Submission to the Journal of Answers to Gollier
Benjamin J. Keys
James M. Sallee
Submitted January 13, 2005
Revised January 25, 2005
Exercise 2 (Ellsberg’s Paradox)
Savage (1954) extended the expected utility theorem to situations where the probabilities are
not objectively known. He introduced the sure thing principle, an axiom that is stronger than
the independence axiom. He showed that this stronger axiom could be used to prove the sub-
jective EU theorem by noting that
there exists a subjective probability measure p,
there exists a real-valued utility function u
such that the decision maker ranks various distributions of consequences ωby using their
subjective expected utility Pspsu(ωs).
The following example is due to Ellsberg (1961). An urn contains 90 colored balls. Thirty
balls are red, and the remaining 60 either black or white; the number of black (white) balls
is not specified. There are 4 lotteries as described in table 1.2:
Table 1.2
Lottery Red Black White
La50 0 0
Lb0 50 0
Ma50 0 50
Mb0 50 50
Given that many people report La> Lband Mb> Ma, show that there exists no pair (p, u)
that sustains such ranking of lotteries. The agents are averse to ambiguous probabilities, a
possibility that is ruled out by the sure thing principle. (For a model with ambiguity aversion,
see Gilboa and Schmeidler 1987.)
Solution 2
The first step in each column uses the expected utility theorem. The rest is algebra:
