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Chapter 18 NAME
Auctions
Introduction. An auction is described by a set of rules. The rules
specify bidding procedures for participants and the way in which the
array of bids made determines who gets the object being sold and how
much each bidder pays. Those who are trying to sell an object by auction
typically do not know the willingness to pay of potential buyers but have
some probabilistic expectations. Sellers are interested in finding rules that
maximize their expected revenue from selling the object.
Social planners are often interested not only in the revenue generated
from an auction method, but also in its efficiency. In the absence of
externalities, an auction for a single object will be efficient only if the
object is sold to the buyer who values it most highly.
18.1 (1) At Toivo’s auction house in Ishpemming, Michigan, a beautiful
stuffed moosehead is being sold by auction. There are 5 bidders in atten-
dance: Aino, Erkki, Hannu, Juha, and Matti. The moosehead is worth
$100 to Aino, $20 to Erkki, and $5 to each of the others. The bidders do
not collude and they don’t know each others’ valuations.
(a) If the auctioneer sells it in an English auction, who would get the
(b) If the auctioneer sells it in a sealed-bid, second-price auction and if
no bidder knows the others’ values for the moosehead, how much should
$5. Who would get the moosehead and how much would he pay?
18.2 (2) Charlie Plopp sells used construction equipment in a quiet
Oklahoma town. He has run short of cash and needs to raise money
quickly by selling an old bulldozer. If he doesn’t sell his bulldozer to a
customer today, he will have to sell it to a wholesaler for $1,000.
Two kinds of people are interested in buying bulldozers. These are
professional bulldozer operators and people who use bulldozers only for
recreational purposes on weekends. Charlie knows that a professional
bulldozer operator would be willing to pay $6,000 for his bulldozer but no
222 AUCTIONS (Ch. 18)
more, while a weekend recreational user would be willing to pay $4,500
but no more. Charlie puts a sign in his window. “Bulldozer Sale Today.”
Charlie is disappointed to discover that only two potential buyers
have come to his auction. These two buyers evidently don’t know each
other. Charlie believes that the probability that either is a professional
bulldozer operator is independent of the other’s type and he believes that
each of them has a probability of 1/2 of being a professional bulldozer
operator and a probability of 1/2 of being a recreational user.
Charlie considers the following three ways of selling the bulldozer:
•Method 1. Post a price of $6,000, and if nobody takes the bulldozer
•Method 2. Post a price equal to a recreational bulldozer user’s buyer
•Method 3. Run a sealed-bid auction and sell the bulldozer to the
(a) What is the probability that both potential buyers are professional
bulldozer operators? 1/4. What is the probability that both are recre-
ational bulldozer users? 1/4. What is the probability that one of them
is of each type? 1/2.
(b) If Charlie sells by method 1, what is the probability that he will be
able to sell the bulldozer to one of the two buyers? 3/4. What is
the probability that he will have to sell the bulldozer to the wholesaler?
(c) If Charlie sells by method 2, how much will he receive for his bulldozer?
(d) Suppose that Charlie sells by method 3 and that both potential
buyers bid rationally. If both bidders are professional bulldozer oper-
ators, how much will each bid? $6,000. How much will Char-
lie receive for his bulldozer? $6,000. If one bidder is a profes-
sional bulldozer operator and one is a recreational user, what bids will
NAME 223
Charlie get for his bulldozer? $4,500. If both bidders are recre-
ational bulldozer users, how much will each bid? $4,500. How
much will Charlie receive for his bulldozer? $4,500. What will be
Charlie’s expected revenue from selling the bulldozer by method 3?
(e) Which of the three methods will give Charlie the highest expected
18.3 (2) We revisit our financially afflicted friend, Charlie Plopp. This
time we will look at a slightly generalized version of the same problem. All
else is as before, but the willingness to pay of recreational bulldozers is an
amount C<$6,000 which is known to Charlie. In the previous problem
we dealt with the special case where C=$4,500. Now we want to explore
the way in which the sales method that gives Charlie the highest expected
revenue depends on the size of C.
(a) What will Charlie’s expected revenue be if he posts a price equal to
(b) If Charlie posts a price equal to the reservation price Cof recreational
bulldozer operators, what is his expected revenue? $C.
(c) If Charlie sells his bulldozer by method 3, the second-price sealed-bid
auction, what is his expected revenue? (The answer is a function of C.)
(d) Show that selling by method 3 will give Charlie a higher expected pay-
224 AUCTIONS (Ch. 18)
(e) For what values of Cis Charlie better off selling by method 2 than by
method 1? C>$(3/4)6,000 + (1/4)1,000 = 4,750.
(f) For what values of Cis Charlie better off selling by method 1 than
18.4 (3) Yet again we tread the dusty streets of Charlie Plopp’s home
town. Everything is as in the previous problem. Professional bulldozer
operators are willing to pay $6,000 for a bulldozer and recreational users
are willing to pay $C. Charlie is just about to sell his bulldozer when a
third potential buyer appears. Charlie believes that this buyer, like the
other two, is equally likely to be a professional bulldozer operator as a
recreational bulldozer operator and that this probability is independent
of the types of the other two.
(a) With three buyers, Charlie’s expected revenue from using method 1
is 5375 , his expected revenue from using method 2 is C,and
(b) At which values of Cwould method 1 give Charlie a higher expected
revenue than either of the other two methods of selling proposed above?
(c) At which values of C(if any) would method 2 give Charlie a higher
expected revenue than either of the other two methods of selling proposed
(d) At which values of Cwould method 3 give Charlie a higher expected
revenue than either of the other two methods of selling proposed above?
18.5 (2) General Scooters has decided to replace its old assembly line
with a new one that makes extensive use of robots. There are two contrac-
tors who would be able to build the new assembly line. General Scooters
does not know exactly what it would cost either of the contractors to do
NAME 225
this job. However its engineers and investigators have discovered that for
either contractor, this cost will take one of three possible values: H,M,
and L,whereH>M>L. The best information that General Scoot-
ers’s investigators have been able to give it is that for each contractor the
probability is 1/3 that the cost is H, 1/3 that the cost is M, and 1/3 that
the cost is Land that the probability distribution of costs is independent
between the two contractors. Each contractor knows its own costs but
thinks that the other’s costs are equally likely to be H,M,orL. General
Scooters is confident that the contractors will not collude.
Accountants at General Scooters suggested that General Scoooters
accept sealed bids from the two contractors for constructing the assembly
line and that it announce that it will award the contract to the low bidder
but will pay the low bidder the amount bid by the other contractor. (If
there is a tie for low bidder, one of the bidders will be selected at random
to get the contract.) In this case, as your textbook shows for the Vickrey
auction, each contractor would find it in his own interest to bid his true
valuation.
(a) Suppose that General Scooters uses the bidding mechanism suggested
by the accountants. What is the probability that it will have to pay L?
1/9 (Hint: The only case where it pays Lis when both contractors have
a cost of L.) What is the probability that it will have to pay Hto get
the job done? 5/9 (Hint: Notice that it has to pay Hif at least one of
the two contractors has costs of H.) What is the probability that it will
have to pay M?1/3. Write an expression in terms of the variables
H,M,andLfor the expected cost of the project to General Scooters.
(b) When the distinguished-looking, silver-haired chairman of General
Scooters was told of the accountants’ suggested bidding scheme, he was
outraged. “What a stupid bidding system! Any fool can see that it is
more profitable for us to pay the lower of the two bids. Why on earth
would you ever want to pay the higher bid rather than the lower one?”
he roared.
A timid-looking accountant summoned up his courage and answered
the chairman’s question. What answer would you suggest that he
226 AUCTIONS (Ch. 18)
(c) The chairman ignored the accountants and proposed the following
plan. “Let us award the contract by means of sealed bids, but let us do it
wisely. Since we know that the contractors’ costs are either H,M,orL,
we will accept only bids of H,M,orL, and we will award the contract
to the low bidder at the price he himself bids. (If there is a tie, we will
randomly select one of the bidders and award it to him at his bid.)”
If the chairman’s scheme is adopted, would it ever be worthwhile for
a contractor with costs of Lto bid L? (Hint: What are the contractor’s
profits if he bids Land costs are L. Does he have a chance of a positive
(d) Suppose that the chairman’s bidding scheme is adopted and that
both contractors use the strategy of padding their bids in the following
way. A contractor will bid Mif her costs are L, and she will bid H
if her costs are Hor M. If contractors use this strategy, what is the
Which of the two schemes will result in a lower expected cost for General
(e) We have not yet demonstrated that the bid-padding strategies pro-
posed above are equilibrium strategies for bidders. Here we will show that
this is the case for some (but not all) values of H,M,andL. Suppose
that you are one of the two contractors. You believe that the other con-
tractor is equally likely to have costs of H,M,orLand that he will bid
Hwhen his costs are Mor Hand he will bid Mwhen his costs are L.
Obviously if your costs are H,youcandonobetterthantobidH.If
your costs are M, your expected profits will be positive if you bid Hand
* The chairman’s scheme might not have worked out so badly for Gen-
eral Scooters if he had not insisted that the only acceptable bids are H,
M,andL. If bidders had been allowed to bid any number between L
and H, then the only equilibrium in bidding strategies would involve the
use of mixed strategies, and if the contractors used these strategies, the
expected cost of the project to General Scooters would be the same as it
is with the second-bidder auction proposed by the accountants.
NAME 227
negative or zero if you bid Lor M. What if your costs are L? For what
values of H,M,andLwill the best strategy available to you be to bid
18.6 (3) Late in the day at an antique rug auction there are only two
bidders left, April and Bart. The last rug is brought out and each bidder
takes a look at it. The seller says that she will accept sealed bids from
each bidder and will sell the rug to the highest bidder at the highest
bidder’s bid.
Each bidder believes that the other is equally likely to value the
rug at any amount between 0 and $1,000. Therefore for any number X
between 0 and 1,000, each bidder believes that the probability that the
other bidder values the rug at less than Xis X/1,000. The rug is actually
worth $800 to April. If she gets the rug, her profit will be the difference
between $800 and what she pays for it, and if she doesn’t get the rug,
her profit will be zero. She wants to make her bid in such a way as to
maximize her expected profit.
(a) Suppose that April thinks that Bart will bid exactly what the rug is
worth to him. If she bids $700 for the rug, what is the probability that
she will get the rug? 7/10. If she gets the rug for $700, what is her
profit? $100. What is her expected profit if she bids $700? $70.
(b) Suppose that Bart will pay exactly what the rug is worth to him.
If April bids $600 for the rug, what is the probability that she will get
the rug? 6/10. What is her profit if she gets the rug for $600?
$200. What is her expected profit if she bids $600? $120.
(c) Again suppose that Bart will bid exactly what the rug is worth to
him. If April bids $xfor the rug (where xis a number between 0 and
1,000) what is the probability that she will get the rug? x/1,000
What is her profit if she gets the rug? $800-x Write a formula for
her expected profit if she bids $x.$(800 −x)(x/1,000).Find
the bid xthat maximizes her expected profit. (Hint: Take a derivative.)
x= 400.
(d) Now let us go a little further toward finding a general answer. Suppose
that the value of the rug to April is $Vand she believes that Bart will
bid exactly what the rug is worth to him. Write a formula that expresses
her expected profit in terms of the variables Vand xif she bids $x.
228 AUCTIONS (Ch. 18)
$(V−x)(x/1,000) . Now calculate the bid $xthat will maximize
her expected profit. (Same hint: Take a derivative.) x=V/2.
18.7 (3) If you did the previous problem correctly, you found that if
April believes that Bart will bid exactly as much as the rug is worth to
him, then she will bid only half as much as the rug is worth to her. If
this is the case, it doesn’t seem reasonable for April to believe that Bart
will bid his full value. Let’s see what would the best thing for April to do
if she believed that Bart would bid only half as much as the rug is worth
to him.
(a) If Bart always bids half of what the rug is worth to him, what is
the highest amount that Bart would ever bid? $500. Why would it
(b) Suppose that the the rug is worth $800 to April and she bids $300 for
it. April will only get the rug if the value of the rug to Bart is less than
$600. What is the probability that she will get the rug if she bids
$300 for it? 6/10. What is her profit if she bids $300 and gets the
rug? $500. What is her expected profit if she bids $300? $300.
(c) Suppose that the rug is worth $800 to April. What is the probability
that she will get it if she bids $xwhere $x<$500? 2x/1,000.Write
a formula for her expected profit as a function of her bid $xwhen the rug
is worth $800 to her. $(800−x)2x/1,000.What bid maximizes
her expected profit in this case? $400.
(d) Suppose that April values the rug at $Vand she believes that Bart
will bid half of his true value. Show that the best thing for April is to
NAME 229
(e) Suppose that April believes that Bart will bid half of his actual value
and Bart believes that April will bid half of her actual value. Suppose also
that they both act to maximize their expected profit given these beliefs.
Will these beliefs be self-confirming in the sense that given these beliefs,
each will take the action that the other expects? Yes.
18.8 (2) Rod’s Auction House in Bent Crankshaft, Oregon, holds sealed-
bid used-car auctions every Tuesday. Each used car is sold to the highest
bidder at the second-highest bidder’s bid. On average, half of the cars sold
at Rod’s Auction House are lemons and half are good used cars. A good
used car is worth $1,000 to any buyer and a lemon is worth only $100.
Buyers are allowed to look over the used cars for a few minutes before
they are auctioned. Almost all of the buyers who attend the auctions can
do no better than random choice at picking good cars from among the
lemons. The only exception is Al Crankcase. Al can sometimes, but not
always, detect a lemon by licking the oil off of the dipstick. To Al, the
oil from a good car’s dipstick invariably has a sweet, lingering taste. On
the other hand, the oil from the dipsticks of 1/3 of the lemons has a sour,
acidic taste, while the oil from the dipsticks of the remaining 2/3 of the
lemons has the same sweet taste as the oil from the good cars. Al attends
every auction, licks every dipstick, and taking into account the results of
his taste test, bids his expected value for every car.
(a) This auction environment is an example of a (common, private)
common value auction.
(b) Suppose that Al licks the dipsticks of 900 cars, half of which are good
cars and half of which are lemons. Suppose that the dipstick oil from all
of the good cars and from 2/3 of the lemons tastes sweet and the oil from
1/3 of the lemons tastes sour to Al. How many good cars will there be
(c) If Al finds that the oil on a car’s dipstick tastes sweet, what is the
(d) If Al finds that the oil on a car’s dipstick tastes sour, what is the
