“Everything should be made as simple as possible, but no simpler.” – Einstein.
This is one of my favorite maxims. It is also a very difficult rule to follow, giving it a certain humor which was probably not lost on Einstein. How simple is too simple? Well, if adding just a little more detail reverses the analysis completely, it was too simple. Here’s an example from this weekend’s diversion, the NFL playoffs.
With 4:08 left in their game against the Packers, the Eagles faced 4th and goal, inches from the goal line. They trailed by 11. To win, they would either need to score a touchdown with two-point conversion and field goal (in either order), then win in overtime, or score two touchdowns. Eagles coach Andy Reid called timeout to think it over. What should he have done?
Announcer Troy Aikman argued that they should kick the field goal, saying that “you have to cut it to a one-possession game.” This is a classic example of what decision theorists call coarse decision-making. Aikman knows that it’s better to cut it to 5 or 3 than 8, but he simplifies the decision by considering 3, 5 and 8 to be similar to one another (in each case, one possession may suffice to tie or win) while they are very different from 11. Is his analysis good enough? No, it turns out he made it too simple, and in the wrong way.
Let me simplify it differently. I won’t even need the language of probability; I can put it purely in terms of verbal logic, perhaps a kind you could even explain on TV (almost). First, assume that if we score a touchdown now, the 2-point conversion will fail and the deficit will be 5 (this goes against the decision I’m advocating, so it’s an acceptable simplification.) Now, since we are going to be down at least 5, assume that we will get the ball back and score a touchdown; otherwise, our decision now is irrelevant. This makes it very easy to see which option is best right now:
If we kick a field goal, victory will depend on later converting the 2-pointer (from the 2-yard line), then winning in overtime.
If we go for it now, victory will depend only on making it now from inches away.
Would you rather try to score from 2 yards out for a tie, or try to score from inches away for a win? Looked at this way, it’s a total no-brainer. And of course, I ignored the fact that you might cut it to 3 if you score now, which gives you an even better chance to win. Andy Reid got this decision right (though one colleague, an Eagles fan, is harsh on him for needing a timeout to think about it, I find this forgivable.) The Eagles indeed scored a touchdown to cut the lead to 5, then were in position to go for the win later but were thwarted by an interception in the end zone.
What went wrong with Aikman’s simplification, or “coarsening”? One way to look at it is that he lumped the wrong scores together. His way of thinking comes very naturally, especially since one-possession vs. two-possession corresponds roughly to alive vs. dead, and this is how we tend to categorize probabilities. But in terms of win probability, an 8-point deficit is actually closer to 11 than it is to 5! If t is the probability of scoring a later touchdown, then if the deficit after this possession is
5, the probability of winning is t
8, the probability of winning is (roughly) .25t
11, the probability of winning is roughly 0.
This is based on a 50/50 chance at a two-point conversion (published figures vary) and a 50-50 chance at winning in overtime. These figures mean that it would be worthwhile to try for a touchdown even if it only worked 25% of the time (the actual figure is over 70%), and even if you never successfully cut the deficit to 3. It turns out it would be better to think of an 8-point deficit as “not quite dead” rather than “alive.”
This situation also relates to my previous entry “Risky Move.” In principle, all strategies are risky, and you should pick the one with the best chance of success. In practice, people think of the decision which resolves more of the uncertainty immediately as “risky,” and are biased against this decision. They sometimes then choose a slower, but more certain, death.