
\documentclass[12pt,epsf]{article}

      %  pdflatex     carried-rasmusen.tex

 

%\includegraphics[width=150mm]{fig07-01.jpg}

\usepackage{graphicx}

\usepackage{amsmath}

 \usepackage{amssymb}


\reversemarginpar

\topmargin  -1in

\oddsidemargin .25in

 \textheight  9.4in

 \textwidth 6.4in

 

         \begin{document}

  \parindent 24pt

\parskip 10pt

 

\baselineskip 16pt


\titlepage
                     \vspace*{12pt}

\begin{center}
{\Large \textbf{Getting Carried Away in  Auctions as Imperfect Value
Discovery } \\ 
}


 \bigskip


 WORKING PAPER--LONG, OLDER VERSION\\ (Comments welcomed)





\bigskip

June 6, 2004. 22 August 2005.  \\ 

\bigskip

Eric Rasmusen \\ 

\bigskip

\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
 {\small \hspace*{20pt} Dan R. and Catherine M. Dalton 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,
http://rasmusen.org.
http://rasmusen.org/papers/carried-rasmusen.pdf.  }

\noindent {\small I thank Michael Baye, Rene Kirkegaard,  Kerry Krutilla,
Yacheng Sun,
and
participants in    seminars at Indiana
University's Kelley School of Business, DePaul University,   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


 \begin{raggedright}

 \parindent 24pt

 \parskip 10pt



\noindent
 \textbf{1. Introduction}

Do people get carried away at auctions? Certainly they do in the sense
of bidding more than they had intended when they arrived at the
auction. What must be determined is why that happens. Most simply,
other people bid more than expected, so a bidder ends up paying more
than he thought he would. But what we usually think of as getting
carried away is that during the auction a bidder raises the maximum
amount he is willing to bid.

Even this often has a simple explanation: that the auction is for an
object with a common value component, so that during the auction our
bidder learns from the high bids of other people that the object is
worth more than he had thought earlier. That, too,  is not
what we think of as getting carried away. Rather, we think of a
private value auction--- of some good that the bidder intends to keep
for personal consumption rather than for production or resale--- in
which the bidder ends up bidding more than his pre-auction estimate of
the value of the object.

   The standard advice to bidders is to avoid getting carried away.
One website puts it like this:

\begin{quotation}
\begin{small}

 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.

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,'' (http:
//www.taxsaleproperties.com/abt\_7.html)

\end{small}
\end{quotation}



It is quite plausible that people make such mistakes (see Malhotra \&
Murnighan [2000] for a persuasive example of irrational bidding, or
the survey  evidence  of
confusion about auction rules on page  14 of  Roth \& Ockenfels [2002]),
and even more
plausible that in
auctions, as in ordinary purchases, people end up  spending
extravagantly on current consumption to their later regret.    I wish
to propose another explanation, however,  for bidders who end up
paying more than their pre-auction maxima: that the bidder rationally
revises   his estimate of the value of the object  upwards during the
course of the auction, so   at the moment of purchase he actually
does value it at more than the price he pays. The model will still be
one with purely private values, since our bidder will not be deriving
any information about his own value from the other players' bids. The
difference from the standard private-value auction will be that it is
costly for him 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''.


	The model that will be used is similar  to the models of
Compte \& Jehiel (2000)  and Rasmusen (2003a): a bidder in a
private-value ascending auction will be uncertain about his value  and
will be
able to pay a fixed amount to improve his information. Unlike in those
models, here his information will still be imperfect after value
discovery, and we will focus on his decision to update his bid rather
than on the auction's payoffs or welfare implications under different
auction rules. Note, too, that the situation is quite unlike that in models such
as Persico (2000) and the articles cited there which examine the incentives to
gather information on the value of objects only before,  not during, an auction.


\bigskip

 \noindent
\textbf{2. The Model}

There are  two possible buyers  in an auction for an object, both risk-
neutral and  with private values which are statistically independent.

Bidder 1's value is $v_1$, which has three components:   $v_1= \mu+u+
\epsilon$.  Bidder 1 does not know  the sizes of $u$, $\epsilon$, or
$v_1$. He does know $\mu$, and he knows that that $u$ and $\epsilon$ are
independently distributed  according to   symmetric densities $f(u)$ and
$g(\epsilon)$ with mean zero and supports  such that $Min (\mu+u+\epsilon)\geq
0$,    so   that $v_1 \geq 0$. As a
result, Bidder 1's initial expectation of $v_1$ equals $\mu$. 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 $\alpha$ with probability $ \theta$
and $\beta $ with probability $(1-\theta)$,  where
$\alpha,$   $ \theta$,
and $\beta $ are common knowledge; and
$\theta \in
(0, 1)$,   $\alpha \in (0, \mu)$,  and $
\beta>\mu$.
We do not need any assumption on the expected value of $v_2$ relative to
$v_1$.   Bidder 2 knows the value   of $v_2$ but not  the value of
$v_1$.

The auction is open and ascending.  The price starts at zero and rises
continuously until one player drops out, at which point the other player
wins the object   and pays that price. A player's bidding strategy is a
choice of a  price at which to drop out (a ``bid ceiling'') in an open-
exit auction such as this one. This setup avoids the technical
untidiness  created when the winnning player must bid a positive
increment higher than the next-highest  bid  in order to win.

\bigskip

\noindent
 \textit{Discussion of the Assumptions}

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

  We assume the  low value for Bidder 2, $ \alpha$,   to be
less than $\mu$ so that if $v_2=\alpha$ Bidder 1 will win the
auction even if he just bids  up to  $\mu$.   The value $ \alpha
 $ is assumed to exceed $ \mu - \underline{u} $ so that   Bidder 1's
ex post payoff  might be negative  if he wins at a price of
$\alpha$.

  We assume the  possible high value for Bidder 2, $ \beta$,
exceeds  $\mu$ so that if $v_2 =\beta$   Bidder 1 will lose the
auction   if he just bids up to  $\mu$.  Note that however high
$\beta$ may be,  under our assumptions
there
is still  some chance that $(\mu+u)$ will be high enough that Bidder 1
will win the auction.



   A  different  way to model this situation would be to assume general
differentiable    distributions for $v_2$ as well as for $u$ and
$\epsilon$. That model will be  used  in Section 4.   The assumption of
a  two-point distribution  used here, however,  will allow for some
interesting comparative statics, and will be heuristically useful.

 In this game it  is especially important   to think of the
probabilities of each  of Bidder 2's types, $v_2 = \alpha$ and
$v_2 =\beta$,   as being the subjective probabilities of the
uninformed player, which are not necessarily the true population
magnitudes. The variable $ \theta$ represents the strength of Bidder 1's
belief that Bidder 2's value is low.  Note that $ \theta$ can be
arbitrarily close to one and the results of the model still hold.  The
model is most  interesting for high values of $\theta$, which represent
situations in which Bidder 1 is surprised to find that he faces tight
competition from Bidder 2.

   Rasmusen (2003a) also models a bidder who begins a private value
auction unsure of his own value but who can pay $c$ to acquire
information. The important differences between that model and this one
are that  here if the uninformed bidder pays $c$ then (a) he acquires
the value information immediately, not after a time lag, and (b) he only
acquires better information about his value, not perfect information.
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.


\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.

The optimal  bidding strategies   are straightforward.  A player should choose a
bid ceiling equal to the expected value of the object being auctioned.
If he bids any less, he could lose even though the winning  price  was
less than his  expected value. If he bids any more, he could win at a
price greater than his  expected value.

Thus, 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 (2003a), 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,\alpha) $, 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^* \geq \alpha  $ and Bidder 2
has failed to drop out, most simply if the bidding reaches Bidder 1's
initial bid ceiling, so $b^*=\mu$.   The strategy of no discovery is
to refuse to pay to discover $u$ regardless of what happens.

  Bidder 1's expected payoff if he chooses never to pay to discover
his value is
 \begin{equation}  \label{e2}
 \pi_1 (no\;discovery) =   \theta (\mu-\alpha) +(1-\theta)
(0),
 \end{equation}
 because with probability $ \theta$ he will win the auction at a price
of $\alpha$ 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 $\alpha$--- is made up of the expected value  of
winning at price $\alpha$ and the expected value of winning
at price $beta$ when $\mu+u>\beta$.
\begin{equation}  \label{e1}
\begin{array}{ll}
\pi_1 (late\; discovery)  & =  \theta (\mu-\alpha)
 + \left[1-\theta \right]  \left[ -c+   \int_{ \beta -\mu}
^{\infty}  [(\mu+u)-\beta] 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 $\alpha$ when
$\mu+u> \alpha$ and $v_2=\alpha $,  plus the
expected value of winning at price $\beta$ when $\mu+u>
\beta$ and $v_2=\beta$.
\begin{equation}  \label{e0}
\begin{array}{ll}
\pi_1  (early \; discovery) & =  \displaystyle{ \theta \left[ -c+
\int_{
\alpha-\mu}
^{\infty}  [(\mu+u)-\alpha] f(u) du     \right]+} \\
  & \displaystyle{  \left[1-
\theta \right]  \left[ -c+   \int_{ \beta-\mu}^{\infty}
[(\mu+u)-\beta] f(u) du     \right]} \\
\end{array}
\end{equation}

\bigskip

 We can now make more precise what happens in equilibrium. If $c$ is
low enough, early discovery is best: Bidder 1 pays $c$ at the start of
the game to avoid the negative payoff from winning  even at the low
price of $ \alpha$  when $v_1$ happens to be low. For
moderate levels of $c$, late discovery is best: he waits until he
observes Bidder 2 bidding more than $\alpha$ and then to pay
$c$ to discover his value. If $c$ is high enough, no discovery is
best; he sticks with his initial bid ceiling  and never pays $c$.  The
optimal value discovery strategies are made precise in Proposition 1,
which uses the following shorthand notation:
\begin{equation}  \label{e4a}
\begin{array}{ll}
A_1  & \displaystyle{  \equiv  \int_{ \alpha-\mu}
^{\infty}  [(\mu+u)-\alpha] f(u) du  } \\
 & \\
 A_2 & \displaystyle{  \equiv \int_{ \beta-\mu}^{\infty}
[(\mu+u)-\beta] f(u) du} \\
 \end{array}
\end{equation}


\noindent
{\bf Proposition 1:} {\it All three value discovery
strategies can be optimal, depending   on the value of the
discovery cost $c$:  }
\begin{equation}  \label{e3}
\begin{array}{ll}
 {\rm Early \;discovery}& if \; c \leq A_1-(\mu-\alpha)\\
 & \\
{\rm Late \;discovery} & if \;c \in [A_1- (\mu-\alpha),
A_2] \\
&  \\
 {\rm No \; discovery } & if \;  c \geq A_2\\
\end{array}
\end{equation}
 {\it If Bidder 2's possible high value  is   farther from $\mu$ than      his
possible low value,   then the middle range is empty and late discovery is not
optimal:
 If $\beta-\mu > \mu- \alpha$ then late discovery will not occur.}








\noindent
{\bf Proof:}   The payoffs in equations  (\ref{e0}), (\ref{e1}),  and
(\ref{e2}) can be written as:
 \begin{equation}  \label{e4}
\begin{array}{lll}
(a) & \pi_1(early \; discovery) &= \theta  (-c+ A_1) + (1- \theta)  (-
c+A_2) \\
& & \\
(b) &\pi_1(late \; discovery) &=\theta(\mu-\alpha) + (1-
\theta)  (-
c+A_2)\\
& & \\
(c) &\pi_1(no \; discovery) &= \theta(\mu-\alpha)  \\
 \end{array}
\end{equation}
   The early discovery payoff in  equation (\ref{e4}a) is greater than
the late discovery payoff in  equation (\ref{e4}b)  whenever $c <  A_1-
(\mu-\alpha)$. The no discovery payoff in  equation (\ref{e4}c) is greater than
the late discovery payoff in  equation (\ref{e4}b)
whenever $c > A_2$.

 If   $A_1- (\mu-\alpha)< A_2$,  then   the
middle range of $c$ values is not empty and   any of the three
strategies might be optimal.    That inequality can be rewritten as
  \begin{equation}\label{e6b}
 \int_{\alpha-\mu}^{\infty} \left( \mu+u-\alpha
\right)
f(u)du - (\mu - \alpha) < \int_{\beta-\mu}
^{\infty}
 ( \mu+u-\beta)f(u)du
\end{equation}
or
  \begin{equation} \label{e7}
\begin{array}{l}
 \displaystyle{ 0 <  -( \mu -\alpha  )[F(\infty) -
F(\alpha-\mu)]
-\int_{\alpha-\mu}^{\beta-\mu} u f(u) du  + (\mu -
\alpha) +(\mu -\beta  )[F(\infty) -
F(\beta- \mu)]} \\
   \\
 \displaystyle{ 0 <   ( \mu-\alpha  )  F(\alpha-\mu) -
\int_{\alpha-\mu}^{\beta-\mu} u f(u) du    +( \mu -
\beta  )[1 -F(\beta-\mu)].}\\
\end{array}
\end{equation}
  Suppose $\mu -\alpha > \beta
 -\mu $.   Then the middle term of the right side of
inequality (\ref{e7}),$ -
\int_{\alpha-\mu}^{\beta-\mu} u f(u) du$, which is the
expected value of $u$ conditional on    being between   $\mu -
\alpha$ and  $ \beta
 -\mu $, is positive. Is the positive first term's magnitude  greater
than the  negative third term's? Yes, for the following reason. Since
the distribution $F$ is symmetric,
 if $   \mu -\alpha =
\beta-\mu  $ (contrary to our assumptions) then
 \begin{equation} \label{e7a}
  ( \mu-\alpha  )  F(\alpha-\mu)
   +( \mu -
\beta  )[1 -F(\beta-\mu)]
\end{equation}
equals zero.
  Suppose now that we increase  $\mu-\alpha$, so that it
exceeds $\beta-\mu$.   Since $d/dz \; zF(z) = F(z) + zf(z)>0$,
this increases the first, positive,  term  of expression (\ref{e7a}) while
leaving the second, negative,  term unaffected. Thus, expression
(\ref{e7a}), when brought into accordance with the assumption that  $\mu-
\alpha>\beta-\mu$, is positive,  expressions
(\ref{e7}) and (\ref{e6b}) are positive,
  and the middle range of $c$
in Proposition 1 is not empty; otherwise, it is, and late discovery is not
optimal.    $\blacksquare$

\bigskip


     Inequality (\ref{e6b}) is true if    $ \mu-
\alpha > \beta -\mu$; that is, if the low Bidder 2
value, $\alpha$, is further from Bidder 1's expected  value,
$\mu$, than is the high Bidder 2 value, $\beta$.  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
$\alpha$ when $\mu+u>\alpha$. But in that case, when
the bid rose to $\alpha$ 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>\beta $.






\bigskip

	Having established the equilibrium, we can now see how Bidder
1's behavior is affected by changes in parameters other than $c$.

\noindent
   {\bf  Proposition 2:} {\it  Bidder 1's    willingness to pay to improve his
estimate of   falls  with  the toughness of competition but is unaffected by the
probability of tough competition: the level of $c$ which makes ``No Discovery''
optimal    is falling in $\beta $  and unchanged in $\theta$.}

\noindent
   {\bf Proof:}  Bidder 1's willingness to pay to improve his estimate
is captured by the bounds in Proposition 1.  In particular, he will
follow the policy of no discovery if and only if $c  \geq A_2,
 $  which is to say, if
\begin{equation}
 c \geq  \int_{ \beta-\mu}^{\infty}
[(\mu+u)-\beta] f(u) du
\end{equation}
   The derivative of $A_2$ with respect to $\beta$ is then
\begin{equation}
 \frac{d A_2}{d  \beta } = -  (\mu+[\beta-\mu] -
beta)
f(\beta-\mu) -  \int_{ \beta-\mu} ^{\infty}
f(u) du,
\end{equation}
 which is negative. The derivative of $A_2$ with respect to $\theta$ is
zero.

Thus,  increases in  $\beta$ expand the parameter range for no
discovery but increases in $\theta$ leave it unchanged. $\blacksquare$

   The intuition behind the first part of  Proposition 2 is that as
$\beta$
increases and becomes further from $\mu$,   it becomes less   likely
that the expected value after discovery, $(\mu+u)$, will be greater than
$\beta$ and Bidder 1 will want to increase his bid and win the
auction. Thus, giving up becomes more attractive, unless the cost $c$ of
discovering $u$ is low. It is interesting to see what happens near $
\beta = \mu$ (though the assumptions of the model   rule out $
\beta \leq \mu$). For $\beta \leq \mu$, Bidder 1 would
not pay even a tiny  $c$ to discover $u$, because   Bidder 2 would have
already dropped out and it would be too late for Bidder 1 to change his
behavior. If, however, $\beta $ is just slightly above $\mu$,
then  the value of information about $u$ is very large because with a
probability of almost .5, discovery of $u$  will lead Bidder 1 to change
his behavior.

  The  key to the second part of Proposition 2 is that the decision
between late and no discovery is deferred until new information arrives
that renders irrelevant
$\theta$,  the probability  that $v_2 = \beta$.  Bidder 1 need
not decide about paying $c$ until he sees that Bidder 2's value  must be
high-- at which point the ex ante probability it is high is moot.

\bigskip
\noindent
   \textit{Interpretation as Getting Carried Away}

This model  provides an interpretation for ``getting carried away'' in  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 $\beta$, the most
Bidder 2 will pay. Bidder 1 thus wins the auction,  at price $\beta$. After the
auction is over, however, he discovers $\epsilon$ and finds that
$\mu+u+\epsilon<\beta $. He says to himself: ``I got carried away and bid too
much. I wish I'd stuck with my original ceiling of $\mu$.''

     This, of course, is only one possible scenario. It is worth exploring the
conditions under which unhappy victories occur. In the   story above, Bidder 1
had an unhappy victory.  With equal likelihood, after the auction is over he
would have discovered  $\epsilon>0$,  so   his consumer surplus would have been
even higher than he  had expected-- a sort of ``extra-happy victory''.

 It is worth comparing the probability of unhappy victories with and without
value discovery.
Even if he does not discover  $u$ and increase his bid ceiling, Bidder 1    will
still   sometimes overpay. Suppose we are in the middle range of costs, so
Bidder 1 is following the policy of late discovery.  If  Bidder 2's value turns
out to be low ($ v_2= \alpha$),  Bidder 1 will win at a price of $\alpha$.
This is less than the expected value of $v_1$, which is $\mu$, but it might be
more than the true value of $v_1$, which is $\mu +u + \epsilon$,  giving rise to
an unhappy victory.  Since  $u$ and $\epsilon$ have symmetric distributions,
however, unhappy victories will occur with probability less than 50\%.  Indeed,
if $\alpha$ is low, unhappy victories may be very rare. Victories without value
discovery-- without ``getting carried away''-- will occur only if the
competition from Bidder 2 is weak, so   the winning price is low and  Bidder 1
will come away with a good chance of sizeable consumer surplus.

   On the other hand,  if Bidder 2's value turns out to be  high ($ v_2=
\beta$),  Bidder 1 will pay to discover $u$, and if $u$ is high enough he will
raise his bid ceiling high enough to win at a price of $\beta$.   This is less
than the expected value of $v_1$, which is $\mu+u$, but it might be more than
the true value of $v_1$, which is $\mu +u + \epsilon$,  giving rise to an
unhappy victory.

 What is the probability of this unhappy victory?
 For the borderline case of $\mu+u = \beta$, an unhappy victory has probability
50\%. For higher values of $u$, the probability of an unhappy victory falls. But
victories following value discovery-- those that happen because the bidder
``gets carried away''--   occur only if the competition from Bidder 2 is strong.
The  winning price is thus high, and  Bidder 1's chance  of coming away with
positive consumer surplus may be very little higher than his chance of coming
away with negative consumer surplus.

 Thus, situations in which Bidder 1 pays to discover his value and then
increases his bid ceiling and wins the auction are worse situations for him than
when he does not pay to discover his value but wins the auction anyway. This is
not because  the strategy of late discovery is suboptimal, though-- it is not.
Rather, it is because actually  carrying through and discovering his value under
that strategy only occurs after bad news-- the news that he is facing tough
bidding competition. The strategy of late discovery is analogous to  a person's
strategy of using chemotherapy if he is diagnosed with cancer. Under that
strategy,   chemotherapy will come to be associated with pain and death, but
that does not lessen its usefulness  in making the best of a bad situation.



\bigskip

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

  In the model above, Bidder 2 had two possible values.  This brings
sharply into relief the  late discovery strategy, in which    Bidder 1
delays discovering his value in the hope that nobody else will have a
high value. Another possible case, not more general, but equally interesting,
is when Bidder 2 has a continuous
distribution for his value.    We will model that in a way  similar to
  Rasmusen (2003a), adapted to the   auction rules  and
instantaneous value discovery assumed in the present paper.  This will
show that the phenomenon of a bidder increasing his reservation bid in
the course of an auction is robust, and, indeed, in the continuous
density  model not only will  value discovery  be optimal  if $c$ is
not too large, but the optimality of   late discovery for moderate levels of $c$
will not require any condition analogous to that stated in Proposition 1.



As in Section 2, let there be two possible bidders, both risk-neutral,
with private values   which are statistically independent.

Our assumption about Bidder 1 will remain the same.
Bidder 1's value is $v_1$, which has three components:   $v_1= \mu+u+
\epsilon$.  Bidder 1 does not know  the sizes of $u$, $\epsilon$, or
$v_1$. He does know $\mu$, and he knows that that $u$ and $\epsilon$ are
independently distributed  according to   symmetric densities $f(u)$ and
$g(\epsilon)$ with mean zero and supports  such that $Min (\mu+u+\epsilon)\geq
0$,    so   that $v_1 \geq 0$. As a
result, Bidder 1's initial expectation of $v_1$ equals $\mu$. 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.






Unlike in Section 2, we will now  assume that Bidder 2's value, $v_2$,
is distributed according to an atomless and   differentiable density
$h(v_2)$  on $[\alpha, \beta]$, where   $0 < \alpha < \mu$ and  $\beta>\mu$ and
where $h(v_2)>0$ 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 equal to $v_2$.  Bidder 1's optimal bid ceiling is $Ev_1$,
which will be either $\mu$ or $(\mu+u)$, depending on whether he has
learned $u$.   Bidder 1 must also choose  a ``discovery level''  $p$--- a bid
level   at which Bidder 1   pays $c$ to discover $u$, where possibly $p <
\alpha$
(early discovery, because Bidder 1 will pay to discover his value before
discovering anything about $v_2$)   or $p > \mu$ (no discovery, because without
discovery it never happens that the price rises above $\mu$).

  To analyze Bidder 1's payoff as a function of $p$, let us start by
supposing (contrary to the assumptions) that Bidder 1 knows $v_2$.
Suppose also that $p \leq \mu$, so that there is positive probability
that Bidder 1 pays $c$ and discovers $u$.

If    $v_2< p  $ then Bidder 1 wins the auction at price $v_2$, for an
expected payoff of $(\mu-v_2)$.

 If    $v_2> p  $ then he pays $c$ to
discover $u$.  He loses the auction if $\mu + u <p$;  otherwise, he
wins. Overall, if $v_2 >p$ his expected payoff is
 \begin{equation} \label{e100}
  \pi_1(v_2| v_2>p)  = -c +  \int_{u= - \infty }^{v_2- \mu } (0) f(u)
du +
  \int_{u=v_2- \mu    }^{\infty } (\mu + u -v_2 ) f(u) du.
\end{equation}



Integrating over the possible values of $v_2$ yields an overall
expected
payoff for Bidder 1 of
 \begin{equation} \label{e101}
\pi_1  =  \int_{v_2=\alpha}^{p} (\mu - v_2) h(v_2) dv_2 +
 \int_{v_2=p}^{\beta} \left( -c + \int_{u=v_2- \mu}^{\infty }
(\mu + u -v_2 ) f(u) du   \right) h(v_2) dv_2
\end{equation}

If, on the other hand, $p > \mu$, then Bidder 1 is following the policy
of no discovery, and his expected  payoff is simply the first part of
equation (\ref{e101}):
 \begin{equation} \label{e101a}
\pi_1 (p>\mu)  =  \int_{v_2=\alpha}^{\mu} (\mu - v_2) h(v_2) dv_2.
\end{equation}



\noindent
 {\bf Proposition 3:}     {\it   In the model with continuous value
densities,  the optimal discovery level,  $p^*$,     rises with $c$, rising
strictly  if $p^* \in (\alpha, \mu)$.
Bidder 1 will  follow a policy of early discovery ($p^* \in [0, \alpha)$)  if
$c$ is low enough, late  discovery ($p^* \in [  \alpha,\mu]$)  for higher levels
of  $c$, and   no discovery ($p^* \in (  \mu, \infty]$) if $c$ is sufficiently
high.      }


\noindent
 {\it Proof:}
 Differentiating  equation (\ref{e101}) with respect to $p$ yields
 \begin{equation} \label{e101b}
\begin{array}{ll}
\frac{d \pi_1 }{dp} & =   (\mu - p) h(p)   - \left( -c + \int_{u=p- \mu}^{\infty
} (\mu + u -p ) f(u) du   \right) h(p)\\
 & \\
 & = \left[  c  + (\mu - p) - \int_{u=p- \mu}^{\infty } (\mu   -p  + u ) f(u) du
\right] h(p)   \\
 & \\
  & = \left[  c  +     -\int_{u=-\infty}^{\infty } (\mu - p+u) f(u) du   -
\int_{u=p- \mu}^{\infty } (\mu   -p  +  u ) f(u) du \right] h(p)   \\
 & \\
  & =  \left[c  +   \int_{u=-\infty}^{p- \mu } (\mu   -p  +  u ) f(u) du
\right] h(p)   \\
 \end{array}
\end{equation}
   If $h(p)$ is positive (which it is between $\alpha$ and $\mu$) and $c$ is
small enough,  then this derivative is negative.
If $c$ is small enough,   $\frac{d \pi_1 }{dp}<0$  for   $p \in [\alpha, \mu]$,
and profit rises if $p$ is reduced to below  $\alpha$ -- that is, to early
discovery. If $p<\alpha$, then $h(p)=0$, so further reductions are unimportant--
early discovery can take the form of any $p$ in the interval $[0, \alpha)$.

 If $c$ is greater, then $\frac{d \pi_1 }{dp} >0$ at $p=\alpha$, and the optimal
$p$ exceeds $\alpha$.
Evaluated at the optimal level of $p$, the  second derivative of profit is
 \begin{equation} \label{e101c}
\begin{array}{ll}
\frac{d^2 \pi_1 }{dp^2}      & = \left( (\mu   -p  +   [p-\mu] ) f(u)   +
\int_{u=-\infty}^{p- \mu } (-1) f(u) du \right) h(p) +  \left[ c+
\int_{u=-\infty}^{p- \mu } (\mu   -p  +  u ) f(u) du    \right] h'(p)   \\
  &\\
 & =  0 -  \left( \int_{u=-\infty}^{p- \mu }   f(u) du \right)  h(p) + (0)h'(p),
\\
 & \\
 &<0,  \\
 \end{array}
\end{equation}
  where we use the fact that   $\frac{d \pi_1 }{dp}=0$  at the optimum to obtain
the term $(0)h'(p)$. Since it is also true that
\begin{equation} \label{e101d}
\begin{array}{ll}
\frac{d^2 \pi_1 }{dp dc}      & =  (1)  h(p)     \\
  &\\
 &>0,  \\
 \end{array}
\end{equation}
 the implicit function theorem tells us that $\frac{d p }{ dc}  >0$ when $h(p) >
0$,  i.e., the optimal discovery level rises continuously with the cost of
discovery. This means that there exist levels of $c$ such that the optimal
discovery level lies within the interval $(\alpha, \mu)$, so late discovery is
optimal. It also means that as $c$ increases, eventually the optimal discovery
level exceeds $\mu$, so that no discovery becomes optimal. $\blacksquare$





\bigskip

  If the discovery cost is low enough, early discovery is best, because the
bidder averts the possibility that he  might pay more than his value by winning
even at the other bidder's lowest possible value. If the discovery cost is
somewhat higher, it is not worth paying it  to avoid that risk, and the bidder
will choose late discovery.  How late depends on the size of the discovery cost,
and the optimal discovery level rises smoothly with the discovery cost, and if
the discovery cost is too high, then no discovery becomes optimal.



Thus, we see that Section 3's conclusion that a bidder may  decide to increase
his bid ceiling in the course of an auction is robust to allowing rival values
to take more than two possible levels.  The  comparative statics of Proposition
2 (on what happens when the probability $\theta$ of a low $v_2$ change) are not
easily adapted to    the continuous model, but a new kind of comparative statics
result  possible only in the continuous model    is be derived in Proposition 4.




\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 strictly increase  everywhere
else, while leaving  the  mean of $u$ unchanged at zero.   }




\noindent
{\bf Proof.} Let us define $X$ as a component of    equation
(\ref{e101b}) (rearranged here   slightly):
  \begin{equation}  \label{e9b}
\begin{array}{lll}
\frac{d\pi_1 }{dp } &  =  &\displaystyle{  \left(  \mu -p  \right)  h(p)
-\left( -c + \int_{u=p- \mu    }^{\infty } (u - [\mu   -p] ) f(u) du
\right)h(p)}\\
  & &\\
  &  = &\left[   \mu -p       +c   \right] h(p)  -   \left(   \int_{u=p- \mu
}^{\infty }  X  f(u) du \right)h(p),    \\
\end{array}
\end{equation}
 where $x \equiv u - [\mu   -p]$.

Changing $f$ affects only the third term, which is always positive  because it
includes only values of $u$ such that  $(u - [\mu   -p] ) \geq 0$. Making $f$
riskier using the conventional definition of risk from Rothschild \& Stiglitz
(1970)  might leave $X$ unchanged (as in Figure 1a, where the density changes in
four regions, all to the right of $p-\mu$), or might  increase it (as in Figure
1b, where the density falls between $r$ and $s$ but increases everywhere else).

Suppose, however, we spread out $f$ using a strict decrease in $f$ on any
interval $[r, s]$ and a strictly increase  everywhere else, while leaving  the
mean of $u$ unchanged at zero. This is the continuous-distribution analog of the
concept of ``pointwise riskiness'' that I explore at greater length in Rasmusen
(2004).  This flattens $f$  because while  the mean stays the same, the density
strictly declines on the  middle interval   and strictly increases on each side
of it.   This forces an increase  in $f$ everywhere to the right of $s$. Since
the spread leaves the unconditional mean of $v_2$ unchanged, it must increase
the mean of $v_2$ conditional upon $v_2$ being above any specific value-- and in
particular, above $p-\mu$, so      $\int_{u=p- \mu    }^{\infty }  X  f(u) du$
must increase.
If  $\int_{u=p- \mu    }^{\infty }  X  f(u) du$  increases, then $p$  must fall
if we are at an interior solution and the derivative is to stay equal  to zero.
$\blacksquare$


 \includegraphics[width=150mm]{carriedfigure1.jpg}



\begin{center}
{\bf Figure 1: Two Kinds of Increase in Risk}
\end{center}


\bigskip

 Proposition 4 is true because value discovery has option value, and
option value increases with the amount of uncertainty. When the
uncertainty   is larger, there is a greater
probability that value discovery will disclose that $\mu+u >
beta$, even though the expected value of $u$ is zero. Thus,
even  a risk-neutral Bidder 1 likes having more uncertainty.

Note that $ \epsilon $, the size of the remaining uncertainty
over $v_1$, is irrelevant to Bidder 1's decision. It could be that
this uncertainty  is far larger than that from $u$, but this
makes no difference to the value to Bidder 1 of information about $u$.
It would make a difference if Bidder 1 were risk averse, but we have
assumed he is risk neutral, and a risk-neutral player only cares about
variance to the extent that it affects option value. The variable
$\overline{\epsilon} $ will enter our analysis in the next subsection.



\bigskip
\noindent
\textbf{5. Concluding Remarks}

	I have suggested an   explanation for the phenomenon
of bidders getting carried away in auctions.  The man on the street
would say that when   a bidder
has increased his bid ceiling from what he had decided before the
auction,   he has been overcome by   emotion.  He might also say that
the bidder bid so high simply because he wanted to win, rather than
because he wanted to own the object, but this variant too must
 rely on economic irrationality,    since a
rational bidder would  factor his utility from winning into his
original bid ceiling. I do not deny that there may be an emotional
explanation; in fact,   formalizing and testing such an explanation
would be worthwhile.  Here, however,
I have proposed an alternative: in the course of the
bidding, the bidder rethinks his private value, and with some
probability his rethinking results in an upwards revision of the
amount he is willing to pay. This revision is rational, and, indeed,
it would be irrational for the bidder to incur too much cost in
determining the maximum he is willing to pay before he knows whether
that maximum will be a binding constraint.

   Our paradigmatic example for the private-value auction is the open-
cry ascending antique auction, but the idea of value discovery applies
to any auction, and, indeed, the present model applies best when time
pressure exists but is not so intense, so that a bidder does have a
chance to reflect on his willingness to pay.  One application with
which many readers  of this article may have experience is in house
purchases.  In buying a house, a person's first aim is to find a house
for which his private value exceed the likely price. Once  he has
found such a house, however, he may well find that other buyers also
are interesting in it, in which case an auction, usually informal but
sometimes formal, begins. At that point, the buyer will think harder
about his private value, and may revise it either up or down, but the
buyer would have been foolish to undergo the emotional strain of such
fine valuation if not forced to by tight competition.


	The value discovery
explanation for bid updates has three empirical implications that
could help to test it.
First, 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.
Second, a  short ``cooling off period'' would presumably  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.\footnote{For an experimental study
of cooling off periods which is aimed at seeing whether they benefit
the seller by encouraging more aggressive bidding, see Asker (2000).
See also the theoretical analysis of  Rothkopf (1991). }
Third, 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.



\bigskip
\noindent
\textbf{References}\newline



Asker, John (2000) ``Bidding Up, Buying Out And Cooling-Off: An Examination Of
Auctions With Withdrawal Rights'' {\it Economic Theory}, 16: 585-611 (November
2000).

Bulow, Jeremy \& Paul Klemperer (2002) ``Prices and the Winner's Curse,'' {\it
RAND Journal of Economics},  33:1-21 (Spring 2002).

 Compte, Olivier \& Philippe Jehiel (2000) ``On the Virtues of the Ascending
Price Auction: New Insights in the Private Value Setting,'' working paper,
CERAS-ENPC, CNRS, Paris (December 2000).



Larson, Kate \& Tuomas Sandholm (2002) ``Costly Valuation Computation in
Auctions,'' Carnegie Mellon University, Dept. of Computer Science, undated.




Malhotra, Deepak \& J. Keith Murnighan (2000) ``Milked for All Their
Worth: Competitive Arousal and Escalation in the Chicago Cow
Auctions.'' working paper, Kellogg School of Management, Northwestern
University, April 2000.

Persico, Nicola (2000) ``Information Acquisition in Auctions,'' {\it
Econometrica}, 68: 135-148 (January 2000).

Rasmusen, Eric (2001) ``Explaining Incomplete Contracts as the Result
of Contract-Reading Costs,'' {\it  Advances in
Economic Analysis and Policy},  Vol. 1: No. 1, Article 2, \newline
Http://www.bepress.com/bejeap/advances/vol1/iss1/art2.

Rasmusen, Eric (2003a) ``Strategic Implications of Uncertainty Over
One's Own Private Value in Auctions,'' January 26, 2003, \\
 Http://www.rasmusen.org/papers/auction.pdf.

Rasmusen, Eric (2004) ``When Does Extra Risk Strictly Increase the
Value of Options?''  May 14, 2004,\\
   Http://www.rasmusen.org/papers/options\_rasmusen.pdf.

Roth, Alvin E.  \&  Axel Ockenfels (2002)  ``Late-Minute Bidding and the Rules
for Ending Second-Price Auctions: Evidence from eBay and Amazon on the
Internet,''    {\it American
Economic Review},   92(4): 1093-1103  (September 2002).


 Rothkopf,
Michael  (1991) ``Auctions With Withdrawable Winning Bids,'' {\it
Marketing
Science,}  10,1: 40-58  (Winter 1991)


 Rothschild, Michael \& Joseph Stiglitz (1970)  ``Increasing Risk
I,''
{\it Journal of Economic Theory},     2: 225-243 (September 1970).


 \end{raggedright}

\end{document}

