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!
8 comments
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February 24, 2011 at 12:44 am
Colin Camerer
Love this post as it captures the essence and cleverness of Kota’s result (and implicitly, our good & intrepid judgment @ Caltech scooping him up!). It also points to something important that psychology (and maybe neuroscience) can lead economic theory on a little, which is non-obvious– that two types of preference could be linked through a deeper primitive. This is *not* an uneconomic idea, it is just an idea that is not especially incubated by thinking about each different type of preferences on choices (risk, ambiguity, time, social) as primitive and distinct. But once some patterns are discovered or hypothesized, economic theory will be extremely useful backing up into more basic axioms that generate interestingly correlated preferences.
February 24, 2011 at 9:04 am
jeff
Colin:
You guys are smart.
February 24, 2011 at 5:35 am
Ryan
Probably just a typo in your post, but Kota’s family name appears to be Saito not Satio. (There is no such Japanese name “Satio.”)
February 24, 2011 at 5:36 am
Ryan
Sorry I see now that the title has “Saito”. I think you just have a typo where you link his name.
February 24, 2011 at 7:08 am
jeff
Thanks. I fixed it.
February 24, 2011 at 10:37 am
k
how would this work in the real world? I mean, once you take a decision – simplified by Ellsberg into the choice of which urn to pick from – you need to make a bet conditional on choosing that urn. The choice can’t be reversed; I mean you can’t go back to the other urn. Or have I misunderstood something here?
February 24, 2011 at 10:27 pm
Prisoners’ Dilemma Everywhere: Competition Among Colleges and Early Admissions Programs « Cheap Talk
[…] sorts of settings. A player can try to cream skim before competitors notice. And so NU student Kota Saito heads off to Caltech without even going on the job market. I know MEDS has dome something similar in the past. Will […]
March 5, 2011 at 12:02 am
Jonathan Weinstein
Ambiguity-hater in the house: You say “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.” Do I understand “neutrality to ex post ambiguity” correctly? I think it means that I’m indifferent to a) seeing an irrelevant ball drawn from an ambiguous urn, then having a fair coin flip decide my fate, vs. b) having a fair coin flip (or known 50-50 urn) decide my fate. Have I misunderstood? Or are there really people who are not indifferent to this? I should read that experimental paper, but the subjects must have either misunderstood, or found the setup so weird and confusing they assumed the experimenter was lying.
I guess the ambiguous ball determines whether heads or tails is a winner for me…and people care? Yikes. Let’s improve our educational system.