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\begin{center}
   {\bf Strategic Implications of Uncertainty Over One's Own
Private Value in Auctions } \\  

 \bigskip
June 23, 2002 \\

\bigskip

Eric Rasmusen \\

\bigskip

{\it Abstract}
\end{center}

Suppose a bidder must decide whether and when to incur the cost of
estimating his own  private value in an auction. This can explain why
a bidder might increase his
bid ceiling in the course of an auction, and why a bidder would like
to know the private values of other bidders.      It also  can explain
sniping--
flurries of bids at the end of    auctions with deadlines-- as the
result of other bidders
 trying to avoid stimulating the uninformed bidder  to examine  his
value.

\noindent
 {\small \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.
Php.indiana.edu/$\sim$erasmuse. Copies of this paper can be found at
Php.indiana.edu/$\sim$erasmuse/papers/auction.pdf. JEL Code: D44.
   }

\noindent
 {\small I thank Michael Fishman, Daniel Levin, Michael Rothkopf,
Jeffrey Stake, and participants at  seminars at  Ohio State  and Tokyo
Universities  for
their comments, and Ariel Kemper for research assistance. I thank
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{center}
 A   STORY OF MISTAKEN BIDDING
\end{center}

\begin{quotation}
 
   Jeff happily awaited the end of the Ebay auction. He'd submitted a
bid ceiling of \$2,100 for a custom-made analog stereo amplifier, and
the highest  anybody else had submitted was \$1,400, so he was sure to
win. Since he'd followed the advice of Ebay and academic auction
theory, submitting his true maximum price, he looked forward to a cool
\$700 in consumer surplus. 

It was five minutes before the auction
deadline. And then disaster struck. The winning bid rose to \$1,800,
and then \$1, 900, and \$2,000. And then it rose to \$2,150, and Jeff
was losing! Worse yet, as he feverishly  thought hard about how much
the   amplifier was worth to him, he realized he actually would have
been willing to pay \$2,500.

 But by then it was too late. The auction
was over. 
 
\end{quotation}


\newpage
  \begin{center}
 VALUE DISCOVERY AND ITS IMPLICATIONS
\end{center}

What if a bidder in a private value auction does now know his own
value precisely, but can discover his value at some cost?
 
Value discovery can explain:
\begin{enumerate}
\item
  Why  bidders    would like
to
know how much other  bidders are going to bid.

\item
 How other bidders  can benefit when an uninformed bidder learns his
value
more precisely.

 \item
 How improved buyer information on the value of the object   hurts
the seller.


\item
   Why bidders update the   bid ceilings they submit  during 
the course of an auction such as those on Ebay and Amazon  that uses
proxy bidding.

\item
   Why bidders use ``pre-emptive bids'', bidding early  in auctions
rather than later.



\item
  Why bidders use ``sniping''---  the practice of  submitting bids at
the last minute.



\end{enumerate}

  \newpage
 \begin{center}
 THE MODEL
\end{center}

  
The  two players  in an auction  are both risk-neutral and have 
private  values which  are statistically independent and distributed
over the same support $[0,z]$, on which the densities are strictly
positive.  Bidder 1 has value $u$ distributed
according to  the atomless density $f(u)$.
Bidder 2 has value $v$
distributed according to  the atomless density $g(v)$.

 Bidder 1  knows neither  $u$
nor $v$.   At any time  he may, unobserved by Bidder 2,  pay $c$ and
learn  $u$ after additional time $\delta$ has passed. Bidder 2 knows
$v$,  but not $u$.

We will look at three sets of auction rules: a sealed-bid second-price
auction,
an Amazon auction, and an Ebay auction. In each, a player submits his
``bid ceiling'', but the winner pays the second-highest bid ceiling
submitted.

The Amazon auction has a ``soft deadline,'' ten minutes after the last
bid ceiling update.

 The Ebay auction has a ``hard deadline'': at a preannounced time. 


  \newpage
\begin{center}
 THE SECOND-PRICE SEALED BID AUCTION
\end{center}
 
 
 
 
 

 \noindent
  {\it Equilibrium.}   Bidder 1
  pays to discover his  value and submits a bid ceiling of  $u$ if $
c$ is sufficiently
low.   Otherwise,  he  submits a bid ceiling of   $Eu$. Bidder 2
submits a bid ceiling of
$v$.

\noindent
 {\it  Bidder 1's  Bidding Strategy}. First, suppose Bidder 1 has paid
$c$ and discovered  his value, $u$.
Once  Bidder 1 knows    $u$,  if  Bidder 2 submits a bid ceiling of
$p$ with probability $m(p)$, Bidder 1's expected payoff is
\begin{equation}  \label{e0b}
\pi_1(u)  =   \int_0^b   (u-p) m(p) dp.
 \end{equation}
 Maximizing by choice of $b$ yields $ (u-b) m(b) =0$, so $b^*=u$.
Bidder 1  should bid his value, $u$.  
 
 Second, suppose Bidder 1 has not discovered his value. 
 His payoff if 
he bids $b$ and Bidder 2 bids  $p$ with probability $m(p)
$  is
\begin{equation}  \label{e3}
\pi_1    =   \int_0^z \left( \int_0^b   (u-p) m(p) dp
\right) f(u) du.
 \end{equation}
Maximizing  by choice of $b$ yields
\begin{equation}  \label{e4a}
  \int_0^z (u-b) m(b)  f(u) du=0, 
 \end{equation}
so 
\begin{equation}  \label{e4b}
  \int_0^z b m(b)  f(u) du=  \int_0^z u m(b)  f(u) du, 
 \end{equation}
and $b^*= \int_0^z u  f(u) du$, the expected value of $u$.   
Bidder 1 should bid his expected value.  

 

\bigskip


\noindent
 {\it   Bidder 2's Bidding  Strategy}.    Bidder
2's payoff if he submits a bid ceiling of
$b$ and Bidder 1 submits a bid ceiling of  $p$ with probability $m(p)$
is
\begin{equation}  \label{e0}
\pi_{sniper} = \int_0^b   (v-p) m(p) dp,
 \end{equation}
    because Bidder 2 wins the value $v$  and pays the price $p$ if his
bid
of
$b$ exceeds Bidder 1's bid of $p$, and otherwise his payoff is
zero.
Maximizing his payoff by choice of $b$ yields
\begin{equation}  \label{e0a}
 (v-b) m(b) =0,
 \end{equation}
so $b^*=v$. Bidder 2 should bid his value.





\newpage
\begin{center}
 THE DECISION OF WHETHER TO PAY TO DISCOVER ONE'S VALUE
\end{center}


    For given  Bidder 2 value  $v$,
Bidder 1's  expected payoff before he actually learns $u$ is
\begin{equation} 
  \pi_1^d (v) = -c + \int_0^v (0) f(u) du + \int_v^z( u-v) f(u) du.
\end{equation} 
  
  

\noindent
{\bf Proposition 1:} {\bf    If the discovery cost $c$ is low enough,  
Bidder 1's expected payoff   is higher if he   learns Bidder 2's
value,  $v$, before he must  decide  whether to pay to learn his own
value,
$u$.}



	If $c$ is low enough,     there are   two  reasons  why Bidder
1
might benefit from knowing $v$. 

  First,  if  $\pi_1^d >  \pi_1^{nd}$,
Bidder 1  would  switch from always paying to
discover his value to  paying only if $\pi_1^d(v)>  \pi_1^{nd}$.


Second,    if  $\pi_1^d < \pi_1^{nd}$,   Bidder 1 would  switch
from never discovering his value to discovering it if  $\pi_1^d(v)>
\pi_1^{nd}$.    


\newpage

\begin{center}
 THE PAYOFF FROM DISCOVERING ONE'S VALUE
\end{center}

\begin{equation} 
  \pi_1^d (v) = -c + \int_0^v (0) f(u) du + \int_v^z( u-v) f(u) du.
\end{equation} 

\begin{equation} \label{disc8}
  \pi_1^d   = -c +  \int_0^z \left(  \int_v^z( u-v) f(u) du \right)
g(v)
dv.
\end{equation}
  
\oddsidemargin -1in
 \begin{equation}  
\begin{array}{ll}
   \pi_1^d   & ={\displaystyle -c +  \int_0^{Eu} \left(  \int_v^z (u-
v) f(u) du
\right)
g(v) +
\int_{Eu}^z \left(  \int_v^z( u-v) f(u) du \right) g(v)dv. }\\
 & \\
  &= -c + A_1 + A_2. \\
\end{array}
\end{equation}

\oddsidemargin -.7in

Now let us find   Bidder 1's payoff if he does not learn $u$. He will
 bid $Eu$.  If  $Eu<v$, then $\pi_1^{nd}  (v) = 0$.   If  $Eu>v$, then  
 \begin{equation} \label{ndisc10}
  \pi_1^{nd}  (v) =   \int_0^v (u-v) f(u) du + \int_v^z( u-v) f(u) du.
\end{equation} 
  

\oddsidemargin -1in
\noindent
Integrating $ \pi_1^{nd}  (v)$ over $v$ yields the expected payoff
 \begin{equation}  
\begin{array}{ll}
   \pi_1^{nd} &  {\displaystyle = \int_0^{Eu}   \left(   \int_0^v (u-
v) f(u) du +
\int_v^z(
u-v) f(u) du \right)  g(v) dv + \int_{Eu}^z (0) g(v) dv }\\
 & \\
 &   { \displaystyle = \int_0^{Eu}    \left(   \int_0^v (u-v) f(u) du
\right) g(v) dv
+
\int_0^{Eu}    \left(  \int_v^z( u-v) f(u) du \right)  g(v) dv }\\
 & \\
 &=   -A_3+A_1. \\
\end{array}
 \end{equation}  
 

  \oddsidemargin -.7in

\newpage
 \begin{center}
 SUMMARY OF THE VALUE OF INFORMATION
\end{center}


 We can summarize the results for small discovery cost $c$
as follows:



 
 \noindent
1. Learning $v$ is useful  to Bidder 1 if  he does not yet know $u$
(Proposition 1).

\noindent 
2. Learning $v$ is useless   to Bidder 1 if  he already    knows $u$. 
 
\noindent
3. Learning $u$ is useful  to Bidder 1 either if he does not know $v$
or if he knows $v$ and   $v>Eu$.


\newpage
 \begin{center}
THE EFFECT OF BIDDER 1'S VALUE DISCOVERY ON BIDDER 2  
\end{center}

 
\noindent
{\bf Proposition 2:}  {\bf  Bidder 2's expected payoff is higher if
Bidder 1
knows $u$ at the
start of the auction  than if Bidder
1 submits a bid ceiling of $Eu$.}

\noindent
{\it Proof:} First, suppose $v <Eu$.  Bidder 2's payoff is zero
without
value discovery, because Bidder 1 will bid $Eu$ and win.  Bidder 2's
payoff is positive with value discovery, because there is probability
$F(v)$ that $u$ will be less than $v$ and Bidder 1 will win. Thus, if
$v <Eu$, Bidder 2 benefits from Bidder 1 knowing $u$.

\noindent
Second, suppose $v>Eu$. 
Without value discovery, Bidder 2's payoff is 
 \begin{equation}
v- Eu,     
 \end{equation}
which can be rewritten as
\begin{equation} \label{ndiscovery}
  \int_0^z v  f(u) du-    \int_0^z   u  f(u) du .
 \end{equation}

\noindent
With value discovery, Bidder 2's payoff is  
 \begin{equation}\label {discovery}
  \int_0^v  (v - u) f(u) du  + \int_v^z  (0) f(u) du  =  v \int_0^v
f(u) du  -  \int_0^v    u  f(u) du.
\end{equation}
 We  need to show that   
 \begin{equation}
      \int_v^z  v  f(u) du  <    \int_v^z    u  f(u) du,   
 \end{equation}
 which is true  because in  the right-hand-side integral $u$ is taking
values that are $v$ or greater.

  
\newpage
 \begin{center}
THE EFFECT OF BIDDER 1'S VALUE DISCOVERY ON  THE SELLER 
\end{center}

 
 
\noindent
 {\bf Proposition 3:} {\bf  The seller prefers that Buyer 1 not know
$u$ at
the start of the auction;    if 
Bidder 1 submits a bid ceiling of $Eu$ instead of $u$, 
 the expected   price  is  higher.}

\noindent
 {\it Proof:} Take a given $v$. First,  suppose $v <Eu$. The winning
price would be $v$.  Value discovery will either keep the winning
price at $v$ (if $u \geq v$), or reduce it to below $v$ (if $u<v$).
Thus, the expected winning price is higher if Bidder 1 does not know
$u$.

Second, suppose $v \geq Eu$. The winning price would be $Eu$ if Bidder
1 does not know $u$.  Value discovery will change the winning price to
$Min\; (u,v)$.
 The winning price is $u$ if $u<v$ and $v$ if $u>v$, so its expected
value is
 \begin{equation} 
 \int_0^v u f(u)  + \int_v^z v f(u). 
\end{equation}  
  This is less than $Eu$ if 
  \begin{equation} 
 \int_0^v u f(u)  + \int_v^z v f(u)  < \int_0^z u f(u),
\end{equation}  
 which is true if 
  \begin{equation} 
   \int_v^z v f(u)  < \int_v^z u f(u),
\end{equation}  
 which is true since in the second integral $u$ ranges from $v$   up
to $z$ whereas in the first integral $v$ is  a constant.

  

\newpage
 \begin{center}
  EXAMPLE 1: SECOND-PRICE SEALED BID    
\end{center}
 
  Bidder 1 has private value $u$ uniformly distributed on
[0,100]. He can take  5 minutes and  pay  amount 5 to discover $u$
  if he wishes; otherwise, his estimate is 50.
  Bidder 2 has value  $v$ of either 10 or 60, with equal
probability.

Bidder 2 will submit his value, either 10 or 60. If Bidder 1
submits a bid ceiling of  50, his expected
payoff is $ \pi_1^{nd}= .5 (Eu-10) + .5 (0) = .5 (50-10) =  
20. $ Bidder 2's payoff is   $ \pi_2 = .5 (60-50) + .5 (0) = 5. $
The expected price would be  .5(10) + .5(50) = 30.

Suppose Bidder 1 decides to pay to discover his value before he
submits his bid ceiling. He will then submit a bid ceiling of $u$.
With probability .5, $v=10$. If that is the case, then $u \geq 10$ and
Bidder 1 wins the auction   with probability .9, and the expectation
of his payoff if he wins is $ Eu|(u \geq 10)-10 $. With probability
.5, $v=60$. If that is the case, then  $u \geq 60$ and Bidder 1 wins
the auction   with probability .4, and the expectation of his payoff
if he wins is $ Eu|(u \geq 60)-60 $.   Bidder 1's overall expected
payoff is therefore  $ \pi_1^{d}= -5 + .5  (.9) [ Eu|(u \geq 10)-10] +
.5  (.4) [
Eu|(u \geq 60)-60] = -5 + .45 [55-10] + .20 [80-60] = -5 + 20.25 + 4 =
19.25.$ Thus, Bidder 2 will choose not to pay to discover his value.
 

\newpage
 \begin{center}
  EXAMPLE 1: SECOND-PRICE SEALED BID, continued.  PROPOSITIONS 1,2,3 
\end{center}

 Bidder 1 has private value $u$ uniformly distributed on
[0,100]. He can take  5 minutes and  pay  amount 5 to discover $u$
  if he wishes; otherwise, his estimate is 50.
  Bidder 2 has value  $v$ of either 10 or 60, with equal
probability.

  If, however, $c=0$ instead of $c=5$, Bidder 2 would pay to discover
his value; for small enough $c$, value discovery would occur. And if
$c=5$ and Bidder 1 knew that $v=60$, he would pay to discover his
value, because his payoff from not discovering it would be 0, whereas
his payoff from discovering it would be  $-5 + .4(Eu|(u \geq 60)-60]
=
-5 + .4(80-60) = 3$. This illustrates Proposition 1, that a bidder
could benefit from knowing the other bidder's value.

 This example can also illustrate Propositions 2 and 3. If Bidder 1
knew his value and bid $u$, Bidder 2's expected payoff would be $.5
(.1)  (10 - Eu|u \leq 10)  + .5  (.6) (60 - Eu|u \leq 60) = .05 (10 -
5) + .30 (60-30) = .25 + 9 = 9.25.$  As Proposition 2 predicts, Bidder
2's payoff of 9.25 when  Bidder 1 knows $u$ is greater than his payoff
of 5 when Bidder 1 does not know $u$.

 The expected price if Bidder 1
bids $u$ is   $.5 [(.1)  (Eu|u \leq 10) + .9 (10)]   + .5   [(.6) (
Eu|u \leq 60) + .4 (60)] = .5 [.5 + 9] + .5 [18 +24] = 25.75.$ This is
less   than the  expected price of 30 if Bidder 1 does not know $u$,
as Proposition 3 predicts.

 


  


\newpage
 \begin{center}
 THE AMAZON AUCTION (SOFT DEADLINE)
\end{center}


\noindent
 {\bf Proposition 4:} {\bf   The phenomenon of bidders increasing
their
bid
ceilings during the auction can be a necessary part of equilibrium.}

  
\noindent
   {\it Equilibrium.}   Bidder 1 submits a bid ceiling of either
$Eu$, if $c$ is high enough, or      $ \overline{p}>0$ otherwise.  If
he has submitted a bid ceiling of  $ \overline{p}$ and the current
winning bid rises to $ \overline{p}$, he pays $c$ to discover $u$ and
then increases his bid ceiling to $u$ if $u> \overline{p}$.  Bidder 2
submits a bid ceiling of $v$.
 

  \noindent
 {\it Explanation.}   
 Bidder 1's
expected payoff conditional on $v$ is 
 \begin{equation}
  \pi_1^d(v)  = \int_0^z (u-v) f(u) du. 
\end{equation}
If $v \geq \overline{p}$, Bidder 1's expected payoff is 
 \begin{equation}
  \pi_1^d(v)  = -c +  \int_0^v (0) f(u) du +  \int_v^z (u-v) f(u) du. 
\end{equation}
 
 \begin{equation} \label{overall}
\pi_1^d =  \int_0^{\overline{p}}  \left( \int_0^z (u-v) f(u) du
\right)
g(v) dv
+ \int_{\overline{p}}^z  \left(  -c +   \int_v^z (u-v) f(u) du \right)
g(v) dv.
\end{equation} 

Our results from the second-price sealed-bid auction tell  us
immediately that    $\pi_1^d > \pi_1^{nd}$ if $c$ is low enough but
not otherwise.  

It can also be shown that $\overline{p} >0$.
  

\newpage
 \begin{center}
  WHY VALUE DISCOVERY WOULD NEVER OCCUR BEFORE  THE START OF THE
AUCTION
\end{center}


\noindent
 {\bf Lemma:} {\it   
Bidder 1 will set  his initial bid ceiling to be strictly   positive,
delaying value discovery: $\overline{p} >0$.}


\noindent
 {\it Proof:}
Differentiating Bidder 1's payoff with respect to $\overline{p}$
(which we can do if the
densities are  atomless) yields
\begin{equation}  \label{e9}
\frac{d\pi_1^d  }{d\overline{p} }   =   \left( \int_0^z  ( u-
\overline{p}) f(u) du   \right)  g(\overline{p})  - -   c g(
\overline{p} ) - \left( \int_{\overline{p}}^z   (u-\overline{p} ) f(u)
du   \right) g(\overline{p})
\end{equation}
  If $\overline{p}=0$, the first and third terms of this derivative
cancel out.  The second term  is positive, however, given our
assumption that the value density is everywhere positive.
Thus the payoff  derivative is positive at  $\overline{p}=0$, in
which case  the optimal value of  $\overline{p}=0$ is positive, which
was to be proved.

  The advantage  of
increasing $\overline{p}$ is that possibly the bidder will win at  a
price $p<\overline{p}$ and not have to pay the discovery cost $c$. The
disadvantage   is that possibly the bidder
will win at  $p<\overline{p}$ such that $p$ exceeds his value:
$p=v>u$.  The
size of this  disadvantage depends on   the  likelihood that Bidder
1's value is below  $\overline{p}$, which is $\int_0^{\overline{p}}
f(u)
du$. If $\overline{p}=0$, this disadvantage vanishes; there is no risk
that Bidder 1's value will be below  $\overline{p}$.   Bidder 1
should increase his initial bid ceiling  until the marginal gain from
avoiding the  discovery cost  equals the marginal loss from winning
when his value is below the price he pays.

 
 
 
 
  
\newpage
 \begin{center}
 THE EBAY AUCTION (HARD DEADLINE)
\end{center}



\noindent
 {\bf Proposition 5:}{\bf  A bidder may purposely bid early, so as to
stimulate value discovery.  }
 
\noindent
{\it Equilibrium.}  If $c$ is low enough, Bidder 1    submits a bid
ceiling of $\overline{b}$ before time $T-\delta$, and then pays to
discover his value if  the current  winning bid  reaches $\overline{b}
$. If the current winning bid does not reach $\overline{b}$, he raises
his bid ceiling to $Eu$. If $c$ is higher, Bidder 1 submits a bid
ceiling of  $Eu$ and never discovers his value.  Bidder 2 submits a
bid ceiling of $\overline{b}$ before time $T- \delta$, and raises his
bid ceiling to $v$ after time $T- \delta$.

\noindent
{\it Explanation.}   Now Bidder 2 can 
wait to bid until after $T-\delta$ and prevent value discovery.
Whether Bidder 2 wants to do this
depends on whether he  wants  to provoke Bidder 1 to discover $u$.

\noindent
  Does he? 

\newpage
 \begin{center}
 IN THE EBAY AUCTION DOES BIDDER 2 WANT TO PROVOKE VALUE DISCOVERY? 
\end{center}

If Bidder 2 has value $v$ and  provokes value discovery by submitting
a bid ceiling of  $\overline{b}$ or more, his expected payoff is made
up of three parts, depending on whether he wins at a price of
$\overline{b}$, wins at a price of $u$, or loses the auction.
  \begin{equation}\label {discovery1}
\pi_2 =  \int_0^{\overline{b}}  (v - \overline{b}) f(u) du + 
\int_{\overline{b}}^v  (v - u)
f(u) du  + \int_v^z  (0) f(u) du.
 \end{equation}
 
\begin{equation}\label {discovery2}
 \frac{d\pi_2}{d \overline{b}} = (v - \overline{b}) f(\overline{b}) -
\int_0^{\overline{b}}    f(u)
du  +   (v - \overline{b}) f(\overline{b}).
 \end{equation}
  If $\overline{b}$ is greater, then Bidder 2 benefits
less from Bidder 1 learning $u$ because Bidder 1  cannot reduce his
bid ceiling below $\overline{b}$ after learning that  $u<\overline{b}
$.

 
   We can deduce from Proposition 2   that there is some positive
value $k$ of $\overline{b}$ low enough that Bidder 2 will benefit from
value discovery, since if $\overline{b} =0$.
In choosing a   discovery threshold,  $\overline{b}
$, Bidder 1 is  constrained to set $\overline{b} \leq k$, because
otherwise Bidder 1  would want to avoid provoking value discovery and
would    delay submitting $v$ as his bid ceiling until
after $T-\delta$.  Thus, $\overline{b}$ will be less than or equal to
the $\overline{p}$ of the Amazon auction.   If $c$ is too high, then
Bidder 1 will prefer not to discover his value and to simply bid $Eu$
at the start of the auction.

 
 
\newpage
\vspace*{-.5in}
 \begin{center}
EXAMPLE 2: EBAY AUCTION
\end{center}

   Bidder 1 has private value $u$ uniformly distributed on
[0,100]. He can take  5 minutes and  pay  amount 5 to discover $u$
  if he wishes; otherwise, his estimate is 50.
  Bidder 2 has value  $v$ of either 10 or 60, with equal
probability.

  If Bidder 1    submits a bid ceiling of  50,  $ \pi_1^{nd}=  
20  $ and    $ \pi_2 =   5. $
 

Suppose Bidder 1 instead follows  the strategy  of submitting a bid
ceiling of $\overline{b} =11$  and then either paying to discover his
value   if the current winning bid rises to  11 or raising his bid
ceiling to 50.   Bidder 2 will  respond  by bidding $v$.

 With probability .5, Bidder 1 will win at $p=10$ because $v=
10$, and with probability .5, Bidder 2 will bid 60 and stimulate value
discovery. If that happens, Bidder 1 incurs the cost 5, and with
probability .4 finds that he wants to overbid Bidder 2 and win at the
$p=60$. With probability .6, he finds that he does not want to
increase his bid past 11, and Bidder 2 will win.  
  $ \pi_1^d =   21.5$ and 
  $\pi_2 =  14.7$.
 Thus,  the payoffs of both bidders are
higher with value discovery.


Example 2  illustrates Proposition 4, that a bidder may
wish to increase his bid ceiling in the course of an auction, and
Proposition 5, that a bidder may wish to stimulate value
discovery by bidding early.

 

 



\newpage
 \begin{center}
 NAIVE BIDDERS
\end{center}



  
  Suppose Bidder 1  is ``naive'': he does not realize that Bidder 2 is
present
(i.e., he
assigns probability 0 to Bidder 2 being present).   After
all, the Ebay instructions say:
 \begin {quotation}
``For example, if the current bid on an item is \$5 and you are
willing
to pay up to \$10, you would enter \$10 as your maximum bid. Your bid
would be shown on the item page as \$5, but if another bidder places a
bid for \$6, then eBay will place a higher bid on your behalf. The bid
would be just above the other member's bid. This would continue until
either you win the auction at or below \$10 or the bidding exceeds the
\$10 you were willing to pay. eBay will notify you via email if you
are
outbid and you can return to place another bid if you like. Your
maximum bid is never disclosed to other bidders or to the seller.''


--EBay Tutorials, ``Place Your Bid''\\
http://pages.ebay.com/education/tutorial/course1/bidding\_3.html (May
25, 2002)
\end{quotation}


\newpage
 \begin{center}
THE RESULT OF NAIVETE
\end{center}

   Bidder 1    would submit a bid
ceiling of  $Eu$  regardless of how low $c$ was, under all three
auction rules.

In the Ebay and Amazon auctions, this hurts Bidders 1 and 2 and helps
the seller.




In the  Ebay auction, if $v >Eu$, then if  parameters are such that
Bidder 1 would pay to discover $u$ if he were aware of Bidder 2 being
present,   Bidder 2  would submit a bid ceiling of $v$  in the time
interval $[T-\delta, T]$. If Bidder 1 would not pay, Bidder 2  could
submit a bid ceiling of $v$ at any time.

\noindent {\bf  Proposition 6:} {\bf  ``Sniping'' can occur in
equilibrium. A bidder may purposely delay submitting a bid ceiling
higher than  the current winning bid until near the  auction deadline.
}

 

\newpage
 \begin{center}
 EXAMPLE 3: EBAY AUCTION WITH NAIVETE
\end{center}

  Let   Bidder 1 have a private value $u$ uniformly distributed on
[0,100]. He can take  5 minutes and  pay  amount 5 to discover his
value
precisely if he wishes; otherwise, his estimate is 50.
Let   Bidder 2  have a value  $v$ of either 10 or 60, with equal
probability.
 

Bidder 1 will  submit  a bid ceiling of   50.
   Bidder 2 
will submit a bid ceiling of  10 if $v=10$, and timing will not
matter. If $v=60$, though, he
should wait  until within 5   minutes of the deadline and submit   a
bid ceiling  of 60. He will win at  a price of 50, for a payoff 
of  10.

What if Bidder 2  were to submit a bid ceiling of    60
earlier?   Bidder 1's payoff from paying 5 to discover $u$
  would be $-5+ .6 (0) + .4[(Eu|u>60)-60] = -5 +
.4(80-60)= -5+ 8 =3,$ compared to  a payoff of 0 from not discovering
his value and retaining his bid ceiling of 50.  Thus, Bidder 1 would
pay to discover his value and with probability
.4 he would increase his bid ceiling to  more  than 60 and win the
auction. $\pi_2=.6(60-50) + .4(0) = 6$,
  less than the 10  Bidder 2 would get  from   delaying his bid
submission
and sniping.


Example 3  illustrates  the value of sniping (Proposition 6). Bidder 2
does not want to bid early, because it would stimulate value
discovery, and that would either increase the price Bidder 2 pays or
cost him victory in the auction.

\newpage
 \begin{center}
 OTHER EXPLANATIONS FOR SNIPING
\end{center}
 
1.  Alvin  Roth and Axel Ockenfels (2000):
  players making bids in the last minute may find the
computer has not been able to get their bids in time. In that case,
players will submit low bids early in the auction and higher bids in
the last minute. There is some chance that none of the high bids will
be accepted, and so some bidder wins with his very low initial bid.

2.  The auction is common value,  and  updating valuations  requires
time.
 
3.  There is no point to submitting a bid early, and possibly you will
discover something about your private value exogenously as time
passes.
 
  

 


 \end{LARGE}

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