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\noindent
26 February 2006  \\
 Eric Rasmusen, Erasmuse@indiana.edu, Http://www.rasmusen.org.

\section*{   13 Auctions}

\bigskip
\noindent
  {\bf  13.1  Values Private and Common, Continuous and Discrete  }

\noindent
 Bargaining and auctions are two extremes in  the many ways to sell goods,  as
  Bulow  \& Klemperer (1996) explain.
  In typical  bargaining, one buyer faces one seller and they make offers and
counteroffers free from the  formal rules we impose in theoretical modelling
(though as in  Bulow and Klemperer,  it is worth considering what happens if
the bargaining does have rules to which the  players commit).  In typical
auctions, many bidders face one seller and make offers according to formal rules
as rigid as those of the theorist. Bargaining is slow but flexible; auctions are
fast but rigid.

  Bargaining models   generate different results with different assumptions, but
since it is usually hard to match the assumptions with particular real
situations, the practical implications  come from the simplest models-- ideas
such as the importance of   avoiding  misunderstanding, determining  and then
concealing one's own  reservation price, bluffing, and manipulating the timing
of offers.

 Auction  models, on the other hand, may  also generate different results with
different assumptions  but it is easier to match the assumptions with particular
real situations, or even to create a real situation to match the model. That is
because auctions vary not only in the underlying preferences of the players, as
in bargaining, but in the specific rules used to play the game, and those rules
are chosen by one of the players, usually with legal commitment to them. Thus,
auction theory lends itself to what Alvin Roth (2002) calls  ``the economist as
engineer'': the  use of technical economic theory to design institutions for
specific situations.  As with other engineering, the result may not be of
general interest, but tailoring the model to the situation is both tricky and
valuable, requiring the same kind of talent and care as developing the general
theory.



  Because auctions are stylized markets with well-defined rules, modelling them
with game theory is particularly appropriate. Moreover, several of the
motivations behind auctions are similar to the motivations behind the asymmetric
information contracts of Part II of this book.  Besides the mundane reasons such
as speed of sale that make auctions important, auctions are useful for a variety
of informational purposes.  Often   bidders know more than the seller about the
value of what is being sold, and the seller, not wanting to suggest a price
first, uses an auction as a way to extract information. Art auctions are a good
example, because the value of a painting depends on the bidder's tastes, which
are known only to himself. Efficient allocation of resources is a   goal
different from profit maximization, but auctions are useful for that too. A good
example is told in Boyes and Happel's 2001 article, ``Auctions as an Allocation
Mechanism in Academia: The Case of Faculty Offices.'' At their business school,
the economics department used an auction to allocate new offices, whereas the
management department used seniority, the statistics department used dice, and
the finance department posted a first-come first-serve sign-up sheet without
warning. The auction has the best chance of coming up with an immediate
efficient allocation. (Why do I say ``immediate''? Why would the long-run
probably stay the same?)

   Auctions are also useful for agency reasons, because they hinder dishonest
dealing.  If the mayor were free to offer a price for building the new city hall
and accept the first contractor who showed up, the lucky contractor would
probably be the one who made the biggest political contribution.  If the
contract is put up for auction, cheating the public is more costly, and the
difficulty of rigging the bids may outweigh the political gain.

  We will spend most of this chapter on the effectiveness of different kinds of
auction rules in extracting surplus from bidders, which will require finding the
strategies with which bidders respond to the rules.  Section 13.1 classifies
auctions based on the relationships between different bidders'   estimates of
the
value of  what is being auctioned.  Section 13.2, a necessarily very long
section,     explains the possible auction rules and the bidding strategies
optimal for each rule.  Section 13.3 compares the outcomes under the various
rules, proving the Revenue Equivalence Theorem and showing how it becomes
invalid if bidders are risk averse.   Section 13.4  shows how to choose an
optimal reserve price  using the similarity between optimal auctions and
monopoly pricing.  Section 13.5   analyzes  common-value auctions, which can
lead bidders into ``the  winner's curse'' if they are not careful. Section 13.6
discusses asymmetric equilibria (The Wallet Game) and  information affiliation.

\bigskip

 \noindent
 {\bf Private-Value and  Common-Value Auctions }

\noindent
  Auctions differ enough for an intricate classification to be useful.  One way
to classify auctions is based on differences in the values bidders put on what
is
being auctioned. We will call the dollar value of the utility that bidder $i$
receives from an object its  {\bf  value }  to him, $v_i$, and we will denote
his estimate of the value by  $\hat{v_i}$.


In a {\bf private-value auction}, a bidder can learn nothing about his value
from
knowing the values of the other bidders. An example is the sale of antique
chairs
to people who will not resell them. Usually a bidder's value equals his  value
estimate in   private-value auction models. If an auction is to be private-
value, it cannot be followed by costless resale of the object. If there were
resale, a bidder's value   would depend on the price at which he could resell,
which would depend on the other bidders' values. What is special about a
private-
value auction is that a bidder cannot extract any information about his own
value
from the value estimates of the other bidders. Knowing all the other values  in
advance would not change his estimate. It might well change      his bidding
strategy, however, so we distinguish between the {\bf independent private-value
auction}, in which knowing his own value tells him nothing about other bidders'
values, and other situations such as the {\bf affiliated private-value auction}
(affiliation being a concept which will be explained later) in which he might be
able to  use  knowledge of his  own value to deduce something about other
players' values.

In a {\bf pure  common-value auction}, the bidders have identical values, but
each
bidder forms his own  estimate   on the basis of his own  private information.
An
example is bidding for U. S. Treasury bills. A bidder's  estimate would change
if
he could sneak a look at the other bidders'   estimates, because they are all
trying to estimate the same true value.

The values in most real-world auctions are a combination of private value and
common value, because the value estimates of the different bidders are
positively
correlated but not identical. As always in modelling, we trade off descriptive
accuracy against simplicity. It is common for economists to speak of mixed
auctions as ``common-value'' auctions, since their properties are closer to
those of common-value auctions. Krishna (2002) has used the term {\bf
interdependent value} for the mixed case.

 The private-value/common-value dichotomy is about what a  bidder knows about
his
own value, but a
  separate dimension of auctions is  what a bidder  knows about  other bidders'
values. One possibility is that the
  values are common knowledge.
 This makes optimal bidding simple. In a private-value auction, the highest-
valuing bidder can bid just above the second-highest value. In a common-value
auction, the bidders will take into account each others' information and their
value estimates will instantly converge to a single common   estimate, the best
one given all available information. In either case, bidders don't have to worry
about cleverly deducing each others' information by observing how the bidding
proceeds.

  It would be odd, however, to observe an auction in the real world in which the
seller  knew  the value estimates.   In that case, the  seller  shouldn't be
using an auction. He should just charge a price equal to the highest value
estimate, a price he knows will be accepted by one of the bidders.

   More commonly,  only the bidder himself knows his value, not other bidders or
the seller.  Most simply, the
  estimated values  are   statistically independent,  the independent private-
value
case that  we will be analyzing for the first half of this chapter.
 If the   estimates are
independent, then a bidder's only source of information about his   value is his
private information and his only source of information about   other bidders'
value estimates is what he observes of their bidding.  If the value estimates
are not independent,   a bidder  can use his  private information to help
estimate  other bidders' values (and thus how they will bid),  and if he does
not
know his own value perfectly he can   use observations of their value estimates
to improve his estimate of his own value--- which takes us to the common value
case.

     Lack of statistical independence can be present even in private-value
auctions, however. Smith might know his own value perfectly, and so would not be
able to
learn anything about it by learning Jones's value. That is what makes the
auction  a  private-value auction. But  if the values are not independent--- if,
say they are positively correlated--- then    if Smith knows that his own value
is
unusually high  he could predict that    Jones's value was also high. This would
affect Smith's bidding strategy, even though it would not affect the maximum he
would be willing to pay for the object being sold. I will not be analyzing
private-value auctions with correlated values here, but as explained in Riley
(1989)  the difference can    be important. Although bidding in the ascending
and
second-price  auction rules that we will soon discuss   is unchanged when values
are correlated,   bidding becomes lower in the descending and first-price
auctions. In contrast, we will see that bidding in a common-value auction is
generally more cautious than in a private-value auction   even in   ascending
and second-price auctions.

 To look at these various auction rules,
we will use the following two games, which I have set up using the language of
mechanism design  since  choosing auction rules fits well into that paradigm.
The Ten-Sixteen Auction  will be our running  example for when values are
discrete ($v=10$ or $v=16$),  and the Continuous-Value Auction will be our
example for when they are on a continuum ($v \sim f(v)$ on  $[\underline{v},
\overline{v}]$).

\begin{center}
{\bf The Ten-Sixteen   Auction }
 \end{center}
{\bf Players:}  One  seller and two bidders.

\noindent
 {\bf Order of Play:}\\
0. Nature chooses   Bidder $i$'s value for the object  to be either $v_i=10$ or
$v_i=16$, with equal probability. (The seller's value is zero.)  \\
1. The seller chooses a mechanism $[G(\tilde{v}_i, \tilde{v}_{-i}) v_i - t
(\tilde{v}_i, \tilde{v}_{-i} )]$ that takes payments $t$ and gives the object
with probability $G$ to    player $i$   (including the seller)  if he announces
that his value is $\tilde{v}_i$ and the other players  announce $\tilde{v}_{-i}
$.   He also chooses the procedure in which bidders  select $\tilde{v}_i$
(sequentially, simultaneously, etc.). \\
  2. Each bidder  simultaneously chooses   to participate in the auction or  to
stay  out. \\
 3. The bidders  and the  seller choose $\tilde{v}$ according to the mechanism
procedure. \\
  4. The  object is allocated and transfers are paid according to the
mechanism.\\

\noindent
 {\bf Payoffs:}\\
 The seller's payoff is
 \begin{equation} \label{e1}
 \pi_s=     \sum_{i=1}^{n} t(\tilde{v}_i, \tilde{v}_{-i}  )
\end{equation}
   Bidder $i$'s payoff is zero if he does not participate, and otherwise is
 \begin{equation} \label{e2}
   \pi_i(v_i)  =   G(\tilde{v}_i, \tilde{v}_{-i}) v_i - t (\tilde{v}_i,
\tilde{v}_{-i} )
 \end{equation}

To get around the open-set problem, we will assume   for any auction rule
that ties are broken in favor of whoever has the highest value, or randomly if
the values are equal. Otherwise, if, for example, we said that ties simply split
the probability of winning, then if  $v_1=10$ and $v_2= 16 $ and this were known
to both bidders, it would not be even a weak equilibrium to have them bid  $p_1=
10$ and $p_2=10$, because Bidder 2 would deviate to a slightly higher bid-- but
the smallest bid strictly greater than 10 does not exist if bid increments can
be infinitesimal.

\bigskip

 The second game will be our example for when values lie on a continuum.

\begin{center}
{\bf The Continuous-Value Auction }
 \end{center}
{\bf Players:}  One  seller and two  bidders.

\noindent
 {\bf Order of Play:}\\
0. Nature chooses   Bidder $i$'s value for the object,  $v_i$, using the
strictly positive, atomless  density $f(v)$ on the interval $[\underline{v},
\overline{v}]$.  \\
1. The seller chooses a mechanism $[G(\tilde{v}_i, \tilde{v}_{-i}) v_i - t
(\tilde{v}_i, \tilde{v}_{-i} )]$ that takes payments $t$ and gives the object
with probability $G$ to    player $i$   (including the seller)  if he announces
that his value is $\tilde{v}_i$ and the other players  announce $\tilde{v}_{-i}
$.   He also chooses the procedure in which bidders  select $\tilde{v}_i$
(sequentially, simultaneously, etc.). \\
  2. Each bidder  simultaneously chooses   to participate in the auction or  to
stay  out. \\
 3. The bidders  and the  seller choose $\tilde{v}$ according to the mechanism
procedure. \\
  4. The  object is allocated and transfers are paid according to the mechanism,
if it was accepted by all bidders.\\

\noindent
 {\bf Payoffs:}\\
 The seller's payoff is
 \begin{equation} \label{e3}
 \pi_s=     \sum_{i=1}^{n} t(\tilde{v}_i, \tilde{v}_{-i}  )
\end{equation}
    Bidder $i$'s payoff is zero if he does not participate, and otherwise is
 \begin{equation} \label{e4}
   \pi_i(v_i)  =   G(\tilde{v}_i, \tilde{v}_{-i}) v_i - t(\tilde{v}_i, \tilde{v}
_{-i} )
 \end{equation}



Many  possible auction procedures   fit the mechanism paradigm, even ones that
are never used in practice.  The  mechanism could allocate the good with 70\%
probability to the high bidder and with 30\%  probability to the lowest bidder,
for example;  or each bidder could be made to pay the amount he bids, even if he
loses; or  $t$ could include  an entry fee; or there could be a ``reserve
price,''   a minimum bid for which the seller will surrender the good.  In this
analysis, the   seller will  choose a direct mechanism    that  satisfies a
participation constraint for each  bidder type $v_i$   (Bidder $i$ will join the
auction, so, for example, the entry fee is not too large), and  an incentive
compatibility   constraint  (the bidder will truthfully reveal his type;
$\tilde{v}_i=v_i$).

\bigskip

\noindent
  {\bf  13.2   Optimal Strategies under Different   Rules  in Private-Value
Auctions  }




\noindent
 Auctions have  the same bewildering variety of rules as poker does. We will
look at five different  auction rules, using the private-value setting  since it
is simplest.    In teaching this material, I ask each student to pick a value
between 80 and 100, after which we conduct the various kinds of auctions.  I
advise the reader to try this. Pick two values and try out sample strategy
profiles for the different auctions as they are described.  Even though the
values are private, it will immediately become clear that the best-response bids
still depend on the strategies the bidder thinks  other bidders have adopted.

\noindent
The five auction rules we will consider  (with   common synonyms  for them) are
:

\noindent
 1 Ascending (English, open-cry, open-exit); \\
2 First-Price (first-price sealed-bid); \\
 3 Second-Price  (second-price sealed-bid, Vickrey); \\
 4 Descending (Dutch) \\
5 All-Pay\\

\bigskip

 \noindent
{\it Ascending (English, open-cry, open-exit)}

\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 bidder's strategy is his series of bids as a function of (1) his value, (2)
his prior estimate of other bidders' values, and (3) the past bids of all the
bidders. 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 ($t=p$ for him and $t=0$
for everyone else).  The losers' payoffs are zero.

  \noindent
{\bf Discussion}\\
    A bidder's dominant strategy in a private-value  ascending auction is to
stay
in the bidding   until  bidding higher would require him to exceed his   value
and then to stop.  This is optimal because he always wants to buy the object if
the price is less than its value to him, but he wants to pay the lowest price
possible.  All bidding ends when the price reaches the   second-highest value of
any bidder present at the auction.  The optimal strategy is independent of risk
neutrality if bidders know their own values with certainty rather than having to
estimate them, although risk-averse bidders who must estimate their values
should
be more conservative in bidding.

  The optimal  private-value  strategy is simple enough that details of the
ascending auction usually do not make much difference, but there are a number of
  possibilities.

\noindent
 (1)  The bidders  offer new  prices  using  pre-specified increments such as
dollars or thousands of dollars. \\
  (2)   The {\bf open-exit} auction, in which the price rises continuously and
bidders show their willingness to pay the price by not dropping out, where a
bidder's dropping out is publicly announced to the other bidders. \\
 (3)   The {\bf silent-exit} auction (my neologism), in which the price rises
continuously and bidders show their willingness to pay the price by not dropping
out, but a bidder's dropping out is not known to the other bidders. \\
 (4) The {\bf Ebay auction}, in which a bidder  submits his  ``bid ceiling,''
the
maximum price he is willing to pay. During the course of the auction the seller
uses the bid ceilings to raise the current winning bid only as  high as
necessary, and the winner is the player whose bid is highest at a prespecified
ending time. \\
  (5) The {\bf Amazon auction}, in which a bidder  submits his   bid ceiling.
During the course of the auction the seller  uses the bid ceilings to raise the
current winning bid only as  high as necessary, and the winner is the player
whose bid is highest at a prespecified ending time or ten minutes after the last
increase in the current winning bid, whichever is later.  \\

   The precise method can be quite important in common-value auctions, where
knowing what other players are doing   alters a bidder's own value estimate. If
the auction is open-exit,  for example, a bidder who observed that most of the
other bidders dropped out at a low price would probably revise his own value
estimate downwards, something he would not know to do in a silent-exit auction.

The
ascending auction   can be seen as a mechanism in which each bidder announces his value (which becomes his bid), the object is awarded to whoever announces the highest value (that is, bids highest), and he pays the second-highest announced value (the second-highest bid).
   In the Continuous-Value Auction, denote the highest announced value by
$\tilde{v}_{(1)}$,  the second-highest by $\tilde{v}_{(2)}$, and so forth.  the highest bidder gets
the object with probability
$G(\tilde{v}_{(1)}, \tilde{v}_{-1}) = 1$ at price $ t
(\tilde{v}_{(1)}, \tilde{v}_{-1} ) = \tilde{v}_{(2)}$, and    for $i \neq
1$,  $G(\tilde{v}_{(i)}, \tilde{v}_{-i}) = 0$ and $ t
(\tilde{v}_{(1)}, \tilde{v}_{-1} )   =0.$  This is incentive compatible, since a
player's value  announcement only matters if his value is highest, and he then
wants to win if and only if the price is less than or equal to his value.  It
satisfies the participation constraint because his lowest possible payoff
following that strategy is zero, and his payoff is higher if he wins and
$\tilde{v}_{(1)}> \tilde{v}_{(2)}$. 

  Since each  bidder's expected payoff is strictly positive, the {\it optimal}
mechanism for the seller would be   more complicated. As we will discuss later,
it would include a {\bf reserve price} $p^*$ below which the object would remain
unsold, changing the   first part of the mechanism to $G(\tilde{v}_{(1)},
\tilde{v}_{-1}) = 1$ and $ t
(\tilde{v}_{(1)}, \tilde{v}_{-1} ) = Max \{ \tilde{v}_{(2)}, p^*\}$ if
$\tilde{v}_{(1)} \geq  p^*$  but $G(\tilde{v}_{(1)}, \tilde{v}_{-1}) = 0$ if
$\tilde{v}_{(1)} < p^*$. We have seen in Chapter 10 that  optimal mechanisms are
not always efficient, and this is an example: the object will go unsold if
$\tilde{v}_{(1)}< p^*$.



In the Ten-Sixteen Auction,   the seller's value is $v_s=0$, and
each of two  bidders' private  values   $v_1$  and $v_2$ is either 10 or 16
with
equal probability,     known only to the bidder himself.
A bidder's optimal strategy in the  ascending auction  would be to set his bid
or
bid ceiling  to $p(v=10) = 10$ and $p(v= 16) = 16$. His expected payoff would be
 \begin{equation} \label{e5}
 \begin{array}{ll}
   \pi (v=10)  &=0 \\ &\\
   \pi (v=16)  &= 0.5(16-10)  +0.5(16-16)  = 3  \\
 \end{array}
\end{equation}
The expected price, the payoff to the seller,  is
\begin{equation} \label{e6}
   \pi_s  = 0.5^2 (10) + 0.5^2 (16) + 2 (0.5)^2(10)  = 2.5 + 4 + 5 = 11.5 \\
\end{equation}


\bigskip
 \noindent
{\bf First-Price  (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.

\noindent
 {\bf Strategies}\\
 A bidder's strategy is his bid as a function of his value.

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

\bigskip


   \noindent
{\bf Discussion}\\
    In the first-price auction what the winning bidder wants to do is to have
submitted a sealed bid  just enough higher than the second-highest bid to win.
If all the bidders' values are common knowledge and he can predict the second-
highest bid perfectly, this is a simple problem.  If the values are private
information, then he has to guess at the second-highest bid, however, and take a
gamble. His tradeoff is between bidding high--thus winning more often--and
bidding low--thus benefiting more if the bid wins.   His optimal strategy
depends on his degree of risk aversion and beliefs about the other bidders, so
the equilibrium is less robust  to mistakes in the assumptions of the model than
the equilibria of ascending and second-price auctions. As we will see later,
however, there are good reasons why sellers so often choose to use  first-price
auctions.


\noindent
{\bf    The First-Price Auction with a Continuous Distribution of Values }
 

   Suppose Nature independently  assigns  values to  $n$ risk-neutral bidders    using the  continuous  density $f(v)>0 $ (with cumulative probability $F(v)$)  on the  support $[0,    \bar{v}]$.

   A bidder's payoff as a function of his value $v$ and his bid function $p(v)$
is,    letting  $G(p(v))$   denote the probability of
winning with a particular $p(v)$:
\begin{equation} \label{e10}
  \pi(v, p(v)) = G(p(v)) [v - p(v)].
  \end{equation}

Let us first prove a lemma. 

 \noindent
{\it   Lemma: If  player's  equilibrium bid function is differentiable, it    is strictly increasing in  his value: $p'(v) >0$.}


\noindent
{\it   Proof}:
 The first-order condition  from payoff (\ref{e10}) is  
  \begin{equation} \label{e10a}
 \frac{d\pi(v)}{dp}  = G' (v-p) -G =0.
\end{equation}
The optimum is an interior solution because  at $p_i=0$ the payoff is
increasing and if $p_i$ becomes large enough, $\pi$ is negative.  Thus, 
  $\frac{d^2\pi(v_i)} {dp_i^2} \leq 0$ at the optimum.  Using the implicit function theorem and
the fact that  $\frac{d^2\pi(v_i)} {dp_i dv_i} = G' \geq  0$ because a higher
bid does not yield a lower probability of winning, we can conclude that
$\frac{dp_i}{dv_i} \geq 0$, at least if the bid function is differentiable. But
it cannot be that  $\frac{dp_i} {dv_i} = 0$, because then there would be values
$v_1$ and $v_2$ such that  $p_1=p_2=p$ and then 
\begin{equation} \label{e10b}
 \frac{d\pi(v_1)}{dp_1}  = G'(p)  (v_1-p )     -G(p) =0 = \frac{d\pi(v_2)}
{dp_2}= G'(p)  (v_2-p )     -G(p),
\end{equation}
 which cannot be true. So  the bidder bids more if his value is   higher. Q.E.D. 

   
 Now let us try to find an  equilibrium bid function. From equation (\ref{e10}), it is 
\begin{equation} \label{e10c}
  p(v) = v - \frac{ \pi(v, p(v))}{   G(p(v))}.
  \end{equation}
 That is not very useful in itself,  since it has $p(v)$ on both sides. We need to find ways to rewrite 
  $\pi$  and  $G$ in terms of just $v$. 

First,  tackle 
  $G(p(v))$.   Monotonicity of the bid function  (from Lemma 1) implies that
  the bidder with the greatest $v$ will bid highest  and win. Thus,   the probability $G(p(v))$  that a  bidder with price $p_i$ will
win is the probability that $v_i$ is the highest value of all $n$ bidders.  The
probability  that a bidder's value $v$ is the highest  is  $F(v)^{n-1}$, the probability
that each of the other $(n-1)$ bidders has a value  less than $v$. Thus,
\begin{equation} \label{e13b}
 G(p(v)) = F(v)^{n-1}.
\end{equation}

   Next   think about  $\pi(v, p(v))$. 
 The Envelope Theorem says that if $\pi(v, p(v))$ is the value of a function
maximized by choice of $p(v)$  then its total derivative with respect to $v$
equals its partial derivative,  because  $\frac{\partial \pi  }
{\partial p}=0$: 
 \begin{equation} \label{e12}
\begin{array}{ll}
  \frac{d \pi(v, p(v))}{d v} & =  \frac{\partial \pi(v, p(v))}{\partial p}
    \frac{\partial p  }{\partial v}  + \frac{\partial \pi(v,
p(v))}{\partial v} 
  =  \frac{\partial \pi(v, p(v))}{\partial v}.\\
\end{array}
  \end{equation}
  We can apply  the Envelope Theorem to equation (\ref{e10}) to see how $\pi$ changes with $v$  assuming $p(v)$ is chosen optimally, which is appropriate because we are  characterizing not just any bid function, but  the optimal bid function. Thus,  
\begin{equation} \label{e13}
 \frac{d \pi(v, p(v))}{dv}  =  G(p(v)).
  \end{equation}
   Substituting from equation (\ref{e13b}) gives us $\pi$'s derivative, if not $\pi$, as a function of $v$:  
\begin{equation} \label{e13a}
 \frac{d \pi(v, p(v))}{dv}  =  F(v)^{n-1}.
  \end{equation}
  To get $\pi(v, p(v))$ from its derivative, (\ref{e13a}),    integrate   over  all possible values  from
zero to $v$   and include the a base value of $\pi(0)$ as the  constant of integration:
\begin{equation} \label{e14}
\begin{array}{ll}
 \pi(v, p(v))&{\displaystyle  = \pi(0) + \int_0^v   F(x)^{n-1} dx}=  \int_0^v   F(x)^{n-1} dx. \\
\end{array}
  \end{equation}
The last step is true because  a bidder with $v=0$ will never bid a
positive amount and so will have a payoff of $\pi(0, p(0))=0$.


We can now return to   the bid function in 
  equation (\ref{e10c})  and substitute for $G(p(v))$ and   $\pi(v, p(v))$ from  equations (\ref{e13b})  
(\ref{e14}):
\begin{equation} \label{e15}
{\displaystyle  
  p(v) = v - \frac{\int_0^v   F(x)^{n-1} dx}{    F(v)^{n-1} }.
}
  \end{equation}
  

Suppose $F(v) = v/\bar{v}, $ the uniform distribution. Then (\ref{e15}) becomes
\begin{equation} \label{e17}
\begin{array}{ll}
p(v) &= {\displaystyle v- \frac{   \int_0^v
\left(\frac{x}{\bar{v}}\right)^{n-1} dx }{ \left( \frac{v}{\bar{v}}\right)^{n-1}   }} \\
  & \\
 & ={\displaystyle  v-       \frac{  \bigg|_{x=0}
^{v}    \left(\frac{1}{\bar{v}  }\right)^{n-1}  \left(\frac{1}{n }\right)     
 x^n   }{ \left( \frac{v}{\bar{v}}\right)^{n-1} }  }  \\
& \\
 &{\displaystyle  = v-  \frac{  \left(\frac{1}{\bar{v}  }\right)^{n-1}  \left(\frac{1}{n }\right)     
 v^n   -0    } { \left( \frac{v}{\bar{v}}\right)^{n-1}   } }  \\
& \\
 &  =  v - \frac{v}{n} =     \left( \frac{n-1 }{n} \right)v.     \\
\end{array}
  \end{equation}
  What a happy  ending to a complicated derivation! If there are 
  two bidders and values are uniform on
$[0,1]$, a bidder should bid $ p   = v /2$, which since he has probability  $v $ of
winning yields an expected payoff of $  v^2/2$. If $n=10$ he should bid
$\frac{9} {10} v$, which since he has probability $v^9$ of winning yields
him an expected payoff of $  v^{10}/10,$ quite close to zero if $v  < 1$.
 
  

\bigskip





\noindent
{\bf  The First-Price Auction:   A Mixed-Strategy Equilibrium  in the 
 Ten-Sixteen   Auction       }

The result   in equations (\ref{e9}) and (\ref{e17})   depended crucially on the
value distribution having a continuous support.  When this is not true, the
equilibrium in a first-price auction may not even  be in pure strategies.   Now
let  each  of two  bidders' private  value  $v$  be  either 10 or 16  with equal
probability  and   known only to himself.

 In a first-price auction,  a bidder's  optimal strategy is to   bid  $p(v=10) =
10$,  and if $v=16$ to use a mixed strategy, mixing   over the support
$[\underline{p},\bar{p}]$, where it will turn out that $\underline{p}=10$ and
$\bar{p}= 13$, and the   expected payoffs will be:
\begin{equation} \label{e18}
 \begin{array}{ll}
   \pi (v=10)  &=0 \\
 &\\
   \pi (v=16)  &=  3  \\
&\\
   \pi_s  &=  11.5.    \\
  \end{array}
\end{equation}
 These are the same payoffs  as in the ascending auction, an equivalence  we
will come back to in a later section.

This will serve as an illustration of how to find  an equilibrium mixed strategy
when bidders mix over  a  continuum of pure strategies rather than just between
two.  The first step is to see why the equilibrium cannot be in pure strategies
(though  some portions of the equilibrium strategy can be pure, e.g. if $v=10$,
a bidder will bid $p=10$ with probability one).

 First, there is no equilibrium in pure strategies, either symmetric or
asymmetric.  In any equilibrium, $ p(v =10) = 10$, because if either bidder used
the bid $p<10$, it  would cause the other player to  deviate to  $(p+\epsilon)$,
and a    bid above 10 exceeds the object's value.

 If $v=16$, however, a player will randomize his bid, as I will now show. 
Suppose the  two  bidders are using the  pure strategies  $p_1(v_1=16) = z_1   $
and
$p_2(v_2=16) = z_2$.  The values of    $z_1 $ and $z_2$  would lie in $(10,16]$
because a bid of exactly 10 would  lose to the positive-probability bid
of   $ p (v =10) = 10$  given our tie-breaking
assumption  and a bid   over 16 would exceed the object's value. 
yield a negative payoff.    Either $z_1= z_2$,  or $z_1 \neq z_2$. If $z_1=
z_2$,  then each bidder has incentive to deviate to $(z_1-\epsilon)$ and win
with probability one  instead of tying.  If $z_1 < z_2$, then  Bidder 2 will
deviate to bid   $(z_1+ \epsilon)$. If he does that, however, Bidder 1 would
deviate to bid  $(z_1+ 2\epsilon)$, so he could win with probability one  at
trivially higher cost.  The same holds true if    $z_2 < z_1$. Thus, there is no
equilibrium in pure strategies.

 The second step is to figure out what pure strategies will be mixed between by
a bidder with $v=16$. It turns out that they form the interval $[10, 13]$.  As
just explained, the bid  $p(v=16)$  will be no less than  10 (so the bidder can
win if his rival's value is 10) and no greater than  16 (which would always win,
but unprofitably). The  pure strategy of $(p=10)|(v=16)$ will win with
probability of at least 0.50 (when the other bidder happens to have $v=10$,
given
our tie-breaking rule),  yielding a payoff of   $0.50 (16-10) =3$.  This rules
out bids in $(13, 16]$, since even if they always win, their payoff is less than
3.
Thus,  the upper bound $\bar{p}$ must be no greater than 13.



The lower bound $\underline{p}$ must be exactly 10. If it were at $(10+\epsilon)
$ then a bid of $(10-2\epsilon)$ would have an equal certainty of winning the
auction, but would have $\epsilon$ higher payoff. Thus, $\underline{p} = 10$.

   The upper bound $\bar{p}$  must be exactly 13. If it were any less, then the
other player would respond by using the pure strategy of $(\bar{p}+\epsilon)$,
which would   win with probability one and yield  a payoff of greater than the
payoff of  3 ($=0.5 (16-10)$) from $p=10$. In a mixed-strategy equilibrium,
though, the payoff from any of the strategies mixed between must be equal. Thus,
$\bar{p}$ cannot be less than 13.

  We are not quite done looking at the  strategies mixed between.
   When a player mixes over a continuum, the modeller must be careful to check
for (a)
  atoms (some particular  point  which has positive   probability, not just
positive density), and (b)  gaps  (intervals  within the mixing range with  zero
probability of  bids). Are there any atoms or gaps  within  the interval
[10,13]? No, it turns out.

\noindent
(a)  Bidder 2's mixing density  does not have an atom  at any  point $a$ in
$[10,
13]$-- no point $a$ has positive probability, as opposed to  positive density.
An example of such an atom would be if the mixing distribution were the density
$m(p)=1/6$ over the interval $[10,13]$ plus   an atom of  probability 1/2 at $p=
13$, so the cumulative probability would be $M(p) = p/6$  over $[10,13)$ and
$M(13)=1$.  Using $M(p)$,   a point such as 11 would have zero probability even
though the interval, say, of  [10.5,  12.5]  would have probability  2/6.

     If there were an atom at $a$,   Bidder  1 would respond   by putting
positive probability on   $(a+\epsilon)$ and zero probability    on   $a$.  But
then  Bidder  2 would respond by putting zero probability on   $a$ and shifting
that probability to  $(a+2\epsilon)$.

\noindent
(b)  Bidder 2's mixing density does not have a gap  $[g,h]$ anywhere  with $g>
10$
and $h<13$.  If it did, then  Bidder  1's payoff from bidding $g$ and $h$ would
be
 \begin{equation} \label{e19}
 \pi_1(g)  =  Prob (p_2 < g) v_1 - g
  \end{equation}
and
 \begin{equation} \label{e20}
 \pi_1(h)  =  Prob (p_2 < h) v_1 - h = Prob (p_2 < g) v_1 - h,
  \end{equation}
 where the second equality in $\pi_1(h)$ is true because there is zero
probability that $p_2$ is between $g$ and $h$.   Bidder 1 will put zero
probability  on $p_1=h$, since its payoff is lower than the payoff from $p_1=g$
and will put zero probability on slightly larger values of $p_1$ too, since by
continuity their payoffs will also be less than the payoff from $p_1=g$. This
creates a gap $[h, h^*]$ in which $p_1=0$.   But then Bidder  2 will want to put
zero probability on $p_2=h^*$ and slightly higher values, by the same reasoning,
which means that our original hypothesis of only a gap $[g,h]$ is false.




Thus, we can conclude that the mixing density $m(p)$ is positive over the entire
interval $[10, 13]$, with no atoms. What will it look like? Let us confine
ourselves to looking for a symmetric equilibrium, in which both bidders use the
same function $m(p)$. We know the expected payoff from any bid $p$ in the
support must equal the payoff from $p=10$ or $p=13$, which is 3.  Therefore,
since if our player has value $v=16$  there is probability 0.5 of winning
because the other player has $v=10$ and probability $0.5 M(p)$ of winning
because the other player has $v=16$ too but bid less than $p$, the payoff is
  \begin{equation} \label{e21}
0.5  (16-p) + 0.5   M(p)   (16-p)  =3.
 \end{equation}
This implies that $  (16-p) +  M(p)  (16-p)  = 6$, so
\begin{equation} \label{e22}
   M(p)    =  \frac{6}{16-p}-1,
\end{equation}
   which has the density
 \begin{equation} \label{e22a}
   m(p)    =     \frac{6}{(16-p)^2}
\end{equation}
 on the support $[10,13]$, rising from $m(10) = \frac{1 }{6}$ to $m(13) =
\frac{4 }{6}$.

 Since each bidder type has the same expected payoff in this first-price auction
as in the ascending auction, and the object is sold with probability one,  it
must be that the seller's payoff is the same, too, equal to 11.5, as we found in
equation (\ref{e6}).

 \includegraphics[width=150mm]{fig13-01.jpg}

\begin{center}
 {\bf Figure  1:   Smoothing a Discrete Distribution}
 \end{center}



You may find it odd that     the general continuous-value auction  has a pure-
strategy equilibrium but  our particular discrete-value auction  does not.
Usually if a game lacks a   pure-strategy equilibrium in discrete type space, it
also lacks one if   we ``smooth'' the  probability distribution by making it
continuous but still putting almost all the weight on the old discrete types,
as in Figure 1.

    This is related to a remarkable feature of private-value auctions with
discrete values: the mixed-strategy equilibria do not necessarily block
efficiency  (and the revenue equivalence we study later).  When players
randomize, it would seem that sometimes  by chance the highest-valuing player
would be unlucky and lose the auction, which would be inefficient.  Not so here.
As explained in Riley (1989) and   Wolfstetter (1999, p. 204), if the values
of each player are distributed discretely over some set $\{0, v_a, v_b, ...,
v_w\}$  then in the  symmetric equilibrium mixed strategy, the supports of the
mixing distributions are $v_a: [0, p_1], . v_b: [p_1, p_2], v_w: [p_{w-1}, p_w]
$, where  $p_1<p_2<... <p_w$.  The supports do not overlap. Each type of bidder
acts as if he was in competition just with his own type (since he will surely
win over the lower types and will surely lose to the higher types) and the
object is allocated to a bidder who values it most. The mixing   only
determines who wins when two players happen to have the same type.



\bigskip
\noindent
 {\bf Second-Price  Auctions  (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.

\noindent
 {\bf Strategies}\\
 A bidder's strategy is his bid as a function of his value.


\noindent
 {\bf Payoffs}\\
The winning bidder's payoff is his value minus the second-highest bid.  The
losing bidders' payoffs are zero. The seller's payoff is the second-highest-bid.

     \noindent
{\bf Discussion}\\
       Second-price auctions are similar to ascending auctions.  They are rarely
used in reality, but are useful for modelling.  Bidding one's value   is a
weakly  dominant strategy: a bidder who bids less is no more likely to win the
auction (and probably less likely, depending on $f(v)$), but he  pays the same
price---  the second-highest-valuing player's bid---  if he does win.  The
structure of the payoffs is reminiscent of the Groves Mechanism of Section 10.4,
because in both games a bidder's strategy affects some major event (who wins the
auction,  or whether the project is undertaken), but his strategy affects his
own payoff only via that event.  In the auction's symmetric equilibrium, each
bidder bids his value and the winner ends up paying the second-highest value. If
bidders know their own values, the outcome does not depend on risk neutrality.

 One  difference between ascending and second-price auctions is that second-
price auctions have peculiar asymmetric equilibria because the actions in them
are simultaneous.
Consider a variant of the Ten-Sixteen Auction,  in which each of two bidders'
values  can be 10 or 16, but  where the realized  values    are  common
knowledge. Bidding one's value is  a {\bf symmetric equilibrium},  meaning that
the bid function $p(v)$ is the same for both bidders: $ \{p(v=10) = 10, p(v=16)=
16\}$. But there are asymmetric equilibria.

\noindent
 Consider the following equilibrium.
\begin{equation} \label{e22b}
\begin{array}{ll}
    p_1(v=10) = 10 & p_1(v= 16)=16\\
 & \\
 p_2(v=10) = 1 & p_2(v=16)=10  \\
\end{array}
\end{equation}

 Since Bidder 1    never bids less than 10,  Bidder 2 knows that if $v_2=10$  he
can never get a positive payoff, so he is willing to choose  $p_2(v=10) = 1$.
Doing so results in a sale price of 1, for any $p_1 >1$,  which is better for
Bidder 1 and worse for the seller than a price of 10, but Bidder 2 doesn't care
about their payoffs.  In the same way, if $v_2=16$, Bidder 2 knows that if he
bids    10 he will win if $v_1=10$, but if $v_2=16$ he would have to pay 16 to
win and would earn a payoff of zero.  He might as well bid   10 and earn his
zero by losing. \footnote{Trembling-hand perfectness, however, would rule out
this kind of equilibrium. If Bidder 1 might tremble and  bid, for example,  4 by
accident, Bidder 2 would not want to ever bid less than 4.  Bidding less than
one's value is weakly dominated by bidding exactly one's value-- but Nash
equilibrium strategies can be weakly dominated, as we saw with the Bertrand Game
in Chapter 3.   }

  Perhaps the seller's fear of asymmetric equilibria like this is why Second-
price auctions are so rare. They have actually    been used, though,  in a
computer operating system. An operating system must assign a computer's
resources to different tasks, and researchers at Xerox Corporation designed the
Spawn system, under which users allocate ``money'' in a second-price   auction
for computer resources. See ``Improving a Computer Network's Efficiency,'' {\it
The New York Times},  p. 35 (29 March   1989).


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

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

\noindent
 {\bf Strategies}\\
 A bidder's strategy is when to stop the bidding as a function of his value.

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


  \noindent
{\bf Discussion}

        The typical descending auction is {\bf strategically equivalent} to the
first-price   auction, which means   there is a one-to-one mapping between the
strategy sets and the equilibria of the two games. The reason for the strategic
equivalence is that no relevant information is disclosed in the course of the
auction, only at the end, when it is too late to change anybody's behavior.  In
the first-price auction a bidder's bid is irrelevant unless it is the highest,
and in the descending auction a bidder's stopping price is   irrelevant unless
it
is the highest.  The equilibrium price is calculated the same way for both
auctions.

 A descending auction does not have to be like a first-price auction as a matter
of logic, though.  Vickrey (1961) notes that a  descending auction could be set
up as a second-price auction. When the first bidder presses his button, he
primes
an auction-ending buzzer that  does not goes off until a second bidder presses
his button.   In that case, the descending auction would be strategically
equivalent to a second-price auction. Economists almost always  mean  ``first-
price descending auctions'' when they use the term, however.

Descending--``Dutch''--auctions have  been  used in the Netherlands  to sell
flowers-- see the Aalsmeer auction website at http://www.vba.nl  for information
and photos. They have also been used in
 Ontario to sell tobacco using a clock four feet in diameter marked with
quarter-cent gradations. Each of six or so bidders has a stop button.  The clock
hand drops a quarter-cent at a time, and the stop buttons are registered so that
ties cannot occur (tobacco bidders need reflexes like race-car drivers). The
farmer sellers  watch from an adjoining room and can later reject the bids if
they feel they are too low (a form of reserve price)     The clock is fast
enough to sell 2,500,000 lb. per day (Cassady [1967, p. 200]).

        Descending auctions are common in less obvious forms. Filene's is one of
the biggest stores in Boston, and Filene's Basement is its most famous
department. In the basement are a variety of marked-down items formerly in the
regular store, each with a price and date attached. The price customers pay at
the register is the price on the tag minus a discount which depends on how long
ago the item was dated. As time passes and the item remains unsold, the discount
rises from 10 to 50 to 70 percent.  The idea of predictable time discounting has
also   been used by bookstores too (``Waldenbooks to Cut Some Book Prices in
Stages in Test of New Selling Tactic,'' {\it  The  Wall Street Journal}, March
29, 1988, p. 34).




\bigskip

  \noindent
  {\bf All-Pay  Auctions}

\noindent
 {\bf Rules}\\
 Each bidder places a bid simultaneously. The bidder with the highest bid wins,
and each bidder pays the amount   he bid.

\noindent
  {\bf Strategies}\\
 A bidder's strategy is his bid as a function of his value.


\noindent
{\bf Payoffs}\\
 The winner's payoff is his value minus his bid. The losers' payoffs are  the
negative of their bids.

\noindent
{\bf Discussion}\\
  The winning bid will be lower in the all-pay auction than  under the other
rules,    because bidders  need a bigger payoff when they do win to make up for
their negative payoffs when they lose.    At the same time, since  even the
losing bidders pay something to the seller  it is not obvious that the seller
does badly (and in fact, it turns out to be just as good an auction rule as the
others, in this simple risk-neutral context).

  I do not know of the all-pay   rule  ever being used in a real auction, but it
is a useful modelling tool   because it models rent-seeking very well. When a
number of companies lobby a politician for a privilege, they are in an all-pay
auction  because even  the losers have paid by incurring the cost of lobbbying.
When a number of companies pursue a patent, it is an all-pay auction because
even the losers have incurred the cost of doing research.



\bigskip

\noindent
{\bf The Equal-Value All-Pay Auction }

  Suppose  each of the $n$ bidders has the same value, $v$. That is not a very
interesting game for most of the auction rules, though it is true that for the
second-price auction there exists the strange asymmetric equilibrium  $ \{v, 0,
0, ....0\}$. Under the all-pay auction rule, however, this game is quite
interesting.
 The equilibrium is  in mixed strategies.   This is easy to see, because in any
pure-strategy profile, either the maximum bid is less than $v$, in which case
someone could deviate to $p=v$ and increase his payoff; or one bidder bids $v$
and the rest bid  at most $p'<v$, in which case the high bidder will deviate to
bid  just above $p'$.



 Suppose we have a symmetric equilibrium, so  all bidders use the same mixing
cumulative distribution $M(p)$. Let us conjecture that $\pi(p) =0$, which we
will
later verify.\footnote{It turns out there is a continuum of asymmetric
equilibria in this game if $n >2$, but a unique equilibrium if $n=2$. See
Kovenock,   Baye \&   de Vries (1996) for a full characterization of all-pay
auctions with complete information.}   The payoff function for each  bidder is
the
probability of winning times the value of the prize minus the bid, which is paid
with probability one,  and if we equate that to zero we get
 \begin{equation} \label{e24}
  M(p)^{n-1}  v =    p,
\end{equation}
 so
  \begin{equation} \label{e25}
   M(p)     = \sqrt[n-1]{\frac{p}{v}},
\end{equation}
 as shown in Figure 2.
   At the extreme bids that a bidder with value $v$ might offer,    $M(0)  =
\sqrt[n-1]{\frac{0}{v}} = 0$ and  $M(v)  = \sqrt[n-1]{\frac{v}{v}} = 1$,  so we
have found a  valid distribution function $M(p)$.  Moreover, since the payoff
from one of the strategies between which it mixes, $p=0$, equals zero, we have
verified  our   conjecture that $\pi(p) =0$ in the equilibrium.


\includegraphics[width=150mm]{fig13-02.jpg}

\begin{center}
 {\bf Figure 2: The Bid Function in an All-Pay Auction with Identical Bidders}
\end{center}



\bigskip

Consider now what happens with the all-pay rule in the Continuous-Value Auction
Game, using  an explanation adapted from  Krishna (2002, ch. 3.2).

\noindent
{\bf The Continuous-Value All-Pay Auction  }

 Suppose  each of the $n$ bidders picks his value $v$ from the same density
$f(v)
$. Conjecture that  the equilibrium is symmetric,  in pure strategies,    and
that the bid function, $p(v)$, is strictly increasing.   The  equilibrium payoff
function for a bidder with value $v$ who pretends he has value  $z$ is
  \begin{equation} \label{e261}
  \pi(v,z) =  F(z)^{n-1} v - p(z),
\end{equation}
 since   if our bidder bids $p(z)$, that is the highest bid only if all $(n-1)$
other bidders have $v<z$, a probability of $F(z)$ for each of them.


 The function   $\pi(v,z) $ is not necessarily concave in $z$, so  satisfaction
of the  first-order condition will not be a sufficient condition for payoff
maximization, but it is  a necessary condition   since the optimal $z$ is not 0
(unless $v=0$) or infinity and from (\ref{e261})  $\pi(v,z) $  is differentiable
in $z$ in our conjectured equilibrium.  Thus, we need to find $z$ such that
 \begin{equation} \label{e27}
   \frac{\partial \pi(v,z)}{\partial z}   =  (n-1) F(z)^{n-2} f(z) v - p'(z)=0
\end{equation}
 In the equilibrium, our bidder does follow the strategy $p(v)$, so $z=v$ and we
can write
 \begin{equation} \label{e28}
 p'(v)=   (n-1) F(v)^{n-2} f(v) v
\end{equation}
Integrating up, we get
 \begin{equation} \label{e29}
 p(v)=  p(0) +  \int_0^v (n-1) F(x)^{n-2} f(x) x dx
\end{equation}
 This is deterministic, symmetric,  and  strictly increasing in $v$, so we have
verified    our conjectures.


We can verify that truthelling is a symmetric equilibrium strategy by
substituting for $p(z)$ from (\ref{e29}) into  payoff equation (\ref{e261}).
\begin{equation} \label{e30}
\begin{array}{ll}
  \pi(v,z) & =  F(z)^{n-1} v - p(z)\\
 &\\
 & {\displaystyle=  F(z)^{n-1} v -p(0) -  \int_0^z (n-1) F(x)^{n-2} f(x) x dx}\\
 &\\
 &  {\displaystyle =  F(z)^{n-1} v -p(0) - F(z)^{n-1} z +   \int_0^z   F(x)
^{n-1}   dx,} \\
\end{array}
   \end{equation}
 where the last step uses integration by parts ($\int g h' = gh - \int h g'$,
where $g = x$ and $h'=(n-1) F(x)^{n-2} f(x)$).   Maximizing  (\ref{e30}) with
respect to $z$ yields
\begin{equation} \label{e31}
   \frac{\partial \pi(v,z)}{\partial z}   =  (n-1) F(z)^{n-2} f(z) (v-z),
\end{equation}
 which is maximized by setting $z=v$. Thus, if  $(n-1)$ of the  bidders are
using
this $p(v)$ function, so will  the remaining bidder, and we have a Nash
equilibrium.

    Let's see what happens with a particular value distribution. Suppose  values
are uniformly distributed over [0,1], so $F(v) =v$. Then equation (\ref{e29})
becomes
 \begin{equation} \label{e31a}
\begin{array}{ll}
 p(v) & =  {\displaystyle  p(0) +  \int_0^v (n-1) x^{n-2}  (1) x dx} \\
 & \\
 & {\displaystyle  =  p(0) +  \bigg|_{x=0}^v    (n-1)  \frac{x^n }{n}   } \\
 & \\
 &{\displaystyle  =  0+  \left( \frac{n-1}{n}  \right) v^n, }
 \end{array}
\end{equation}
 where we can tell that $p(0)=0$ because  if $p(0)>0$ a bidder with $v=0$  would
have a negative expected payoff.  If there were $n=2$ bidders,  a bidder with
value $v$ would bid $ v^2/2$, win with probability $v$, and  have expected
payoff $\pi = v(v) -  v^2/2 = v^2/2$.  If there were $n=10$ bidders,   a bidder
with value $v$ would bid $ (9/10) v^{10} $, win with probability $v^9$, and
have expected payoff $\pi = v(v^9) -  (9/10)v^{10}  =  v^{10}/(10)$.   As we
will see when we discuss the Revenue Equivalence Theorem, it is no accident that
this is the same payoff as for the   first-price auction when values were
uniformly distributed on [0,1], in equation (\ref{e9}).



\bigskip
\noindent
  {\bf  The Dollar Auction }

   A famous example of an auction in which not just the winner pays is the
dollar auction of
 Shubik (1971). This is an ascending   auction  to sell a  dollar bill in which
the players offer higher and higher bids, and the highest bidder wins-- but both
the   first- and   second-highest bidders pay their bids.   If the players begin
with infinite wealth, the game illustrates why equilibrium might not exist if
strategy sets are unbounded. Once one bidder has started bidding against
another,
both of them do best by continuing to bid,  so as to win the dollar as well as
pay the bid. (If there are three or more players, all but the top two will be
happy to stop bidding early in the game.)    This auction may seem absurd, but
as a variant on the all-pay auction it  has considerable similarity to patent
races  and arms races. See Baye  \&     Hoppe  (2003) for more on the
equivalence between innovation games and auctions.

   The  all-pay  auction and the  dollar auction are just two  examples of
auctions  in which a player must pay something even though he loses. Chapter 3's
War of Attrition  is another example, which is something like  a second-price
all-pay auction.    Even odder is the {\bf loser-pays auction}, a two-player
auction in which only the loser pays.   As we have seen with the all-pay auction
and its revenue equivalence to other auction forms, however, the fact that an
auction's rules are strange does not mean it is necessarily  worse for bidders.

  All-pay auctions are a standard way to model rentseeking: imagine that $n$
players each exert $e$ in effort simultaneously to get a prize worth $V$, the
winner being whoever's effort is highest.
 Another common way to model rentseeking is as an auction in which the highest
bidder has the best chance to win  but lower bidders might win instead. Tullock
(1980) started a literature on this in an  article  which was valuable despite a
mistaken claim that   the expected amount paid by the bidders  might exceed the
value of the prize. (See Baye, Kovenock \& de Vries  (1999) for a more recent
analysis of this  {\bf rent dissipation}).  There is no obvious way to model
contests, and the functional form  does matter to behavior, as  Jack
Hirshleifer (1989)  tells us.  In the  most popular functional form,   $P_1$ and
$P_2$ are the probabilities of winning of the two players,  $e_1$ and $e_2$ are
their efforts,  and $R$ and $\theta$ are parameters which can be used to
increase the probability that the high bidder wins   or  to give one player an
advantage over the other. The victory function is then assumed to be
 \begin{equation} \label{e32}
 P_1 = \frac{ \theta  e_1^R}{ \theta e_1^R+ e_2^R} \;\;\; {\rm and} \;\;\;P_2 =
\frac{  e_2^R}{ \theta e_1^R+ e_2^R}
  \end{equation}
 If $\theta =1$ and $R$ becomes large, this becomes close to the simple all-pay
auction, because neither player has an advantage and the highest bidder wins
with
probability near one.

  Once we depart from true auctions, however, the modeller must be careful about
some seemingly obvious assumptions. Two bidders can simply  refuse to enter  the
dollar auction, for example, but two countries have a harder time refusing to
enter a situation in which an arms race is tempting, though they can try to
collude in keeping the bids small.   It is also  often plausible that the size
of the prize rises or falls with the bids-- for example, when  the contest is a
mechanism used by a  team to motivate  its members to produce more (see Chung
[1996]) or   when the prize shrinks with effort because rent-seeking hurts the
economy (see Alexeev   \&  Leitzel [1996]).

\vspace{1in}



 \noindent
 {\bf  13.3 Revenue Equivalence,  Risk Aversion, and Uncertainty}

     We can now collect together the outcomes of these various auction rules and
compare them. We have seen that the first-price and descending auctions are
strategically equivalent, so the payoffs to the bidders and seller will be the
same under each rule regardless of whether values are private or common and
whether the players are risk neutral or risk averse.

    When values are  private  and  independent, the second-price   and ascending
auctions are the same in the sense that the bidder who values the object most
highly wins and pays the  second-highest of the values of all the bidders
present,    but the strategies are different in the two auctions.  In all five
kinds of  auctions, however,   the seller's expected revenue is the same.  This
is the biggest result in auction theory: the {\bf Revenue Equivalence Theorem}
of Vickrey (1961).  This has been variously generalized from Vickrey's original
statement (e.g., Klemperer (2004, p. 40), so there is no  single   Revenue
Equivalence Theorem, but they all share the same idea of payoffs   being the
same under various auction rules. We will look at two versions, a general one
for auctions with particular properties and a more specific one-- a corollary,
really-- for the five auction rules just  analyzed.


\noindent
   THE  REVENUE EQUIVALENCE THEOREM. {\it
 Let  all players be risk-neutral with private values drawn independently from the same atomless, strictly increasing 
distribution $F(v)$   on $[\underline{v},
\bar{v}]$.   If  under either     Auction  Rule $A_1$ or
  Auction Rule $A_2$  it is true that:    \\
 (a)  The winner of the object is the player with the highest value; and     \\
 (b) The lowest bidder type, $v=\underline{v}$, has an expected payment of zero;
    \\
   then the symmetric equilibria of the two auction rules have  the same expected  payoffs for each type of bidder and for the seller.}


\noindent
{\it  Proof.}
     Let us  represent the auction as the truthful equilibrium of a direct mechanism in which each bidder  sends a message $z$ of his type $v$ and then pays an expected amount $p(z)$. (The Revelation Principle says that we can do this.)  By assumption (a), the  probability that a player   wins the object given that he chooses message $z$ equals  $F(z)^{n-1}$, the probability that all $(n-1)$ other players have values $v<z$.  Let us denote this winning probability by $G(z)$, with density $g(z)$. Note that $g(z)$ is well defined because we   assumed that $F(v)$ is atomless and everywhere increasing. 

 
 The  expected payoff of any  player of type $v$ is the same, since we are restricting ourselves to symmetric equilibria. It equals     
 \begin{equation} \label{e33}
 \pi(z, v) =  
   G(z) v  - p(z). 
 \end{equation}
  The  first-order condition     with respect to the player's choice of type message $z$  (which we can use because neither $z=0$  nor $z=\bar{v}$ is the optimum  if  condition (a)    is to be true) is 
 \begin{equation} \label{e34}
 \frac{ d \pi(z; v)}{d z }  = g(z) v - \frac{ d p(z)}{d z } =0,    
 \end{equation}
 so 
 \begin{equation} \label{e34a}
\frac{ d p(z)}{d z } = g(z) v.      
 \end{equation}
  We are looking at a truthful equilibrium, so we can replace $z$ with $v$: 
\begin{equation} \label{e34b}
\frac{ d p(v)}{d v} = g(v) v.      
 \end{equation}
 
 Next, we integrate (\ref{e34b}) over all values from zero to $v$, adding $p( \underline{v})$ as the  constant of integration: 
\begin{equation} \label{e33a}
  p(v)  = p( \underline{v}) + \int_{\underline{v}}^v  g(x) x dx.      
 \end{equation}
 We can use (\ref{e33a} to substitute for $p(v)$ in the payoff equation (\ref{e33}), which becomes, after replacing $z$ with $v$  and setting $p( \underline{v})=0$ because of assumption (b),  
 \begin{equation} \label{e35}
 \pi(v, v) =  
   G(v) v  - \int_{\underline{v}}^v  g(x) x dx. 
 \end{equation}
    Equation (\ref{e35}) says  the expected payoff of a bidder of type $v$ depends only on the $G(v)$ distribution, which in turn depends only  on the $F(v)$ distribution, and not on the $p(z)$ function or other details of the particular auction rule. But if the bidders' payoffs do not depend on the auction rule, neither does the seller's.   Q.E.D. 


   


\bigskip

 There are many versions of the revenue equivalence theorem,  and the name of the theorem comes from a version that  just says that the seller's revenue is the same across auction rules rather than including bidders too. The version proved above is adapted from Proposition 3.1 of Krishna (2002, p. 30). Other versions, which use different proof approaches, can be found in  Klemperer (1996, p. 40),   and  Milgrom (2004, p.
74).  Two assumptions that are standard across versions are that the bidders are risk neutral and that their values are drawn from the same distribution. 
 
 
   It is only when we apply the Revenue Equivalence Theorem   to
the diverse auction rules we laid out earlier that its remarkable nature   can be appreciated. The symmetric equilibria of the   ascending, first-price,  second-price,   descending, and all-pay auctions with continuous values all satisfy the two conditions of the Theorem: (a) the winner is the bidder with the highest value, and (b) the  lowest type makes an expected payment of zero. Thus, the following corollary is true. 



  \noindent
 A REVENUE EQUIVALENCE COROLLARY. {\it
 Let  all players be risk-neutral with private values drawn from the same
strictly increasing, atomless distribution $F(v)$.  The symmetric equilibria of
the  ascending, first-price,  second-price,   descending, and all-pay auctions
all   have the same  expected  payoffs for each type of bidder and for the
seller.}

  
 


Although the different auctions have the same expected payoff  for the seller,
they do not have the same realized payoff. In the first-price auction, for
example, the winning bidder's payment depends entirely on his own  value. In the
second-price  auction the winning bidder's payment depends entirely on     the
second-highest value,  which is sometimes close to his own value  and sometimes
much less.  Thus, we will see that  first-price auctions are better if players
are risk averse.


 Remember, too, that the revenue equivalence theorem requires not just that the bidders have private values, not common, but that the private values be independent. To see why, consider what happens if there are two bidders, both with values drawn uniformly from [0,10], but interdependently, with $v_2=10-v_1$. If we put aside   equilibria with  weakly dominated strategies (e.g., for a player to bid 0 if his value is less than 5), the second-price auction yields revenue equal to $p=v_{(2)}$, the second-highest value. The seller can extract more revenue, however, by using the auction rule that the winner is the highest bidder, and he pays 10 minus the second-highest bid. One equilibrium under that rule has both players bidding their values, and $p= 10-v_{(2)}= v_{(1)}> v_{(2)}$.



\bigskip

\noindent
  {\bf  Risk Aversion in Private-Value Auctions  }\footnote{This explanation is
adapted from Chapter 8 of the  English draft of the Chinese version of
Wolfstetter (1999). }

When bidders are risk averse, the Revenue Equivalence Theorem fails.  Consider
Bidder 1 in The   Ten-Sixteen Auction   when he knows his own  value is  $v_1=
16$
but does not know $v_2$.  In the  second-price auction,  he has an equal chance
of a payoff of either 0 (if $v_2=16$) or 6 (if $v_2=10$), regardless of whether
the bidders are risk averse or not, because bidding    one's value  is a weakly
dominant strategy.

Compare that with his payoff in the first-price  auction, in which the
equilibrium is in mixed strategies.   If the bidders are risk neutral, then  as
we found earlier,  if the   bidder has value 16    he wins using  a bid in  the
mixing support  [10,13]  and achieves a payoff in [3,6]    with probability
0.75,  and    he loses and earns payoff of zero with probability 0.25.  The
(0,6) gamble of the second-price auction is riskier than the  (0, 3 to 6) gamble
of the first-price auction.  The  (0,6) gamble is simpler, but it has more
dispersion.

If the bidders are risk averse, then the optimal strategies in the first-price
auction change. It remains true that the bidders mix on an  interval
$[10,\bar{p}]$. We derived $\bar{p}$ and the optimal mixing distribution  by
equating expected payoffs, however, and a certain win at a price of 10 will now
be worth more to a bidder than a 50\% chance of winning at a price of 13. Let us
denote the concave utility function of each bidder by $U(v-p)$ and normalize by
defining $U(0) \equiv 0$. The expected payoff from $p=10$, which wins with
probability 0.5,   must  equal the  expected payoff from the upper bound
$\bar{p}$ of the mixing support, so
 \begin{equation} \label{e44a}
 0.5 U(6) =  U( 16- \bar{p}).
 \end{equation}
   Since $0.5   U(6) < U( 16- 13)$   by concavity of $U$, it must be that
$\bar{p} >13.$  We found the mixing distribution function $M(p)$ by equating
$\pi(p)$ to the payoff from bidding 10, which is $0.5 U(6)$,  so
 \begin{equation} \label{e44b}
  \pi(p) =  0.5 U(16-p) +  0.5 M(p) U(16-p) = 0.5 U(6),
 \end{equation}
 which can be solved to yield
\begin{equation} \label{e44c}
   M(p)  = \frac{U(6)} {U(16-p)} -1,
 \end{equation}
 which has the density
\begin{equation} \label{e44d}
     m(p)  = \frac{U(6)} U'(16-p) {U(16-p)^2},
 \end{equation}
 compared with the risk-neutral density $ m(p)     = \frac{6}{(16-p)^2}  $ from
equation (\ref{e22a}). Thus,  risk aversion of the bidders actually spreads out
their equilibrium bids (the support is broader than  [10,13]), but it remains
true that the first-price auction is less risky than the second-price auction.

What happens in the Continuous-Value Auction? In  the  second-price auction ,the
optimal strategies are unchanged, so seller revenue does not change if bidders
are risk averse. To solve for the equilibrium of the first-price auction, let us
look at a given bidder's incentive to report his true type $v$ as $z$ in an
auction in which the payment is $p(z)$ and the probability of winning the object
is $G(z)$. The bidder maximizes by choice of $z$
 \begin{equation} \label{e45}
\begin{array}{ll}
 \pi(v,z)& =
   G(z ) U[v  - p(z)]  \\
 & \\
 &= F(z)^{n-1} U[v  - p(z)], \\
 \end{array}
 \end{equation}
where   $\pi(v,0) = 0$ because $F(0) =0$.   At the optimum,
\begin{equation} \label{e46}
\frac{ \partial \pi(v,z ) }{\partial z} =
 (n-1)  F(z)^{n-2}  f(z)   U[v  - p(z)] +  F(z)^{n-1} U'[v  - p(z)] [ -p'(z)]=0,
 \end{equation}

 In equilibrium,   $z=v$. Using that fact, for all $v>\underline{v}$ (since
$F(\underline{v}) =0$) we can solve  equation (\ref{e46}) for
 $p(z)$ to get
\begin{equation} \label{e47}
 p(v) =  \left( \frac{(n-1)     f(v)}{ F(v) } \right) \left(  \frac{ U[v  -
p(v)] }{  U'[v  - p(v)]}\right)
 \end{equation}
  Now let's  look at the effect of risk aversion on  $p(v)$. If $U$ is linear,
then
\begin{equation} \label{e47a}
\frac{ U[v  - p(v)] }{  U'[v  - p(v)]}= v  - p(v),
\end{equation}
but if  the bidder is risk averse, so $U$ is strictly concave,
\begin{equation} \label{e47b}
\frac{ U[v  - p(v)] }{  U'[v  - p(v)]}> v  - p(v).
\end{equation}
 Thus, for a given $v$,  the bid function in (\ref{e47})  makes the    bid
higher if
the bidder is risk averse than if he is not.  The bid for every value of $v$
except $v=\underline{v}$  increases  ($p(\underline{v}) =\underline{v}$,
regardless of risk aversion).  By
  increasing his bid from the level optimal for a risk-neutral bidder, the risk-
averse bidder insures himself. If he wins, his surplus is slightly less because
of the higher price, but he is more likely to win and avoid a surplus of zero.


As a result the seller's revenue is greater in the first-price   than in the
second-price auction if bidders are risk averse. But since under risk neutrality
the   first-price  and second-price auctions yield  the same revenue,   under
risk aversion the first-price auction must yield greater revenue, both in
expectation and conditional on the highest $v$ present in the auction. The
seller, whether risk neutral or risk averse,  will prefer the first-price
auction when
bidders are risk averse.



  \noindent
 {\bf     Uncertainty over   One's Own Value }

\noindent
      We have seen that when bidders are risk averse, revenue equivalence fails
because the second-price  auction is riskier than the first-price auction. By
the same reasoning, the ascending auction is riskier, and by strategic
equivalence the descending auction is the same as the first-price.   Risk
aversion matters for different reasons if the bidders do not know their values
precisely. As we will see later, in the context of common values, uncertainty
over one's own value will generate  conservative bidding behavior for the
strategic reason of the ``Winner's Curse''--- a reason which applies whether
bidders
are risk averse or not--- and because of the ``Linkage Principle'' of Milgrom \&
Weber (1982).   Value  uncertainty also has a simpler effect that is
driven by risk aversion and applies even in independent  private-value auctions:
buying a
good of uncertain value is instrinsically risky, whether buying it by auction or
by a posted price.

\noindent
  Consider  the following question:

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

  Let us  assume that the seller can   precommit to  reveal both favorable
information and   unfavorable information, since of course he would like best to
reveal only information that raises bidder estimates of the object's value
(though the unravelling effect of Chapter 10 might undo such a strategy).  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
auction.  Each measures his private value  $v$ with an independent error
$\epsilon>0$. This error is with equal probability $-x, +x$ or 0.   The bidders
have diffuse priors, so they take all values of $v$ to be equally likely, ex
ante.  Let us denote a bidder's  measured value by $\hat{v}= v + \epsilon$,
which
is an unbiased estimate of $v$. In the ascending auctions we have been studying
so far, where $\epsilon=0$, the optimal bid ceiling was $v$. Now, when $\epsilon
>0$,  what bid ceiling should be used by  a  bidder with utility function $U(v-
p)
$?




 If the  bidder  wins the auction and pays $ p$ for the object, his  expected
utility  at that point is
\begin{equation} \label{e43}
 \pi ( p )  =  \frac{U([\hat{v}-x] - p)  }{3}+ \frac{ U (\hat{v}  - p) }{3}
+\frac{U([\hat{v}+x]  - p)    }{3}
\end{equation}

If he is risk neutral, this yields him a payoff of zero if $p = \hat{v}$, and
winning at any lower price would yield a positive payoff. This is true   because
we can write
$U(v-p)=v-p$ and
\begin{equation} \label{e42}
  \pi (risk\; neutral, p = \hat{v}) =  \frac{  ([\hat{v}-x] - \hat{v}) } {3} +
\frac{ (\hat{v}  - \hat{v}) }{3} + \frac{ (\hat{v}+x - \hat{v}) }{3}  =  0.
 \end{equation}
 Thus, under risk neutrality uncertainty over one's own value does not affect
the optimal strategy, except that the bidder's bid ceiling is his expected value
for the object rather than his value. Adverse selection aside, there is no
reason for the seller to try to improve bidder information.



 If the bidder  is risk averse, however,   then the utility function $U$  is
concave and
\begin{equation} \label{e44}
\frac{ U([\hat{v}-x] - p)  }{3}  +\frac{U( [\hat{v}+x] - p)  } {3}< \left(
\frac{2}{3} \right) U(\hat{v}-p),
\end{equation}
so his expected payoff  in equation (\ref{e43}) is less than $U( \hat{v}-p)$,
and if $p=\hat{v}$ his payoff is less than $U(0)$.
A risk-averse bidder   will have a negative expected payoff from paying his bid
ceiling  unless it is strictly less than his  estimated value.



 Notice  that  the seller does not have control over all the elements of the
model. The seller can often choose the auction  rules unilaterally. This
includes not just  how bids are made, but such things as whether the bidders get
to know how many potential bidders are in the auction, whether the seller
himself
is allowed to bid,  and so forth. Also, the seller can  decide how much
information to release about the goods.      The seller cannot, however,  decide
whether the bidders are risk averse or not, or whether they have common or
private values,  no more than he can choose what their values are for the good
he is selling.  All of those assumptions concern the utility functions of the
bidders.   At best, the seller can do things such as choose to produce goods to
sell at the auction which have common values instead of private values.

An  error I have often seen is to think that the presence of uncertainty  over
one's value always causes the Winner's Curse that we will shortly examine. It
does not, unless the auction is  in  common values; uncertainty over one's value
is a necessary but not sufficient condition for the Winner's Curse. It is true
that  risk-averse bidders should not bid as high as their value estimates if
they
are uncertain about them, even if the auction is in  private values.  That
sounds a  lot like a Winner's Curse, but the reason for the discounted bids is
completely different, depending as it does on risk aversion.  If bidders are
uncertain about value estimates but they are risk neutral, their dominant
strategy is still to bid up to their value estimates.  If  the Winner's Curse is
present, even if   a bidder  is  risk-neutral  he discounts his bid  because  if
he wins, on average his   estimate will be greater than the value.


\bigskip
\noindent
  {\bf 13.4  Reserve Prices and  the  Marginal Revenue    Approach  }

 A {\bf reserve price} $p^*$ is a bid put in by the seller, secretly or openly,
before the auction begins, which commits him not to sell the object if nobody
bids more than  $p^*$.  The seller will often find that a reserve price can
increase his payoff.  If he does,  it turns out that he will choose a reserve
price strictly greater than his own value: $p^*>v_s$. To see this, we will use
the {\bf marginal revenue approach}  to auctions, an approach developed  in
Bulow \& Roberts (1989) for risk-neutral private values, and  Bulow \& Klemperer
(1996) for common values and risk aversion.  This approach compares the seller
in an auction to an ordinary monopolist who sells using a posted price.  We
 start with  an auction to just one bidder, then  extend the idea to an auction
with multiple bidders, and finally return to the surprising  similarity between
an auction with one bidder and a monopoly selling to a continuum of bidders
using
a posted price.

\noindent
 {\bf 1.   One Bidder.} If there is just one bidder,  the seller will do badly
in
any of the auction rules we have discussed so far.   The single bidder would bid
$p_1=0$ and win.

  The situation is really better suited to bargaining  or simple monopoly than
to an auction.  The seller could use an auction, but a standard auction yields
him zero revenue. so posting a price offer to the bidder makes more sense.   If
the auction has a reserve price, however, it can be equivalent to posting a
price, just as in    bargaining the making of a    single take-it-or-leave-it
offer of $p^*$ is equivalent to posting a price.

\noindent
 What should the offer $p^*$  be? Let the bidder have value distribution $F(v)$
on $[\underline{v}, \bar{v}]$ which is differentiable and strictly increasing,
so the density  $f(v)$ is always positive. Let the seller value the object at
$v_s \geq  \underline{v}$.  The seller's payoff is
\begin{equation} \label{e36}
\begin{array}{ll}
 \pi(p^*) & =     Pr (  p^*< v) (p^* - v_s) +Pr (  p^*> v) (0) \\
 & \\
 & =    [1-F(p^*)] (p^* - v_s).\\
\end{array}
 \end{equation}
 This has first-order-condition
 \begin{equation} \label{e37}
 \frac{d\pi(p^*)}{d p^*}  =  [1-F(p^*)]  - f(p^*) [p^* - v_s]=0.
 \end{equation}
 On  solving (\ref{e37}) for for $p^*$ we get
 \begin{equation} \label{e38}
{\displaystyle p^*=  v_s + \left( \frac{1-F(p^*)}{f(p^*)} \right).  }
 \end{equation}
 The optimal take-it-or-leave-it offer, the  ``reserve price'' $p^*$   satisfies
equation (\ref{e38}).  The reserve price is strictly greater than the seller's
value  for the object ($p^*> v_s$) unless the solution is such that $F(p^*)=1$
because the optimal reserve price is the greatest possible bidder value, in
which
case the object has probability zero of being sold. One reason to use a reserve
price is so the seller does not sell an object for a price worth less than its
value to him, but that is not all that is going on.\footnote{The second-order
condition for the problem is  $\frac{d^2\pi(p^*)}{d p^*^2}  =  -2 f(p^*)
+f'(p^*) [p^* - v_s] \leq 0.$  This might well be false, and in any case several
values of $p^*$ might satisfy  equation (\ref{e38})  so it is only a necessary
condition, not a sufficient one. Another way to see that $p^*>v_s$ is to observe
that $\frac{d\pi(p^*=v_s)}{d p^*}  =  [1-F(v_s)  - f(v_s) [v_s - v_s]>0$, so
$p^*$ should be increased beyond $v_s$.  }



\noindent
 {\bf 2.  Multiple Bidders.}  Now let there be $n$ bidders, all  with  values
distributed independently by   $F(v)$. Denote the bidders with the highest and
second-highest values as Bidders 1 and 2.
The seller's payoff in a second-price auction  is
\begin{equation} \label{e39}
\begin{array}{ll}
 \pi(p^*) & =  Pr (  p^*> v_1) (0)+   Pr ( v_2< p^*< v_1 ) (p^*  - v_s) +  Pr (
p^*< Ev_2 < v_1) (v_2 - v_s)     \\
 & \\
 &  {\displaystyle =  \int_{v_1=\underline{v}}^{p^*}  f(v_1) (0) dv_1 +
\int_{v_1=p^*}^{\bar{v}}   \left( \int_{v_2=\underline{v}}^{p^*} (p^* - v_s)
f(v_2)dv_2     +   \int_{v_2=p^*}^{v_1} (v_2 - v_s) f(v_2)dv_2  \right) f(v_1)
dv_1   } \\
 & \\
\end{array}
 \end{equation}
  This expression integrates over two random variables. First, it matters
whether $v_1$ is greater than or less than $p^*$,  the outer integrals. Second,
it matters whether  $v_2$  is less than  $p^*$ or not, the inner integrals.



  Now differentiate equation (\ref{e39}) to find the optimal reserve price
$p^*$  using   Leibniz's integral  rule (given in the Mathematical  Appendix):
 \begin{equation} \label{e40}
\begin{array}{ll}
{\displaystyle \frac{d \pi(p^*)}{d p^*}    }     & {\displaystyle  =0 +  -f
(p^*) \left(  \int_{v_2=\underline{v}}^{p^*} (p^* - v_s) f(v_2) dv_2  +
\int_{v_2=p^*}^{p^*} (v_2 - v_s)  f(v_2) dv_2  \right) } \\
 & \\
 &  {\displaystyle +   \int_{v_1=p^*}^{\bar{v}}    \left(    (p^* - v_s)f(p^*)-
(p^* - v_s)f(p^*)  +  \int_{v_2=\underline{v}}^{p^*}   f(v_2)dv_2   \right)
f(v_1)dv_1 }   \\
 & \\
   & {\displaystyle = -f (p^*) \left(  \int_{v_2=\underline{v}}^{p^*}  (p^* -
v_s)  f(v_2) dv_2 + 0   \right) +   \int_{v_1=p^*}^{\bar{v}}   \left(
\int_{v_2=\underline{v}}^{p^*}   f(v_2)dv_2   \right) f(v_1)dv_1 }   \\
 & \\
   & = -f (p^*) F( p^*)  (p^* - v_s) +  (1- F( p^*))  F( p^*)    = 0      \\
\end{array}
 \end{equation}





\noindent
Dividing by $F$, the last line of expression (\ref{e40}) implies that
 \begin{equation} \label{e40a}
 p^*=  v_s + \frac{1-F(p^*)}{f(p^*)},
 \end{equation}
 just what we found in    equation (\ref{e38}) for the one-bidder case.
Remarkably,  the optimal reserve price is unchanged!  Moreover, equation
(\ref{e40a}) applies to any number  of bidders, not just $n=2$. Only Bidders 1
and
2 show up in the equations we used in the derivation, but that is because they
are the only ones to affect the result in a second-price auction.

   In fact, only the highest-valuing   bidder  matters to the optimal reserve
price.   The reserve price only affects the winning price if the second-highest-
valuing bidder  happens to have    a rather low value. Conditioning on that,
Bidder 1 would bid low if there were no reserve price-- it is almost as if he
faced no competition. But   the seller, conditioning on that value being low,
would want to set a reserve price so that  Bidder 1 must pay more. Since the
reserve price only matters if all but one of the bidders have low values, it
doesn't matter whether ``all but one'' is 99 or 0. Conditioning on all but one
bidder having low values, the seller is conditioning on the game   having only
one     bidder who matters.


  \bigskip

\noindent
{\bf 3. A Continuum of Bidders: The Marginal Revenue Interpretation }
  Now think of  a firm with a constant marginal cost of $c$  facing  a
continuum of bidders      along the same distribution $F(v)$ that we have been
using.  The quantity of bidders with values above $p$ will be $(1-F(p))$, so the
demand equation  is
\begin{equation} \label{e40b}
 q(p)   = 1-F(p)
\end{equation}
 and
\begin{equation} \label{e40c}
Revenue \equiv  pq  = p(1-F(p))
\end{equation}
 The marginal revenue is then (keeping in mind that $\frac{dq}{dp} = -f(p)$)
\begin{equation} \label{e41}
 \begin{array}{ll}
Marginal \; Revenue \equiv  \frac{dR}{dq} & = p + \left( \frac{dp}{dq} \right)
q\\
 & \\
  &=   p + \left( \frac{1}{  \frac{dq}{dp}} \right)q\\
 & \\
 &  =  p+    \left( \frac{1}{   -f(p)} \right)  (1-F(p))\\
 & \\
& =  p-    \frac{ 1-F(p)}{   f(p)}     \\
\end{array}
\end{equation}
  Setting marginal revenue to marginal cost, the profit-maximizing  monopoly
price is the one at which the marginal revenue  in  (\ref{e41}) equals $c$.

Does equation (\ref{e41}) look  familiar?\footnote{As with monopoly in general,
it might  happen here  that
marginal revenue   equals  marginal cost at more than one quantity.  The
$MR=MC$ rule is only a necessary condition, not a sufficient one, for profit
maximization.}  Equating (\ref{e41}) to marginal cost, thinking of marginal cost
as $v_s$, the seller's opportunity cost,  and moving $p$ to the left-hand-side
yields   the  optimal price equation  we found for one bidder in  equation
(\ref{e38}).  That is because the mathematics of the problem is identical
whether the seller is facing a continuum of bidders on distribution $F(v)$ or
one
one bidder  drawn randomly from  the continuum  $F(v)$.   The problem is just
like that  in     a take-it-or-leave-it-offer  bargaining model where the
bidder's type   is unknown to the seller. In all three situations-- the
continuum
of bidders, the auction to one seller with a reserve price, and the single offer
to a single bidder, the seller is in effect using  the basic monopoly pricing
rule of setting quantity so that marginal revenue equals marginal cost.  The
difference is in interpretation. In the auction and bargaining contexts,  the
marginal change in  the number of units of   quantity  becomes the marginal
change in
the probability of selling one unit.  The seller is still picking quantity, but
he is picking it in the interval [0,1] instead of $[0,\infty]$ when he is
selling just one unit to one bidder.   To increase that probability, and thus
the
expected number of units sold, the seller must reduce his price,      just
as an ordinary monopolist must reduce his price to increase the number of units
he sells.

  Figure  3a shows this.    In the auction context, $c$
could represent the seller's production cost, or it could be any other kind of
opportunity cost that creates the minimum price at which the seller would part
with the good,    $v_{seller}$. The auction seller    should act like a
monopolist with constant marginal cost of $v_{seller} $ (constant because he is
producing just one unit),  facing the demand curve based on $f(v)$, which means
he should set a reserve price for  the quantity where marginal revenue equals
marginal cost.

\includegraphics[width=150mm]{fig13-03a.jpg}


\begin{center}
 {\bf Figure 3a:  Auctions and Marginal Revenue: Reserve Price Needed}
 \end{center}



\includegraphics[width=150mm]{fig13-03b.jpg}




\begin{center}
 {\bf Figure  3b:  Auctions and Marginal Revenue: No Reserve Price Needed}
 \end{center}

\newpage

The optimal  reserve price will always be positive.  In Figure 3a, even if $v_s
$  were zero  instead of   positive, the curves are such that the reserve
price should   be positive, even though the    probability of making a sale
would then equal one. This corresponds to the idea  that a monopolist will
always raise the price above marginal cost, even though in certain situations
(such as the curves in Figure 3b) he will not reduce output below the
competitive level and a reserve price is redundant.

    If output is reduced below the competitive level, the outcome is
inefficient, something true both in conventional monopoly and here.  Here,
output is inefficiently low if no sale takes place of the one unit even though
$v> v_s$ for some bidder.  In that case, what the seller has done is to
inefficiently reduce the expected output to below the one unit he has available,
resulting in an expected welfare loss equal to the area of a triangle, just as
in conventional monopoly.

 Unlike a conventional monopoly, there is a possibility of inefficient
``overproduction''  in an auction. That happens if the sale takes place even
though no bidder values the good as much as the seller: $v<v_s$ for the winning
bidder.  A positive reserve price, therefore, can help efficiency rather than
hurt it.
All the five auction forms --- first-price, second-price, descending, ascending,
and all-pay---  can be efficient in a private-value setting, but  only  if the
reserve price is set  not at the profit-maximizing level  but at $p^*= v_s$.  We
have also shown, however, that without a reserve price greater than $v_s$, none
of the five auction rules is optimal for the seller. With the addition of an
optimal   reserve price, though,  it can be shown (though we will not do so
here) that in simple settings  the seller need use no more    complicated
auction rules than one of  the five we have studied.

 In more complicated settings, of course,  things do get more complicated, and
there is the possibility of inefficiency  not just because the object is not
sold at all, but because it might be sold to the ``wrong'' bidder (that is, not
to the bidder who values it most). I have already mentioned that this can
happen in asymmetric auctions, where the bidders have values drawn from
different distributions instead of just one $F(v)$.   It could happen that for
two bidders,   $ \left( \frac{ 1-F_1(v )}{f_1(v )} \right) > \left( \frac{ 1-
F_2(v )} {f_2(v )} \right)$, in which case our rule for setting $p^*$ becomes
ambiguous-- which bidder's $F$ function should we use?  In such cases,   the
seller does best if he biases the auction rules in    favor of Bidder 2,  who
will then  sometimes win even if $v_2 < v_1$.  A similar problem can arise if
$F$ is an unusual distribution, for which higher $v$ does not imply that $(v -
\frac{ 1-F(v )}{f(v )}) $ is higher.  For more on the intricacies of such
situations, see
Myerson (1981) or   Bulow \& Roberts (1989).






\bigskip
 \noindent
 {\bf Hindering Bidder Collusion}

\noindent
  The choice of auction rule can matter for another reason which comes to mind
now that we have been discussing the seller as monopolist: the bidders may act
as
a monopsonist by colluding. If the bidders can cooperate as if they were one
bidder, we are  in the situation just described of an auction to one bidder,
where
use of a reserve price is critical and where the information-eliciting benefit
of an auction to the seller evaporates. If the bidders try to collude but remain
selfish, however, the choice of an auction rule can make the difference between
successful collusion and failure for them.
 Some auction rules are more vulnerable to  collusion than others.

  Robinson (1985)  has  pointed out that whether the auction is private-value or
common-value, the first-price  auction is superior to the second-price   or
ascending auctions for deterring collusion among bidders. (See, too,  Graham \&
Marshall [1987]).
  Consider a bidder's cartel in which bidder Smith has a private value of 20,
the
other bidders' values are each 18, and they agree that everybody will bid 5
except Smith, who will bid 6. (We will not consider the rationality of this
choice of bids, which might be based on avoiding legal penalties.) In an
ascending auction this is self- enforcing, because if somebody cheats and bids
7, Smith is willing to go all the way up to 20 and the cheater will end up with
no gain from his deviation. Enforcement is also easy in a second-price
auction, because the cartel agreement can be that Smith bids 20 and everyone
else bids 6, and if anyone cheats and bids higher than 6 he still loses unless
he bids 20 or more.

 Unlike in a ascending or second-price auction, however, in
 a first-price   auction  the   bidders have a strong temptation to cheat. The
bid $p'$ that  the colluders would choose for Smith would be lower than $p'=20$,
since he would have to pay his bid, but if  $p'$
  is  anything less than the other bidders' value of 18  any one of them could
gain by  deviating to bid more than $p'$ and win.

Thus, the seller should use a first-price auction if he fears collusion. Even
then,  cheating will be a problem if the game is repeated, and where  collusive
``bidding rings'' are  most often found is in markets where the same bidders
meet over and over in auctions for similar objects. Thus, antique auctions are
notorious for collusion among  bidders who are professional antique dealers (and
thus  are ``regulars'' at the auctions and meet repeatedly).  A similar problem
arises when the same highway  contractors repeatedly compete with each other in
bidding for who will carry out a government job    at the lowest cost.   A
similar example, described in the book by Sultan (1974),  is the Electric
Conspiracy of the 1950's, in which antitrust authorities found and prosecuted
collusion among executives  in  a few large companies  in bidding for electrical
generating equipment contracts.




\bigskip



\vspace{1in}


 \noindent
 {\bf  13.5   Common-Value Auctions and the Winner's Curse}

\noindent
  In Section 13.1 we distinguished private-value auctions from common-value
auctions, in which the bidders all have the same value for the object but
their
value  estimates   may differ.  All five  sets of rules discussed there  can be
used for common-value auctions, but the optimal strategies change. In common-
value auctions, each bidder can extract useful information about the object's
value to himself from the bids of the other bidders.  Surprisingly enough, a
bidder can use the information from other bidders' bids even in a sealed-bid
auction, as will be explained below.


    A common-value auction in which all the bidders knew the value would not be
very interesting--or very different from a private-value auction---  but more
commonly  the bidders must estimate the common value.  The obvious strategy,
especially following our discussion of private-value auctions, is for a risk-
neutral  bidder to bid up to his unbiased estimate of the value.  But this
strategy makes the winner's payoff negative, because the winner is the bidder
who
has made the largest positive error in his   estimate.  The bidders who
underestimated the number of pennies   lose the auction, but their payoff
equals zero, which they would receive even if the true value were common
knowledge.  Only the winner suffers from his estimation error: he has stumbled
into the {\bf Winner's Curse,} a phenomenon first described in Rothkopf (1969)
and  Wilson  (1969).


 When some bidders are better informed than others,  the Winner's Curse becomes
even more severe. Naturally if a bidder knows that he is the  worst informed, he
should be cautious.   Anyone, for example, who outbids   50 experts on the value
of an object to win an auction  should worry about why  all the experts bid
less. But the presence of the poorly informed bidder also  increases the danger
for the experts. Any experts who wins not just against 49 equally well-informed
experts but also against a naive bidder who might well  have made a large
overestimation error and bid too high should worry too.

Once bidders recognize the possibility of the Winner's Curse and adjust their
bidding strategies, the winner will no longer have to regret his victory.
Having adjusted by scaling down their bids to be lower than their unbiased
estimates, the  winner may still be the bidder with the biggest overestimation
error, but the winning bid can still be less than the true value. Thus, the
problem is to decide how much less than one's value estimate to bid.

  The mental process is a little like deciding how much to bid in a private-
value, first-price  auction.  There,  a bidder wants to bid less than his value,
but he  also wants to win if he can do so cheaply enough. He therefore tries to
estimate  the  value of the second-highest bid  conditional upon himself having
the highest value and winning.   In a common-value auction  a bidder's first
step
is to  use  similar mathematics, but to estimate his own value conditional upon
winning the auction, not the second-highest value.

  One way to think about a bidder's conditional estimate is to think about it as
  a conditional bid. The bidder
   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.  Intead, he would
like to submit a bid of    [{\it $X$ if I lose, but $(X-Y)$ if I win}], where
$X$ is his   value estimate conditional  on losing and $(X-Y)$ is his
estimate conditional  on winning--- a lower estimate, since winning implies his
overestimation error was the biggest of anybody's. 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  when he loses, the size of his bid is irrelevant.

Another  way to look at  the Winner's Curse  is based on   the Milgrom
definition of ``bad news'' (Milgrom [1981b], appendix b). Suppose   the
government is auctioning off the mineral rights to a plot of land with common
value $v$ and that bidder $i$ has value estimate $\hat{v}_i$. Suppose also that
the bidders are identical in everything but their value estimates, which are
based on the various information sets Nature has assigned them, and that the
equilibrium is symmetric, so the equilibrium bid function $p(\hat{v}_i)$ is the
same for each bidder. If Bidder 1 wins with a bid $p(\hat{v}_1)$ that is based
on
his prior value estimate $\hat{v}_1$, his posterior value estimate $\tilde{v}_1$
is
 \begin{equation} \label{e47c}
 \tilde{v}_1 = E(V | \hat{v}_1, p(\hat{v}_2) < p(\hat{v}_1), \ldots, p(\hat{v}
_n) < p(\hat{v}_1)).
 \end{equation}
    The news that $p(\hat{v}_2) < \infty$ would be neither good nor bad, since
it conveys no information.  The information that $p(\hat{v}_2) < p(\hat{v}_1)$,
however,  is bad news, since it rules out values of $p$ more likely to be
produced by large values of $\hat{v}_2$. In fact, the lower the winning  value
of
$p(\hat{v}_1)$, the worse is the news of having won.  Hence,
  \begin{equation} \label{e48}
\tilde{v}_1 < E(V | \hat{v}_1) = \hat{v}_1,
 \end{equation}
   and if Bidder 1 had bid $p(\hat{v}_1) = \hat{v}_1$ 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 bidders scale down.   In a second-price
auction
a bidder calculates the value of $\tilde{v}_1$ using equation (\ref{e47c}), but
that equation hides considerable complexity under the disguise of the term
$p(\hat{v}_2)$, which is itself calculated as a function of $p(\hat{v}_1)$ using
an equation like (\ref{e47c}).

\bigskip





 \noindent
 {\bf Oil Tracts and the Winner's Curse}

 \noindent
 The best known example of the Winner's Curse is from bidding for offshore oil
tracts. Offshore drilling can be unprofitable even if oil is discovered, because
something must be paid to the government for the mineral rights. Physicists
Capen, Clapp  \& Campbell   suggested in a 1971 paper in the {\it Journal of
Petroleum Engineering} that bidders' ignorance of what they termed ``the
Winner's
Curse''  caused overbidding in US government auctions of the 1960s.  If the oil
companies had bid close to what their engineers estimated the tracts were worth,
rather than scaling down their bids, the winning companies would have lost on
their investments. The hundredfold difference in the sizes of the bids in the
sealed-bid auctions shown in Table 1 lends some plausibility to the view that
this is what happened.

 \begin{center}
 {\bf Table 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}

 Later studies such as Mead, Moseidjord \& Sorason (1984) that actually looked
at profitability concluded  that the rates of return from offshore drilling were
not abnormally low, so perhaps the oil companies did scale down their bids
rationally.  The spread in bids is surprisingly wide, but that does not mean
that the bidders did not properly scale down their estimates. Although expected
profits are zero under optimal bidding, realized profits could be either
positive or negative.  With some probability, one bidder makes a large
overestimate which results in too high a bid even after rationally adjusting for
the Winner's Curse. The knowledge of how to bid optimally does not eliminate bad
luck; it only mitigates its effects.

  Another consideration is the  rationality of the other bidders.  If bidder
Apex
has figured out the Winner's Curse, but bidders Brydox and Central have not,
what
should Apex do?  Its rivals will overbid, which affects Apex's best response.
Apex should scale down its bid even further than usual, because the Winner's
Curse is intensified against overoptimistic rivals. If Apex wins against a rival
who usually overbids, Apex has very likely overestimated the value.


 Risk aversion affects bidding in a surprisingly similar way.  If all the
bidders
are equally risk averse, the bids would be lower, because the asset is a gamble,
whose value is lower for the risk averse.  If Smith is more risk averse than
Brown, then Smith should be more cautious for two reasons.  The   direct reason
is  that the gamble is worth less to Smith-- the reason analyzed above in the
private-value setting. The    indirect reason is  that when Smith wins against a
rival like Brown who regularly bids more, Smith probably overestimated the
value. Parallel reasoning holds if the bidders are risk neutral, but the private
value of the object differs among them.


In fact, Capen, Clapp, and Campbell all  worked for the oil company Arco,  and
developed a bidding strategy for oil leases that took into account the Winner's
Curse. Company executives realized that they were bidding too high, but were not
clear about whether the solution was to reduce the geological estimates, raise
the discount rate, or something else, before the three worked out their
strategy.    There was debate within the company as to whether to make the idea
of the Winner's Curse public. The idea was a private advantage for Arco-- but
other oil company's ignorance of it meant that they were bidding too high, which
may have hurt them most, but hurt Arco too. After some years, the company
decided to let them reveal the secret
(See ``The Tale of the 'Winner's Curse' '' at
http://www.aapg.org/explorer/2004/12dec/capen.cfm)



 The Winner's Curse crops up in situations seemingly far removed from auctions.
An employer should beware of hiring a worker passed over by other employers.
Someone renting an apartment should worry that he is  the first potential renter
who arrived when the neighboring trumpeter was asleep.  A firm considering a new
project should be concerned that the project has been considered and rejected by
competitors.  The Winner's Curse can even be applied to political theory;
certain proposals for innovations  keep  reappearing in the political arena over
time.  Will the first electorate to adopt them fall prey to the Winner's Curse?


   On the other hand, if information is revealed in the course of an auction,
the fact that values are common can lead to higher bidding, not lower.
 ``Getting carried away'' may be a rational feature of a common-value auction.
Suppose that the setting is not pure common value, but a mix of a common value
and private values.  If a bidder has a high private value and then learns in
the
course of the bidding that the common value is larger than he thought, he may
well end up  bidding more than he had planned  and  winning, but  he would not
regret it afterwards. Especially in these mixed situations, when bidders have
different private values and when some but not all of each bidder's information
is revealed by his bidding, common-value auctions can become very complicated
indeed.




 \bigskip
\noindent
  {\bf  Strategies in Common-Value Auctions}

Milgrom \& Weber (1982) found that when there is a common-value element in an
auction  and signals are ``affiliated''    then  revenue
equivalence fails. The first-price and descending auctions are still identical,
but they raise less revenue than the ascending or second-price auctions. If
there are more than two bidders, the ascending auction raises more revenue than
the second-price auction. (In fact, if signals are affiliated  then even in a
private value auction, in which each bidder knows his own value with certainty,
the first-price and descending auctions will do worse.)

 


   We will not prove the    ranking of revenues by auction  generally, but we
will go through an example
with estimation errors that are uniformly distributed. This example will be
tractable because of some special properties of optimal estimates when errors
are uniformly distributed, so let's start with discussion of those properties.

Suppose   $n$ signals are independently drawn from the  uniform distribution  on
$ [\underline{s},  \overline{s} ]$. Denote the $j^{th}$  highest signal by
$s_{(j)}$.
The expectation of the $k$th highest value happens to be
\begin{equation} \label{e49}
 E s_{(k)} =  \underline{s} +\left(  \frac{ n+1-k}{n+1} \right) \left(
\overline{s} - \underline{s} \right)
 \end{equation}
      This means the expectation of the very  highest value is
\begin{equation} \label{e49a}
\begin{array}{ll}
 E s_{(1)}& =
  \underline{s} +\left(  \frac{ n }{n+1} \right) \left(  \overline{s} -
\underline{s} \right) \\
\end{array}
   \end{equation}
The expectation of the
second-highest value is
 \begin{equation} \label{e50}
\begin{array}{ll}
E s_{(2)}& =   \underline{s} +\left( \frac{ n-1}{n+1} \right) \left(
\overline{s} - \underline{s} \right)\\
\end{array}
     \end{equation}
The  expectation of the lowest value, the $n$'th highest,  is
 \begin{equation} \label{e51}
\begin{array}{ll}
E s_{(n)} &=   \underline{s} +\left( \frac{  1 }{n+1} \right) \left(
\overline{s} - \underline{s} \right) . \\
\end{array}
   \end{equation}

Let $n$ risk-neutral bidders, $i= 1,2,...n$ each receive a signal $s_i$
independently drawn from the uniform distribution on $ [v- m, v+m]$, where $v$
is the true value of the object to each of them. Assume that they have ``diffuse
priors'' on $v$, which means they think any value   from $v=-\infty$ to $v=
\infty$ is equally likely and we do not need to make use of bayes's rule.
The best estimate of the value given the set of $n$ signals  is
 \begin{equation} \label{e52}
  E v | (s_1, s_2, ..., s_n) =     \frac{ s_{(n)}+ s_{(1)}}{2}.
  \end{equation}
The estimate depends only on two out of the $n$   signals---   a remarkable
property of the uniform distribution.   If there were five signals $\{6,7,7, 16,
24\}$, the expected value of the object  would be 15 (=[6+24]/2), well above the
mean  of 12 and the median of 7, because only the extremes of 6 and 24 are
useful information.  A density that had a peak, like the normal density, would
yield a different result, but here all we can tell from the data is that all
values of $v$ between  $(6+m)$ and $(24-m)$ are equally probable.

 Figure  4 illustrates why this is true.   Someone who saw just signals $s_{(n)}
$ and $s_{(1)}$ could deduce that $v$ could not be less than $(s_{(1)}- m)$ or
greater than  $(s_{(n)}+m)$. Learning the signals in between--- would be
unhelpful, because  the only information that, for example,  $s_{(2)}$
conveys is that $v  \leq   (s_{(2)}+m)$ and $v  \geq   (s_{(2)}-m)$, facts which
our observer had already figured  out  from $s_{(n)}$ and $s_{(1)}$.

\includegraphics[width=150mm]{fig13-04.jpg}


\begin{center}
 {\bf Figure  4: Extracting Information From Uniformly Distributed Signals }
 \end{center}


\bigskip

 Now let us go to the game itself, which is based on the introductory chapter of   Paul Klemperer's 2000
book. 


\begin{center}
{\bf  The Uniform-Signal  Common-Value Auction}
 \end{center}
{\bf Players:} \\
  One  seller and $n$ bidders.

\noindent
 {\bf Order of Play:}\\
0. Nature chooses  the common value for the object $v$ using the uniform density
on $[-\infty, \infty]$ (the limit of $[-x,x]$ as $x$ goes to infinity), and sends signal $s_i$  to Bidder $i$ using the uniform
distribution on $ [v- m, v+m]$.\\
1. The seller chooses a mechanism  that allocated the object and payments based
on each player's choice of $p$.   He also chooses the procedure in which bidders
select $p$ (sequentially, simultaneously, etc.). \\
  2. Each bidder  simultaneously chooses   to participate in the auction or  to
stay  out. \\
 3. The bidders  and the  seller choose value of $p$ according to the mechanism
procedure. \\
  4. The  object is allocated and transfers are paid according to the
mechanism.\\

\noindent
 {\bf Payoffs:}\\
   Payoff depends on the particular rules, but if the object is sold,  the
payoff to the seller is the sum of all the payments  and the value to a bidder
is
the value of the object, if he wins, minus his payments.



\noindent
  What are the  strategies in symmetric equilibria for the different auction
rules? (we will ignore possible  asymmetric equilibria.)

\bigskip
\noindent
 {\bf  The Ascending Auction (open-exit)  }

\noindent
{\bf  Equilibrium:}  If no bidder has quit yet,  Bidder $i$ should drop out
when
the price rises to  $s_i$. Otherwise, he should  drop out when the price rises
to $p_i = \frac{p_{(n)} +s_i}{2}$, where $p_{(n)}$ is the price at which the
first dropout occurred.

\noindent
{\bf  Explanation:} If no other bidder   has quit yet,     Bidder $i$  is safe
in
agreeing to pay his signal, $s_i$. Either  (a) he has the lowest signal, or  (b)
everybody else has  the same signal value  $s_i$ too, and they will all drop out
at the same time. In case (a), having the lowest signal,  he will lose anyway.
In case
(b),  the best estimate of the value is $s_i$, and that is where he  should drop
out.

 Once one bidder has dropped out at $p_{(n)}$, the other bidders can deduce that
he had the lowest signal, so they know  that  signal $s_{(n)}$ must equal
$p_{(n)}$.   Suppose Bidder $i$ has signal $s_i > s_{(n)}$.  Either (a) someone
else has a higher signal and  Bidder $i$ will lose the auction anyway and
dropping out too early does not matter, or (b)  everybody else  who has not yet
dropped out has signal $s_i$ too, and they will all drop out at the same time,
or (c) he  would be the last to drop out, so  he will win. In cases (b) and (c),
his estimate of the value is $p_{(i)}= \frac{p_{(n)} +s_i}{2}$, since $p_{(n)}$
and $s_i$ are the extreme signal values and the signals are uniformly
distributed, and that is where he  should drop out.

  The price paid by the winner will be the price at which the second-highest
bidder drops out, which is $\frac{s_{(n)} +s_{(2)}}{2}$. The  expected values
are, from equations  (\ref{e50}) and (\ref{e51}), 
\begin{equation} \label{e53}
\begin{array}{ll}
E s_{(n)}& =  (v-m) +\left(  \frac{ n+1-n}{n+1} \right) \left( (v+m)-  (v-m)
\right) \\
 & \\
&  =v +\left(  \frac{ 1-n}{n+1} \right)  m \\
\end{array}
 \end{equation}
  and
\begin{equation} \label{e54}
\begin{array}{ll}
 E s_{(2)} &=  (v-m) +\left(  \frac{ n+1-2}{n+1} \right) \left( (v+m) -  (v-m)
\right)\\
 &\\
 & =   v +\left(  \frac{ n-3}{n+1} \right)  m.\\
\end{array}
 \end{equation}

  Averaging them yields the expected winning price,
 \begin{equation} \label{e55}
\begin{array}{ll}
E p_{(2)} &=   \frac{\left[  v +\left(  \frac{ 1-n}{n+1} \right) m \right]  +
\left[v +\left(  \frac{ n-3}{n+1} \right)  m \right]   } {2} \\
 &\\
 & =   v -    \left( \frac{1}{2} \right)\left( \frac{  1}{n+1} \right)   2m. \\
\end{array}
\end{equation}
 If $m=50$ and $n=4$, then
\begin{equation} \label{e57a}
 E p_{(2)} =v -   \left(  \frac{1}{10}  \right) (100) = v-10 .
\end{equation}

Expected seller revenue increases in $n$, the number of bidders (and thus of
independent signals) and falls in the
 uncertainty $m$ (the inaccuracy of the signals).  This will be true for all
three auction rules we examine here.

 It is not always true that the bidders can deduce the lowest signal in an
ascending auction and use that to form their bid.    Their
ability to  discover $s_{(n)}$ depended crucially on the open-exit feature of
the auction--- that the player with the lowest signal had to openly drop out,
rather than lurk quietly in the background.   A secret-exit ascending auction
would  behave   like a second-price auction instead.

\bigskip

\noindent
 {\bf  The   Second-Price Auction   }

\noindent
{\bf  Equilibrium:}  Bid $p_i = s_i  - \left( \frac{n-2}{n}\right) m $.

\noindent
{\bf  Explanation:} In forming his strategy, Bidder $i$  should think  of
himself
as being tied for winner with one other bidder, and so having to pay  exactly
his
bid.  Thus, he imagines himself as  the highest of $(n-1)$ bidders drawn from
$[v-m, v+m]$ and tied with one other. On   average,  if this happens,
  \begin{equation} \label{e55a}
\begin{array}{ll}
 s_i &=   (v-m) +\left( \frac{([n-1]+1- 1)} {[n-1] +1}  \right)([v+m]- [v-m])\\
 & \\
 & =(v-m) + \left(\frac{n-1}{n} \right)(2m)\\
 & \\
 & =   v + \left( \frac{n-2}{n}  \right)(m).
\end{array}
 \end{equation}
   He will bid the value $v$ which solves equation (\ref{e55a}), yielding the
optimal strategy,     $p_i = s_i - \left( \frac{n-2}{n} \right) (m) $.

On average, the  second-highest bidder actually has the signal $E s_{(2)} = v
+\left(  \frac{ n-3}{n+1} \right) m$, from equation (\ref{e54}). So the expected
price, and hence the expected revenue from the auction,  is
 \begin{equation} \label{e56}
\begin{array}{ll}
 E p_{(2)}& =    [v +\left(  \frac{ n-3}{n+1} \right)  m ]- \left(\frac{n-2}{n}
\right)  (m)\\
  & \\
  &=  v + \left(  \frac{ n (n-3) - (n+1) (n-2)}{(n+1)n} \right) m, \\
  & \\
& =  v- \left(\frac{ n-1}{n}\right)\left( \frac{ 1}{n+1}\right) 2m.\\
\end{array}
 \end{equation}
 If $m=50$ and $n=4$, then
\begin{equation} \label{e57b}
 E p_{(2)} =v - \left( \frac{3}{4} \right)  \left(  \frac{1}{5}\right) (100) =
v-15.
\end{equation}

  If there are at least three bidders, expected revenue is  lower in the second-
price auction. (We  found revenue of  $(v-10)$ with $n=4$ in the
 ascending auction.) 
 If $n=2$, however,  the expected price is the same in the second-
price  and ascending auctions. Then,  $v_{(n)}= v_{(2)}$, so the winning price is based on the same information in both auctions. 




\bigskip

\noindent
 {\bf  The  First-Price Auction  }

\noindent
{\bf  Equilibrium:}   Bid $(s_i - m)   $.

\noindent
{\bf  Explanation:} This is the simplest strategy of the three, but the hardest
to derive.  Bidder $i$ bids $(s_i-z)$ for some amount $z$ that does not
depend on his signal, because given the assumption of diffuse priors, he does
not know whether his signal is a high one or a low one.   Define $T_i$ to be how
far the signal $s_i$ is above its minimum possible value, $(v-m)$, so
\begin{equation} \label{e57c}
 T_i  \equiv s_i - (v - m)
 \end{equation}
and   $s_i \equiv v - m +T_i$. Bidder $i$ has the highest signal and wins the
auction if  $T_i$ is big enough, which has probability $\left(\frac{T_i}{2m}
\right)^{n-1} $, which we will define as $G(T_i)$,  because it is the
probability that the $(n-1)$ other signals are all less than $s_i = v -m +T_i$.
He earns $v$  minus his bid of $ (s_i- z)$ if he wins, which equals $(z+m-
T_i)$.

 If, instead, Bidder $i$ deviated and bid   a small amount $\epsilon$ higher,
he
would win    with a higher probability, $G(T_i+\epsilon)$, but he would lose
$\epsilon$ whenever he would have won with the lower bid.     Using a Taylor
expansion, $G(T_i+\epsilon) \approx  G(T_i) + G'(T_i) \epsilon$, so
 \begin{equation} \label{e58}
 G(T_i+\epsilon)-  G(T_i)  \approx  (n-1) T_i^{n-2}  \left(   \frac{1  }{2m}
\right)^{n-1}  \epsilon.
 \end{equation}
  The  benefit from bidding higher is the higher probability,
$[G(T_i+\epsilon)-  G(T_i)]$ times the  winning surplus $(z+m-T_i)$. The
loss from    bidding higher is that the bidder  would pay an additional
$\epsilon$ in the $\left(   \frac{T_i }{2m} \right)^{n-1}$ cases in which he
would have won anyway. In equilibrium, he is indifferent about this
infinitesimal deviation, taking the expectation across all possible values of
his ``signal height'' $T_i$,    so
 \begin{equation} \label{e59}
\int_{T_i=0}^{2m}  \left[ \left( (n-1) T_i^{n-2}  \left(   \frac{1  } {2m}
\right)^{n-1}  \epsilon   \right)   \left( z+m-T_i  \right)   - \epsilon \left(
\frac{T_i }{2m} \right)^{n-1}  \right] dT_i =0.
 \end{equation}
 This implies that
 \begin{equation} \label{e60}
\epsilon  \left(   \frac{1 }{2m} \right)^{n-1}     \int_{T_i=0}^{2m} \left[
\left( (n-1) T_i^{n-2} \right)   \left( z+m  \right)   -  (n-1) T_i^{n-1} -
T_i^{n-1}  \right] dT_i =0.
\end{equation}
which in turn implies that
 \begin{equation} \label{e61}
{\displaystyle  \epsilon \left(   \frac{1  }{2m} \right)^{n-1}     \bigg|_{T_i=
0}
^{2m}  \left( T_i^{n-1}       \left( z+m \right)   -    T_i^n  \right) =0},
 \end{equation}
so $ (2m)^{n-1}(z+m) -(2m)^n -0+0 =0$ and  $z=m$. Bidder $i$'s optimal strategy
in the symmetric equilibrium is to bid $p_i = s_i - m$.

 The winning bid is set by the bidder with the highest signal, and that highest
signal's expected value is
\begin{equation} \label{e62}
 \begin{array}{lll} E s_{(1)}& = &   \underline{s}  + \left(\frac{ n+1-1}{n+1}
\right) \left(    \overline{s}- \underline{s}  \right)\\ & & \\ &= &  v-m +
\left(\frac{ n }{n+1}\right) \left(   (v+m)- (v-m) \right)\\ & & \\ &= &  v-m +
\left(\frac{ n }{n+1}\right) \left(2m  \right)\\
\end{array}
 \end{equation}
 The expected revenue is therefore
 \begin{equation} \label{e63}
 E p_{(1)} =       v -  \left(1 \right)  \left(  \frac{  1}{n+1} \right)2 m.
  \end{equation}
 If $m=50$ and $n=4$, then
\begin{equation} \label{e64}
E p_{(1)} =v -    \left( \frac{1}{5}  \right) (100) = v-20.
\end{equation}
 Here, the revenue is even lower than in  the second-price auction, where it was
$(v-15)$ (and the revenue is lower even if $n=2$).    


The revenue ranking is thus that the ascending open-exit auction has the highest expected revenue for the seller, the second-price auction is in the middle, and the first-price auction is lowest.   The revenue depends on    how  intensely bidders compete up the price under  each auction rule, which in turn depends on how much of an informational advantage the highest-signal-bidder has. In the ascending auction, all the bidders come to know $s_{(n)}$, and the winning price and who wins depends on $s_{(2)}$ and $s_{(1)}$, so the  bidder with the highest signal  has a relatively small advantage. In the second-price auction, the winning price and who wins depends on $s_{(2)}$ and $s_{(1)}$,  and that information comes to be known only to those two bidders. In the first-price auction, the winning price and who wins depend  only on $s_{(1)}$, so the  bidder with the highest signal  has the only relevant information. Thus, his   informational rent is greatest under that auction rule. Paradoxically, the bidders prefer an auction rule which makes it harder for them to pool their information and accurately estimate $v$.  Or perhaps this is not paradoxical. What the seller would like best would be for every bidder to truthfully announce his signal publicly, because then every bidder would have the same estimate,   that would be the amount each would bid, and the informational rent would fall to zero.  



\vspace{1in}

 \noindent
{\bf  13.6  Asymmetric Equilibria, Affiliation, and Linkage: The Wallet Game }




\noindent
 {\bf Asymmetric Equilibria in Common-Value Auctions}

Besides the symmetric equilibria I have been discussing so far, asymmetric
equilibria are  typical,  robust, and plausible in common-value auctions. That
is because the severity of the winner's curse facing   Bidder   $i$   depends
on
the bidding behavior of the other bidders. If other bidders bid aggressively,
then
if $i$ wins anyway, he must have a big overestimate of the value of the object.
So the more aggressive are the other bidders, the more conservative ought Bidder
$i$ to be--- which in turn will make the other bidders more aggressive. The
Wallet Game  of Klemperer (1998) illustrates this.



\begin{center}
 {\bf The Wallet Game }
\end{center}
\noindent
 {\bf Players }\\
Smith and Jones.

\noindent
 {\bf Order of Play} \\
 (0) Nature  chooses the amounts $s_1$ and $s_2$ of the money in each  player's
wallet  using density functions $f_1(s_1)$ and $f_2(s_2)$. Each player observes
only his own wallet's contents.  \\
 (1)  Each player chooses a bid ceiling   $p_1$ or $p_2$.   An auctioneer
auctions off the two wallets by gradually raising the price until  either $p_1$
or $p_2$ is reached.

\noindent
  {\bf Payoffs:} \\
The player who bids less has a payoff of zero. The winning player pays the bid
ceiling of the loser and hence  has a payoff
of
\begin{equation} \label{e64a}
  s_1+s_2 -  Min (p_1, p_2)
\end{equation}

A symmmetric equilibrium is for Bidder $i$ to choose bid ceiling $p_i = 2s_i$.
This is an equilibrium because if he wins at exactly that price,  Bidder $j$'s
signal must be $s_j   = s_i $ and the value of the wallets is    $2s_i$.
If  Bidder $i$ bids any lower,  he might pass up a chance to buy the wallet for
less than its value. If he bids any higher,  he would only  win  if $p>2s_j$
too, which implies that $p>s_i+s_j$.

 This   equilibrium  clearly illustrates   how bidders   should base their
strategy on the strategy they expect the other bidders  to use.  Note that
Bidder's $i$'s strategy is unrelated to his prior beliefs about Bidder $j$'s
value.  It might be, for example, that using $f_2(s_2)$, the expected value of
$s_2$ is 100, but if Bidder 1's  wallet contains   $s_1=8$, he should just bid
16.  So doing, he will probably lose, but if he bids 108 and wins, it will only
be because Bidder 2's wallet contains 54 or less. That is fine if it contains 8
or less, but if it contains, for example, $s_2=9$, then Bidder 2 will bid up to
18 and stop, Bidder 1 will win with his bid ceiling of 108, and his payoff will
be $8+9-18 = -1$.

 The Wallet Game  has both independent values and pure common values,  a
curious combination.  The value of the two
wallets is the same for both bidders, but the signal each receives is
independent. Knowing $s_1$ is useless in predicting $s_2$, though it is useful
in predicting the common value.

 The independence of the signals makes a bidder's  optimal strategy particularly
sensitive to what he thinks the other bidder's strategy is, since his own
signal tells him nothing about the other player's signal and he must rely on the
other player's bidding for any information about it.
Thus, there are many  asymmetric equilibria. 
One of them is $(p_1 = 10s_1, p_2 = \frac{10}{9} s_2$). If the two players tie,
having choosen $p= p_1=p_2$, then $10 s_1 = \frac{10}{9} s_2$, which implies
that $s_1 = \frac{1}{9} s_2$, so   $s_1 + s_2 = 10s_1= p$, and $v=p$. This is a
bad equilibrium   for  Bidder 2  because he hardly ever wins  and when he does
win it's because $s_1$ was very low--- so there is hardly any money in  Bidder
1's wallet.  Being the aggressive bidder in an equilibrium is valuable.  If
there
is a sequence of auctions, this means establishing a reputation for
aggressiveness can be worthwhile, as shown in   Bikhchandani (1988).


  Asymmetric equilibria can even arise when the players are identical. Second-
price, two-person, common-value auctions usually have many asymmetric equilibria
besides the symmetric equilibrium we have been discussing (see Milgrom [1981c]
and Bikhchandani [1988]). Suppose that Smith and Brown have identical payoff
functions, but Smith thinks Brown is going to bid aggressively.  The winner's
curse is intensified for Smith, who would probably have overestimated if he won
against an aggressive bidder like Brown, so Smith bids more cautiously.  But if
Smith bids cautiously, Brown is safe in bidding aggressively, and there is an
asymmetric equilibrium.  For this reason, acquiring a reputation for
aggressiveness is valuable.

    Oddly enough, if there are three or more bidders the   second-price, common-
value auction has a unique equilibrium, which is also symmetric. The open-exit
ascending auction is different: it has asymmetric equilibria, because after one
bidder drops out, the two remaining bidders know that they are alone together in
a
subgame which is a two-bidder auction.  Regardless of the number of bidders,
first-price   auctions do not have this kind of asymmetric equilibrium. Threats
in a first-price auction are costly because the high bidder pays his bid even if
his rival decides to bid less in response. Thus, a bidder's aggressiveness is
not
made safer by intimidation of another bidder.




 \bigskip

\noindent
{\bf  Affiliation, the Monotone Likelihood-Ratio Property, and the Linkage
Principle}\footnote{Much of this discussion is based on Cramton (undated),
Klemperer (2004, p. 51) and
Wolfstetter (1999).}


Milgrom \& Weber (1982) introduced the   idea of {\bf affiliation}: a formal
definition of   two variables tending to move upwards together that is useful in
the auction context. Suppose bidder $1$ has a value $v_1$ which is a increasing
function of a private signal $x_1$ that he receives,  and  which might also
depend on the private signal of  bidder  2, $x_2$. What we need to know to
analyze the auction is what happens to bidder $1$'s estimate of his value as he
observes or deduces  more about  bidder 2's signal. A simple and plausible
situation is that whenever he  learns that    $x_2$ takes a large value, his
estimate of his    value $v_1$ rises. Thus, if he observes bidder 2 bid more and
he deduces that $x_2$ is large, he should increase his estimate of $v_1$.
Affiliation is defined so this will happen if the signals $x_1$ and $x_2$    are
strongly affiliated, and if they are weakly affiliated, bidder 1's estimate of
$v_1$ at least will not fall as $x_2$ rises.  For simplicity, I will define
affiliation for the case of two signals.

 
\noindent
 {\it Definition.} The signals $x_1$ and $x_2$ are  affiliated  if for all
possible realizations  $Small < Big$ of $x_1$  and $Low < High$ of $x_2$, the
joint probability  $f(x_1, x_2)$ is such that
$z_1$ and $z_2$,
 \begin{equation} \label{e65}
 f(x_1=Small, x_2=Low) f(x_1=Big, x_2=High) \geq  f(x_1=Small, x_2=High)f(x_1=
Big, x_2=Low).
\end{equation}
   Thus, affiliation says that  the probability  the values of $x_1$ and $x_2$
move in the same direction is
greater than the probability   they move oppositely. Notice that this
allows a  joint probability distribution  such as  the following, in which even
if  $x_2$ is $Low$, $x_1$
is probably $Big$.

\noindent
$Prob (x_1=Big, x_2=High)=0.5$\\
$  Prob (x_1=Small, x_2=Low) =0.1$\\
 $Prob (x_1=Big, x_2=Low) =0.2$\\
 $Prob (x_1=Small, x_2=High) =0.2$\\

 Imagine that  Bidder 3, who is ignorant of both $x_1$ and $x_2$,  
 can deduce  about $x_1$ if he learns that $x_2=Low$. Bidder 3's prior   is
 \begin{equation} \label{e67}
\begin{array}{ll}
 Prob (x_1=Big)  & =  Prob (x_1=Big, x_2=High)  +Prob (x_1=Big, x_2=Low)\\
 &\\
 & =  0.5 + 0.2 = 0.7.   \\
\end{array}
\end{equation}

 Suppose Bidder 3 finds out that $x_2$ is Low. We will use the same kind of
manipulation as in bayes's rule, though not bayes's rule itself. Since
 \begin{equation} \label{e68}
Prob (x_1=Big, x_2=Low)= Prob (x_1=Bigh|x_2=Low)    Prob (x_2=Low),
\end{equation}
we can write Bidder 3's posterior as
\begin{equation} \label{e69}
\begin{array}{ll}
 Prob (x_1=Big|x_2=Low) &= \frac{Prob (x_1=Big, x_2=Low)}{  Prob (x_2=Low) }\\
& \\
& = \frac{0.2}{  Prob (x_1=Big, x_2=Low) +  Prob (x_1=Small, x_2=Low) }\\
 & \\
 &= \frac{0.2}{  0.2 +  0.1 } \\
 \end{array}
\end{equation}
  Thus, observing $x_2=Low$ leaves  Bidder 3 still thinking that probably $x_1=
Big$, but  the  probability   has fallen from 0.7 to 0.66.  $x_2=Low$ is   bad
news about the value of $x_1$. 

 The big implication of two signals being affiliated is that the expected value
of the winning bid conditional on the signals is  increasing in all the signals.
When one signal rises, that has the positive direct effect of increasing the
bid of the player who sees it, and non-negative indirect effects  once the other
players see his bid increase and deduce that he had a high signal.

  This is very much like the Monotone Likelihood-Ratio Property, which is the
same thing  expressed in terms of the conditional densities, the posteriors.


 \noindent
 {\it Definition.} The conditional probability $g(x_1|x_2)$ satisfies the
Monotone
Likelihood Ratio Property  if  the likelihood ratio is weakly decreasing in
$x_1$, that is,  for all  possible realizations  $Small < Big$ of $x_1$  and
$Low < High$ of $x_2$,
 \begin{equation} \label{e70}
   \frac{ g(Big| Low)} {g(Big| High)}  \leq  \frac{ g(Small| Low)} {g(Small|
High)}.
\end{equation}

 The Monotone Likelihood Ratio Property says that as $x_2$  goes from Low to
High,  the   Big value of $x_1$ becomes relatively more likely. It can be shown
that this implies that for any value $z$,    the conditional cumulative
distribution of $x_1$ up to $x_1=z$ given $x_2$ weakly increases with $x_2$,
which is to say that the distribution $G(x_1|x_2)$  conditional on a larger
value of $x_2$ stochastically dominates the  distribution conditional  on a
smaller value of $x_2$.


 This no doubt leaves  the reader's head spinning quite as much as it does the
author's, despite my attempt at simplification. A final, equivalent definition
of affiliation, applicable when   the signals are distributed according to a
joint density $f(x_1, x_2)$  that is continuous
and  twice differentiable  is that    $x_1$ and $x_2$  are
affiliated if  
\begin{equation} \label{e65a}
  \frac{ \partial  log(f)^2}{\partial x_1 \partial x_2} \geq 0.
\end{equation}
    
    
   \bigskip


 One of the rewards of establishing that bidders' signals are affiliated is 
  {\bf  the linkage principle}, which  says that when the amount of   affiliated
information available to bidders increases, the equilibrium sales price  becomes
greater.  Thus, the seller should have a policy of disclosing    any affiliated
information  he possesses. Also, auction rules which reveal affiliated
information in the course of the auction (e.g., open-exit auctions)  or use it
in determining the winner's payment  (e.g., the second-price auction) will
result in
higher prices. We saw  this in the Uniform-Signal Common-Value
Auction examples.   The result is not restricted to common-value auctions,
however; seller revenue rises when affiliated information is released even in an
affiliated private value auction, in which the bidders each know their own
private values but not those of other bidders. The intuition behind the linkage
principle is   hard to grasp, because it applies even when there is no winner's
curse (as in the affiliated private value auction), and it does not always apply
in common value auctions (when there are just two bidders, the ascending auction
is not superior to the second-price auction, as we saw above). The best
intuition I have seen is that on page 128 of Klemperer (2004): that bidder
profits arise from their private information, and release of affiliated
information reveals something about the private information of each bidder,
including, especially, the one who will win, and thus heightens competition.  

   
   


 The linkage principle   provides a reason why sellers may wish to extend
auctions
over time in multiple rounds, why they should encourage active bidding
throughout rather than let bidders  ``lurk'' and suddenly bid near the end,
why they should lay out their own information as clearly and early as possible,
and why they should let bidders know each others' identities.  The idea is a
slippery one, howevever,   and
Perry  \&     Reny (1999) show that the linkage principle actually can fail if
more than one unit of the good is being auctioned, so players submit bids for
the possible purchase of multiple units.  Multiple-unit auctions and   ``package
auctions,'' in which not just multiple units but multiple objects are sold  in a
single auction, are active areas of research.





 \newpage

\begin{small}

\noindent
 {\bf Notes}

\noindent
{\bf N13.1} {\bf  Values Private and Common, Continuous and Discrete}
 \begin{itemize}
 \item
 Milgrom \& Weber (1982) is a classic article that covers many aspects of
auctions.
 McAfee \& McMillan (1987) is an excellent older  survey  of  
 auction theory  which takes some pains to relate it  to models of
asymmetric information. More recent is  Maskin (2004).
Klemperer (2000) collects many of the most important articles in an edited
volume.    Vijay Krishna's 2002  {\it Auction Theory}, Paul Klemperer's  2004
{\it Auctions: Theory and Practice} (which is relatively nontechnical), and
Paul Milgrom's 2004 {\it  Putting Auction Theory to Work} (which is very good on
the mathematical assumptions)  are good textbook treatments. I particularly like
the auction chapters in Elmar Wolfstetter's 1999 {\it Topics in Microeconomics:
Industrial Organization, Auctions, and Incentives}.
  Paul Milgrom's consulting firm, Agora Market Design, has a website with many
good working papers that can be found via http:\\www.market-design.com.


\item
Cassady (1967) is an excellent source of pre-Web institutional detail. The
appendix to his book includes advertisements and sets of auction rules, and he
cites numerous newspaper articles. The  rise of the Web, an ideal setting for
auctions, fortuitously occurred at the same time as the rise of auction theory.
See  Bajari  \&  Hortacsu's 2004   survey,
``Economic Insights from Internet Auctions.''


\end{itemize}




 \noindent
{\bf N13.2 Optimal Strategies under Different   Rules  in Private-Value Auctions
}
\begin{itemize}
\item
  Many (all?) leading auction theorists were involved in the   seven-billion
dollar spectrum auction by the United States government in 1994, either helping
the government choose an auction rule to  sell spectrum or helping bidders
decide
how to buy it.   Paul Milgrom's 1999 book,  {\it   Auction Theory for
Privatization}, tells the story. See also McAfee \& McMillan (1996).
Interesting institutional details have come in the spectrum auctions and
stimulated new theoretical research.     Ayres \& Cramton (1996), for example,
explore the possibility that affirmative action provisions designed to help
certain groups of bidders  may have actually increased the revenue raised by the
seller by increasing the amount of competition in the auction.

 \item
One might think that an ascending second-price, open-cry auction would come to
the same results as an ascending  first-price, open-cry auction, because if the
price advances by $\epsilon$ at each bid, the first and second bids are
practically the same. But the second-price auction can be manipulated. If
somebody initially bids \$10  for something worth  \$80, another bidder could
safely bid \$1,000. No one else would bid more, and he would pay only the second
price: \$10.





\item Auctions are especially suitable for empirical study because  they are so
stylized and generate masses of data. Hendricks \& Porter (1988) is a classic
comparison of auction theory with data.   See  
 Bajari,   Hong \&  Ryan (2004)  or the Athey \& 
  Haile (2005) and Hendricks \& Porter (forthcoming)  surveys  of empirical work
on auctions.





\item After the last bid of an open-cry art auction in France, the
representative of the Louvre has the right to raise his hand and shout ``pre-
emption de l'etat,'' after which he takes the painting at the highest price bid
({\it The Economist}, May 23, 1987,  p. 98). How does that affect the
equilibrium strategies? What would happen if the Louvre could resell?

 \newpage

\item
 {\bf Share Auctions.} In a share auction each bidder submits a bid for both a
quantity and a price.  The bidder with the highest price receives the quantity
for which he bid at that price.  If any of the product being auctioned remains,
the bidder with the second-highest price takes the quantity he bid for, and so
forth. The rules of a share auction can allow each bidder to submit several
bids,
often called a {\bf schedule} of bids. The details of share auctions vary, and
they can be either first-price or second-price. Modelling is    complicated; see
Wilson (1979).

\end{itemize}

 \noindent
 {\bf N13.3  Revenue Equivalence,  Risk Aversion, and Uncertainty}
 \begin{itemize}
\item
 Che \& Gale (1998) point out  that  if bidders differ in their willingness to
pay in a private value auction because of budget constraints rather than tastes
then the revenue equivalence theorem can fail.  The following example from page
2 of their paper shows this.  Suppose   two budget-constrained bidders are
bidding for one object.  In Auction 1,  each bidder has a budget of 2 and  knows
only his own value, which is   drawn  uniformly from [0,1].    The budget
constraints are never binding, and it turns out  that  the expected price is 1/3
under either a first-price or a second-price auction. In Auction 2, however,
each bidder  knows only his own budget, which is    drawn uniformly from [0,1],
and both have values for the object  of  2.    The budget constraint  is always
binding, and  the equilibrium  strategy   is to bid one's entire budget under
either set of auction rules.  The expected price is  still 1/3 in  the second-
price auction, but now it is 2/3 in the first-price auction.  The seller
therefore  prefers to use a first-price auction.

\item
{\bf  A Mechanism To Extract All the Surplus. }
Myerson (1981) shows that if the bidders' private information is correlated, the
seller can  construct something akin to a   Maskin matching scheme mechanism of
the kind discussed in Chapter 10 that extracts all the information and all the
surplus. In the Uniform-Signal Common-Value Auction, where signals are uniform
in $[v-m, v+m]$
 ask  Bidder $i$  to bid  $s_i$,   allocate the good to the high bidder at the
price $\frac{s_{(1)} + s_{(n)}}{2}$, which is an unbiased estimate of $v$, and
ensure truthtelling by the boiling-in-oil punishment of   a large negative
payment   if the reports are such that $ s_{(n)}   <   s_{(1)}-m$, which cannot
possibly occur if all bidders tell the truth.

\end{itemize}

\newpage

 \noindent
{\bf N13.4} {\bf   Common-Value Auctions and the Winner's Curse }
\begin{itemize}

\item[]
    Rothkopf (1969) and  Wilson (1969) 
seem  to the be first published accounts of the Winner's Curse.   An article on
Edward Capen,  ``The
Tale of the 'Winner's Curse' '' at
http://www.aapg.org/explorer/2004/12dec/capen.cfm  says that the term   was
first
published in Capen, Clapp \& Campbell (1971). I recommend the article for its
tale of the use  of theory in business practice.

 \item
  The Winner's Curse  and the idea of common values versus private values  have
broad application. The Winner's Curse is related to the idea of ``regression to
the mean''  discussed in Section  2.4 of this book. Kaplow \& Shavell (1996)
use the idea to discuss property versus liability rules, one of the standard
rule choices in law-and-economics.    If someone violates a property rule, the
aggrieved party can undo the violation, as when a thief is required to surrender
stolen property. If someone violates a liability rule, the aggrieved party can
only get monetary compensation, as when  someone who   breaches a contract is
required to pay damages to  the aggrieved party.  Kaplow and Shavell argue that
if a good has independent values, a  liability rule is best because it gives
efficient incentives for rule violation; but if it has common value and courts
make errors in measuring the common value, a  property rule may be better.  See
especially   page 761 of their article.

 \end{itemize}


 \noindent
{\bf N13.6  Asymmetric Equilibria, Affiliation, and Linkage: The Wallet Game }
\begin{itemize}
 \item
Even if value estimates are correlated, the optimal bidding strategies can still
be the same as in private-value auctions if the values are independent. If
everyone overestimates their values by 10 percent, a bidder can still extract no
information about his value by seeing other bidders' value estimates.


\end{itemize}

\newpage

\noindent
{\bf Problems}

 \bigskip \noindent
 {\bf 13.1.  Rent Seeking} (medium)\\
  Two risk-neutral neighbors in sixteenth century England, Smith and Jones, have
gone to court and are  considering bribing a judge.  Each of them makes a gift,
and the one whose gift is the largest is awarded property worth \pounds 2,000.
If  both bribe the same amount, the chances are 50 percent for each of them to
win the lawsuit. Gifts must be either \pounds 0, \pounds 900, or \pounds 2,000.

    \begin{enumerate}

    \item[(a)]
 What is the unique pure-strategy equilibrium for this game?

    \item[(b)]  Suppose that it is also possible to give a \pounds 1500 gift.
Why does there no longer exist a pure-strategy equilibrium?

    \item[(c)]  What is the symmetric mixed-strategy equilibrium for the
expanded game?  What is the judge's expected payoff?

     \item[(d)]  In the expanded game, if the losing litigant gets back his
gift, what are the two equilibria? Would the judge prefer this rule?

  \end{enumerate}

\bigskip \noindent
 {\bf 13.2. The Founding of Hong Kong} (medium) \\
 The Tai-Pan and Mr. Brock are bidding in an ascending auction for a parcel of
land on a knoll in Hong Kong. They must bid integer values, and the Tai-Pan bids
first. Tying bids cannot be made, and bids cannot be withdrawn once they are
made. The direct value of the land is 1 to Brock and 2 to the Tai-Pan, but the
Tai-Pan has said publicly that he wants it, so if Brock gets it, he receives 5
in ``face'' and the Tai-Pan loses 10.  Moreover, Brock hates the Tai-Pan and
receives 1 in utility for each 1 that the Tai-Pan pays out to get the land.

\begin{enumerate}
 \item[(a)]
 First suppose there were no ``face'' or ``hate'' considerations, just the
direct values. What are the equilibria if the Tai-pan bids first?

  \item[(b)]
 Continue supposing    there were no ``face'' or ``hate'' considerations, just
the direct values. What are the  three possible equilibria if  Mr. Brock bids
first?   (Hint: in one of them, Brock wins; in the other two, the Tai-pan wins.)

    \item[(c)]
 Fill in the entries in Table  2, including the ``face'' and ``hate''
considerations.

 \begin{center} {\bf Table  2:   The Tai-Pan Game}

\begin{tabular}{|l |rrr rrr rrr rr r |}
\hline Winning bid: & 1 & 2 & 3 & 4 & 5 & 6 & 7    & 8 & 9 & 10 & 11 & 12\\
\hline If Brock wins: &  &  &  &  &  &  &        &   &   &   &   & \\
$\pi_{Brock}$  &   & &  &  &  &  &         &   &   &   &   & \\ $\pi_{Tai-Pan }
$ &  &  & &  &  &  &        &   &   &   &   &  \\ \hline If Brock loses:  &  &
&  &  &  &  &        &   &   &   &   &  \\ $\pi_{Brock}$ &  & &  &  &  & &
&   &   &   &   &  \\
 $\pi_{Tai-Pan } $ & & &  &  & &  &        &   &   &   &   &  \\ \hline
\end{tabular}
 \end{center}

       \item[(d)]
In  equilibrium, who wins,  and at what bid?

       \item[(e)]
What happens if the Tai-Pan can precommit to a strategy?

       \item[(f)]
What happens if the Tai-Pan cannot precommit, but he also hates Brock, and gets
1 in utility for each 1 that Brock pays out to get the land?

  \end{enumerate}

\bigskip

 \noindent
 {\bf 13.3. Government and Monopoly} (medium) \\
 Incumbent Apex and potential entrant Brydox are bidding for government favors
in the widget market.  Apex wants to defeat a bill that would require it to
share its widget patent rights with Brydox.  Brydox wants the bill to pass.
Whoever offers the chairman of the House Telecommunications Committee more
campaign contributions wins, and the loser pays nothing.  The market demand
curve for widgets is $P = 25-Q$, and marginal cost is constant at 1.

     \begin{enumerate}

     \item[(a)] Who will bid higher if duopolists follow Bertrand behavior?  How
much will the winner bid?

     \item[(b)] Who will bid higher if duopolists follow Cournot behavior? How
much will the winner bid?

     \item[(c)] What happens under Cournot behavior if Apex can commit to giving
away its patent freely to everyone in the world if the entry bill passes?  How
much will Apex bid?

  \end{enumerate}

\bigskip
  \noindent
  {\bf  13.4.  An Auction with Stupid Bidders } (hard)\\
 Smith's value for an object has a private component equal to 1 and another
component $Z$ that is
common with Jones and Brown.  Jones's and Brown's private components both equal
zero. Each bidder estimates   $Z$ independently. Bidder
$i$'s estimate is either $x_i$ above the true value or $x_i$ below, with equal
probability.  Jones and Brown are naive and always bid their value estimates.
The auction is ascending. Smith knows all three values of $x_i$, but not whether
his estimate is too
high or too low.

 \begin{enumerate}

 \item[(a)] If $x_{Smith}=0$, what is Smith's dominant strategy if his estimate
of $Z$ is 20?

   \item[(b)] If $x_i=8$ for all bidders and Smith estimates that $Z = 20$, what
are
the probabilities that he puts on different possible values of $Z$?

   \item[(c)] If $x_i =8$  for Jones and Brown   but $x_{Smith}=0$,  and  Smith
knows that $Z=12$ with certainty, what are
the probabilities he puts on the different combinations of bids by Jones and
Brown?

\item[(d)] Why is 9 a better upper limit on bids for Smith than 21, if his
estimate of $Z$ is 20, and $x_i = 8$ for all three bidders?

 \item[(e)] Suppose Smith could pay amount 0.001 to explain optimal bidding
strategy to his rival bidders, Jones and Brown. Would he do so?

 \end{enumerate}

\newpage

\bigskip
\noindent
 {\bf  13.5.  A Teapot  Auction with Incomplete Information } (easy)\\
 Smith believes that Brown's value $v_b$ for a teapot  being sold at auction is
0 or 100 with equal probability.  Smith's value of $v_s=400$ is known by both
bidders.

 \begin{enumerate}

 \item[(a)] What are the bidders' equilibrium strategies in an open cry auction?
You may assume that in case of ties, Smith wins the auction.

   \item[(b)] What are the bidders' equilibrium strategies in a first-price
sealed-bid  auction? You may assume that in case of ties, Smith wins the
auction.

   \item[(c)] Now let $v_s =102$ instead of 400.  Will Smith use a pure
strategy? Will Brown? You need not find the exact strategies used.

 \end{enumerate}



 \newpage

\begin{center}
 {\bf  Auctions:  A Classroom Game for  Chapter 13}
 \end{center}


 The instructor will  bring to class   a glass jar of pennies to be auctioned
off. There are between 0 and 100 pennies in the jar, the number being drawn from
a uniform distribution. Twenty percent of the  students in the class  must close
their eyes while the jar is being displayed. The instructor will pass the jar
around to the other students in the class to let them try to figure out how many
pennies are inside. He will then auction off the jar five times, using five
different sets of rules. (The instructor will decide for himself whether to play
for real money or not.)

   The first auction will be  a first-price auction.
 Each student submits a bid, and also record his estimate of  the number of
pennies.

 The second auction will be  a  second-price auction.
 Each student submits a bid, and also record his estimate of  the number of
pennies.

The third auction will be   an all-pay auction.
  Each student submits a bid, and also record his estimate of  the number of
pennies.

 The fourth auction will be  a descending auction.
 After the auction, each student submits his  estimate of  the number of
pennies.


 The fifth auction  will be an ascending auction.
  After the auction, each student submits his  estimate of  the number of
pennies.


  At this point, the instructor will announce the number of pennies in the jar,
and the results of each auction.



\end{small}

\end{document}
