You probably know the Ellsberg urn experiment. In the urn on the left there are 50 black balls and 50 red balls. In the urn on the right there are 100 balls, some of them red and some of them black. No further information about the urn on the right is given. Subjects are allowed to pick an urn and bet on a color. They win $1 if the ball drawn from the urn they selected is the color they bet.

Subjects display aversion to ambiguity: they strictly prefer to bet on the left urn where the odds are known than on the right urn where the odds are unknown. This is known as Ellsberg’s paradox, because whatever probabilities you attach to the distribution of balls in the right urn, there is a color you could bet on and do at least as well as the left urn. This experiment revealed a new dimension to attitudes towards uncertainty that has the potential to explain many puzzles of economic behavior. (The most recent example being the job-market paper of Gharad Bryan from Yale who studies the extent to which ambiguity can explain insurance market failures in developing countries.)

Decades and thousands of papers on the subject later, there remains a famous critique of the experiment and its interpretation due to Raiffa. The subjects could “hedge” against the ambiguity in the right urn by tossing a coin to decide whether to bet on red or black. To see the effect of this note that if there are n black balls and (100-n) red balls, then the coin toss means that with 50% probability you bet on black and win with n% probability and with 50% probability you bet you red and with with (100-n)% probability giving you a total probability of winning equal to 50%. Exactly the same odds as the left urn no matter what the actual value of n is. Given this ability to remove ambiguity altogether, the choices of the subjects cannot be interpreted as having anything to do with ambiguity aversion.

Kota Saito begins with the observation that the Raiffa randomization is only one of two ways to remove the ambiguity from the right urn. Another way is to randomize ex post. Hypothetically: first draw the ball, observe its color, and then toss a coin to decide whether to bet on red or black. Like the ex-ante coin tossing, this strategy guarantees that you have a 50% chance of winning. Kota points out that theories that formalize ambiguity assume that these two strategies are viewed equivalently by decision-makers. If a subject is ambiguity averse, then he prefers either form of randomization to the right urn and he views either of them as indifferent to the left urn.

But the distinct timing makes them conceptually different. In the ex ante case, after the coin is tossed and you decide to bet on red, say, you still face ambiguity going forward just as you would have if you chosen to bet on red without tossing a coin. In the ex post case, all of the ambiguity is removed once you have decided how to bet. (There is an old story about Mark Machina’s mom that relates to this. See example 2 here.)

Kota disentangles these objects and models a decision-maker who may have distinct attitudes to these two ways of mixing objective randomization with subjectively uncertain prospects. In particular he weakens the standard axiom which requires that the order of uncertainty resolution doesn’t matter to the decision-maker. With this weaker assumption he is able to derive an elegant model in which a single parameter encodes the decision-makers pattern of ambiguity attitudes. Interestingly, the theory implies that certain patterns will not arise. For example, any decision-maker who satisfies Kota’s axioms and who displays neutrality toward ex post ambiguity must also display neutrality toward ex ante ambiguity. All other patterns are possible. As it happens, this is exactly what is found in an experimental study by Dominiak and Shnedler.

What’s really cool about the paper is that Kota uses exactly the same setup and axioms to derive a theory of fairness in the presence of randomization. A basic question is whether the “fair” thing to do is to toss a coin to decide who gets a prize, or to give each person an identical, independent lottery. Compared to theories of uncertainty attitudes, our models of preferences for fairness are much less advanced and have barely touched on this kind of question. Kota’s model brings that literature very far very fast.

Kota is a Northwestern Phd (to be) who just defended his dissertation today. You could call it a shotgun defense because Kota’s job market was highly unusual. As a 4th year student he was not planning to go on the market until next year, but CalTech discovered him and plucked him off the bench and into the Big Leagues. He starts as an Assistant Professor there in the Fall. Congratulations Kota!

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