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.

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8 Responses to “The Ellsberg Paradox and 4th Down”

  1. JTapp says:

    Awesome analysis, I think you're on the money with the maximin. Don Shula supposedly said that one day there will come a coach who just won't punt. We'll be waiting forever.

    You could really publish a great follow-up to Romer on this.

  2. JTapp says:

    Also, looking at your previous work (The Passing Paradox) it stands to reason that teams that go for it regularly on 4th down should be more risk-averse on 3rd down. This should greatly affect the team's playcalling mix.

  3. Brian Burke says:

    I would agree. A team with a 4-down mentality would turn 3rd down into a 2nd down. There would be far more runs.

    The CFL is a 3-down only league. From my vague memory of the Baltimore Stallions, the standard series was pass-pass-punt, with an occasional draw play sprinkled in to keep defenses constrained. It's as if they started with a 2nd and 10 on every series.

    The fewer the downs, the more teams will tilt towards the pass. And the more downs allowed, the more attractive running becomes.

  4. JTapp says:

    Canadian teams only have a 20-second play clock as well (vs. 40 for the NFL). They run more plays-per-game.

  5. JackLean says:

    The problem with your analysis is the unspecified assumption that both jars yield a 50/50 chance of picking a red bal. Your contention is "There is a 50/50 chance of winning $100 from either jar." But you don't state that the random jar has a 50/50 mix...which is critical to your analysis. If the random jar has 90 blue balls and 10 red balls, then the probability of chosing a red ball isn't 50/50 and obviously the non-random jar is your best bet. Without knowing the exact ratio of red/blue in the random jar, it is always "more reasonable" to choose the jar where the ratio is known...meaning no paradox. This is an age old problem in mathematical must state your givens very precisely.

  6. Ryan says:

    Jack I think when he says "random" when referring to the random jar, he's talking about a random selection of 100 blue and red balls that were put into a jar. Because there are two balls, there is a 50/50 chance that each ball selected and put into the random jar will be red. In theory, if this random process is carried out 100 times with a 50/50 probability, there should be 50 red balls and 50 blue balls, or at least a close ratio in the random jar. I think that is the assumption that is being made here.

  7. Ryan says:

    I meant to say there are only two colors when referring to there being two balls

  8. Patrick Feihle says:

    I have to admit that i didnt look too much into how these 4th down probabilities are actually calculated. i just can assume that they track all 4th down attempts and log down & distance and whether or not it was successful.

    so the numbers are skewed. for example, there was an article where it said that the chargers should have tried more 4th down conversions in wk17 and in the WC weekend.

    the example was a 4th-and-3 or so deep in your own territory. now, the chances to convert this arent that bad, when you look at it statistically. this is obvious because in which situations are teams going for it on 4th down in their own half? its late in the game, probably even under 2 minutes, one/no timeout left. the defense obviously would like to stop them, but is concerned with the game-winning long play that puts them in FG range/gives them the TD as well.

    just think about it. 4th-and-3 at your own 15, 1:20 left, FG wins it. lets say the offense brings 5 WRs in an empty set, what do you do as a defense? cover-0 and try to stop them right here but if one guy gets beat, the game is over? no, you probably sit back, cover deep and the sidelines first and maybe you can make a play on the draw/short pass.

    the chances of going for it on 4th-and-3 in the first quarter on your own 15 and the implications for the likelihood to win the game are a totally different animal. you cannot compare the two. this is why i think that coaches are generally right to punt as often as they do, since the statistics from converted 4th down attempts, especially in your own half, come from situations which arent transferrable to every single drive.

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