. Although the two-point strategy is slightly worse than the one-point strategy, it's interesting that there isn't even a discussion given how close the two expected payoffs are to each other.