Violations of the Hypothesis of Probabilistic Sophistication

Violations of the Hypothesis of Probabilistic Sophistication
Savage’s (1954) joint axiomatization of expected utility risk preferences
and probabilistic beliefs, employing an expected utility function
for the risk preference function, has been justly termed “the crowning glory of choice theory” (Kreps 1988, p.120). However, the violations of expected utility first observed by Allais were soon matched by violations of probabilistic sophistication, even in situations involving the simplest forms of subjective uncertainty. The most famous of these examples, known as the Ellsberg Paradox (Ellsberg 1961, 2001), involves drawing a ball from an urn containing 30 red balls and 60 black or yellow balls in an unknown proportion. The following
table illustrates four subjective acts defined over the color of the drawn ball, when the entries in the table are payoffs or outcomes:
28 Machina
30 balls
60 balls




red
black
yellow
ƒ1(⋅)
$100
$0
$0
ƒ2(⋅)
$0
$100
$0
ƒ3(⋅)
$100
$0
$100
ƒ4(⋅)
$0
$100
$100
When faced with these choices, most subjects prefer act ƒ1(⋅) over ƒ2(⋅), on the grounds that the probability of winning $100 in ƒ1(⋅) is guaranteed
to be 1/3, whereas in ƒ2(⋅) it could range anywhere from 0 to 2/3. Similarly, most subjects prefer ƒ4(⋅) over ƒ3(⋅), on the grounds that the probability of winning $100 in ƒ4(⋅) is guaranteed to be 2/3, whereas in ƒ3(⋅) it could range anywhere from 1/3 to 1. Although this reasoning may well be sound, it is inconsistent with the hypothesis of probabilistic
beliefs. That is, there is no triple of subjective probabilities {μ(red), μ(black), μ(yellow)} that can simultaneously generate a preference for ƒ1(⋅) over ƒ2(⋅) and for ƒ4(⋅) over ƒ3(⋅), since a probabilistically sophisticated
individual would only exhibit the former ranking when μ(red) > μ(black), and only exhibit the latter ranking when μ(red) < μ(black).
Ellsberg also presented what many feel to be an even more fatal example, involving two urns:
50 balls
50 balls
100 balls
���� � �
���� � �




red
black
red
black
g1(⋅)
$100
$0
g3(⋅)
$100
$0
g2(⋅)
$0
$100
g4(⋅)
$0
$100
In this example, most subjects are indifferent between g1(⋅) and g2(⋅), are indifferent between g3(⋅) and g4(⋅), but strictly prefer either of g1(⋅) or g2(⋅) to either of g3(⋅) or g4(⋅). It is straightforward to verify that there exist no pair of subjective probabilities {μ(red), μ(black)} for the right-hand urn—50:50 or otherwise—that can generate this set of preference rankings. Such examples illustrate the fact that in situations (even simple
situations) where some events come with probabilistic information
States of the World and the State of Decision Theory 29
and some events (termed ambiguous events) do not, subjective probabilities
do not always suffice to fully encode all aspects of an individ-ual’s uncertain beliefs. Since most real-world events do not come with such probabilistic information, Ellsberg’s Paradoxes and related phenomenon
deal a serious blow to the hypothesis of probabilistic sophistication.