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 \titlepage
 \vspace*{12pt}
 
 \begin{center}
 {\large {\bf Correction to an  Auction Model  in Chapter 12
of {\it Games and Information}, Third Edition } }
 
 \bigskip  November 10, 2000 \\[0pt]
 \bigskip
 
 Eric Rasmusen \\[0pt]
 \bigskip
 
 {\it Abstract}
 \end{center}
 
   The Third Edition of {\it Games and Information} is
unclear about  the underlying assumptions used in the model
of private-value auctions with uncertainty over a player's
own valuation of the object being auctioned. These notes
clarify.
 
 {\small   \noindent 
 \hspace*{20pt} Rasmusen: Professor of Business Economics and
Public
Policy and Sanjay Subhedar Faculty Fellow, Indiana University,
Kelley School of Business,
BU 456, 1309 East 10th Street, Bloomington, Indiana, 47405-
1701. (812) 855-9219. Fax: 812-
855-3354. Erasmuse@indiana.edu. Http:
//Php.indiana.edu/$\sim$erasmuse. Copies of these
notes can be found at \newline Http:
//Php.indiana.edu/$\sim$erasmuse/GI/pvauctions.pdf. }

 \newpage
 
 (1)  On page 330, the quick fix is to change

   ``Suppose there are $N$ bidders, each with a private
value, in an ascending  open cry auction.''

by adding a clarification:

 ``Suppose there are $N$ bidders, each with a private value,
in an ascending  open cry auction and 
diffuse priors on the other bidders' values   (that is,
all values on  the real line are equally likely).''
 
 Diffuse priors  have two properties: 1. no value has
more probability than any other value. 2. there are no bounds
on the support.

 If that is not true, then the example needs to be more
complicated, because the expected value is not simply the
signal that the bidder observes, but will involve  regression
to the mean.

(2)  The term ``diffuse priors'' is from Bayesian decision
theory  and
is based on the viewpoint of the decisionmakers rather than
on the world having  reality that is reflected in their
viewpoints. I  feel more comfortable doing the following:

 Each bidder's value is $v_i$,but what he observes is a
signal of that value, $y_i$. The true values  $v$ are
distributed uniformly on  [0,100]. If the bidder's signal
$y_i$ is more than 100, he knows the true value is $y_i-x$.
If it is less than 0, he knows it is $y_i+x$. If it is
between $100-x$ and 100, his estimate is $y_i-x/2$, since he
knwos it cannot be a negative error. If $y_i$ is between 0
and $x$, his estimate is $y_i+x/2$.

For my example, suppose the observed value $y$ happens to be
between $x$ and $100-x$. (The other cases are a bit more
complicated, but the conclusion that risk is bad for the
seller is the same.)

(4) Using a non-uniform distribution involves something like
regression to the mean
(exactly that, if it is single-peaked). Take a weighted average
of $y_i- x$, $y_i$ and
$y_i+x$, with the weights being the   probabilities (or
densities, if it is a continuous distribution) of the three
values. Those
probabilities are derived using Bayes's Rule, since we know
that the true value must take
one of those three values, even though we don't know which one.
Thus, we get 

\begin{equation} \label{new22} 
\begin{array}{lll}
    \hat{v} & = &\frac{ Prob(v=y_i-x)} {Prob(v=y_i-x)+ Prob(v=
y_i )+Prob(v=y_i+x)}(y_i-x) +\\
   & & \\
 & &  \frac{ Prob(v=y_i )} {Prob(v=y_i-x)+ Prob(v=y_i )
+Prob(v= y_i+x)} (y_i ) +\\
 & & \\
 & &  \frac{Prob(v=y_i+x) } {Prob(v=y_i-x)+ Prob(v=y_i )
+Prob(v=y_i+x)} (y_i+x) 
\end{array}
\end{equation}

Note that bidder $i$'s valuation will not change even if he is
told that he is the highest bidder. He will adjust his
valuation down from his signal if the signal is high, but he
will do that whether he knows he is the highest bidder or not.
Thus, there is no  Winner's Curse here, just regression to the
mean.

 Equation (\ref{new22}) also applies to diffuse priors and the
uniform distribution, but there the complicated probability
terms are all equal (at least for the middle value if the
distribution is uniform), and so  $\hat{v} =y$.

 The rest of the example needs to be modified if the
distribution is not uniform. Bidder $i$'s expected utility is,
if he bids $b$,

\begin{equation} \label{new33}  
   \begin{array}{lll} 
    E (utility) & = & \frac{ Prob(v=y_i-x)}{Prob(v=y_i-x)+
Prob(v=y_i )+Prob(v=y_i+x)}
U(y_i-x-b) + \\
 & & \\
  &&\frac{ Prob(v=y_i )}{Prob(v=y_i-x)+ Prob(v=y_i )+Prob(v=
y_i+x)} U(y_i-b ) +\\
 & & \\
 && \frac{  Prob(v=y_i+x) }{Prob(v=y_i-x)+ Prob(v=y_i )+Prob(v=
y_i+x)}U(y_i+x-b) \\
   \end{array}
   \end{equation}

 Will he bid $b   =\hat{v}$? Yes, if he is risk neutral, but
not if he is risk averse.  If he is risk neutral, do $U(z) =
z$,  then equation ({\ref{new33}) boils down to
\begin{equation} \label{new34}  
   \begin{array}{lll} 
    E (utility) & = & \frac{ Prob(v=y_i-x)}{Prob(v=y_i-x)+
Prob(v=y_i )+Prob(v=y_i+x)}
 (y_i-x ) + \\
 & & \\
  &&\frac{ Prob(v=y_i )}{Prob(v=y_i-x)+ Prob(v=y_i )+Prob(v=
y_i+x)}  (y_i ) +\\
 & & \\
 && \frac{  Prob(v=y_i+x) }{Prob(v=y_i-x)+ Prob(v=y_i )+Prob(v=
y_i+x)} (y_i+x ) - b  \\
   & & \\
 &= & \hat{v} -b
  \end{array}
   \end{equation}

In that case, the bidder's expected utility is zero if he wins
with a bid of $b=\hat{v}$ and positive for any smaller bid, so
he is willing to bid all the way up to $\hat{v}$. The effect of
regression to the mean is that he bids more than his signal if
his signal is below the mean, but less if it is above the mean.
An interesting implication is that the winning bid is likely to
be less than the signal of the winning bidder, since the
winning bidder probably got an unusually high signal (and high
true  value too).

   What if the bidder is risk averse? Then $U''<0$ and we can't
use the simplification in
equation ({\ref{new34}).  The utility of  fair gamble-- such as
bidding $b=\hat{v}$-- is  negative for a risk averse person.
This is true even if the fair  gamble is unbalanced, giving
him a greater probability of winning  than of losing, but a
bigger loss when he does lose, as is the case when the signal
$y$ is above the mean and  so $\hat{v} <y$.  Thus, the bidder
should never bid all the way up to $b=\hat{v}$. The bid is
lower, to the detriment of the seller,  than it would be if the
bidder knew his value with certainty. This is true whether the
signal $y$ is above or below the mean of $v$.



\newpage

  HERE IS WHAT IS IN THE BOOK, 3RD EDITION:

    In a private value auction, does it matter what the seller
does, given the Revenue
Equivalence Theorem? Yes, because of risk aversion, which
invalidates the Theorem. Risk
aversion makes it important which auction rule is chosen,
because the seller should aim to
reduce uncertainty, even in a private value auction. (In a
common value auction, reducing
uncertainty has the added benefit of ameliorating the winner's
curse.)

  Consider  the following question:

 {\it  If the seller can reduce bidder uncertainty over the
value of the object being
auctioned, should he do so? }

   We must assume that this is a precommitment on the part of
the seller, since otherwise
he would like to reveal favorable information and conceal
unfavorable information.  But it
is often plausible that the seller can set up an auction system
which reduces
uncertainty-- say, by a regular policy of allowing bidders to
examine the goods before the
auction begins.  Let us build a model to show the effect of
such a policy.

  Suppose there are $N$ bidders, each with a private value, in
an ascending  open cry
auction.  Each measures his private value $v$ with an
independent error. This error is
with equal probability $-x, +x$ or 0.   Let us denote the
measured value by $\hat{v}$,
which is an unbiased estimate of $v$.  What should bidder $i$
bid up to?

 If bidder $i$ is risk neutral, he should bid up to $\hat{v} $.
His expected utility is,
if he pays$ \hat{v}$,

  \begin{equation} \label{new1} 
  E w   =  \frac{  (\hat{v}+x - \hat{v})
}{3} +  \frac{ (\hat{v} - \hat{v}) }{3} + \frac{ (\hat{v}-x -
\hat{v}) }{3}  =  0.
\end{equation}

 If bidder  $i$ is risk averse, however,  and wins with bid
$v_{bid}$, his expected
utility is

 \begin{equation} \label{new2}
  EU (w)  =
\frac{U(\hat{v}+x - v_{bid})  }{3}+
\frac{ U (\hat{v}  - v_{bid}) }{3} +\frac{U(\hat{v}-x -
v_{bid})    }{3}
 \end{equation}
Note that if  the utility function $U$  is concave,
\begin{equation} \label{new3}
  \frac{
U(\hat{v}+x - v_{bid})  }{3}  +\frac{U(\hat{v}-x - v_{bid})  }
{3}<  \frac{2}{3} U(\hat{v}-
v_{bid}). 
\end{equation} 
  
The implication is that   a fair
gamble of $x$ has less utility
than no gamble. This means that the middle term in equation
(12.\ref{new2}) must be
positive if it is to be true that $EU(w)= U(0)$, which means
that $\hat{v} - v_{bid} >0$.
In other words,  bidder $i $   will have a negative expected
payoff unless his maximum bid
is strictly less than his valuation.

\end{document}