CHAPTER 30: Game Applications
MULTIPLE CHOICE
1. (See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them
will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second
best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of
outcome C is $0, outcome B is $1, and outcome A is $3. The money value to Bertha of outcome C is
$0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha
both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha
vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not
allowed to communicate with each other, and each decides to use a mixed strategy in which each
randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and
Bertha in this game?
a.
Arthur and Bertha each votes for A with probability 1/2 and for B with probability 1/2.
b.
Arthur votes for A with probability 3/7 and for B with probability 4/7. Bertha votes for A
with probability 4/7 and for B with probability 3/7.
c.
Arthur votes for A with probability 3/4 and for B with probability 1/4. Bertha votes for A
with probability 1/5 and for B with probability 4/5.
d.
Arthur votes for A with probability 1/5 and for B with probability 4/5. Bertha votes for A
with probability 3/4 and for B with probability 1/4.
e.
Arthur votes for A and Bertha votes for B.
2. (See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them
will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second
best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of
outcome C is $0, outcome B is $1, and outcome A is $4. The money value to Bertha of outcome C is
$0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha
both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha
vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not
allowed to communicate with each other, and each decides to use a mixed strategy in which each
randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and
Bertha in this game?
a.
Arthur votes for A with probability 4/8 and for B with probability 4/8. Bertha votes for A
with probability 4/8 and for B with probability 4/8.
b.
Arthur votes for A with probability 4/5 and for B with probability 1/5. Bertha votes for A
with probability 1/5 and for B with probability 4/5.
c.
Arthur votes for A with probability 1/5 and for B with probability 4/5. Bertha votes for A
with probability 4/5 and for B with probability 1/5.
d.
Arthur and Bertha each votes for A with probability 1/2 and for B with probability 1/2.
e.
Arthur votes for A and Bertha votes for B.
3. (See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them
will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second
best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of
outcome C is $0, outcome B is $1, and outcome A is $4. The money value to Bertha of outcome C is
$0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha
both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha
vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not
allowed to communicate with each other, and each decides to use a mixed strategy in which each
randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and
Bertha in this game?
a.
Arthur votes for A with probability 4/5 and for B with probability 1/5. Bertha votes for A
with probability 1/5 and for B with probability 4/5.
b.
Arthur and Bertha each votes for A with probability 1/2 and for B with probability 1/2.
c.
Arthur votes for A with probability 1/5 and for B with probability 4/5. Bertha votes for A
with probability 4/5 and for B with probability 1/5.
d.
Arthur votes for A with probability 4/8 and for B with probability 4/8. Bertha votes for A
with probability 4/8 and for B with probability 4/8.
e.
Arthur votes for A and Bertha votes for B.
4. (See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them
will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second
best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of
outcome C is $0, outcome B is $1, and outcome A is $5. The money value to Bertha of outcome C is
$0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha
both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha
vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not
allowed to communicate with each other, and each decides to use a mixed strategy in which each
randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and
Bertha in this game?
a.
Arthur and Bertha each votes for A with probability 1/2 and for B with probability 1/2.
b.
Arthur votes for A with probability 5/6 and for B with probability 1/6. Bertha votes for A
with probability 1/5 and for B with probability 4/5.
c.
Arthur votes for A with probability 1/5 and for B with probability 4/5. Bertha votes for A
with probability 5/6 and for B with probability 1/6.
d.
Arthur votes for A with probability 5/9 and for B with probability 4/9. Bertha votes for A
with probability 4/9 and for B with probability 5/9.
e.
Arthur votes for A and Bertha votes for B.
5. (See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them
will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second
best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of
outcome C is $0, outcome B is $1, and outcome A is $3. The money value to Bertha of outcome C is
$0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha
both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha
vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not
allowed to communicate with each other, and each decides to use a mixed strategy in which each
randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and
Bertha in this game?
a.
Arthur and Bertha each votes for A with probability 1/2 and for B with probability 1/2.
b.
Arthur votes for A with probability 3/4 and for B with probability 1/4. Bertha votes for A
with probability 1/5 and for B with probability 4/5.
c.
Arthur votes for A with probability 1/5 and for B with probability 4/5. Bertha votes for A
with probability 3/4 and for B with probability 1/4.
d.
Arthur votes for A with probability 3/7 and for B with probability 4/7. Bertha votes for A
with probability 4/7 and for B with probability 3/7.
e.
Arthur votes for A and Bertha votes for B.
6. (See Problem 3.) Two players are engaged in a game of “chicken.” There are two possible strategies,
Swerve and Drive Straight. A player who chooses to Swerve is called “chicken” and gets a payoff of
zero, regardless of what the other player does. A player who chooses to Drive Straight gets a payoff of
84 if the other player Swerves and a payoff of −36 if the other player also chooses to Drive Straight.
This game has two pure strategy equilibria and
a.
a mixed strategy equilibrium in which each player swerves with probability 0.30 and
drives straight with probability 0.70.
b.
two mixed strategies in which players alternate between swerving and driving straight.
c.
a mixed strategy equilibrium in which one player swerves with probability 0.30 and the
other swerves with probability 0.70.
d.
a mixed strategy in which each player swerves with probability 0.15 and drives straight
with probability 0.85.
e.
no mixed strategies.
7. (See Problem 3.) Two players are engaged in a game of “chicken.” There are two possible strategies,
Swerve and Drive Straight. A player who chooses to Swerve is called “chicken” and gets a payoff of
zero, regardless of what the other player does. A player who chooses to Drive Straight gets a payoff of
15.43 if the other player Swerves and a payoff of −36 if the other player also chooses to Drive Straight.
This game has two pure strategy equilibria and
a.
two mixed strategies in which players alternate between swerving and driving straight.
b.
a mixed strategy in which each player swerves with probability 0.35 and drives straight
with probability 0.65.
c.
a mixed strategy equilibrium in which one player swerves with probability 0.70 and the
other swerves with probability 0.30.
d.
a mixed strategy equilibrium in which each player swerves with probability 0.70 and
drives straight with probability 0.30.
e.
no mixed strategies.
8. (See Problem 3.) Two players are engaged in a game of “chicken.” There are two possible strategies,
Swerve and Drive Straight. A player who chooses to Swerve is called “chicken” and gets a payoff of
zero, regardless of what the other player does. A player who chooses to Drive Straight gets a payoff of
48 if the other player Swerves and a payoff of −48 if the other player also chooses to Drive Straight.
This game has two pure strategy equilibria and
a.
a mixed strategy equilibrium in which each player swerves with probability 0.50 and
drives straight with probability 0.50.
b.
a mixed strategy in which each player swerves with probability 0.25 and drives straight
with probability 0.75.
c.
a mixed strategy equilibrium in which one player swerves with probability 0.50 and the
other swerves with probability 0.50.
d.
two mixed strategies in which players alternate between swerving and driving straight.
e.
no mixed strategies.
9. (See Problem 3.) Two players are engaged in a game of “chicken.” There are two possible strategies,
Swerve and Drive Straight. A player who chooses to Swerve is called “chicken” and gets a payoff of
zero, regardless of what the other player does. A player who chooses to Drive Straight gets a payoff of
4 if the other player Swerves and a payoff of −36 if the other player also chooses to Drive Straight.
This game has two pure strategy equilibria and
a.
a mixed strategy equilibrium in which one player swerves with probability 0.90 and the
other swerves with probability 0.10.
b.
a mixed strategy in which each player swerves with probability 0.45 and drives straight
with probability 0.55.
c.
a mixed strategy equilibrium in which each player swerves with probability 0.90 and
drives straight with probability 0.10.
d.
two mixed strategies in which players alternate between swerving and driving straight.
e.
no mixed strategies.
10. (See Problem 3.) Two players are engaged in a game of “chicken.” There are two possible strategies,
Swerve and Drive Straight. A player who chooses to Swerve is called “chicken” and gets a payoff of
zero, regardless of what the other player does. A player who chooses to Drive Straight gets a payoff of
3 if the other player Swerves and a payoff of −12 if the other player also chooses to Drive Straight.
This game has two pure strategy equilibria and
a.
a mixed strategy equilibrium in which each player swerves with probability 0.80 and
drives straight with probability 0.20.
b.
a mixed strategy equilibrium in which one player swerves with probability 0.80 and the
other swerves with probability 0.20.
c.
a mixed strategy in which each player swerves with probability 0.40 and drives straight
with probability 0.60.
d.
two mixed strategies in which players alternate between swerving and driving straight.
e.
no mixed strategies.
11. (See Problem 6.) Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the
Trough. If both pigs choose Wait, both get 3. If both pigs Press the Button, then Big Pig gets 9 and
Little Pig gets 1. If Little Pig Presses the Button and Big Pig Waits at the Trough, then Big Pig gets 10
and Little Pig gets 0. Finally, if Big Pig Presses the Button and Little Pig Waits, then Big Pig gets 4
and Little Pig gets 1. In Nash equilibrium,
a.
Little Pig will get a payoff of 1 and Big Pig will get a payoff of 4.
b.
Little Pig will get a payoff of 1 and Big Pig will get a payoff of 9.
c.
both pigs will wait at the trough.
d.
Little Pig will get a payoff of zero.
e.
the pigs must be using mixed strategies.
12. (See Problem 6.) Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the
Trough. If both pigs choose Wait, both get 4. If both pigs Press the Button, then Big Pig gets 7 and
Little Pig gets 3. If Little Pig Presses the Button and Big Pig Waits at the Trough, then Big Pig gets 10
and Little Pig gets 0. Finally, if Big Pig Presses the Button and Little Pig Waits, then Big Pig gets 2
and Little Pig gets 5. In Nash equilibrium,
a.
both pigs will wait at the trough.
b.
Little Pig will get a payoff of zero.
c.
Little Pig will get a payoff of 3 and Big Pig will get a payoff of 7.
d.
Little Pig will get a payoff of 5 and Big Pig will get a payoff of 2.
e.
the pigs must be using mixed strategies.
13. (See Problem 6.) Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the
Trough. If both pigs choose Wait, both get 1. If both pigs Press the Button, then Big Pig gets 6 and
Little Pig gets 4. If Little Pig Presses the Button and Big Pig Waits at the Trough, then Big Pig gets 10
and Little Pig gets 0. Finally, if Big Pig Presses the Button and Little Pig Waits, then Big Pig gets 6
and Little Pig gets 2. In Nash equilibrium,
a.
Little Pig will get a payoff of 4 and Big Pig will get a payoff of 6.
b.
Little Pig will get a payoff of 2 and Big Pig will get a payoff of 6.
c.
both pigs will wait at the trough.
d.
Little Pig will get a payoff of zero.
e.
the pigs must be using mixed strategies.
14. (See Problem 6.) Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the
Trough. If both pigs choose Wait, both get 3. If both pigs Press the Button, then Big Pig gets 9 and
Little Pig gets 1. If Little Pig Presses the Button and Big Pig Waits at the Trough, then Big Pig gets 10
and Little Pig gets 0. Finally, if Big Pig Presses the Button and Little Pig Waits, then Big Pig gets 7
and Little Pig gets 4. In Nash equilibrium,
a.
Little Pig will get a payoff of 4 and Big Pig will get a payoff of 7.
b.
Little Pig will get a payoff of zero.
c.
both pigs will wait at the trough.
d.
Little Pig will get a payoff of 1 and Big Pig will get a payoff of 9.
e.
the pigs must be using mixed strategies.
15. (See Problem 6.) Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the
Trough. If both pigs choose Wait, both get 1. If both pigs Press the Button, then Big Pig gets 9 and
Little Pig gets 1. If Little Pig Presses the Button and Big Pig Waits at the Trough, then Big Pig gets 10
and Little Pig gets 0. Finally, if Big Pig Presses the Button and Little Pig Waits, then Big Pig gets 5
and Little Pig gets 1. In Nash equilibrium,
a.
Little Pig will get a payoff of zero.
b.
both pigs will wait at the trough.
c.
Little Pig will get a payoff of 1 and Big Pig will get a payoff of 9.
d.
Little Pig will get a payoff of 1 and Big Pig will get a payoff of 5.
e.
the pigs must be using mixed strategies.
16. (See Problem 7.) The old Michigan football coach had only two strategies, Run the Ball to the Left
Side of the line, and Run the Ball to the Right Side. The defense can concentrate either on the left side
or the right side of Michigan’s line. If the opponent concentrates on the wrong side, Michigan is sure
to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be
stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan
would still gain at least 5 yards with probability .70. It is the last play of the game and Michigan needs
to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan
would
a.
be sure to run to the right side.
b.
run to the right side with probability .77.
c.
run to the right side with probability .87.
d.
run to the two sides with equal probability.
e.
run to the right side with probability .70.
17. (See Problem 7.) The old Michigan football coach had only two strategies, Run the Ball to the Left
Side of the line, and Run the Ball to the Right Side. The defense can concentrate either on the left side
or the right side of Michigan’s line. If the opponent concentrates on the wrong side, Michigan is sure
to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be
stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan
would still gain at least 5 yards with probability .60. It is the last play of the game and Michigan needs
to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan
would
a.
run to the right side with probability .71.
b.
run to the right side with probability .83.
c.
be sure to run to the right side.
d.
run to the two sides with equal probability.
e.
run to the right side with probability .60.
18. (See Problem 7.) The old Michigan football coach had only two strategies, Run the Ball to the Left
Side of the line, and Run the Ball to the Right Side. The defense can concentrate either on the left side
or the right side of Michigan’s line. If the opponent concentrates on the wrong side, Michigan is sure
to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be
stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan
would still gain at least 5 yards with probability .50. It is the last play of the game and Michigan needs
to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan
would
a.
run to the two sides with equal probability.
b.
be sure to run to the right side.
c.
run to the right side with probability .67.
d.
run to the right side with probability .80.
e.
run to the right side with probability .50.
19. (See Problem 7.) The old Michigan football coach had only two strategies, Run the Ball to the Left
Side of the line, and Run the Ball to the Right Side. The defense can concentrate either on the left side
or the right side of Michigan’s line. If the opponent concentrates on the wrong side, Michigan is sure
to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be
stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan
would still gain at least 5 yards with probability .70. It is the last play of the game and Michigan needs
to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan
would
a.
run to the right side with probability .77.
b.
run to the right side with probability .87.
c.
run to the two sides with equal probability.
d.
be sure to run to the right side.
e.
run to the right side with probability .70.
20. (See Problem 7.) The old Michigan football coach had only two strategies, Run the Ball to the Left
Side of the line, and Run the Ball to the Right Side. The defense can concentrate either on the left side
or the right side of Michigan’s line. If the opponent concentrates on the wrong side, Michigan is sure
to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be
stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan
would still gain at least 5 yards with probability .70. It is the last play of the game and Michigan needs
to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan
would
a.
run to the right side with probability .77.
b.
be sure to run to the right side.
c.
run to the right side with probability .87.
d.
run to the two sides with equal probability.
e.
run to the right side with probability .70.
21. Suppose that in the Hawk-Dove game discussed in Problem 8, the payoff to each player is −9 if both
play Hawk. If both play Dove, the payoff to each player is 5, and if one plays Hawk and the other
plays Dove, the one that plays Hawk gets a payoff of 9 and the one that plays Dove gets 0. In
equilibrium, we would expect hawks and doves to do equally well. This happens when the proportion
of the total population that plays Hawk is
a.
.31.
b.
.15.
c.
.08.
d.
.65.
e.
1.
22. Suppose that in the Hawk-Dove game discussed in Problem 8, the payoff to each player is −7 if both
play Hawk. If both play Dove, the payoff to each player is 4, and if one plays Hawk and the other
plays Dove, the one that plays Hawk gets a payoff of 7 and the one that plays Dove gets 0. In
equilibrium, we would expect hawks and doves to do equally well. This happens when the proportion
of the total population that plays Hawk is
a.
.08.
b.
.30.
c.
.65.
d.
.15.
e.
1.
23. Suppose that in the Hawk-Dove game discussed in Problem 8, the payoff to each player is −7 if both
play Hawk. If both play Dove, the payoff to each player is 5, and if one plays Hawk and the other
plays Dove, the one that plays Hawk gets a payoff of 9 and the one that plays Dove gets 0. In
equilibrium, we would expect hawks and doves to do equally well. This happens when the proportion
of the total population that plays Hawk is
a.
.36.
b.
.68.
c.
.09.
d.
.18.
e.
1.
24. Suppose that in the Hawk-Dove game discussed in Problem 8, the payoff to each player is −3 if both
play Hawk. If both play Dove, the payoff to each player is 1, and if one plays Hawk and the other
plays Dove, the one that plays Hawk gets a payoff of 5 and the one that plays Dove gets 0. In
equilibrium, we would expect hawks and doves to do equally well. This happens when the proportion
of the total population that plays Hawk is
a.
.14.
b.
.57.
c.
.79.
d.
.29.
e.
1.
25. Suppose that in the Hawk-Dove game discussed in Problem 8, the payoff to each player is −8 if both
play Hawk. If both play Dove, the payoff to each player is 3, and if one plays Hawk and the other
plays Dove, the one that plays Hawk gets a payoff of 8 and the one that plays Dove gets 0. In
equilibrium, we would expect hawks and doves to do equally well. This happens when the proportion
of the total population that plays Hawk is
a.
.19.
b.
.69.
c.
.38.
d.
.10.
e.
1