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\begin{LARGE}

\begin{center} {\LARGE \textbf{Getting Carried Away in  Auctions as Imperfect
Value Discovery } \\[0pt] }

  May 14, 2004 \\[0pt]



Eric Rasmusen \\[0pt]



\textit{Abstract} \end{center}

Bidders in auctions  must   decide whether  and when to incur the cost of
estimating the most they are willing to pay. This can explain why people seem to
get carried away, bidding higher than  they had planned before the auction   and
then finding they had paid more than the object was worth to them. Even when
such  behavior is rational, ex ante,   it may be  perceived as irrational  if
one ignores  other situations in which people revise their bid ceilings upwards
and are happy when that enables them to win the auction.

\noindent
 {  \hspace*{20pt} Indiana University Foundation Professor, Department  of
Business Economics and Public Policy, Kelley School of Business,Indiana
University,  BU 456, 1309 E. 10th Street, Bloomington, Indiana, 47405-1701.
Office: (812) 855-9219. Fax: 812-855-3354. Erasmuse@indiana.edu.\newline
Mypage.iu.edu/$\sim$erasmuse, Mypage.iu.edu/$\sim$erasmuse/papers/carried-
rasmusen.pdf.  }

\noindent
 { I thank Michael Baye, Rene Kirkegaard,  Kerry Krutilla, Yacheng Sun, and
participants in    seminars at Indiana University's Kelley School of Business
and   the University of Tokyo's economics department, and the 2004 International
Industrial Organization Conference for their comments, and Harvard Law School's
Olin Center and the University of Tokyo's Center for International Research on
the Japanese Economy for their hospitality. }

\newpage

 Rule No. 3: Set a ceiling for your bid.

``People can make emotional decisions. In the cool of the preview, think of how
much you want to spend and set a limit, rather than getting caught up in the
frenzy of bidding and spending more,'' Keane says.

``When I go to an auction, I do set a ceiling,'' says auction lover Renee
Alexander. ``I've been known to go beyond that. I've also been the object of
dealers upping the price. I bought a brass headboard and footboard, and they
were bidding me up and I got carried away and paid more than I should have.''

But at \$200, the brass bed was still a steal, she says.

It's easy, when you're bidding, to lose track of everything but the object of
your desire.

``What happens in an auction situation is that people's sense of self is on the
line,'' says psychologist Dick Geist, president of the Institute of Psychology
and Investing in Newton. ``Once they start bidding, they're holding themselves
up for the validation or confirmation of whatever money means to them, be it
self-worth or power. . . . You're proving that you can have an impact on the
worth of this item.''

\newpage

 \noindent \textbf{2. The Model}

The two possible bidders in an auction, both risk-neutral, have private values
which are statistically independent.

Bidder 1's value is $v_1$, which has three components:   $v_1= \mu+u+ \epsilon$.
He knows the value of  $\mu$, and he knows that that $u$ and ${\epsilon}$ are
independently  distributed  with  mean zero and  differentiable densities  on
$[-\overline{u},\overline{u} ]$ and $[- \overline{\epsilon},
\overline{\epsilon}]$, where $\mu-\overline{u}- \overline{\epsilon}>0$ so
$v_1$ is never negative.  If he wishes, at any time he can pay $c$ and learn the
value of $u$ immediately. He cannot discover the other component, ${\epsilon}$,
however, until after the auction.

Bidder 2's value, $v_2$, is $\underline{v_2}$ with probability $ \theta$ and
$\overline{v_2} $ with probability $(1-\theta)$, with $\theta \in (0,1)$; and
with $\underline{v_2} \in (\mu-\overline{u} ,\mu)$ and $  \overline{v_2} \in
(\mu, \mu+\overline{u})$.   Figure 1 illustrates the distribution of values.
Bidder 2 knows the value of $v_2$ but not $v_1$.




\newpage

\noindent
{\bf Discussion of the Assumptions}

 Our purpose is to model a situation in which a bidder---Bidder 1 in the
model---  is uncertain about (a) his value and (b) whether there exists any
other bidder who has a higher value.  The model's focus is on his  decision  on
whether to incur the cost of learning more about his value.

   The   low value for Bidder 2, $ \underline{v_2}$, is assumed  to be less than
$\mu$ so that if $v_2$ does take that value Bidder 1 will win the auction even
if he just bids  up to his initial expected value. Bidder 2's low value is
chosen to be greater than $(\mu- \overline{v_2} )$ so that  if Bidder 1 does win
at a price of $\underline{v_2}$, Bidder 1's ex post payoff  might be negative.

  The possible high value for Bidder 2, $ \overline{v_2}$, is assumed to exceed
$\mu$ so that if $v_2$  takes that value Bidder 1 will lose the auction   if he
just bids up to  his initial expected value. The value of $\overline{ \epsilon}$
is less than $(\mu+\overline{u})$ so that if Bidder 1 does discover $u$, there
is still  some chance that $(\mu+u)$ will be high enough that Bidder 1 will win
the auction.



 \newpage

\bigskip \noindent
  \textbf{3. The Equilibrium }

Each bidder must decide on a bid ceiling. Bidder 1 must also decide at what bid
level, if any, to pay $c$ to discover $u$, after which he may wish to revise his
bid ceiling.

If Bidder 1 does not acquire any information about his value, his best strategy
is to bid  up to $\mu$, the expected value of the object to him. If he does
discover $u$, his optimal strategy is to bid up to $(\mu+u)$, his updated
estimate of $v_1$. Bidder 2's optimal strategy is  to choose a bid ceiling of
$v_2$. Note that there is  no benefit to Bidder 2 in changing  his bid ceiling
in order to affect the timing of Bidder 1's value discovery; value discovery is
instantaneous, so timing is unimportant in this model, unlike in Rasmusen
(2002), where  value discovery cannot take place late in the auction.

Bidder 1 has three value discovery strategies  that might be optimal in
equilibrium:    early discovery,  late discovery, and no discovery. The  early
discovery  strategy is to pay to discover $u$   when the bid level reaches some
value $b^* \in [0,\underline{v_2}) $, most simply at the start of the auction,
so $b^*=0$.     The  late discovery  strategy is to pay to discover $u$   if the
bid level reaches  some level $b^* \in [ \underline{v_2}, \mu+ \overline{u}] $
and    Bidder 2 has failed to drop out, most simply if the bidding reaches
Bidder 1's initial bid ceiling, so $b^*=m$.   The strategy of no discovery is to
refuse to pay to discover $u$ regardless of what happens.

 \newpage

  Bidder 1's expected payoff if he chooses never to pay to discover his value is
the simplest, \begin{equation}  \label{e2} \pi_1 (no\;discovery) =   \theta
(\mu-\underline{v_2}) +(1-\theta)(0), \end{equation} because with probability $
\theta$ he will win the auction at a price of $\underline{v_2}$ and with
probability $(1-\theta)$ he will lose the auction.

 Bidder 1's expected payoff from late discovery--- paying $c$ and discovering
$v_{1}$  if and only if Bidder 2 does not drop out at the price of
$\underline{v_2}$--- is made up of the expected value  of winning at price
$\underline{v_2}$ and the expected value of winning at price $\overline{v_2}$
when $\mu+u>\overline{v_2}$. \begin{equation}  \label{e1} \begin{array}{ll}
\pi_1 (late\; discovery)  & =  \theta (\mu-\underline{v_2}) \\
& + \left[1-\theta \right]  \left[ -c+   \int_{ \overline{v_2} -\mu} ^{\infty}
[(\mu+u)-\overline{v_2}] f(u) du     \right] \\ \end{array}
\end{equation}

 Bidder 1's payoff from early discovery is made up of  the cost $c$ plus the
expected value of winning at  price $\underline{v_2}$ when $\mu+u>
\underline{v_2}$ and $y=\underline{v_2} $,  plus the expected value of winning
at price $\overline{v_2}$ when $\mu+u>\overline{v_2}$ and $y=\overline{v_2}$.
\begin{equation}  \label{e0} \begin{array}{ll}
 \pi_1  (early \; discovery) & = \theta \left[ -c+  \int_{ \underline{v_2}-\mu}
^{\infty}  [(\mu+u)-\underline{v_2}] f(u) du     \right]\\
 & + \left[1- \theta \right]  \left[ -c+   \int_{ \overline{v_2}-\mu}^{\infty}
[(\mu+u)-\overline{v_2}] f(u) du     \right] \\ \end{array}
 \end{equation}

\newpage

   Proposition 1 uses the following shorthand notation:
 \begin{equation}  \label{e4a}
\begin{array}{ll}
 A_1  &  \equiv  \int_{ \underline{v_2}-\mu} ^{\infty}
[(\mu+u)-\underline{v_2}] f(u) du   \\ & \\
 A_2 & \equiv \int_{ \overline{v_2}-\mu}^{\infty} [(\mu+u)-\overline{v_2}] f(u)
du \\
  \end{array}
 \end{equation}

\noindent {\bf Proposition 1:} {\it  Which of the three value discovery
strategies is optimal depends on $c$   as follows:} \begin{equation}  \label{e3}
\begin{array}{ll} {\rm No \; discovery } & if \;  c \geq A_2 \\ &  \\ {\rm Late
\;discovery } & if \;  c  \in  [A_1- (\mu-\underline{v_2}), A_2 ] \\ &  \\ {\rm
Early \;discovery } & if \;  c \leq A_1  - (\mu-\underline{v_2}) \\ \end{array}
\end{equation}

 \newpage

\noindent {\it Lemma: The middle range is not empty: }
  \begin{equation} \int_{\underline{v_2}-\mu}^{\infty} \left(
\mu+u-\underline{v_2}\right) f(u)du - (\mu- \underline{v_2}) <
\int_{\overline{v_2}-\mu}^{\infty} ( \mu+u-\overline{v_2})f(u)du. \end{equation}

    This is true if and only if  $(\mu-\underline{v_2})> \overline{v_2} -\mu$;
that is, if the low Bidder 2 value, $\underline{v_2}$, is further from Bidder
1's expected  value, $\mu$, than is the high Bidder 2 value, $\overline{v_2}$.
This is true by assumption in the model. If this were not true, then as the
discovery cost $c$ increased, Bidder 1 would simply jump from early discovery to
no discovery.   Bidder 1 would choose no discovery if  $c$ became too big to
justify paying it to avoid overpaying $\underline{v_2}$ when $\mu+u>
\underline{v_2}$. But in that case, when the bid rose to $\underline{v_2}$ and
Bidder 2 was still in the auction, Bidder 1 would find {\it a fortiori} that $c$
was too big to justify paying it to gain the chance of winning  the auction when
$\mu+u>\overline{v_2} $.





\newpage

\noindent {\bf  Proposition 2:} {\it   As  the toughness of competition (the
level of $\overline{v_2}$) increases, Bidder 1's willingness to pay to improve
his estimate of $v_1$ falls, but that willingness is unaffected by the
probability of tough competition ($\theta$).   }

\noindent {\bf Proof:}

REPLACE IN HERE



 The derivative of $A_1$ with respect to $\theta$ is zero.

Thus,  increases in  $\overline{v_2}$ expand the parameter range for No
Discovery  but increases in $\theta$ leave it unchanged. $\blacksquare$

\newpage

\noindent
 {\bf  Proposition 3:} {\it  As   $\overline{u}$, the degree of uncertainty over
his private value,  increases,  Bidder 1's willingness to pay to  improve his
estimate of his value increases, as does his payoff if he chooses early or late
discovery.  }

\noindent {\bf Proof.}  The derivative of $A_1$ with respect to $\overline{u}$
is
 \begin{equation}
\label{e8} \frac{d A_1}{d  \overline{u} } =  (\mu+\overline{u}
-\underline{v_2}) f(\overline{u})  >0.
\end{equation}
 Thus, from Proposition 1 his willingness to pay $c$ increases.  Also,
\begin{equation}
 \label{e8s} \frac{d A_2}{d  \overline{u} } =  (\mu+\overline{u}
-\overline{v_2}) f(\overline{u})  >0.
 \end{equation}
Together, these inequalities imply that Bidder 1's payoff increases if he
chooses early or late discovery (it is unchanged  if he chooses No Discovery).
$\blacksquare$

\newpage

\noindent \textit{Interpretation as Getting Carried Away}

 The main interest of this model is that it provided an interpretation for
``getting carried away'' in the course of an auction. Suppose we see a bidder
winning an auction at a price higher than the most he entered the auction being
willing to pay, and that he later regrets having won at that high price--what I
will call an ``unhappy victory.''   At the start of the auction, $\mu$ was the
most Bidder 1 intended to bid. The auction begins, and the bidding rises to
$\mu$. Now, however, he reconsiders, and raises his bid ceiling to $\mu+u$. This
new ceiling is greater than $\overline{v_2}$, the most Bidder 2 will pay. Bidder
1 thus wins the auction at price $\overline{v_2}$. After the auction is over,
however, he discovers ${\epsilon}$ and finds that $\mu+u+y$ is less than
$\overline{v_2}$, so he regrets having won. He says to himself: ``I got carried
away and bid too much. I wish I'd stuck with my original ceiling of $\mu$.''

\newpage

\noindent
 \textbf{4.  A  Model with Continuous Densities}

Unlike in Section 2, we will now  assume that Bidder 2's value, $v_2$, is
distributed according to an atomless and   differentiable density $g(v_2)$  on
$[0,k]$, where $k>\mu$ and where $g(v_2)>0$ for all $v_2$ on that interval.
Bidder 2 does not know $v_1$, but he does know   $  v_2$. All parameters are
common knowledge.

  As in Section 3, Bidder 2's  optimal strategy is to choose a bid ceiling of
$v_2$, Bidder 1's optimal bid ceiling is $Ex$, which will be either $\mu$ or
$\mu+u$, depending on whether he has paid $c$ to discover $u$.   Bidder 1 must
also    decide  at what bid level $p$ to pay $c$ to discover $u$, where possibly
$p=0$ (early discovery)   or $p = k$ (no discovery).

\newpage

 If Bidder 1 chooses the policy of submitting a bid ceiling of $ p$, his
expected payoff  is, if $v_2< p, $
 \begin{equation}
\pi_1(v_2)  = \int_{\mu-\overline{v_2} }^{\mu+\overline{u}} (\tilde{v_1}-v_2)
f(\tilde{v_1}) d\tilde{v_1}.
 \end{equation}

\noindent
If $v_2 \geq p$, Bidder 1's expected payoff is
 \begin{equation} \pi_1 (v_2)  = -c +  \int_{\mu-\overline{v_2} }^{v_2} (0)
f(\tilde{v_1}) d\tilde{v_1} +  \int_{v_2}^{\mu+\overline{u}} (\tilde{v_1}-v_2)
f(\tilde{v_1}) d\tilde{v_1}.
 \end{equation}

Integrating over the possible values of $v_2$ yields an overall expected payoff
for Bidder 1 of
\begin{equation} \label{overall}
 \begin{array}{ll}
\pi_1  &=  \int_0^{p}  \left( \int_{\mu-\overline{v_2} } ^{\mu+\overline{u}}
(\tilde{v_1}-v_2) f(\tilde{v_2}) d\tilde{v_1} \right) g(v_2) dv_2\\
 & + \int_{p}^{\mu+\overline{u}}  \left(  -c +   \int_{v_2} ^{\mu+\overline{u}}
(\tilde{v_1}-v_2) f(\tilde{v_1}) d\tilde{v_1} \right) g(v_2) dv_2.\\
\end{array}
\end{equation}

\noindent
 {\bf Proposition 3:}     {\it   In the model with continuous value
densities,
Bidder 1 will  follow a policy of late  discovery if $c$ is low
enough  and otherwise no discovery, but he will never follow a policy
of early discovery: (i)   $p>0$,  if   $c$ is low
enough then
$0<p< \mu$, and if $c$ is high enough then $p > \mu$. }




\noindent
 {\bf  Proposition 4:} {\it  As the degree of uncertainty over his
private value   increases,  Bidder 1 becomes more willing to pay to
discover his value: if $p^*< \mu$ , then  $p^*$ falls if  we spread
density $f(u)$   using a strict decrease in $f$ on any interval $[r, s]$
and a
strict increase  everywhere else  while leaving the  mean of $u$
unchanged at zero.   }






\newpage

\begin{quotation}

``Never go beyond a predetermined limit when bidding. Base this limit on the
information you have gathered. Avoid becoming obsessed with an item. Doing so
will lead you to bid more than the property (or merchandise) is worth. If you
are bidding on a tax sale property, you might bring a certified cheque for the
maximum amount you intent to bid. This should ensure that you do not get carried
away with the bidding process. If you are the successful bidder and the property
is sold for less than the amount on your cheque, the clerk/treasurer will issue
a refund for the difference.''(``Tax Sale Properties/Auction Guidelines''
website)

 \vspace{.8in}

``Avoid catching auction fever. This happens when bidders get carried away with
the process; they will bid on anything and everything that is being auctioned
and often will end up being the owner of things they did not even want and
paying far too much for these items. The opposite of auction fever is auction
paralysis. This occurs when the bidder is paralysed with fear and thus is unable
to make a bid. Apparently such a state is often due to a fear of overpaying. If
you don't overcome it you will never get started. Often, if you fail to do your
homework, you will not have the confidence to bid.'' (``Tax Sale
Properties/Auction Guidelines'' website)

\end{quotation}

\newpage

My story: The bidder rationally revises   his estimate of the value of the
object  upwards during the course of the auction, so that at the moment of
purchase he actually does value it at more than the price he pays.

  The difference from the standard private-value auction will be that it is
costly for the bidder to discover his own private value, so he defers doing so
until the middle of the auction.

 At that point he might revise it upwards--``auction fever''-- or he might
revise it downwards-- ``auction paralysis''.

Compte \& Jehiel (2000)  and Rasmusen (2002):  a bidder in a private- value
ascending auction  is   uncertain about his value and will be able to pay a
fixed amount to improve his information.

    \newpage \noindent \textbf{Differences from My Other Auction Paper}

  The important differences  from Rasmusen (2002) is that  here if the
uninformed bidder pays $c$ then \begin{enumerate} \item He acquires the value
information immediately, not after a time lag, and\\ \item He    only acquires
better information about his value, not perfect information. \end{enumerate}

The absence of a time lag means that the informed bidder has no incentive to
strategically delay bidding, the ``sniping'' phenomenon at the heart of my other
article. The imperfection of the information means that even if the uninformed
bidder makes optimal decisions ex ante, ex post he may regret having made them.

 \newpage

 EMPIRICAL IMPLICATIONS

1. In the value discovery model the carried-away  winner would regret  having
won less than half of the time--- less than half, because  even  though  his
revised value is still an overestimate, he usually will not have to pay the
entire amount to win the auction.

2. A short ``cooling off period'' would   affect an emotional winner more than
a value-discovering winner,   although even in the value discovery model,  the
winner would, after thinking more, wish to return the object a significant
fraction of the time.

3. The value discovery model implies that if the value is more uncertain, the
bidder will be more likely to increase his bid ceiling in the course of the
auction, because the option value of value discovery is higher. An emotional
explanation might or might not have this implication.

\end{LARGE}

\end{document}


