Ellsberg paradox

The Ellsberg paradox is a paradox in decision theory in which people's choices violate the postulates of subjective expected utility. It is generally taken to be evidence for ambiguity aversion. The paradox was popularized by Daniel Ellsberg, although a version of it was noted considerably earlier by John Maynard Keynes.

The basic idea is that people overwhelmingly prefer taking on risk in situations where they know specific odds rather than an alternative risk scenario in which the odds are completely ambiguous—they will always choose a known probability of winning over an unknown probability of winning even if the known probability is low and the unknown probability could be a guarantee of winning. That is, given a choice of risks to take (such as bets), people "prefer the devil they know" rather than assuming a risk where odds are difficult or impossible to calculate.

Ellsberg actually proposed two separate thought experiments, the proposed choices which contradict subjective expected utility. The 2-color problem involves bets on two urns, both of which contain balls of two different colors. The 3-color problem, described below, involves bets on a single urn, which contains balls of three different colors.

Suppose you have an urn containing 30 red balls and 60 other balls that are either black or yellow. You don't know how many black or how many yellow balls there are, but that the total number of black balls plus the total number of yellow equals 60. The balls are well mixed so that each individual ball is as likely to be drawn as any other. You are now given a choice between two gambles:

Also you are given the choice between these two gambles (about a different draw from the same urn):

This situation poses both Knightian uncertainty – how many of the non-red balls are yellow and how many are black, which is not quantified – and probability – whether the ball is red or non-red, which is 1/3 vs. 2/3.

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