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\section*{   12 AUCTIONS} 

\noindent
 April 13, 1998

   

\noindent
{\bf Private, Common, and Correlated Values}

\noindent
   We will call the
dollar value of the utility that player $i$ receives from an object
its {\bf value} to him, $V_i$, and we will call his {\it estimate} of
its value his {\bf valuation}, $\hat{V_i}$.

              In a {\bf private-value} auction, each player knows his
value with certainty, although he may still have to estimate the
values of the other players. 

 
    
   In a {\bf common-value} auction, the players have identical
values, but each player forms his own valuation by estimating with
his private information.  

  The {\bf correlated-value} auction is a general category which
includes the common-value auction as an extreme case.  

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\newpage

 \noindent
 {\bf Auction Rules and Private-Value Strategies}

 
  The types of auctions to be described are:

\noindent
 (1) English.\\ 
 (2) First-price sealed-bid.\\
 (3) Second-price sealed-bid.\\
 (4) Dutch.  

\bigskip
\noindent
 {\bf (1) English (first-price open-cry)}


\noindent
 {\bf Rules.} Each bidder is free to revise his bid upwards.  When no
bidder wishes to revise his bid further, the highest bidder wins the
object and pays his bid.

\samepage{
\noindent
  {\bf Strategies.} A player's strategy is his series of bids as a
function of (1) his value, (2) his prior estimate of other players'
valuations, and (3) the past bids of all the players. His bid can
therefore be updated as his information set changes.

\noindent
 {\bf Payoffs.} The winner's payoff is his value minus his highest
bid.
 

     
\bigskip
\noindent
{\bf (2) First-price sealed-bid}


\noindent
         {\bf Rules.} Each bidder submits one bid, in ignorance of
the other bids.  The highest bidder pays his bid and wins the object.
 \nopagebreak

\noindent
  {\bf Strategies.} A player's strategy is his bid as a function of
his value and his prior beliefs about other players' valuations.

\noindent
 {\bf Payoffs.} The winner's payoff is his value minus his bid.

\newpage
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\noindent
{\bf (2) First-price sealed-bid, PRIVATE VALUES: A Neat Case}
 
         Suppose that there are $N$ risk-neutral bidders, and that
Nature assigns them values independently using a uniform density from
0 to some amount $\bar{v}$. Denote player $i$'s value by $v_i$, and
let us consider the strategy for player 1. If some other player has a
higher value, then in a symmetric equilibrium player $1$ is going to
lose the auction anyway, so we can ignore that possibility in finding
his optimal bid.  Player $1$'s equilibrium strategy is to bid epsilon
above his expectation of the second-highest value, conditional on his
bid being the highest (i.e., assuming that no other bidder has a
value over $v_1$).

  If we assume that $v_1$ is the highest value, the probability that
player $2$'s value, which is uniformly distributed between 0
and $v_1$, equals $v$ is $1/v_1$, and the probability that $v_2$ is
less than or equal to $v$ is $v/v_1$.  The probability that $v_2$
equals $v$ and is the second-highest value is 
  \begin{equation} \label{e12.1}
  Prob(v_2 = v) Prob(v_3 \leq v)Prob(v_4 \leq v) \cdots Prob(v_N \leq
v), 
 \end{equation}
 which equals
\begin{equation} \label{e12.2}
 \left( \frac{1}{v_1} \right) \left(\frac{v}{v_1} \right)^{N-2}.
 \end{equation}
 Since there are $N-1$ players besides player 1, the probability that
one of them has the value $v$, and $v$ is the second-highest is $N-1$
times expression (12.\ref{e12.2}).  The expectation of $v$ is the integral of
$v$ over the range 0 to $v_1$,
 \begin{equation} \label{e12.3}
\begin{array}{ll}
 E(v) & = \int_0^{v_1} v (N-1) (1/v_1) [v/v_1]^{N-2}dv\\
 & = (N-1) \frac{1}{v_1^{N-1}} \int_0^{v_1}  v^{N-1} dv\\
 & = \frac{(N-1)v_1}{N}.
 \end{array}
 \end{equation}
  Thus we find that player 1 ought to bid a fraction $\frac{N-1}{N}$
of his own value, plus epsilon.  

 \newpage
\noindent
{\bf (2) First-price sealed-bid, PRIVATE VALUES: A Not-so-Neat Case}

  The previous example is an elegant result, but it is not a general
rule. Suppose  Smith knows that Brown's value is 0 or 100 with
equal probability, and Smith's value of 400 is known by both players.
Brown bids either 0 or 100 in equilibrium, and Smith always bids
$(100 + \epsilon)$, because his value is so high that winning is more
important than paying a low price. 

 If Smith's value were 102 instead of 400, the equilibrium would be
much different. Smith would use a mixed strategy, and while Brown
would still offer 0 if his value were 0, if his value were 100 he
would use a mixed strategy too.  No pure strategy can be part of a
Nash equilibrium, because if Smith always bid a value $x < 100$,
Brown would always bid $x+ \varepsilon$, in which case Smith would
deviate to $x + 2\varepsilon$, and if Smith bid $x \geq 100$ he would
be paying 100 more than necessary half the time.

\newpage
 
 
\noindent
 {\bf (3) Second-price sealed-bid (Vickrey)}

\noindent
 {\bf Rules.} Each bidder submits one bid, in ignorance of the other
bids.  The bids are opened, and the highest bidder pays the amount of
the second-highest bid and wins the object.  
\nopagebreak

\noindent
   {\bf Strategies.} A player's strategy is his bid as a function of
his value and his prior belief about other players' valuations.

\noindent
 {\bf Payoffs.} The winner's payoff is his value minus the
second-highest bid that was made.


         
\bigskip
\noindent
 {\bf (4) Dutch  (Descending)}

\noindent
 {\bf Rules.} The seller announces a bid, which he continuously
lowers until some buyer stops him and takes the object at that price.


\noindent
 {\bf Strategies.} A player's strategy is when to stop the bidding
as a function of his valuation and his prior beliefs as to other
players' valuations.

\noindent
 {\bf Payoffs.} The winner's payoff is his value minus his bid.


                             The Dutch auction is {\bf strategically
equivalent} to the first-price sealed-bid auction, which means that
there is a one-to-one mapping between the strategy sets and the
equilibria of the two games. 

  
        \newpage
 
 
\bigskip
\noindent
{\bf Equivalence Theorems}

\noindent
   When one mentions auction theory to an economic theorist, the
first thing that springs to his mind is the idea that various kinds
of auctions are the same in some sense. Milgrom \& Weber (1982) give
a good summary of how and why this is true.  Regardless of the
information structure, the Dutch and first-price sealed-bid auctions
are the same in the sense that the strategies and the payoffs
associated with the strategies are the same.  That equivalence does
not depend on risk neutrality, but let us assume that all players are
risk neutral for the next few paragraphs.

    In private, independent-value auctions, the second-price
sealed-bid and English auctions are the same in the sense that the
bidder who values the object most highly wins and pays the valuation
of the second-highest valuer, but the strategies are different in the
two auctions.  In all four kinds of private independent-value
auctions discussed, the seller's expected price is the same.  This
fact is the biggest result in auction theory: the {\bf revenue
equivalence theorem} (Vickrey [1961]). 

  The revenue equivalence theorem does not imply that in every
realization of the game all four auction rules yield the same price,
only that the expected price is the same. The difference arises
because in the Dutch and first-price sealed-bid auctions, the winning
bidder has estimated the value of the second-highest bidder, and that
estimate, while correct on average, is above or below the true value
in particular realizations. The variance of the price is higher in
those auctions because of the additional estimation, which means that
a risk-averse seller should use the English or second-price auction.

   
   
\newpage
   
\subsection{Common-Value Auctions and the Winner's Curse} %section 12.4

 
   When I teach this material I bring a jar of pennies to class and
ask the students to bid for it in an English auction.  All but two of
the students get to look at the jar before the bidding starts, and
everybody is told that the jar contains more than 5 and less than 100
pennies. Before the bidding starts, I ask each student to write down
his best guess of the number of pennies.  The two students
who do not get to see the jar are like ``technical analysts,'' those
peculiar people who try to forecast stock prices using charts showing
the past movements of the stock while remaining ignorant of the
stock's ``fundamentals.''

   \newpage
 
         To avoid the winner's curse, players should scale down their
estimates to form their bids.  The mental process is a little like
deciding how much to bid in a private-value, first-price sealed bid
auction, in which bidder Smith estimates the second-highest value
conditional upon himself having the highest value and winning.  In
the common-value auction, Smith estimates his own value, not the
second-highest, conditional upon himself winning the auction.  He
knows that if he wins using his unbiased estimate, he probably bid
too high, so after winning with such a bid he would like to retract
it.  Ideally, he would submit a bid of [{\it $X$ if I lose, but
$(X-Y)$ if I win}], where $X$ is his valuation conditional upon
losing and $(X-Y)$ is his lower valuation conditional upon winning.
If he still won with a bid of $(X-Y)$ he would be happy; if he lost,
he would be relieved. But Smith can achieve the same effect by simply
submitting the bid $(X-Y)$ in the first place, since the size of
losing bids is irrelevant.

Another explanation of the winner's curse can be devised from the
Milgrom definition of ``bad news'' (Milgrom [1981b], note N7.5).
Suppose that the government is auctioning off the mineral rights to a
plot of land with common value $V$, and bidder $i$ has valuation
$\hat{V}_i$. Suppose also that the bidders are identical in
everything but their valuations, which are based on the various
information sets Nature has assigned them, and that the equilibrium
is symmetric, so the equilibrium bid function $b(\hat{V}_i)$ is the
same for each player. If Bidder 1 wins with a bid $b(\hat{V}_1)$ that
is based on his prior valuation $\hat{V}_1$, his posterior valuation
$\tilde{V}_1$ is 
 \begin{equation} \label{e12.4}
  \tilde{V}_1 = E(V | \hat{V}_1, b(\hat{V}_2) < b(\hat{V}_1),
\ldots, b(\hat{V}_n) < b(\hat{V}_1)).
 \end{equation}
  The news that $b(\hat{V}_2) < \infty$ would be neither good nor
bad, since it conveys no information, but the information that
$b(\hat{V}_2) < b(\hat{V}_1)$ is bad news, since it rules out values
of $b$ more likely to be produced by large values of $\hat{V}_2$. In
fact, the lower the value of $b(\hat{V}_1)$, the worse is the news of
having won.  Hence, 
 \begin{equation} \label{e12.5}
  \tilde{V}_1 < E(V | \hat{V}_1) = \hat{V}_1,
 \end{equation}
  and if Bidder 1 had bid $b(\hat{V}_1) = \hat{V}_1$ he would
immediately regret having won. If his winning bid were enough below
$\hat{V}_1$, however, he would be pleased to win.

  Deciding how much to scale down the bid is a hard problem because
the amount depends on how much all the other players scale down.  In
a second-price auction a player calculates the value of $\tilde{V}_1$
using equation (12.\ref{e12.4}), but that equation hides considerable
complexity under the disguise of ``$b(\hat{V}_2)$,'' which is itself
calculated as a function of $b(\hat{V}_1)$ using an equation like
(12.\ref{e12.4}).

\newpage
 
\begin{center}
{\bf Table 12.1  Bids by Serious Competitors in Oil Auctions} 

 \begin{tabular}{|llll|}
 \hline
 {\bf Offshore} & {\bf  Santa Barbara} &  {\bf Offshore}  &  {\bf  Alaska} \\ 
{\bf Louisiana} & {\bf   Channel} &  {\bf Texas}  &  {\bf  North Slope} \\ 
{\bf 1967} & {\bf 1968} &  {\bf  1968}  &  {\bf 1969} \\ 
  Tract SS 207 &  Tract 375  &  Tract 506    & Tract 253   \\ 
   &   &     &    \\ 
   \hline
  32.5 & 43.5  &  43.5   &    10.5\\ 
   17.7 & 32.1  & 15.5    & 5.2   \\ 
  11.1 & 18.1 & 11.6    &2.1   \\ 
 7.1  &10.2   &8.5     & 1.4   \\ 
  5.6 & 6.3  & 8.1    &0.5   \\ 
  4.1 &   &   5.6  &   0.4 \\ 
  3.3 &   &   4.7  &    \\ 
  &   &    2.8  &    \\ 
  &   &   2.6  &    \\ 
  &   &   0.7  &    \\ 
    &   &   0.7  &    \\ 
  &   &   0.4  &    \\ 
     \hline
\end{tabular} 
 \end{center}


    
 
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