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230 AUCTIONS (Ch. 18)
(e) Assuming that Al always bids his expected value for a car, given the
result of his taste test, how much will Al bid for a car that tastes sweet?
(f) Consider a naive bidder at Rod’s Auction House, who knows that half
of the cars are good and half are lemons, but has no clue at all about which
ones are good. If this individual bids his expected value for a randomly
(g) Given that Al bids his expected value for every used car and the naive
bidders bid the expected value of a randomly selected car, will a naive
(h) What is the expected value of cars that naive bidders get if they
always bid the expected value of a randomly selected car? $100. Will
naive bidders make money, lose money, or break even if they follow this
(i) Suppose that bidders other than Al realize that they will get only the
cars that taste sour to Al. If they bid the expected value of such a car,
(j) Suppose that bidders other than Al believe that they will only get
cars whose oil tastes sour to Al and suppose that they bid their expected
value of such cars. Suppose also that for every car, Al bids his expected
value, given the results of his taste test. Who will get the good cars and
at what price? (Recall that cars are sold to the highest bidder at the
(k) What will Al’s expected profit be on a car that passes his test?
18.9 (3) Steve and Leroy buy antique paintings at an art gallery in
Fresno, California. Eighty percent of the paintings that are sold at the
gallery are fakes, and the rest are genuine. After a painting is purchased,
it will be carefully analyzed, and then everybody will know for certain
whether it is genuine or a fake. A genuine antique is worth $1,000. A
fake is worthless. Before they place their bids, buyers are allowed to
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inspect the paintings briefly and then must place their bids. Because
they are allowed only a brief inspection, Steve and Leroy each try to
guess whether the paintings are fakes by smelling them. Steve finds that
if a painting fails his sniff test, then it is certainly a fake. However, he
cannot detect all fakes. In fact the probability that a fake passes Steve’s
sniff test is 1/2. Leroy detects fakes in the same way as Steve. Half of
the fakes fail his sniff test and half of them pass his sniff test. Genuine
paintings are sure to pass Leroy’s sniff test. For any fake, the probability
that Steve recognizes it as a fake is independent of the probability that
Leroy recognizes it as a fake.
The auction house posts a price for each painting. Potential buyers
can submit a written offer to buy at the posted price on the day of the
sale. If more than one person offers to buy the painting, the auction house
will select one of them at random and sell it to that person at the posted
price.
(a) One day, as the auction house is about to close, Steve arrives and
discovers that neither Leroy nor any other bidders have appeared. He
sniffs a painting, and it passes his test. Given that it has passed his test,
what is the probability that it is a good painting? (Hint: Since fakes are
much more common than good paintings, the number of fakes that pass
Steve’s test will exceed the number of genuine antiques that pass his test.)
(b) On another day, Steve and Leroy see each other at the auction, sniffing
all of the paintings. No other customers have appeared at the auction
house. In deciding how much to bid for a painting that passes his sniff
test, Steve considers the following: If a painting is selected at random and
sniffed by both Steve and Leroy, there are five possible outcomes. Fill in
the blanks for the probability of each.
(c) On the day when Steve and Leroy are the only customers, the auction
house sets a reserve price of $300. Suppose that Steve believes that Leroy
will offer to buy any painting that passes his sniff test. Recall that if
Steve and Leroy both bid on a painting, the probability that Steve gets
it is only 1/2. If Steve decides to bid on every painting that passes his
232 AUCTIONS (Ch. 18)
own sniff test, what is the probability that a randomly selected painting is
genuine and that Steve is able to buy it? .1. What is the probability
that a randomly selected painting is a fake and that Steve will bid on it
and get it? .3. If Steve offers to pay $300 for every painting that
passes his sniff test, will his expected profit be positive or negative?
18.10 (2) Every day the Repo finance company holds a sealed-bid,
second-price auction in which it sells a repossessed automobile. There
are only three bidders who bid on these cars, Arnie, Barney, and Carny.
Each of these bidders is a used-car dealer whose willingness to pay for
another used car fluctuates randomly from day to day in response to the
variation in demand at his car lot. The value of one of these used cars
to any dealer, on any given day is a random variable which takes a high
value $Hwith probability 1/2andalowvalue$Lwith probability 1/2.
The value that each dealer places on a car on a given day is independent
of the values placed by the other dealers.
Each day the used-car dealers submit written bids for the used car
being auctioned. The Repo finance company will sell the car to the dealer
with the highest bid at the price bid by the second-highest bidder. If there
is a tie for the highest bid, then the second-highest bid is equal to the
highest bid and so that day’s car will be sold to a randomly selected top
bidder at the price bid by all top bidders.
(a) How much should a dealer bid for a used car on a day when he places
a value of $Hon a used car? $H.How much should a dealer bid for a
used car on a day when he places a value of $Lon a used car? $L.
(b) If the dealers do not collude, how much will Repo get for a used car
on days when two or three dealers value the car at $H?$H.How
much will Repo get for a used car on days when fewer than two dealers
value the car at $H?$L.
(c) On any given day, what is the probability that Repo receives $Hfor
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(d) If there is no collusion and every dealer bids his actual valuation for
every used car, what is the probability on any given day that Arnie gets
a car for a lower price than the value he places on it? (Hint: This will
happen only if the car is worth $Hto Arnie and $Lto the other dealers.)
1/8. Suppose that we measure a car dealer’s profit by the difference
between what a car is worth to him and what he pays for it. On a
(e) The expected total profit of all participants in the market is the sum
of the expected profits of the three car dealers and the expected revenue
realized by Repo. Used cars are sold by a sealed-bid, second-price auction
and the dealers do not collude. What is the sum of the expected profits of
18.11 (3) This problem (and the two that follow) concerns collusion
among bidders in sealed-bid auctions. Many writers have found evidence
that collusive bidding occurs. The common name for a group that prac-
tices collusive bidding is a “bidding ring.”*
Arnie, Barney, and Carny of the previous problem happened to meet
at a church social and got to talking about the high prices they were
paying for used cars and the low profits they were making. Carny com-
plained, “About half the time the used cars go for $H, and when that
happens, none of us makes any money.” Arnie got a thoughtful look and
then whispered, “Why don’t we agree to always bid $Lin Repo’s used-car
auctions?” Barney said, “I’m not so sure that’s a good idea. If we all bid
$L, then we will save some money, but the trouble is, when we all bid the
same, we are just as likely to get the car if we have a low value as we are
to get it if we have a high value. When we bid what we think its worth,
then it always goes to one of the people who value it most.”
(a) If Arnie, Barney, and Carny agree to always bid $L,thenonanygiven
day, what is the probability that Barney gets the car for $Lwhen it is
actually worth $Hto him? 1/6. What is Barney’s expected profit per
(b) Do the three dealers make higher expected profits with this collusive
* Our discussion draws extensively on the paper, “Collusive Bidder Be-
havior at Single-Object, Second-Price, and English Auctions” by Daniel
Graham and Robert Marshall in the Journal of Political Economy, 1987.
234 AUCTIONS (Ch. 18)
(c) Calculate the expected total profits of all participants in the market
(including Repo as well as the three dealers) in the case where the dealers
(d) The cars are said to be allocated efficiently if a car never winds up
in the hands of a dealer who values it less than some other dealer values
it. With a sealed-bid, second-price auction, if there is no collusion, are
18.12 (2) Arnie, Barney, and Carny happily practiced the strategy of
“always bid low” for several weeks, until one day Arnie had another idea.
Arnie proposed to the others: “When we all bid $L, it sometimes happens
that the one who gets the week’s car values it at only $Lalthough it is
worth $Hto somebody else. I’ve thought of a scheme that will increase
profits for all of us.” Here is Arnie’s scheme. Every day, before Repo holds
its auction, Arnie, Barney and Carny will hold a sealed-bid, second-price
preauction auction among themselves in which they bid for the right to
be the only high bidder in that day’s auction. The dealer who wins this
preauction bidding can bid anything he likes, while the other two bidders
must bid $L. A preauction auction like this is known is a “knockout.”
The revenue that is collected from the “knockout” auction is divided
equally among Arnie, Barney, and Carny. For this problem, assume that
in the knockout auction, each bidder bids his actual value of winning the
knockout auction.*
(a) If the winner of the knockout auction values the day’s used car at
$H, then he knows that he can bid $Hfor this car in Repo’s second-price
sealed-bid auction and he will get it for a price of $L. Therefore the value
of winning the knockout auction to someone who values a used car at $H
must be $H−L.The value of winning the knockout auction to
someone who values a used car at $Lis 0
(b) On a day when one dealer values the used car at $Hand the other
two value it at $L, the dealer with value $Hwill bid $H−Lin
* It is not necessarily the case that this is the best strategy in the
knockout auction, since one’s bids affect the revenue redistributed from
the auction as well as who gets the right to bid. Graham and Marshall
present a variation on this mechanism that ensures “honest” bidding in
the knockout auction.
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the knockout auction and the other two dealers will bid 0In this
case, in the knockout auction, the dealer pays 0for the right to
be the only high bidder in Repo’s auction. In this case, the day’s used
car will go to the only dealer with value $Hand he pays Repo $Lfor
(c) We continue to assume that in the knockout auction, dealers bid
their actual values of winning the knockout. On days when two or more
buyers value the used car at $H, the winner of the knockout auction pays
(d) If Arnie’s scheme is adopted, what is the expected total profit of each
of the three car dealers? (Remember to include each dealer’s share of the
18.13 (2) After the passage of several weeks during which Repo never got
more than one high bid for a car, the Repo folks guessed that something
was amiss. Some members of the board of directors proposed hiring a hit
man to punish Arnie, Barney, and Carny, but cooler heads prevailed and
they decided instead to hire an economist who had studied Intermediate
Microeconomics. The economist suggested: “Why don’t you set a reserve
price $Rwhich is just a little bit lower than $H(but of course much larger
than $L)?Ifyougetatleastonebidof$R,sellitfor$Rto one of these
bidders, and if you don’t get a bid as large as $R, then just dump that
day’s car into the river. (Sadly, the environmental protection authorities
in Repo’s hometown are less than vigilant.) “But what a waste,” said a
Repo official. “Just do the math,” replied the economist.
(a) The economist continued. “If Repo sticks to its guns and refuses to
sell at any price below $R, then even if Arnie, Barney, and Carny collude,
the best they can do is for each to bid $Rwhen they value a car at $H
and to bid nothing when they value it at $L.” If they follow this strategy,
(b) Setting a reserve price that is just slightly below $Hand destroying
cars for which it gets no bid will be more profitable for Repo than setting
236 AUCTIONS (Ch. 18)
18.14 (1) Three boys, Alan, Bill, and Charlie, and three girls, Alice,
Betsy, and Clara, are planning dates for the senior prom. Their prefer-
ences over possible partners are given in this table:
First Second Third
Person Choice Choice Choice
Alan Alice Betsy Clara
Bill Betsy Alice Clara
Charlie Alice Clara Betsy
Alice Bill Alan Charlie
Betsy Alan Bill Charlie
Clara Alan Charlie Bill
For example, Alan likes Alice best, Betsy second, and Clara third, while
Alice likes Bill best, Alan second, and Charlie third.
(a) Suppose that partners are assigned by the deferred acceptance algo-
rithm, where the boys ask the girls. On the first round, Alice will be
(b) Suppose that instead they assign partners by the deferred acceptance
algorithm, with the girls asking the boys. What is the final assignment
of partners? Alice is matched with Bill , Betsy is matched with
(c) In this example, which group is in general better off, the proposing
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(d) Suppose that partners are assigned by the deferred acceptance algo-
rithm, with the boys proposing. Alice knows everybody’s preferences and
believes that everybody else will act according to their true preferences.
Can she get a better outcome by a bit of deception? Suppose that Al-
ice chooses Charlie over Alan, what will happen in the second round?
18.15 (1) Four students, Art, Bob, Cal, and Dan, must be assigned
rooms in a college dorm. The dean of housing would like to assign them
in stable partnerships, so that no two students would prefer each other to
their assigned partners. The students’ preferences are as follows:
238 AUCTIONS (Ch. 18)
First Second Third
Person Choice Choice Choice
Art Bob Cal Dan
Bob Cal Art Dan
Cal Art Bob Dan
Dan Art Bob Cal
(a) Is there any stable assignment? Explain your answer. (Hint: Note
that every student except Dan is the first choice of somebody other
(b) The humorist James Thurber once wrote a book called Is Sex Nec-
essary? In what sense does this example offer an answer to Thurber’s
