Notes on R. Preston McAfee and J. McMillan
"Auctions and Bidding"
Prepared by L. Tesfatsion

Prepared by: Leigh Tesfatsion
Date:        10 October 1999
For:         Electricity Project Meeting, October 11

                  NOTES ON FIRST PART OF

  R. Preston McAfee and J. McMillan, "Auctions and Bidding," Journal
        of Economic Literature XXV (June 1987), 699-738.

The notes below summarize, and in places comment upon, Sections I-VII
(pages 699--716) of the McAfee/McMillan article on auctions and bidding.


I. WHY STUDY AUCTIONS (pages 699-701) and

Some of the most exciting of the recent advances in microeconomic
theory have been in the modeling of strategic behavior under
asymmetric information.

   Comment:  STRATEGIC BEHAVIOR occurs if one agent recognizes
             that the utility (net benefits) he will receive from
             undertaking various actions depend on the actions
             undertaken by other agents as well.  Consequently, it
             is to the agent's possible benefit to either estimate
             what the actions of these other agents will be, or
             even to attempt to influence what the actions of these
             other agents will be.
                 Consequently, strategic behavior can have both
             backward-looking and forward-looking (anticipatory) aspects.
             Backward: If the other agent does this, what should I do?
             Forward:  If I do this, what will the other agent do?
                 ASYMMETRIC INFORMATION means that different agents
             condition their expectations and actions on different
             information sets.  For example, I may know my own true
             valuation for an object up for sale but not the true
             valuation that some other potential buyer places on
             the object.

An AUCTION is a market institution with an explicit set of rules
determining resource alloation and prices on the basis of bids
to buy and/or offers to sell from the market participants.

The study of auctions provides one way of approaching the question
of price formation in markets in which market participants have
asymmetric information and behave strategically on the basis of
their information.  A more practical reason for studying auctions
is that auctions are of considerable empirical importance.

      REAL-WORLD EXAMPLES: (pp. 701--702)

          Artwork, books, antiques, agricultural produce,
          mineral rights, U.S. Treasury bills, corporations,
          and gold are examples of items sold by auction.

          Government procurement -- for many government contracts,
          firms submit sealed bids, and the contract is required
          by law to be submited to the highest bidder.

          Procurement by firms of inputs from other firms through
          sealed-bid tenders.

          Auctioning of rights to import (with an import quota
          providing an upper bound on these rights to import).

          Auctioning of rights to produce noxious wastes (with an
          overall limit on the total amount of wastes that can
          be emitted providing an upper bound on these rights).

          Auctioning of airport time slots to competing airlines.

          Auctioning of frequency bandwidths for airwave transmission.

Why are auctions used rather than other selling devices such as
posting a fixed price?  Main reason given:

    Some products have no standard value, so that the value of
    the product has to be discovered through a bidding process.

DOUBLE AUCTIONS are auctions in which several buyers and sellers
submit bids and offers simulataneously.  In contrast, for a
ONE-SIDED AUCTION, either bids are solicited from buyers by a
monopolist seller for items to be sold, or offers are solicited
from sellers by a monopsonist buyer for items to be bought.


     What is the best form of auction for a monopolist seller to use
     in any particular set of circumstances?

     Should the monopolist seller impose a reserve price?  If so, at
     what level?

     Can the monopolist seller design the auction so as to achieve
     price discrimination among the bidders (thus enabling the
     seller to extract more surplus)?

     Is it ever in the monopolist seller's interest to require
     payment from unsuccessful bidders (as well as the successful

     Is it both feasible and desirable to make payment depend on
     something other than buyer bids?  In particular, on something
     correlated with the true value of the item as perceived by
     the successful bidder (e.g., use royalties)?

     Should the monopolist seller release any information he has
     about the item's "true" value?

     What can the monopolist seller do to counter collusion among
     the bidders?


The ENGLISH AUCTION is the auction form most commonly used for the
selling of goods.  In the English auction, the price is successively
raised until only one bidder remains.  This can be done by having an
auctioneer announce prices, or by having biddersf call the bids
themselves, or by having bids submitted electronically with the
current best bid posted.  The essential feature of the English
auction is that, at any point in time, each bidder knows the level
of the current best bid.  The final remaining bidder receives the
item in return for paying the current best bid for the item.

The DUTCH auction is the converse of the English auction.  The
auctioneer calls an initial high price and then successively lowers
the price until a bidder accepts the current price.

With the FIRST-PRICE SEALED-BID AUCTION, each potential buyer
simultaneously submits a single sealed bid for the item, and the
highest bidder is awarded the item for the price he bid.

potential buyer simultaneously submits a single sealed bid for the
item, and the highest bidder is awarded the item but pays a price
for the item equal not to his own bid but rather to the second-highest
bid received.  [In general, this gives each bidder an incentive to bid
his true valuation for the item; for his bid only determines
whether or not he wins the bid, not what he pays to the seller.]

In practice, many variations of these four basic auction forms are
in use.  For example, the seller sometimes imposes a RESERVE PRICE,
discarding all bids if they are too low.  Bidders may be allowed
only a LIMITED TIME for submitting bids.  The auctioneer may charge
bidders an ENTRY FEE for the right to bid.  Payment may be made to
depend not only on bids but also on something correlated with the
true value of the item, e.g., ROYALTIES.  In an English auction, the
auctioneer may set a MINIMUM ACCEPTABLE INCREMENT to the highest
existing bid.  The seller, instead of selling the item as a unit,
may offer for sale SHARES in the item.


    In theoretical studies of single-sided auctions organized by
monopolist sellers, bargaining problems are typically side-stepped
by presuming that the monopolist seller has all of the bargaining
power.  More precisely, it is assumed that the organizer of the
auction has the ability to commit himself in advance to a set of

    The advantage of commitment is that procedures can be adopted
that induce the bidders to bid in ways deemed desirable by the
auction organizer (the monopolist seller).

    There are several ways in which an auction organizer can achieve

    publicly available book of rules;

    pledge of his own reputation -- the cost of reneging on a
    current commitment might be the inability to commit oneself
    credibly in future transactions, and hence the loss of future
    bargaining power.

Nevertheless, it does not follow from the fact that one party has
the ability to make commitments that he can extract all of the
gains from trade.  What limits his bargaining power is the asymmetry
of information.

The seller can exploit competition among the bidders to drive up the
price; but usually the seller will not be able to drive the price up
so far as to equal the valuation of the bidder who values the item
the most, because the seller does not know what this valuation is.


     Asymmetry of information is the crucial element of an auction
problem.  Asymmetry of information means that both the seller and
the bidders are uncertain about how other agents in the auction
process will behave.

     How the seller and bidders respond to uncertainty depends on
their attitude towards risk.  IT IS ASSUMED THROUGHOUT THE REMAINDER

     COMMENT:  An agent is RISK NEUTRAL if he is indifferent between
      participating or not participating in "fair lotteries," i.e.,
      in lotteries which have zero expected return.  [In this case
      the utility function of the agent is linear.]
            An agent is RISK AVERSE if the utility he attaches to
      the expected return of a lottery, received as a lump sum payment,
      is STRICTLY GREATER than the utility he attaches to actual
      participation in the lottery.  [In this case the agent would
      be willing to pay a small "premium" to avoid participation in
      the lottery, i.e., to avoid risk, and his utility function is
      strictly concave.]
            An agent is RISK LOVING if the utility he attached to
      expected return of a lottery, received as a lump sum payment,
      is STRICTLY LESS than the utility he attaches to actual
      participation in the lottery.  [In this case the agent would
      be willing to pay a small "premium" in order to participate in
      the lottery, i.e., to take on risk, and his utility function
      is strictly convex.]

Differences among the bidders' valuations of an item can arise from
uncertainty for a variety of reasons.  Consider the following two
polar cases (p.705):


       At one extreme, suppose each bidder knows precisely how
       highly he values the item.  However, he does not know
       anyone else's valuation, and he perceives any other
       bidder i's valuation as a draw from some probability
       distribution F_i (and "knows" this is how other bidders
       perceive his own valuation).


       At the other extreme, suppose the item being bid for has a
       single objective value V -- namely, the amount the item is
       worth on the market.  However, no one knows this true value V.
       Rather, each bidder i has a perceived value v_i for the item
       modelled as an independent draw from some conditional
       distribution H(v_i|V).  All agents know the general form H(.|.)
       of the conditional distribution function, but they do not
       observe V.

A further choice to be made by the modeller of an auction regarding
sources of uncertainty for market participants is the extent to
which bidders are recognizably different from each other.  The
following two cases will be considered below (p. 706):


        In the independent-private values model, each agent i is
        modelled as drawing his valuation v_i from a DIFFERENT
        probability distribution F_i, known to all agents.


        In the independent-private values model, each agent i is
        modelled as drawing his valuation v_i from the SAME
        probability distribution F, known to all agents.

Finally, another modelling consideration arising from uncertainty is
that the amount of payment can only be made contingent upon
variables that are observable to both buyer and seller.  For
example, in mineral-rights auctions, royalties make the payment
depend upon the amount of oil extracted as well as the winning bid.

Consequently, an auction organizer needs to consider the
feasibility of such contingent payments (whether commonly observed
variables exist) as well as the desirability of such contingent


    A1.  There are a finite number n of bidders, and each bidder is
         risk neutral (together with the monopolistic seller).

    A2.  The independent-private values assumption applies.  That
         is, the item valuation v_i of bidder i is an independent
         drawing from a distribution F_i, i = 1,...,n.

    A3.  The bidders are symmetric.  That is, in A2, F_i = F for all i.

    A4.  Payments by bidders to the seller are a function only of bids.

The analysis below starts with the benchmark model and then examines
various weakenings of the benchmark model assumptions.  The
following assumptions are maintained throughout this analysis.


    (a) Each of the n bidders knows the rules of the auction that
    the seller has chosen and committed himself to.

    (b) Bidder i knows his own valuation v_i, i = 1,...,n .

    (c) Each bidder i is assumed to know the number of bidders n, their
    risk attitudes, and the probability distributions of valuations.

    (d) Each bidder i knows that every other bidder j knows that he
    knows the auction aspects listed in (a)-(c), and that everyone else
    knows that bidder i knows that everyone else knows this, and so on (the
    so-called "common knowledge" assumption).

         COMMENT:  Assumption (d) implies each agent knows the form
         of each other agent's information set, but not the precise
         values of the elements of this information set -- e.g., each
         bidder j knows that bidder i knows his own valuation v_i, but
         bidder j may not know what precise value v_i takes on.

    (e) The notion of "equilibrium" that all agents are assumed to
    use is "Bayes-Nash equilibrium."

     COMMENT: Roughly defined, a STRATEGY for a particular player i
participating in a game is a rule (contingency plan) that specifies what
action player i should take in each situation he might feasibly encounter.  A
configuration (s_1,...,s_n) of strategies for players 1,...,n in an n-player
game is called a NASH EQUILIBRIUM if for each given i, and given s_j for all
j not equal to i, player i has no incentive to deviate from the strategy s_i.
     Consider an auction in which n bidders are bidding for a single item
from a monopolist seller under the maintained assumptions (a) through (d),
above.  A BIDDING STRATEGY for this game is a rule that specifies a feasible
bid b_i for bidder i for each possible valuation v_i that bidder i might have,
conditional on his beliefs regarding the valuation of each other bidder j.  A
BAYES-NASH EQUILIBRIUM for this game is then a configuration (B_1,...,B_n) of
bidding strategies for bidders 1,...,n that satisfies the following two
properties: (i) (B_1,...,B_n) is a Nash equilibrium; and (ii) all bidder
beliefs about the valuations of other bidders are correct.
     See, for example, Robert Gibbons, "An Introduction to Applicable Game
Theory," _Journal of Economic Perspectives_ 22 (Winter 1997), pages 127--149.


     Which of the four simple auction types (English, Dutch,
first-price sealed bid, second-price sealed bid) should the
monopolistic seller choose when organizing his auction?

     One result can be mentioned immediately: THE DUTCH AUCTION
is because the situation facing a bidder is exactly the same in each
auction:  The bidder must choose how high to bid without knowing the
other bidders' decisions; if he wins, he pays a price equal to his
own bid.  Because of this, the Dutch auction does not need to be
separately considered from the first-price sealed bid auction.

     More interesting is the following famous revenue-equivalence
theorem.  For a proof of this theorem, see pages 707--710.


     Under the maintained assumptions (a)-(e) and the benchmark
model assumptions A1-A4, the **expected** revenue of the
monopolistic seller under the English auction, the Dutch auction,
the first-price sealed bid auction, and the second-price sealed bid
auction are precisely the same.


     COMMENT 1 (page 710):  It is important to note that the revenue
equivalence theorem does not imply that the actual outcomes of the
four auction forms will always be exactly the same.  Rather, it
simply says that these four auction forms have the same **expected**
revenue prior to implementation under the stated assumptions.  Roughly
speaking then, each of these four auction forms would generate the
same revenue **on average** for the seller if they were implemented
repeatedly with different random valuation draws for the n bidders
each time.

    COMMENT 2 (page 710):  Note, also, another practical distinction
between these four auction forms.
     In either an English auction or a second-price sealed bid
auction, it is easy for the bidder to decide how high to bid.  In
the first case, the bidder should remain in the bidding until the
current best price reaches his valuation.  In the second case, the
bidder should submit a bid equal to his true valuation.
     In contrast, in a Dutch auction or first-price sealed bid auction,
the bidder must submit a sealed bid, and typically it is optimal to
submit a bid that is some amount less than one's true valuation.
Exactly how much less depends upon the probability distribution F
perceived to govern other bidders' valuations and the exact number
of competing bidders.  See, e.g., the analytic representation for
the optimal bidding rule for the first-price sealed bid auction (and
Dutch auction) derived in equation (5), page 709.

     COMMENT 3 (footnote 13, page 710): Given the assumption that
the monopolistic seller is risk neutral, the revenue-equivalence
theorem implies that the seller will be indifferent among the four
auction forms under the stated assumptions:  they each have the same
expected revenue, and the seller does not care about variance
     On the other hand, it can be shown that the **variance** of
revenue is lower in an English auction or second-price sealed bid
auction than in a first-price sealed bid auction or Dutch auction
under the assumptions of the revenue-equivalence theorem.  Consequently,
if the seller were instead assumed to be risk averse, he would prefer
the English or second-price sealed bid auction.


Under the assumptions of the revenue equivalence theorem:

    1. Increasing the number n of bidders increases the
       expected revenue of the seller.  This is because,
       as the number of bidders increases, the bids
       submitted by these bidders increase.  [Note that
       it is assumed here that bidders do not incur a
       cost in preparing their bids, and the seller does
       not incur a cost in checking the credentials of
       bidders -- i.e., there are no organizational costs
       to consider as n increases.]

    2. As the number n of bidders approaches infinity,
       the price paid to the seller tends to the
       highest possible valuation, implying that all gains
       from trade go to the seller.

            COMMENT:  The authors claim this is true because,
            as n approaches infinity, the second-highest valuation
            approaches the highest possible valuation.  But this
            need not be the case, even if the distribution function
            has compact support.  Consider a highest valuation v(n)
            = 1+e for some positive e and a sequence of other valuations
            v(i) = 1 - 1/i, i = 1,...,n-1.  As n goes to infinity,
            the second-highest valuation v(n-1) approaches 1 from
            below, hence remains strictly bounded below v(n).

    3. For particular valuation distributions F, such as normal and
       uniform, an increase in the variance of the valuations v_i
       increases both the expected revenue of the seller and the
       rents of successful bidders.

            difference between his valuation and what he pays
            to the seller.


     An AUCTION MECHANISM for a single-sided auction organized by a
monopolistic seller is any process that takes as inputs the bids of
the bidders and produces as its output the decision as to which
bidder receives the item and how much any of the bidders will be
required to pay.

     An auction mechanism is DIRECT if each bidder is simply asked to
report his valuation of the item.  An auction mechanism is INCENTIVE
COMPATIBLE if the mechanism is structured such that each bidder finds
it in his interest to report his valuation honestly.

     Given that the monopolist seller has the power to choose any
auction mechanism, and that bidders cannot collude among themselves,
a more fundamental question than revenue equivalence for the four
basic auction mechanisms is as follows:


The tool used to address this question is the following principle.


     For any auction mechanism for an auction organized by a
monopolistic seller with n participating bidders, there is a direct,
incentive-compatible auction mechanism with the same outcome.


The revelation principle achieves honest revelation in the direct
mechanism by designing the payoff structure in such a way that it is
in the bidders' interests to be honest.

     The revelation principle shows that the modeller can limit his
search for an optimal auction mechanism to the class of direct,
incentive-compatible mechanisms.  The optimal direct mechanism is
founds as the solution to a mathematical programming problem
involving two kinds of constraints (p. 712):

     (i)  incentive-compatibility constraints, which state that a
          bidder cannot gain by misrepresenting their valuations;

     (ii) individual rationality or "free exit" constraints, which
          state that the bidders would not be bettr off if they
          refused to participate in the auction.

The following important implication has been obtained as a corollary to the
revelation principle.


     Suppose the maintained assumptions (a)-(e) and the benchmark
model assumptions A1-A4 hold.  Define a function J(v) by

              J(v) =  v   -   [1 - F(v))]/f(v),

where f(v) is the density function corresponding to the valuation
distribution function F(v).  Suppose J(v) is an increasing function
of v. [Note: This requires F(v) to be sufficiently concave.]
     Also, let v_L denote the lowest feasible valuation, and define
a function B(v) by

                       /v       n-1
                      |   [F(w)]   dw
     B(v)  =  v  -    ----------------

Finally, let v_0 denote the valuation that the seller, himself,
attaches to the item for sale.
     Then the seller's expected revenue is maximized by the
following direct, incentive-compatible auction mechanism:

    (a) The seller asks each bidder i to report his valuation v_i,
        i = 1,...,n.

    (b) If J(v_i) < v_0 for i = 1,...,n, then the seller refuses
        to sell the item;

    (c) Otherwise, the item goes to the bidder i whose valuation
        v_i is highest in return for a payment to the seller
        equal to B(v_i).


    COMMENTS:  Let v_1 denotes the highest valuation of all n
bidders.  As shown on page 708, given the maintained and benchmark
model assumptions, J(v_1) is the amount paid to the seller by the
successful bidder under an English auction, equal to the successful
bidder's valuation, v_1, minus the successful bidder's rent,
[1 - F(v_1)]/f(v_1)
    Also, given the maintained and benchmark model assumptions, it
is shown on page 709 that B(v) is the Bayes-Nash equilibrium bidding
strategy for each bidder in a first-price sealed bid auction.
    Since (as can be shown) the function J(v) satisfies J(v) < v
for all v, the "optimal" mechanism described in the Corollary may
not be efficient.  That is, it is possible by (b) that the seller
keeps the item despite the existence of a bidder i whose valuation
v_i is greater than the seller's valuation v_0.
    Condition (a) in effect establishes a "reserve price" for the
seller.  If condition (a) is not binding, then the optimal auction
is equivalent to an English auction.


     Each of the basic four auction forms -- English, first-price
sealed bid auction, Dutch auction, or second-price sealed auction --
becomes an optimal auction mechanism in the sense of the preceding
revelation principle corollary if it is supplemented by a reserve
price r satisfying J(r) = v_0.


    This optimal auction mechanism design finding described above is
a powerful result.  No restriction has been placed on the types of
auction mechanisms that the seller could use.  The seller could have
several rounds of bidding, charge entry fees, etc.  But the optimal
auction mechanism design finding states that none of these more
complicated auction mechanisms would increase his expected revenue
above the level of the optimal auction mechanism described above.

     This completes the analysis of the benchmark model.  In the
next sections, the effects of relaxing the benchmark assumptions
one at a time is examined.


    Instead of assuming that all bidders appear the same to the
seller and to each other (assumption A3), suppose that the bidders
can be partitioned into two recognizably different classes:

     n_1 bidders with valuation drawn from a distribution F_1
     with corresponding density function f_1

     n_2 bidders with valuation drawn from a distribution F_2
     with corresponding density function f_2

When bidders are asymmetric in this fashion, the first-price sealed
bid auction can yield a different expected revenue for the seller
than the English auction, hence the Revenue Equivalence Theorem
breaks down.

For the English auction, the bids still rise until the price reaches
the second-highest valuation.  Since the highest-valuation bidder
is still the successful bidder, the outcome is still efficient.

However, the first-price sealed bid auction is no longer guaranteed
to be efficient.  While it remains the case that, within a bidding
class, the higher-valuation agents bid higher, this is in general
not the case across classes since bidders from different classes
perceive themselves to be facing different degrees of bidding
competition.  This results in differences in their optimally
calculated bidding functions.

The expected revenue for the seller in an English auction can be
either higher or lower than for a first-price sealed bid auction,
depending on exact parameter values.

Moreover, neither the English auction nor the first-price sealed bid
auction (augmented with a reserve price) is optimal for the seller
when the bidders are asymmetric.  Rather, let v_i_k denote the
valuation of bidder i of type k, k = 1,2, and define

                             [1  - F_k(v_i_k)]
    J_k(v_i_k)  =  v_i_k  -  -----------------

    r_k  =  J_k(v_0)  =  reservation price for class k.

The following can then be shown (p. 715)

     For the auction mechanism that maximizes the seller's expected
     revenue, the seller awards the item to the bidder i in class k
     with the highest value of J_k(v_i_k), assuming that v_i_k is
     such that J_k(v_i_k) exceeds r_k.

Consequently, the optimal auction mechanism for the seller is now
discriminatory, in the sense that there is a possibility that one
bidder succeeds despite another bidder's having a higher valuation.
This is so because asymmetry means that F_1 differs from F_2 so that
J_1(v) differs from J_2(v).  Suppose, for example, that classes 1
and 2 each contain precisely one bidder, and that

       v_1_2   is greater than   v_1_1  ;

       J_1(v_1_1)  is greater than  J_2(v_1_2) ,

Then the bidder in class 1 with valuation v_1_1 is the successful bidder
despite the fact his valuation is not as high as the valuation v_1_2
of the bidder in class 2.

Which type of bidder receives preferential treatment depends on the
relative shapes of the distribution functions F_1 and F_2.  In
particular, one has the following surprising finding (p. 715):

     If F_1 and F_2 are identical apart from their means (one is a
     translation of the other), then the class of bidders with the
     lower mean valuation is favored by the optimal auction.

How can this be?  By favoring the low-valuation bidders, the seller
raises the probability of awarding the item to someone other than
the bidder who values the item the most, which raises the probability
the seller will receive a lower payment.   

On the other hand, the benefit of this favoritism to the seller is
that the favoritism forces the bidders from the higher valuation
class to bid higher than they otherwise would have, thus driving up
the price on average.

BOTTOM LINE CAUTION:  An ostensibly non-discriminatory sealed bid
auction results in ad-hoc discrimination when the bidders are
asymmetric.  One cannot always judge the degree of discrimination
inherent in an auction mechanism by its superficial aspects.