978-0393123982 Chapter 18 Solution Manual Part 2

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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.

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.

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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)

(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

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

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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