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 \begin{Large}

 {\bf  Career Concerns and Ambiguity Aversion  }  \\

\end{Large}


\bigskip



 \bigskip


March 9,2010 [Economics Letters, 108(2): 175--177 (August 2010)
]\\

\bigskip

Eric Rasmusen \\

\bigskip

{\it Abstract}

\end{center}

  \vspace*{-12pt}

  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.

\bigskip

\begin{small}

\noindent
  \hspace*{20pt} Dan R. and Catherine M. Dalton Professor, Department
of
Business
Economics 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. \href{mailto:erasmuse@indiana.edu}{
erasmuse@indiana.edu},  \hspace{5 pt} \url{http://www.rasmusen.org}.
This paper:
 \url{http://www.rasmusen.org/papers/ambiguity-rasmusen.pdf}.

 




\noindent
Keywords and Codes: D81,   G31, L21,  ambiguity, Ellsberg paradox,
career concerns

\noindent
  I thank    Nuffield College, Oxford University for its hospitality and Ashley Johnson for research assistance. 


\end{small}

\newpage

 \noindent
{\bf Introduction}

   Under our standard theory of expected utility, only expected
utility matters, not how its probabilities are  computed. A person
should be indifferent between 
  a gamble with a known, objective,  probability of success to one
with the same expected probability  but  greater dispersion.  If he 
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, as opposed to well-known, ``objective'' probabilities. 

   Most of the literature on ambiguity aversion (or ``uncertainty aversion'')   has tried to   define and axiomatize it, e. g.    Schmeidler (1989), or, for a recent representative, Klibanoff,  Marinacci and   Mukerji (2005).    Yet some cases of ambiguity aversion  would seem to fit into the usual framework of risk aversion, allowing ``strategy-based explanations'' of the kind used in Harbaugh (forthcoming) to mimic the predictions of prospect theory.    Halevy   and  Feltkamp (2005)   explain  it as the result of  the possibility of positive correlation among estimate errors when there are bundled risks   or  Morris (1997), which bases it on rule rationality and the distrust of experimenters or other strategic actors.    




  I, too, will also use a strategy-based explanation      to explain  ambiguity aversion. The approach is   
  in the spirit of the career concerns
literature, in which    agents take actions not just to
maximize current performance, but to protect their reputation for high
ability.  Career concerns will  make the agent
averse  to  ambiguity if he is already averse to risk.    Agents 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 method choice, what is to be
learned is the agent's ability to influence the probability of
success, e.g. Milbourn,      Shockley and   Thakor (2001).     Here,
it will be the agent's ability to estimate the probability of
success. An  agent's  ability will  become  better known if observers
learn something about the agent's subjective probability of success
and something about the objective probability of success.

In the model, a risk-averse agent will choose one of two methods to achieve some goal.   An example would be a doctor choosing either an old medical technique of known risk or a  new one whose probability of success is only slowly becoming known.    The agent will prefer   a   known  probability of success because choosing the  ambiguous method  will reveal information on his ability to estimate probabilities. This information affects his payoff either directly, through his self-esteem, or indirectly, through his future career. In either case, risk aversion makes him prefer not to learn his ability. 

  This story depends on the agent not knowing his own ability.  If he knows he is high-ability, then he will actually be ambiguity-loving, wanting a chance to reveal his ability.   This may be what drove the results in one well-known experiment: Heath and Tversky (1990) found that   subjects actually preferred ambiguous choices when  they felt  knowledgeable about the probabilities being estimated.  

 

\bigskip

\noindent
{\bf  The Model}

An unambiguous method, A, is successful with probability
$p_a$. An ambiguous method, B, is successful   with probability
$p_b$,  which equals with equal  probability either  $\beta$  or
 $\alpha > \beta$.   Profit is 0 if the  chosen
method is a failure,
1 if a success.

The  agent gets a private signal  $Alpha$  or $Beta$  of   method
B's success probability.     With probability .5  he is untalented ($T=0$)  and
his
signal is correct   with probability $\theta=\underline{\theta}$.
With
  probability .5   he is talented ($T=1$)  and his signal is correct
with probability $\theta=\overline{\theta}$, where $.5 \leq
\underline{\theta}< \overline{\theta} \leq 1$.

 Neither   market
nor  agent knows if he is talented.
Denote the market's estimate of  the probability that he  is
talented by $\widehat{T}$. Denote  the    agent's estimate of $p_b$ by $
\widehat{p}_b $.



The  agent's  payoff  function  is
\begin{equation} \label{a1}
  u(profit)  +  v(\widehat{T}),
 \end{equation}
   where $u', v'>0$ and $u'', v''<0$.

 The agent might care about $\widehat{T}$ simply from pride, or because his future compensation   depends  on whether the market
thinks he is good at estimating probabilities. A 
talented agent will have higher expected profit than an untalented
one, and so would be paid more if his talent were known.





 The agent's expected payoffs from      methods  A   and   B are
thus
\begin{equation} \label{a2}
 [ p_a u(   1) +     (1-p_a) u(0)]   +  v(.5)
 \end{equation}
  and
\begin{equation} \label{a3}
 [ \widehat{p}_b  u(   1) +     (1-\widehat{p}_b ) u(0)  ]   + [
\widehat{p}_b v(\widehat{T}|S)  +     (1-\widehat{p}_b ) v(\widehat{T}|F) ]
 \end{equation}

   These assumptions imply that the  agent is risk averse and  would
prefer a safe method with a profit of $p$ to a gamble. Nonetheless,
the ambiguity of  method B's probability of success   does not matter
in the profit part of the agent's  utility function.
 If  $p_a$  and  $\widehat{p}_b $  both equal some number $p$, then
the expected payoffs are
\begin{equation}  \label{a4}
 [  p  u(   1) +     (1-p) u(0) ] +  v(.5)
 \end{equation}
and
\begin{equation}  \label{a4a}
 [  p  u(   1) +     (1-p) u(0) ] +  [ p v(\widehat{T}|S)  +     (1-p  )
v(\widehat{T}|F) ]
 \end{equation}


\bigskip

\noindent
{\bf Reputation}



When the agent receives the signal $Alpha$ or $Beta$ he does not
thereby learn anything about his  ability. Both qualities of method B
have equal probability, so he is equally likely to receive each
signal.



 The agent's expected probability of success for method B if  his
private signal is $Beta$ is
\begin{equation} \label{a5}
\begin{array}{ll}
 \widehat{p}_b  =  Pr(S|Beta) & = .5 (\underline{\theta} \beta +
(1-\underline{\theta})
 \alpha) + .5 (\overline{\theta}  \beta + (1-\overline{\theta})\alpha )
\\
\end{array}
\end{equation}
because with  probability .5 he  is untalented, in which case the
$Beta$ signal is correct with probability  $\underline{\theta}$  so
$p_b=\beta$, and incorrect with probability $1-\underline{\theta}$ so
$p_b= \alpha$.  With probability .5 he is  talented, in which case the
 $Beta$ signal is correct with probability  $\overline{\theta}$ so
$p_b= \beta$, and incorrect with probability $1-\overline{\theta}$ so
$p_b= \alpha$.

As a running example, let $ \beta  =.3, \alpha =.7 , \underline{\theta}
=.6,  \overline{\theta} =.9$. Then
\begin{equation} \label{a5a}
\begin{array}{ll}
 \widehat{p}_b  =   .5
((.6) (.3)  + (.4)
(.7)) + .5 ((.9) (.3)+ (.1)(.7)) =  .40.\\
\end{array}
\end{equation}







Similarly, if  his
private signal is  $Alpha$,  
\begin{equation}  \label{a6}
\begin{array}{ll}
 \widehat{p}_b  = Pr(S|Alpha)&  =  .5 ((1-\underline{\theta})
 \beta + \underline{\theta}
\alpha) + .5 ( (1-\overline{\theta}) \beta+ \overline{\theta}\alpha) \\
 & \\
 & =.5
((.4) (.3)  + (.6)
(.7)) + .5 ((.1) (.3)+ (.9)(.7)) =  .60.\\
\end{array}
\end{equation}



If $p_a< E(p_b|Beta) $,   the  agent would adopt   method B
regardless of his signal.  The market could not learn anything about
his talent, regardless of whether it  observed the success or failure
of the method, or even the true probability $p_b$.


If  method B is chosen and
  $E(p_b|Beta)< p_a< E(p_b|Alpha)$, the  market can deduce that    the
signal was $Alpha$.  This has no immediate use in learning the
agent's talent, but in conjunction with observing $S$ or
$F$ it will be useful.



   After observing $S$ or $F$, the market forms its
posterior belief about the agent's talent. If the method is
successful and  the signal was $Alpha$, then  using Bayes's Rule, the
belief is
\begin{equation} \label{a7}
\begin{array}{ll}
\widehat{T} (S) = Pr(T=1|S, Alpha  ) &=  \frac{Pr
(S|T=1 , Alpha ) Pr (T=1  )}{Pr (S|Alpha)}\\
  & \\
   & =     \frac{ (.5 (1-\overline{\theta}) \beta+ .5
\overline{\theta}
\alpha) (.5)}{  ( .5(1-\overline{\theta})\beta+.5 \overline{\theta}
\alpha)  (.5)  +   (.5(1-\underline{\theta}) \beta+.5
\underline{\theta}
\alpha)   (.5)     }  \\
 & \\
&       =     \frac{[(.5)(.1) (.3)+ (.5)(.9)(.7) ] (.5 )}{[(.5)(.1)
(.3)+ (.5)(.9)(.7) ] (.5 ) + [(.5)(.4) (.3)+ (.5)(.6)(.7) ] (.5 )} =
.55.\\
\end{array}
\end{equation}
 or
\begin{equation} \label{a8}
\begin{array}{ll}
\widehat{T} (F) = Pr(T=1|F, Alpha ) &=  \frac{Pr
(F|T=1 ,  Alpha   ) Pr (T=1  )}{Pr (F| Alpha)}\\
 & \\
   & =     \frac{(.5  \overline{\theta}  (1-\alpha)  +  .5
(1-\overline{\theta})  (1-\beta)  )  (.5)}{ (.5  \overline{\theta}
(1-\alpha)  +  .5 (1-\overline{\theta})  (1-\beta)  )  (.5) + (.5
\underline{\theta}  (1-\alpha)  +  .5 (1-\underline{\theta})
(1-\beta)  )  (.5)     }  \\
 & \\
&    =     \frac{[  .5(.9) (.3) + .5 (.1) (.7)]  (.5)   }{ [  .5(.9)
(.3) + .5 (.1) (.7)]  (.5)    +   [  .5(.6) (.3) + .5 (.4) (.7)]  (.5)
}   = .425.   \\
\end{array}
\end{equation}


\noindent
  The expected value of $\widehat{T}$ is
\begin{equation} \label{a9}
\begin{array}{lll}
E T   &=  &  Pr (S|Alpha)  Pr(T=1|S, Alpha ) \\
 & & + Pr (F|Alpha)  Pr(T=1|F, Alpha   )\\
 & &\\
 & =   &  \overline{\theta}(.5 )+    (1-\overline{\theta})(.5 ) = .5\\
\end{array}
\end{equation}
 where the second line is obtained by substituting   from  (\ref{a7})
and (\ref{a8}).

  If the agent was  risk-neutral in reputation, his decision to
accept the method would be unaffected by the market learning about
his ability, but we  have assumed he is risk averse.  By Jensen's
Inequality,   $E(v(\widehat{T}))< v(E(\widehat{T}))$ because  $v$ is concave and $\widehat{T}$ is
stochastic. If   $\widehat{p}_b  = p_a$,  the agent would
choose method 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.
The agent's principal  should  design compensation   to
encourage the agent to take on ambiguous methods: (1)   to get him
to  choose the most profitable method,  since otherwise he might
prefer method A even if $p_a < \widehat{p}_b $; and (2)  to learn his
ability to decide whether to keep him or hire a new agent.  This
second advantage   would make the principal actually ambiguity loving.

\bigskip

\noindent
 {\bf Fifty-Fifty Hindsight}

   ``Hindsight bias''  is the tendency of  people to  fool themselves
into believing    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 agent in a way
that looks like   hindsight bias but   is  actually  unbiased.

 In the  last section's model, when the market observes $S$ it
deduces that the agent, having observed   $Alpha$,  was more likely
than not talented and the true probability was $\alpha$.  Now assume
that  once the method is completed the market learns whether $p_b =
\alpha$ directly---  regardless of actual success, the ex ante
probability of success  becomes common knowledge.  This is even better
information for learning about the agent's ability.

If the market observes $p_b$ directly,   it does not need to use the
imperfect signal of $S$ or $F$.  Rather, $\widehat{T}$  becomes
\begin{equation} \label{a10}
\begin{array}{ll}
 \widehat{T} (p_b=\alpha) &= Pr(T=1| Alpha, p_b= \alpha )  =  \frac{Pr (p_b=
\alpha|T=1 ,Alpha  ) Pr (T=1  )}{Pr (p_b=\alpha|Alpha)}\\
  & \\
   & =     \frac{\overline{\theta}(.5 )}{\overline{\theta}(.5 )
+\underline{\theta}(.5 )}   = .6.\\
\end{array}
\end{equation}
 or
\begin{equation} \label{a11}
\begin{array}{ll}
\widehat{T} (p_b=\beta) &= Pr(T=1| Alpha, p_b=\beta   )  =  \frac{Pr (p_b=
\beta|T=1 ,Alpha ) Pr (talented  )}{Pr (p_b=\beta|Alpha)}\\
  & \\
    &=     \frac{(1-\overline{\theta})(.5 )}{(1-\overline{\theta})(.5 )
+(1-\underline{\theta})(.5 )}  \\
 & \\
    &=     \frac{(.1)(.5 )}{(.1)(.5 )+ (.4)(.5 )} = \frac{.05}{.05+
.20}   = .2.\\
\end{array}
\end{equation}

The gap between   $ \widehat{T} (p_b=\beta)$ and  $\widehat{T} (p_b=\alpha)$ is   bigger
than the gap  between      $ \widehat{T} (F)$ and  $ \widehat{T} (S)$ because
\begin{equation} \label{a12}
\widehat{T} (p_b=\alpha) =   \frac{\overline{\theta}(.5 )}{\overline{\theta}(.5
)+\underline{\theta}(.5 )}  =   \frac{\overline{\theta}\alpha}
{\overline{\theta}\alpha+\underline{\theta}\alpha}>
\frac{\overline{\theta}\alpha + W_1}{\overline{\theta}\alpha + W_1
+\underline{\theta}\alpha + W_2}
\end{equation}
 if $W_1<W_2$. From equation (\ref{a7}), however, $\widehat{T} (S)$ equals
that right-hand expression in \eqref{a12}, however, if we set  $W_1=
(1-\overline{\theta}) \beta$ and $W_2=(1-\underline{\theta}) \beta$,
so  $\widehat{T} (p_b=\alpha) > \widehat{T} (S)$ not just for our numerical example
but in general. It can  similarly be shown  that  $\widehat{T} (p_b=\beta) <
\widehat{T} (F)$.

 Thus, if the market has 50-50 hindsight and observes $p_b$ directly
instead of just observing $S$ or $F$, the agent's risk
from the ambiguous method rises.  He has reason to prefer a method
whose  success probability     is objectively known to the public
rather than a method whose success probability he has estimated, even
if the probabilities turn out the same. Furthermore, the agent 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 agent's
ability.






\noindent
 {\bf References}
 


 Ellsberg, D., 1961, Risk, ambiguity, and the Savage axioms, Quarterly
Journal of Economics 75, 643--669.



 
Halevy, Y. and V. Feltkamp, 2005,   A Bayesian approach to uncertainty aversion,  Review of Economic Studies  72, 449--466.

Harbaugh, R., forthcoming, Prospect theory or skill signaling? American Economic Review. 


 Klibanoff, P., M. Marinacci and S. Mukerji, 2005, A smooth model of decision making under ambiguity,
  Econometrica   73, 1849--1892. 

    Milbourn, T.,   R. Shockley and  A. Thakor,  2001,    Managerial 
career concerns and investments in information,
  RAND Journal of Economics   32,  334--351.

 
 
Morris, S.,   1997,  Risk, uncertainty and hidden information,  Theory and Decision 42, 235--269.

 


 Schmeidler,  D.,1989,  Subjective probability and expected utility without
additivity,
Econometrica  57, 571--587.


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