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U4. (a) If the team owners can identify the type of each fan, they will sell blue-collar fans an
(b) The incentive-compatibility constraints are
: 12 15 3
: 22 14 8
b
w
IC X Y Y X
IC Y X Y X
- ³ - Þ ³ +
- ³ - Þ £ +
(c) The participation constraints are
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
: 12
: 22
b
w
P X
P Y
£
£
(d) Given the constraints in parts (b) and (c), the owners will set the prices X = 12, Y = 20 to
(e) If the owners sell to only white-collar fans, they offer luxury seats at a price of Y = $22
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(g) From the owner’s perspective, the information asymmetry costs them $10,400 – $9,600 =
U5. (a) With full information the owners sell blue-collar fans an ordinary seat for $14 and
(b) The incentive-compatibility constraints and their graphical representation are exactly as
(e) With only 10% blue-collar fans, selling to only white-collar fans by setting prices X >
(f) Here the owners opt to sell only to white-collar fans, since πw > π2. Nevertheless, πw < π*;
the owners earn less than the potential profit under full information.
(g) Unlike Exercise U4, part (g), the forgone profit for the owners ($13,400 – $12,600 =
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
U6. (a) Under full information the owners sell blue-collar fans an ordinary seat for $14 and
(c) Regardless of B, the participation constraints remain the same as in Exercise U4, part (c).
(d) Under partial information, when selling to both types of fans the owners set the same
(f) The optimal pricing policy for the owners under partial information depends on the value
of B. When π2(B) > πw(B), that is, when
$12,000 $4,000 $14,000 $14,000 $10,000 $2,000 0.2,B B B B- > - Þ > Þ >
the owners will set X = $12, Y = $20 and earn a profit of $12,000 – $4,000B. When π2(B) < πw(B)—that is,
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
Either pricing scheme under partial information yields lower profit to the owners than the
(g) Whenever B ≥ 0.2 and the owners sell tickets to both types of fans there will be no social
cost from the information asymmetry—the profit forgone by the owners, ($14,000 – $6,000B) – ($12,000
U7. (a) A tournament compensation scheme may offer a firm a relatively cheap way to motivate
workers. Multiple workers may work more than they would otherwise in pursuit of the higher-level
prizes, so that in the aggregate the increased productivity is worth much more to the firm than the value of
(b) A tournament scheme may suffer from a number of potential pitfalls. If competing
employees or groups are doing the exact same task or project, there may be unnecessary duplication of
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
(c) Since compensation is based on relative rather than absolute performance under a
tournament scheme, tournament theory would predict that in highly competitive fields where winning
U8. (a) To satisfy the manager’s participation constraint for the low effort, the firm offers a flat
(b) The firm’s expected profit when it optimally induces low effort is the expected revenue
under low effort minus the cost of compensating the manager:
(c) The probability of a successful project is 0.4 with high effort, 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
0.4 0.6 0.1.
H
EU y x= + -
His expected utility under low effort, with a probability of success of only 0.24, is
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
0.24 0.76 .
L
EU y x= +
The participation and incentive-compatibility constraints for exerting a high effort are
1.25 1.5 0.625 0.625 2.5 0.25 .X X X X- ³ + Þ ³ Þ ³
To see what value of X the firm would like to set, look at the firm’s expected profit when the manager
exerts high effort:
2 2
2 2
0.4(1 ) 0.6
0.4 0.4 0.6
0.4 0.4 0.6
0.4(1 ) 0.6
H
E y x
y x
Y X
Y X
p= - -
= - -
= - -
= - -
Substituting for the binding participation constraint, we have
2 2
2 2
2
0.4(1 ) 0.6
0.4(1 (1.25 1.5 ) ) 0.6
0.225 1.5 1.5
1.5 (1 ) 0.225
H
E Y X
X X
X X
X X
p= - -
= - - -
=- + -
= - -
The quantity X(1 – X) is maximized when X = 1 – X = 0.5, so to maximize its profits the firm should set X
= 0.875.
(d) The firm’s expected profit (measured in millions) from inducing high effort is
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
0.4(1 0.765625) 0.6(0.0625)
.05625
H
Ep= - -
=
That is, the firm’s expected profit from inducing high effort is $56,250.
(e) The firm’s expected profit is higher when inducing low effort from its manager ($80,000)
than when inducing high effort ($56,250), so the firm will choose to induce a low effort.
U9. The student knows the true probability p. If he answers x, his expected payoff is F(x) = p log(x) +
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
