The Dark (Frank) Knight

Frank Knight wrote the great book Risk, Uncertainty and Profit, in which he described the distinction between risk and uncertainty. (The book won second prize in a 1917 competition, sponsored by Hart, Schaffner and Marx, intended to “draw the attention of American youth to the study of economic and commercial subjects.” First prize was awarded to E.E. Lincoln, The Results of Municipal Electric Lighting in Massachusetts.) We are operating under risk if an event may or may not happen in the future, and we know the chances that it will happen. For example, flipping a fair coin is a game of risk. We do not know whether the coin will come up heads, but we know that the probability of this event is 1 out of 2, or 50%. An event is uncertain if it may or may not happen in the future, and we do not know the chances that it will happen. (Knight would require that we “cannot” know this chances that it will happen, though this is perhaps too strong; I have an excellent discussion of the do not know/cannot know issue, but this blog post is too small to contain it.) For example, I do not know whether McCain will win the next presidential election, and, unlike the situation with the coin, I also do not, and cannot, know the probability that he will win, because this election is a one-off event.

So what does this have to do with the new Batman movie, The Dark Knight? I have put the explanation after the jump, because it contains minor spoilers. (Or major spoilers, if you are totally unfamiliar with the Batman story.) Repeat: there are spoilers after the jump. Do not read the rest of this post if you do not want a few Dark Knight spoilers. Don’t! Seriously!

Ok, I warned you. Don’t complain!

In The Dark Knight, we see Harvey Dent, the crime-fighting DA, flip a coin three times. The first time, he flips the coin to determine who will be lead counsel at the trial of a mafia leader. “Heads it’s me, tails it’s you,” he tells Rachel Dawes. The coin comes up heads. “Would you really leave something like that up to luck?” she asks him. “I make my own luck,” he replies. The second time, he is trying to get a bad guy to give him information. He points the gun at the guy’s head and says, “Heads you live, tails I shoot you.” The coin comes up heads. He then begins to flip it again. The bad guy screams, terrified, “I don’t know! I don’t know!” (Either the bad guy has changed his mind about whether a 50/50 chance of death is terrifying, or, displaying a basic misunderstanding of probability, he thinks the odds are greater that the coin will come up tails on the second flip because it came up heads on the first flip.) Again, someone asks Dent something along the lines of, “Would you really have shot him?” And Dent says something like, “Not really.” We eventually see, after a third, similar exchange, that the coin has two identical sides: both are heads. The outcome of the coin flips was never in doubt.

Then Harvey Dent in involved in a terrible incident in which half his face is burned. He is no longer Harvey Dent, the crime-fighting DA; he is now Two-Face, a villain. The coin is also burned in the incident, and now the two sides are not identical. Two-Face continues to flip the coin, but now, each time he flips the coin to determine whether someone lives or dies, there is actually a 50% chance that the person will die. This version of coin flipping (“fair” flipping) is associated with the bad guys in this film. The Joker says that there are no rules, only chaos, and Two-Face makes the same point: “The only mercy in a cruel world is chance: unbiased, unprejudiced, fair,” he says, before flipping a coin to determine whether he will kill a small boy.

Dent’s game nicely illustrates Knight’s distinction between risk and uncertainty, as well as the distinction’s instability. When Dent flips the coin, the people who observe him think they know the probability of the coin coming up heads: they think the probability is 50%. They think the outcome is a question of risk, not uncertainty. (Dent, of course, knows that the actual probability of the coin coming up heads is 1–that is, he knows that this is a decision under certainty.) But why do people think that the probability of the flipped coin coming up heads is 50%? Quite simply, because it is a coin, and it is flipping. But actually, there are lots of possibilities. They’ve seen one side of the coin, so they know it’s not a coin with two tails, but it could be a weighted coin, so that the chances of the coin coming up heads are very, very small. Or, it could be, and is, a coin with two heads. Or, for that matter, it could be a special exploding coin (in the world in which the movie occurs, this actually wouldn’t be so far-fetched)–why aren’t they diving under the table when he throws that thing in the air?

Yes, if it is a fair coin the chances of heads are 50%. But how does Dawes, or the bad guy, know it is a fair coin? Take Dawes, who knows Dent well. Given her dubiousness that Dent would let her chair the trial, she might doubt her belief that the coin is fair. And suddenly she would be faced with uncertainty: “If the coin is fair, my chance of chairing the trial is 50%. But Dent would never let me chair the trial…I wonder if the coin isn’t fair? I would guess there’s a 25% chance the coin isn’t fair…but I wonder how unfair it is?”

In other words, few probabilities, if any, are known with certainty. Thus, assuming that we know the probability of a situation may be only that, as Sven Ove Hansson has written, we have “chosen to simplify our description…by treating [the situation] as [a] case[] of known probabilities.” It’s not that risk v. uncertainty is a useless distinction; decisions may be more or less risky, as opposed to uncertain. Moreover, to operate in the world, we have to simplify things away. (In short, heuristics rock! Well, at least sometimes.) But believing with certainty and without investigation that a coin flip is fair, and that one is in a risky situation and not an uncertain one, is indeed a simplification, as Dent’s game shows.

(Note: I’ve only seen the movie once at this point, so if I have any of the quotes wrong, or have mischaracterized what happens in the movie, please do let me know in the comments.)

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2 Responses

  1. A.J. Sutter says:

    As you probably know, there is a very old pedigree for this issue, namely the debate between the frequentist and Bayesian interpretations of probability.

    Under the frequentist interpretation, a probability is a limit of the relative frequency of an event in a large number of trials. Most techniques of statistical inference are based on this interpretation, at least implicitly. But it’s hard to know this of a particular coin. All the less so of the many other things we deal with in life, where (i) many outcomes are not often, or ever, repeated, and (ii) there may also be threshold issues just in defining those outcomes. (E.g., are lawsuits always “won” or “lost” in a clear-cut binary way?)

    Without this knowledge, how can you have “risk”? Maybe one of the more interesting results of these definitions is that (subjective) risk first requires some level of trust.

  2. Edward Swaine says:

    Check plus for the title of your post. I’m being boring, but I think Two-Face’s coin illustrates a more basic risk. In situation # 1 (who will chair the trial), Rachel Dawes may be questioning the fairness of the coin, but I think it’s equally plausible that she is expressing doubt that Dent would abide by any adverse result. Same for situation #2: it’s not that bad guy has changed his mind, or misunderstood probability, but rather that Dent’s behavior indicates that he will not abide by any result favorable to the bad buy (i.e., that Dent will keep flipping until it comes up tails). These are simply commitment problems, reinforced by his disclosure that he actually would not have abided by tails either.

    I don’t remember the third incident, but the final situation speaks more to your issue. As I recall from the comics, once one side of the coin is scarred (I think he does the defacing himself in at least one of the versions), Two-Face is in fact quite good about abiding by the result. Indeed, I think that in his psycho-world, it is of the utmost importance that he NOT influence the result; were that understood, and coupled with the fact that the coin’s original construction obviated any need to weight one side relative to the other, the kid should be pretty confident that the flip would be fair.

    The kid doesn’t know any of this, however. And he also doesn’t have a choice, which is relevant to the fairness assessment — not just because we think forcing such games on people is unfair, but also in a way that bears on the fair coin issue. Two-Face chooses the occasions on which he flips the coin, and does not solicit advance approval. This is pretty awful, in most respects, but if we suppose (as the kid would, though it is not in fact consistent with Two-Face’s character) that Two-Face would have been at liberty to do the worst thing possible without even subjecting matters to a flip — to “cheat” with total likelihood of the bad result prevailing, not 51% or 52% — we should be relatively comfortable with the probable fairness of the coin. More so than if you and I were trying to decide who should spring for lunch.

    If you think this is bad, don’t get me started on Dark Knight’s ending.