General Interests: Accountability Ambiguity Aversion Private Interests: My Pictures

Ambiguity aversion is the name given
by Ellsberg (1961) to a decisional pattern by which clear probabilities
are preferred to vague ones for lotteries that would be considered
equivalent under subjective expected utility (SEU, Savage 1954). In its simplest formulation the problem goes as follows
(Ellsberg two-color problem). Imagine you are given a chance of winning
100 by drawing a ball from one of two urns. The first urn contains 100
balls, 50 of which are black and 50 of which are red. The second urn
also contains 100 balls that are either black or red, but the proportion
of the colors is unknown. You now have to choose an urn and to announce
a color. If the ball you draw from your chosen urn is of the color you
announced, you win 100; if it is of the other color you win nothing.
Which urn do you choose? Would you change your choice if the prize to be
won from the urn with the known color proportion were 99 against a
prize of 100 from the urn with the unknown color proportion?
In experiments carried out to test this proposition, a
majority of subjects is consistently found to prefer the urn with the
known proportion of colors (risky urn) over the urn with the
unknown proportion of colors (ambiguous urn). Per se there is
nothing wrong with such a choice pattern, given that people are
generally bad at randomizing and that a null hypothesis of equally many
people choosing the one and the other urn may thus be unwarranted. Things however change when the
risky urn contains only 49 balls of the winning color and 51 of the
losing color, or when the prize to be won with the ambiguous urn is 100
while the prize to be won with the risky urn is 99. In that case, it
does indeed become irrational to choose the risky urn over the ambiguous
urn, since the latter is then objectively worse. If you are not
convinced of the equivalence of the two urns, consider the following
argument by Raiffa (1961): if you simply throw a (fair) coin to
determine the color you want to bet on, then you should see that you are
back at a 50% chance of winning, whatever the actual distribution of
colors in the urn may be.
The choice pattern described above is
also referred to as the Ellsberg paradox. To see what is so paradoxical
about this behavior however, we need to put the problem in a different
way and add a few facts. Imagine the choice is put to you in the
following way. First you are offered to draw from the risky urn and you
are asked which color you would rather bet on. Most people declare to be
indifferent between the two colors. Then you are offered to draw from
the ambiguous urn and are asked which color you would like to bet on. Again
most people are indifferent between the two colors. Since the
probabilities of the separate events (a black ball is drawn; a red ball
is drawn) must add up to one in each urn, this means that
and
,
where p_{b} and p_{r} represent the
probability of drawing a black ball and the probability of drawing a red
ball respectively, and the superscripts R and A represent
the risky and ambiguous urn.
Now imagine you are offered the chance to bet on black and given a
choice between the two urns. Most people express a strict preference for
the risky urn, which tells us that their subjective probability of winning with black
in the risky urn is strictly bigger than their subjective probability of
winning with black in the ambiguous urn, i.e.
.
Since we know that ,
this means that the subjective probability of winning with a black ball
from the ambiguous urn must be smaller than ½. Taken together with the
equalities in the previous paragraph, this means also that the
probability of winning with a red ball from the ambiguous urn must be
bigger than ½. If asked to choose between betting on red in the risky
urn and red in the ambiguous urn, however, people again declare to
strictly prefer to bet on red in the risky urn. This in turn implies
that ,
and hence that .
Probabilities in the ambiguous urn can thus be shown to add up to less
than one assuming such choice behavior, and hence the paradox.
**What causes ambiguity aversion?**
Nobody would arguably state explicitly that the
total probability of winning in the risky urn is less than one. What
then explains the observed behavior? Before attempting to answer this
question let us take a closer look at the nature of the problem at hand.
The first thing we will notice is the somewhat unnatural nature of the
task. There are two urns, and in one of them some information seems to
be missing. Indeed, the absence of information that appears as salient
for the decision making process (Frisch and Baron 1988) for the
ambiguous urn would seem to play a central role in this issue. This
absence of information that might be salient for the decision making
process (but is not according to SEU) is especially highlighted by the
direct comparison of the ambiguous urn with an urn for which all conceivable
information is known.
The latter point may be worth some
closer consideration. Since the classical Ellsberg task involves a
choice between two lotteries, a direct confrontation of the two is
inevitable. Ambiguity aversion can however also be studied in other
ways, for instance by asking subjects how much they would be willing to
pay to play the lottery in question. Making use of exactly this method,
Fox and Tversky (1995) elicited subjects' willingness to pay (wtp) for
the two urns jointly (in a comparative setting) and separately (in a
non-comparative setting, between subject). Whereas in joint
evaluation people give a much higher wtp for the risky urn than for the
ambiguous one and are thus ambiguity averse, when the two urns are
evaluated in isolation the resulting wtp for the two urns is is not
significantly different - people are no longer ambiguity averse. The
comparative nature of the task, be it in choice or joint evaluation,
seems thus to be a necessary condition for ambiguity aversion to occur.
Once this condition is fulfilled it is
however still not exactly clear why people would shy away from the urn
about which they think to know less. Many different explanations have
been proposed over the years. Curley, Yates, and Abrams (1986) test six
such explanations and find what they call other-evaluation - the fact
that others observe their decision and may judge them for it - to be the
only significant one. In the same line,
Trautmann, Vieider,
and Wakker
(2008) find that eliminating the possibility that decision maker be
judged by others by making her preferences unknown to potential
onlookers is enough for ambiguity aversion to disappear. They also find
that people with a higher score on the Leary (1983) scale measuring fear
of negative evaluation (Watson and Friend 1969) are more likely to
choose the risky urn when exposed to the potential evaluation of
others.
In sum it seems thus that ambiguity
aversion is mainly due to accountability
pressures. When called upon to make a decision which they might be
required to justify in front of an audience who's views on the matter
are unknown (Lerner and Tetlock 1999), people usually take the
decision which they deem most easily justifiable (Shafir et al. 1993).
When it comes to deciding which of the two urns to choose in the Ellsberg
task, there is a common presumption that choosing the risky urn is more
justifiable given that all information is known for it. In other words,
a choice of the ambiguous urn seems hard to justify when an otherwise
perfectly equivalent urn with more information is available.
There is a further methodological point
to be made here. As we have seen, ambiguity aversion may be investigated
both through choice tasks and through valuation tasks.
Trautmann,
Vieider, and Wakker (2009) however find preference reversals to
occur between choice and matching tasks. In particular, they find that
subjects who choose the ambiguous gamble very often give a higher wtp
for the risky gamble. Combining these findings with previous findings in
the literature one could now construct a scale of ambiguity aversion
whereby comparative evaluation tasks produce the strongest ambiguity
aversion; certainty equivalents elicited though choice produce the
second strongest ambiguity aversion due to the asymmetric strength of
valuation (but less than WTP because of the absence of preference
reversals); choice produces somewhat less pronounced, but still strong
ambiguity aversion (depending on the strenght of other-evaluation); and
finally separate evaluations do not produce any ambiguity aversion.
References
Curley, Shawn P., J.
Frank Yates and Richard A. Abrams (1986). Psychological Sources of
Ambiguity Avoidance. Organizational Behavior and Human Decision
Processes 38, 230-256.
Ellsberg,
Daniel (1961). Risk, Ambiguity and the Savage Axioms. The Quarterly
Journal of Economics 75(4), 643-669
Fox, Craig R. and Amos Tversky
(1995). Ambiguity Aversion and Comparative Ignorance. The Quarterly
Journal of Economics.
Frisch, Deborah and Jonathan Baron (1988). Ambiguity and Rationality.
Journal of Behavioral Decision Making 1, 149-157
Leary, Mark R. (1983). A Brief Version of the Fear of Negative
Evaluation Scale. Personality and Social Psychology Bulletin 9(3),
371-375
Lerner, Jennifer S. and Philip E.
Tetlock (1999). Accounting for the Effects of Accountability.
Psychological Bulletin 125, 255-275.
Raiffa, Howard (1961). Risk,
Ambiguity and the Savage Axioms: Comment. The Quarterly Journal of
Economics, vol. 75, no.. 4, pp. 690-694.
Savage, L. J. (1954). The
Foundations of Statistics. Wiley, New York.
Shafir, Eldar,
Itamar Simonson and Amos Tversky (1993). Reason-based Choice. Cognition
49, 11-36.
Trautmann, Stefan T., Ferdinand
M. Vieider, and Peter P. Wakker (2008).
Causes of Ambiguity Aversion: Known
versus Unknown Preferences. Journal of Risk and Uncertainty 36, 225-243.
Trautmann, Stefan T., Ferdinand M.
Vieider, and Peter P. Wakker (2009).
Preference
Reversal under Ambiguity. Working Paper.
Watson, David and Ronald Friend (1969). Measurement of Social-Evaluative
Anxiety. Journal of Consulting and Clinical Psychology 33 (4), 448-457

Ambiguity Aversion