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S7. (a) As in Exercise S3, part (a), we can conclude that there are only three sensible prices to
consider for popcorn: $1.50, $3.50, or $4.00. These three prices yield the following quantities sold, given
individuals’ values for popcorn:
So the profit-maximizing price for popcorn is $3.50.
Similarly, the best prices to choose for soda are $2.50, $3.00, or $3.50. These three prices yield
the following quantities sold:
So the profit-maximizing price for soda is $2.50.
(b) We look at the table of valuations to determine who buys. At these prices, Cameron-type
and Jessica-type customers buy both popcorn and soda. Sean-type consumers buy only soda, not popcorn.
(c) For a bundle, the valuations for each type of customer are
So there are just two sensible prices for Sticky Shoe to consider for a combo: $4.00 or $6.50.
(d) With the combo, Cameron and Jessica buy popcorn and soda, but Sean buys nothing. The
difference from part (b) is that now Sean no longer buys soda.
(e) With separate pricing, Cameron values the popcorn and soda at a total of $6.50 and pays
The same is true for Jessica. She buys both popcorn and soda in either case, but with separate
Sean buys nothing with combo pricing. He buys soda at a price of $2.50 under separate pricing,
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(f) With the combo, Sticky Shoe can get Cameron and Jessica to pay their maximum
willingness for popcorn and soda, earning $6.50 per customer instead of the mere $6.00 it earns through
separate pricing. So combo pricing produces superior profits from Cameron and Jessica. But it loses
profits on Sean-type customers, because it fails to sell them soda.
Note that Sticky Shoe could offer soda separately at a price of up to $3.50 and get Sean-type
Also note that neither Cameron nor Jessica is willing to buy soda at a price of $3.50. So the
incentive-compatibility constraint is automatically satisfied for these two types of consumers. They would
not consider switching from buying the combo to buying just soda, because this would make them strictly
worse off.
Should Sticky Shoe consider selling popcorn separately to Sean-type customers as well? The
answer turns out to be no. The maximum price Sticky Shoe could charge the Sean-type customers is
$1.50. But at that price, both Cameron and Jessica would want to buy popcorn instead of a combo,
How can Sticky Shoe get the customers to buy the products it intends? By setting prices
appropriately. In the language of this chapter, Sticky Shoe wants to satisfy incentive-compatibility
Therefore, Sticky Shoe wants to sell combos to Cameron- and Jessica-type customers, and soda
(g) We saw in part (f) that the optimal price for the combo is $6.50, and the optimal price for
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
customer types. Since Jessica has the greatest willingness to pay for popcorn at $4.00, this means
charging an individual popcorn price greater than $4.00. So, for example, the prices could be
(h) Separate pricing produces more profit than combo pricing, but less profit than
combo-and-separate pricing (also known as “mixed bundling”). The combo allows Sticky Shoe to earn
S8. (a) If the company wants only low effort from the manager, it doesn’t need to worry about an
incentive-compatibility constraint, because the manager prefers low effort to high effort. All the firm
(b) The expected profit is the expected revenue under low effort minus the cost of
compensating the manager:
(c) Now we have both a participation constraint and an incentive-compatibility constraint to
worry about, because we want the manager to prefer high effort to no effort at all.
With high effort, the probability of a successful project is 0.5. So with payment y for a successful
project and x for an unsuccessful project (with amounts measured in millions of dollars), the manager’s
expected utility under high effort is
EU =0.5 y+0.5 x-0.1
.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
His expected utility under low effort, with a probability of success of only 0.25, is
EU =0.25 y+0.75 x
.
So the participation constraint and the incentive-compatibility constraints are
The firm wants to minimize the amount paid to the manager while still getting the manager to
exert high effort. So it wants to minimize both X and Y subject to the above constraints.
In order to keep both X and Y as low as possible, we must also keep the sum Y + X as low as
possible, which means PC must hold with equality
1
1
Y X
Y X
+ =
= -
Substituting this result into the IC constraint, we obtain
1 0.4
2 0.6
0.3.
X X
X
X
- - ³
- ³ -
£
Is it a good idea to set X at its maximum? Or is it better to set a lower value of X? To answer this
question, we need to look at the firm’s profit function to see where it might be maximized. Recall that if
the project is successful (with probability 0.5), the firm earns $1 million. It pays y to the manager for a
successful project, and x to the manager for an unsuccessful project. So the expected profit is
2 2
2 2
Profit 0.5(1 ) 0.5
0.5 0.5 0.5
0.5 0.5 0.5
0.5(1 ).
y x
y x
Y X
Y X
= - -
= - -
= - -
= - -
Substituting in our previous result Y = 1 – X, we have
2 2
2 2
Profit 0.5(1 [1 ] )
0.5(1 [1 2 ] )
0.5(2 )
.
X X
X X X
X
X
= - - -
= - - + -
=
=
(d) The expected profit is
Profit 0.5(1 ) 0.5
0.5(1 0.49) 0.5(0.09)
0.21.
y x= - -
= - -
=
(e) Since the profit is higher in part (d) than in part (b), we know the firm wishes to pay the
manager in a way that gets him to exert high effort.
S9. (a) The insurance company would like the manufacturer to implement the fire-prevention
However, moral hazard arises because the manufacturer has an incentive to save money by not
actually implementing the program, knowing that the insurer cannot tell the difference and will cover its
losses anyway. The source of the moral hazard is the unobservability of the manufacturer’s decision.
(b) The insurer can try to mitigate the moral-hazard problem by making the manufacturer
incur some of the costs in the event of a loss. The fire-prevention program decreases the probability of
For example, suppose the insurer specified a deductible of $10,000, so that the insurer would pay
only $290,000 in case of a loss. Then the manufacturer loses $10,000 in the event of a fire, making its
Of course, if the manufacturer wants to purchase insurance in the first place, it may well be risk
averse, caring not just about expected losses but also about the utility of its expected losses. Faced with a
S10. The most obvious asymmetric information in this case is the amount of salary that Mozart would
be willing to accept in order to work for the emperor. If Mozart applied for a position, then he would
Does this make sense from the point of view of the theory of signaling? Recall that for a signal to
be credible, it must be somewhat costly, and less costly to the type that can signal than to the type that
Another potential source of asymmetric information in this situation is that Mozart may not truly
know how much the emperor really wants him to be a court musician. By waiting for the emperor to
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(d) If BMA has low cost, then the benefit would be
B
L
=2Q
1
-M
1
.
If BMA has high cost, then the benefit would be
B
H
=2Q
2
-M
2
.
So the expected benefit is
1 2
1 1 2 2
0.4 0.6
0.4(2 ) 0.6(2 ).
B B B
Q M Q M
= +
= - + -
(e) First assume that IC1 binds. This means that the low-cost type gets just enough payoff
from choosing contract 1 in order to induce it to choose contract 1 over contract 2:
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Next, assume that PC2 binds. This means that the high-cost type gets just enough payoff to
convince it to accept contract 2 instead of no contract at all:
We just derived a lower bound for M2. Next, substituting the PC2 equation into the IC1 equation,
we have
So our minimum required payments are, in terms of Q1 and Q2:
(f) First, let’s consider PC1:
Substituting for M1 from part (e), we have
Since we know the quantity Q2 is not negative, we see that PC1 is automatically satisfied.
Next, let’s consider IC2:
Substituting for M1 and M2 using the results derived in part (e), we have
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Thus, as long as the final contracts specify that the low-cost firm should produce a higher
quantity than the high-cost firm, which we should expect, then the constraint IC2 will be automatically
satisfied.
(g) Substituting our results from part (e) into our results from part (d):
1 1 2 2
1 1 2 2 2
1 1 2 2 2
1 1 2 2
0.4(2 ) 0.6(2 )
0.4(2 [0.1 0.06 ]) 0.6(2 0.16 )
0.8 0.04 0.024 1.2 0.096
0.8 0.04 1.2 0.12
B Q M Q M
Q Q Q Q Q
Q Q Q Q Q
Q Q Q Q
= - + -
= - + + -
= - - + -
= - + -
(h) This function must be maximized with respect to both Q1 and Q2. The first-order
conditions for Q1 is
1
1
2
1
1
2
1
1
1
1
0
0 0.8(1 / 2) 0.04
0.04 0.4
1
0.1
10
100.
dB
dQ
Q
Q
Q
Q
Q
-
-
=
= -
=
=
=
=
The first-order conditions for Q2 is
2
1
2
2
1
2
2
2
0
0 1.2(1/ 2) 0.12
0.12 0.6
1
0.2
5
dB
dQ
Q
Q
Q
Q
-
-
=
= -
=
=
=
(j) The expected net benefit is, from part (d):
1 1 2 2
0.4(2 ) 0.6(2 )
0.4(2[10] 11.5) 0.6(2[5] 4)
0.4(8.5) 0.6(6)
7.
B Q M Q M= - + -
= - + -
= -
=
(k) The main issue to be addressed by our screening mechanism is that the low-cost firm may
wish to pretend to be the high-cost type, in order to try to earn higher profit.
An important principle of screening is that the low-cost contract has to be designed with extra
profit to induce it not to choose the high-cost contract; we see this in the IC1 constraint. The low-cost firm
earns more profit than would be necessary if its type could be observed directly; this is the “carrot”
referred to in the chapter.
A second feature is to use a “stick” to make the high-cost contract less attractive to the low-cost
firm. As we will see in part (d) of Exercise S12 below, if Oceania knew that BMA had high cost, it would
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
