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Solutions to Chapter 16 Exercises
SOLVED EXERCISES
S1. The painter can compare her estimated cost with a job’s true cost only when she does the job. But
the painter does a job only when she agrees (through the bidding process) to do it for less than anybody
S2. If you turn out to be the lowest bidder and therefore fail to get the object, this must be because all
of the others got a higher estimate of the value of the object than you did. Therefore, you have reason to
believe that you got an exceptionally low estimate—one with a large and negative error. This is a “loser’s
S3. (a) If you offer $3,000, the current owner sells only if the car’s true value to him is less than
(b) Suppose that your offer equals B. The current owner will sell you his car if its value to
S4. (a) If your opponent always bids half her value, and her value is uniformly distributed on [0,
1], then her bid will be uniformly distributed on [0, 0.5].
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
If you bid b = 0.6, you cannot be outbid by your opponent, because 0.6 is greater than her
maximum possible bid. Your probability of winning is 1.
(e) We see that the best response to an opponent’s bidding half her value is to bid half your
S5. (a)
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) As seen in the graph in part (a), the bids will be decreasing in n for lower values of v and
increasing in n for higher values of v. The threshold value of v for this example occurs when v2/2 = 2v3/3,
that is, when v = 3/4.
Thus, when the number of bidders equals n, adding another bidder would decrease the equilibrium bid
function if v < 1 – 1/n2 and increase the equilibrium bid function if v > 1 – 1/n2.
(c) To illustrate, consider the case in which n = 3. Assume that your two opponents are
bidding according to the proposed equilibrium strategy b = 2v3/3.
When you bid b, your probability of winning against one opponent bidding according to the
Your expected profit in an all-pay auction is
The first-order condition is
which implies
The derivation of the general case, where there are n bidders, follows precisely the same pattern.
UNSOLVED EXERCISES
U1. True. Risk aversion will make bidders want to win if possible without overbidding. We expect
risk-averse bidders to shade down their bids less in a first-price sealed-bid auction (compared with
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
U2. (a) If a bidder continues to the point where the bid equals his willingness to pay, then Bidder
(b) All bid truthfully, so Bidder 3 wins with a bid of $16 but pays $14 and gets a profit of $2.
(c) The best-case scenario for the seller is if each bidder shades by just $1, in which case
(d) If risk aversion leads to minimal (or no) shading down of bids in the first-price sealed-bid
U3. (a) Since the company’s current value is uniformly distributed from $10 million to $110
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(b) The probability that a bid of $50 million succeeds is (50 – 10)/(110 – 10) = 0.4. If the bid
(c) In terms of a bid level X, where X is in millions of dollars and 10 ≤ X ≤ 110, the expected
profit from X is
( ) ( )
2
10 10
1.75 0.01 0.1 8.75 0.125 0.00125 0.1 0.875.
110 10 2
X X X X X X Xp- +
æ ö
= × - = - - =- + -
ç ÷
-è ø
Expected profit is maximized when X = 40. With a bid of $40 million, the expected profit is $1.125
million.
U4. (a) With truthful voting, the defendant will be found Not guilty more often when the
unanimity rule holds relative to the majority rule. For example, if all jurors vote in accordance with their
private signals, and three of them vote Not guilty, this will result in a Not guilty verdict under unanimity
but a Guilty verdict under the majority rule. The juror’s curse is that the verdict hinges on the juror with
the lowest private estimate of the defendant’s guilt, arguably resulting in too many Not guilty verdicts.
(b) When jurors make adjustments to account for the juror’s curse, each of them is slightly
(c) The unanimity rule in jury trials is designed to give the defendant the maximum benefit
of the doubt. Its intention is to cause juries to err on the side of voting Not guilty. However—depending
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Another line of argument is that the rationale underlying the unanimity rule may be flawed. While
the unanimity rule is intended to give the defendant the benefit of the doubt, it can be argued that this
built-in bias away from false positive verdicts of Guilty and toward false negatives may not be socially
U5. (a) If every bidder is using the bid function b(v) = v(n – 1)/n, the probability that one rival
bidder’s bid is less than 0.1, where F(x) is the cumulative distribution function, then
1
Pr 0.1 Pr 0.1 0.1 .
1 1
n n n
v v F
n n n
-
æ ö æ ö æ ö
< = < =
ç ÷ ç ÷ ç ÷
- -
è ø è ø è ø
Since every bidder’s value is distributed uniformly from 0 to 1, F(x) = x. Thus the probability that one
(b) By part (a), the probability that one rival’s bid is less than your bid amount b is
b * n/(n – 1).
(c) Since all bids are made independently, the probability that n – 1 rival bidders bid less
than b is the product of the probabilities of each of the rival bidders bidding less than b:
1See Timothy Feddersen and Wolfgang Pesendorfer, “Convicting the Innocent: The Inferiority of Unanimous Jury
Verdicts under Strategic Voting,” American Political Science Review, vol. 92, no. 1 (March 1998), pp. 23–35.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
1
Pr(win) Pr(all other bids less than ) 1
n
n
b b
n
-
æ ö
= =ç ÷
-
è ø
(d) The expected profit is the probability of winning times the profit of winning, namely the
value minus the bid:
( )
1
1
n
n
E b v b
n
p
-
æ ö
= -
ç ÷
-
è ø
(e) To find the value of b that maximizes expected profit, take the partial derivative of the
expression found in part (d) and solve the first-order condition
( ) ( )
1 1
2 1
1 0 1 0.
1 1
n n
n n
E n n
n vb n b n v nb
b n n
p- -
- -
¶æ ö æ ö
= - - = Þ - - =
ç ÷ ç ÷
¶ - -
è ø è ø
This implies that as a function of the value v, the bid that maximizes expected profit is b(v) = v(n – 1)/n.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
