The Ellsberg Paradox and 4th Down

The Romer paper and other research provide fairly conclusive evidence that NFL coaches should go for it on 4th down more often than they currently do. The Ellsberg Paradox might help explain why.

Say there are two jars of 100 balls of which some are red and some are blue. Jar A has 50 red balls and 50 blue balls. Jar B has a random unknown mix of red and blue balls. You'll be given $100 if you pick a red ball from a jar. Which jar would you choose to pick from?

In clinical experiments, people almost universally choose jar A. This is the Ellsberg Paradox, a violation of the utility theory in economics. The expected value of each choice is equal. There is a 50/50 chance of winning $100 from either jar, so we wouldn't expect one option to be significantly preferable to the other.

The Ellsberg Paradox demonstrates the difference between risk and uncertainty. Risk is measurable but uncertainty is not. People almost always prefer a known risk to an unknown uncertainty, even if the expected results are equal.

People prefer Jar A according to the equation above. U() is the utility function.

Punting seems a lot like Jar A, for which the risks and potential outcomes are known. Going for the first down seems more like Jar B, for which the potential outcomes are vague and hard to measure. So at the equilibrium point between going for it and punting, where each decision provides equal chances of ultimately winning, coaches would be heavily biased toward punting. Even beyond the equilibrium point, where going for it would be favorable, coaches would still be biased toward the relatively certain (but less favorable) outcome of the standard 40-net-yard punt.

In a strict analogy, the $100 would be a win, and the red balls would represent the probability of winning the game. There would actually be some uncertainty in each strategy, but far more uncertainty in the go-for-it strategy--perhaps something like 40 to 60 red balls in the punt jar and 20 to 80 balls in the go-for-it jar. The Ellsberg Paradox suggests coaches would naturally prefer punting, the less uncertain option. Only when the advantage of going for it is beyond obvious would a coach choose to go for the 1st down--say 10 to 20 red balls for punting and 15 to 60 red balls for going for it.

I think NFL coaches typically employ the maximin strategy. In game theory the maximin strategy is one that selects the alternative with the best worst-case-scenario. It maximizes the minimum possible payoff. This is a conservative strategy in comparison to the maximax strategy, which selects the alternative with the greatest maximum payoff.

Continuing the jar and red ball analogy, compare jar X with 10 to 90 red balls and jar Y with 30-40 red balls. Utility theory would suggest the rational option is jar X with a higher overall chance of success. The maximin choice however, would be jar Y because it has a higher minimum chance of success.

Calculating the probability distributions of a football game's outcome given the combinations of score, time remaining, field position, etc. is far more complex than being told how many red balls are in a jar. It would be overwhelming for a human brain even to attempt it. In such a situation, coaches, like everyone else, use heuristic shortcuts such as the maximin strategy. Punting on every 4th down is a known risk, especially because coaches can count on opposing coaches to follow the same strategy (which suggests that always punting is a Nash Equilibrium). Punting usually presents the best worst-case-scenario despite being a sub-optimum decision.