Why do people have ambiguity aversion, preferring, a gamble with a 50% chance of success to one whose expected probability of success is 50% but where that 50% is an unbiased estimate? The answer modelled here, in the spirit of the career concerns literature, is learning: a risk-averse person does not wish observers to learn whether he is good or bad at estimating probabilities. He therefore prefers a gamble with objective probabilities.time inconsistency, hyperbolic discounting

Eric Rasmusen

English

Research Papers in Economics

Career Concerns and Ambiguity Aversion
5 May 2008
Eric Rasmusen
Abstract
Why do people have ambiguity aversion, preferring, a gamble with a
50% chance of success to one whose expected probability of success is 50%
but where that 50% is an unbiased estimate? The answer modelled here, in
the spirit of the career concerns literature, is learning: a risk-averse person
does not wish observers to learn whether he is good or bad at estimating
probabilities. He therefore prefers a gamble with objective probabilities.
Dan R. and Catherine M. Dalton Professor, Department of Business Eco-
nomics and Public Policy, Kelley School of Business, Indiana University. BU 438,
1309 E. 10th Street, Bloomington, Indiana, 47405-1701. Oﬃce: (812) 855-9219.
Fax:812-855-3354. erasmuse@indiana.edu, http://www.rasmusen.org. This pa-
per: http://www.rasmusen.org/papers/ambiguity-rasmusen.pdf.
Keywords and Codes: D81, G31, L21, ambiguity, Ellsberg paradox, career concerns
I thank Nuﬃeld College,Oxford University for its hospitality.Introduction
Under our standard theory of expected utility, only expected utility mat-
ters, not how its probabilities are computed. A person should be indiﬀerent
between a a gamble with a known, objective, probability of success to one
with the same expected probability but greater dispersion. If he is prefers the
known probability we say he exhibits “ambiguity aversion.” One variant of
ambiguity aversion is to prefer a gamble with a single probability of success
(e.g., .7) to one with a compound probability (e.g., 50% chance of .6, 50%
chance of .8). The Ellsberg Paradox is an example (Ellsberg, 1961). Most
people prefer a gamble in which they win if a red ball is drawn from urn A
with 100 balls, 50 of which are red and 50 blue; to the same gamble with
urn B which has 100 balls of an unknown color mix. It cannot be that they
guess that there are less than 50 red balls in the second urn, because they
also prefer urn A if they win when a blue ball is drawn. A second variant is
to dislike making decisions that rely on estimated probabilities. Businesses
dislike venturing into situations in which probabilities are based on poor in-
formation, e.g., providing earthquake insurance when a large earthquake is a
rare event.
Such preferences are the subject of a large literature, and have been
axiomatized in two of its best known articles, Gilboa and Schmeidler (1989)
and Schmeidler (1989). I wish here to use instead a structural approach
in the spirit of the career concerns literature. In that literature, managers
take actions not just to maximize current performance, but to protect their
reputation for high ability. This can make them either more averse to risk
or less than they otherwise would be.
In the present context, career concerns will not make the manager averse
to risk, but to ambiguity. Managers dislike ambiguity because it can make
them look stupid. If their utility is concave in reputation, they will avoid a
choice which displays their ability. In most models of career concerns and
project choice, what is to be learned is the manager’s ability to inﬂuence the
probability of success, e.g. Milbourn, Shockley and Thakor (2001). Here, it
will be the manager’s ability to estimate the probability of success. A man-
ager’s ability will become better known if observers learn something about
the manager’s subjective probability of success and something about the ob-
jective probability of success.
2The Model
The unambiguous project, A, is successful with probability pa. The
ambiguous project, B, is successful with probability pb, which is β or α > β
with equal probability. Proﬁt is 0 if the chosen project is a failure, 1 if a
success.
The manager gets a private signal Alpha or Beta of Project B’s success
probability. With probability .5 he is untalented and his signal is correct
with probability θ = θ (e.g. his signal is Alpha and the project’s success
rate is α). With probability .5 he is talented and his signal is correct with
probability θ = θ, where .5 ≤ θ < θ ≤ 1.
The manager’s payoﬀ function is
u(profit) + v(T), (1)
where u0,v0 > 0 and u00,v00 < 0.
The manager might care about T simply from pride, but he might also
care because his future compensation will depend on whether the market
thinks he is good at estimating probabilities. In this model, a talented man-
ager will have higher expected proﬁt than an untalented one, and so would
be paid more if his talent were known.
Neither market nor manager knows if he is talented. Denote the market’s
estimate of the probability that he is talented by T. Denote the manager’s
estimate of pb by b pb.
The manager’s expected payoﬀs from Project A and B are thus
[pau(1) + (1 − pa)u(0)] + v(.5) (2)
and
[b pbu(1) + (1 − b pb)u(0)] + [b pbv(T|Success) + (1 − b pb)v(T|Failure)] (3)
These assumptions imply that the manager is risk averse and would
prefer a safe project with a proﬁt of p to a gamble. Nonetheless, the ambiguity
of project B’s probability of success does not matter in the proﬁt part of the
3manager’s utility function. If pa and b pb both equal some number p, then the
expected payoﬀs are
[pu(1) + (1 − p)u(0)] + v(.5) (4)
and
[pu(1) + (1 − p)u(0)] + [pv(T|Success) + (1 − p)v(T|Failure)] (5)
Reputation
When the manager receives the signal Alpha or Beta he does not thereby
learn anything about his ability. Both qualities of project B have equal
probability, so he is equally likely to receive each signal.
The manager’s expected probability of success for Project B if his private
signal is Beta is
b pb = Pr(Success|Beta) = .5(θβ + (1 − θ)α) + .5(θβ + (1 − θ)α) (6)
because with probability .5 he is untalented, in which case the Beta signal
is correct with probability θ so pb = β, and incorrect with probability 1 − θ
so pb = α. With probability .5 he is talented, in which case the Beta signal
is correct with probability θ so pb = β, and incorrect with probability 1 − θ
so pb = α.
As a running example, let β = .3,α = .7,θ = .6,θ = .9. Then
b pb = .5((.6)(.3) + (.4)(.7)) + .5((.9)(.3) + (.1)(.7)) = .40. (7)
Similarly, if his private signal is Alpha,
b pb = Pr(Success|Alpha) = .5((1 − θ)β + θα) + .5((1 − θ)β + θα)
= .5((.4)(.3) + (.6)(.7)) + .5((.1)(.3) + (.9)(.7)) = .60.
(8)
If pa < E(pb|Beta), the manager would adopt project B regardless of
his signal. The market could not learn anything about his talent, regardless
4of whether it observed the success or failure of the project, or even the true
probability pb.
If Project B is chosen and E(pb|Beta) < pa < E(pb|Alpha), the market
can deduce that the signal was Alpha. This has no immediate use in learning
the manager’s talent, but in conjunction with observing Success or Failure
it will be useful.
After observing Success or Failure, the market forms its posterior belief
about the manager’s talent. If the project is successful and the signal was
Alpha, then using Bayes’s Rule, the belief is
T(Success) = Pr(Talented|Success,Alpha) =
Pr(Success|Talented,Alpha)Pr(Talented)
Pr(Success|Alpha)
=
(.5(1−θ)β+.5θα)(.5)
(.5(1−θ)β+.5θα)(.5)+(.5(1−θ)β+.5θα)(.5)
=
[(.5)(.1)(.3)+(.5)(.9)(.7)](.5)
[(.5)(.1)(.3)+(.5)(.9)(.7)](.5)+(.5)(.4)(.3)+(.5)(.6)(.7)](.5) = .550.
(9)
or
T(Failure) = Pr(Talented|Failure,Alpha) =
Pr(Failure|Talented,Alpha)Pr(Talented)
Pr(Failure|Alpha)
=
(.5θ(1−α)+.5(1−θ)(1−β))(.5)
(.5θ(1−α)+.5(1−θ)(1−β))(.5)+(.5θ(1−α)+.5(1−θ)(1−β))(.5)
=
[.5(.9)(.3)+.5(.1)(.7)](.5)
[.5(.9)(.3)+.5(.1)(.7)](.5)+[.5(.6)(.3)+.5(.4)(.7)](.5) = .425.
(10)
The expected value of T is
ET = Pr(Success|Alpha)Pr(Talented|Success,Alpha)
+Pr(Failure|Alpha)Pr(Talented|Failure,Alpha)
= θ(.5) + (1 − θ)(.5) = .5
(11)
where the second line is obtained by substituting from (9) and (10)
If the manager’s were risk-neutral in reputation, his decision to accept
the project would be unaﬀected by the market learning about his ability,
but we have assumed he is risk averse. By Jensen’s Inequality, E(v(T)) <
5v(E(T)) because v is concave and T is stochastic. Thus, if b pb = pa, the
manager would choose Project A, exhibiting ambiguity aversion.
Thus, we can explain ambiguity aversion as the result of career concerns
without any need to resort to irrationality or direct disutility. There are two
reasons why the ﬁrm’s owners should design compensation to encourage the
manager to take on ambiguous projects: (1) to get him to choose the most
proﬁtable project, since otherwise he might prefer Project A even if pa < b pb;
and (2) to learn his ability to decide whether to keep him or hire a new
manager. This second advantage would make the owners actually ambiguity
loving.
Fifty-Fifty Hindsight
“Hindsight bias” is the tendency of people to fool themselves into be-
lieving ex post that they would have made a diﬀerent decision ex ante. If
we increase the ex post information available to the market, the market will
punish or reward the manager in a way that looks like hindsight bias but is
actually unbiased.
In the last section’s model, when the market observes Success it de-
duces that the manager, having observed Alpha, was more likely than not
talented and the true probability was α. Now assume that once the project
is completed the market learns whether pb = α directly– regardless of actual
success, he ex ante probability of success becomes common knowledge. This
is even better information for learning about the manager’s ability.
If the market observes pb directly, it does not need to use the imperfect
signal of Success or Failure. Rather, T becomes
T(pb = α) = Pr(Talented|Alpha,pb = α) =
Pr(pb=α|Talented,Alpha)Pr(talented)
Pr(pb=α|Alpha)
=
θ(.5)
θ(.5)+θ(.5) = .6.
(12)
6or
T(pb = β) = Pr(Talented|Alpha,pb = β) =
Pr(pb=β|talented,Alpha)Pr(talented)
Pr(pb=β|Alpha)
=
(1−θ)(.5)
(1−θ)(.5)+(1−θ)(.5)
=
(.1)(.5)
(.1)(.5)+(.4)(.5) = .05
.05+.20 = .2.
(13)
The gap between T(pb = β) and T(pb = α) is bigger than the gap
between T(Failure) and T(Success) because
T(pb = α) =
θ(.5)
θ(.5) + θ(.5)
=
θα
θα + θα
>
θα + W1
θα + W1 + θα + W2
(14)
if W1 < W2. From euation (9), however, T(Success) equals that right-hand
expression in (14), however, if we set W1 = (1 − θ)β and W2 = (1 − θ)β, so
T(pb = α) > T(Success) not just for our numerical example but in general.
It can be similarly be shown that T(pb = β) < T(Failure).
Thus, if the market has 50-50 hindsight and observes pb directly instead
of just observing Success or Failure, the manager’s risk from the ambiguous
project rises. He has reason to prefer a project whose success probability is
objectively known to the public rather than a project whose success proba-
bility he has estimated, even if the probabilities turn out the same. Further-
more, the manager is not just reacting to a behavioral bias on the part of
the market: if what would have been the best decision ex ante becomes clear
ex post, the market can use the information to better learn the manager’s
ability.
References
Ellsberg, D., 1961, Risk, Ambiguity, and the Savage axioms, Quarterly
Journal of Economics 75, 643-669.
Gilboa, I. and D. Schmeidler, 1989, Maxmin expected utility with non-
unique prior, Journal of Mathematical Economics, 18, 141-153.
7Milbourn, T., R. Shockley and A. Thakor, 2001, Managerial career con-
cerns and investments in information, RAND Journal of Economics 32, 334-
351.
Schmeidler, D., Subjective probability and expected utility without ad-
ditivity, Econometrica 57, 571-587.
8

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