CHAPTER 29: Game Theory
TRUE/FALSE
1. A situation where everyone is playing a dominant strategy must be a Nash equilibrium.
2. In a Nash equilibrium, everyone must be playing a dominant strategy.
3. In the prisoner’s dilemma game, if each prisoner believed that the other prisoner would deny the
crime, then both would deny the crime.
4. A general has the two possible pure strategies, sending all of his troops by land or sending all of his
troops by sea. An example of a mixed strategy is where he sends 1/4 of his troops by land and 3/4 of
his troops by sea.
5. While game theory predicts noncooperative behavior for a single play of the prisoner’s dilemma, it
would predict cooperative tit-for-tat behavior if the same people play prisoner’s dilemma together for,
say, 20 rounds.
6. A two-person game in which each person has access to only two possible strategies will have at most
one Nash equilibrium.
7. A dominant strategy equilibrium is a set of choices such that each player’s choices are optimal
regardless of what the other players choose.
8. In Nash equilibrium, each player is making an optimal choice for herself, given the choices of the
other players.
9. If a game does not have an equilibrium in pure strategies, then it will not have an equilibrium in mixed
strategies either.
10. A game has two players and each has two strategies. The strategies are Be Nice and Be Mean. If both
players play Be Nice, both get a payoff of 5. If both players play Be Mean, both get a payoff of 23. If
one player plays Be Nice and the other plays Be Mean, the player who played Be Nice gets 0 and the
player who played Be Mean gets 10. Playing Be Mean is a dominant strategy for both players.
MULTIPLE CHOICE
1. A game has two players. Each player has two possible strategies. One strategy is Cooperate, the other
is Defect. Each player writes on a piece of paper either a C for cooperate or a D for defect. If both
players write C, they each get a payoff of $100. If both players write D, they each get a payoff of 0. If
one player writes C and the other player writes D, the cooperating player gets a payoff of S and the
defecting player gets a payoff of T. To defect will be a dominant strategy for both players if
a.
S + T 100.
b.
T 2S.
c.
S 0 and T 100.
d.
S T and T 100.
e.
S and T are any positive numbers.
2. In the game matrix below, the first payoff in each pair goes to player A who chooses the row, and the
second payoff goes to player B, who chooses the column. Let a, b, c, and d be positive constants.
Player B
Left
Right
Player A
Top
a,1
b,1
Bottom
1,c
1,d
If player A chooses Bottom and player B chooses Right in a Nash equilibrium, then we know that
a.
b 1 and d 1.
b.
c 1 and b 1.
c.
b 1 and c d.
d.
b c and d 1.
e.
a 1 and b d.
3. In the town of Torrelodones, each of the N 2 inhabitants has $100. They are told that they can all
voluntarily contribute to a fund that will be evenly divided among all residents. If $F are contributed to
the fund, the local K-Mart will match the private contributions so that the total amount to be divided is
$2F. That is, each resident will get back a payment of $2F/N when the fund is divided. If the people in
town care only about their own net incomes, in Nash equilibrium, how much will each person
contribute to the fund?
a.
$0
b.
$10
c.
$20
d.
$50
e.
$100
4. Frank and Nancy met at a sorority sock hop. They agreed to meet for a date at a local bar the next
week. Regrettably, they were so fraught with passion that they forgot to agree on which bar would be
the site of their rendezvous. Luckily, the town has only two bars, Rizotti’s and the Oasis. Having
discussed their tastes in bars at the sock hop, both are aware that Frank prefers Rizotti’s to the Oasis
and Nancy prefer the Oasis to Rizotti’s. In fact, the payoffs are as follows. If both go to the Oasis,
Nancy’s utility is 3 and Frank’s utility is 2. If both go to Rizotti’s, Frank’s utility is 3 and Nancy’s
utility is 2. If they don’t both go to the same bar, both have a utility of 0.
a.
This game has no Nash equilibrium in pure strategies.
b.
This game has a dominant strategy equilibrium.
c.
There are two Nash equilibrium in pure strategies and a Nash equilibrium in mixed
strategies where the probability that Frank and Nancy go to the same bar is 12/25.
d.
This game has two Nash equilibria in pure strategies and a Nash equilibrium in mixed
strategies where each person has a probability of 1/2 of going to each bar.
e.
This game has exactly one Nash equilibrium.
5. George and Sam have taken their fathers’ cars out on a lonely road and are engaged in a game of
Chicken. George has his father’s Mercedes and Sam has his father’s rattly little Yugoslavian-built
subcompact car. Each of the players can choose either to Swerve or to Not Swerve. If both choose
Swerve, both get a payoff of zero. If one chooses Swerve and the other chooses Not Swerve, the one
who chooses Not Swerve gets a payoff of 10 and the one who chooses Swerve gets zero. If both
choose Not Swerve, the damage to George’s car is fairly minor and he gets a payoff of −5, while for
Sam the results are disastrous and he gets a payoff of −100.
a.
This game has a dominant strategy equilibrium in which George does not swerve and Sam
swerves.
b.
This game has two pure strategy Nash equilibria and no mixed strategy equilibrium.
c.
This game has three different Nash equilibria, two of which are pure strategy equilibrium
and one of which is a mixed strategy equilibrium in which George is more likely to swerve
than Sam is.
d.
The one and only Nash equilibrium in this game is where George does not swerve and
Sam swerves.
e.
This game has two pure strategy equilibria and a mixed strategy equilibrium in which Sam
randomizes his strategy and George chooses Not Swerve with certainty.
6. Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough. If both
pigs choose Wait at the Trough, both get 3. If both pigs choose Press the Button, then Big Pig gets 8
and Little Pig gets 2. 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 at the trough, then
Big Pig gets 2 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 2.
b.
Little Pig will get a payoff of 2 and Big Pig will get a payoff of 8.
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.
7. Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough. If both
pigs choose Wait at the Trough, both get 2. If both pigs choose Press the Button, then Big Pig gets 5
and Little Pig gets 5. 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 at the trough, then
Big Pig gets 6 and Little Pig gets 2. In Nash equilibrium,
a.
Little pig will get a payoff of zero.
b.
Little Pig will get a payoff of 5 and Big Pig will get a payoff of 5.
c.
both pigs will wait at the trough.
d.
Little Pig will get a payoff of 2 and Big Pig will get a payoff of 6.
e.
the pigs must be using mixed strategies.
8. Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough. If both
pigs choose Wait at the Trough, both get 2. If both pigs choose Press the Button, then both pigs get 5.
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 at the trough, then Big Pig gets 3 and Little
Pig gets 2. In Nash equilibrium,
a.
both pigs will wait at the trough.
b.
Little Pig will get a payoff of 2 and Big Pig will get a payoff of 3.
c.
Little pig will get a payoff of zero.
d.
Little Pig will get a payoff of 5 and Big Pig will get a payoff of 5.
e.
the pigs must be using mixed strategies.
9. 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 32 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 .60 and drives
straight with probability .40.
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 .60 and the
other swerves with probability .40.
d.
a mixed strategy in which each player swerves with probability .30 and drives straight
with probability .70.
e.
no mixed strategies.
10. 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 432 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 .10 and drives
straight with probability .90.
b.
a mixed strategy in which each player swerves with probability .05 and drives straight
with probability .95.
c.
a mixed strategy equilibrium in which one player swerves with probability .10 and the
other swerves with probability .90.
d.
two mixed strategies in which players alternate between swerving and driving straight.
e.
no mixed strategies.
11. 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 36 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 .50 and drives
straight with probability .50.
b.
a mixed strategy equilibrium in which one player swerves with probability .50 and the
other swerves with probability .50.
c.
a mixed strategy in which each player swerves with probability .25 and drives straight
with probability .75.
d.
two mixed strategies in which players alternate between swerving and driving straight.
e.
no mixed strategies.
12. A famous Big Ten 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 forces on the left side or the right side. If the
opponent concentrates on the wrong side, his offense is sure to gain at least 5 yards. If the defense
defended the left side and the offense ran left, the offense gained only 1 yard. If the opponent defended
the right side when the offense ran right, the offense would still gain at least 5 yards with probability
.30. It is the last play of the game and the famous coach’s team is on offense. If it makes 5 yards or
more, it wins; if not, it loses. Both sides choose Nash equilibrium strategies. In equilibrium the offense
a.
is sure to run to the right side.
b.
will run to the right side with probability .59.
c.
will run to the right side with probability .74.
d.
will run to the two sides with equal probability.
e.
will run to the right side with probability .70.
13. A famous Big Ten 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 forces on the left side or the right side. If the
opponent concentrates on the wrong side, his offense is sure to gain at least 5 yards. If the defense
defended the left side and the offense ran left, the offense gained only 1 yard. If the opponent defended
the right side when the offense ran right, the offense would still gain at least 5 yards with probability
.70. It is the last play of the game and the famous coach’s team is on offense. If it makes 5 yards or
more, it wins; if not, it loses. Both sides choose Nash equilibrium strategies. In equilibrium the offense
a.
will run to the two sides with equal probability.
b.
will run to the right side with probability .87.
c.
is sure to run to the right side.
d.
will run to the right side with probability .77.
e.
will run to the right side with probability .70.
14. A famous Big Ten 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 forces on the left side or the right side. If the
opponent concentrates on the wrong side, his offense is sure to gain at least 5 yards. If the defense
defended the left side and the offense ran left, the offense gained only 1 yard. If the opponent defended
the right side when the offense ran right, the offense would still gain at least 5 yards with probability
.50. It is the last play of the game and the famous coach’s team is on offense. If it makes 5 yards or
more, it wins; if not, it loses. Both sides choose Nash equilibrium strategies. In equilibrium the offense
a.
will run to the right side with probability .67.
b.
will run to the right side with probability .80.
c.
will run to the two sides with equal probability.
d.
is sure to run to the right side.
e.
will run to the right side with probability .50.
15. Suppose that in a Hawk-Dove game similar to the one discussed in your workbook, the payoff to each
player is −6 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.
.45.
b.
.23.
c.
.11.
d.
.73.
e.
1.
16. Suppose that in a Hawk-Dove game similar to the one discussed in your workbook, the payoff to each
player is −9 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 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.
.10.
b.
.55.
c.
.05.
d.
.03.
e.
1.
17. Suppose that in a Hawk-Dove game similar to the one discussed in your workbook, the payoff to each
player is −6 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 6 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.
.25.
b.
.63.
c.
.06.
d.
.13.
e.
1.
18. If the number of persons who attend the club meeting this week is X, then the number of people who
will attend next week is 63 + 0.30X. What is a long-run equilibrium attendance for this club?
a.
63
b.
210
c.
126
d.
90
e.
27
19. If the number of persons who attend the club meeting this week is X, then the number of people who
will attend next week is 80 + 0.20X. What is a long-run equilibrium attendance for this club?
a.
80
b.
400
c.
160
d.
100
e.
20
20. If the number of persons who attend the club meeting this week is X, then the number of people who
will attend next week is 90 + 0.40X. What is a long-run equilibrium attendance for this club?
a.
225
b.
150
c.
180
d.
90
e.
60
21. Professor Binmore has a monopoly in the market for undergraduate game theory textbooks. The
time-discounted value of Professor Binmore’s future earnings is $2,000. Professor Ditt is considering
writing a book to compete with Professor Binmore’s book. With two books amicably splitting the
market, the time-discounted value of each professor’s future earnings would be $200. If there is full
information (each professor knows the profits of the other), under what conditions could Professor
Binmore deter the entry of Professor Ditt into his market?
a.
Professor Binmore threatens to cut his price so that Professor Ditt would loose $200. In so
doing, Professor Binmore would loose $20 over time.
b.
Professor Binmore threatens to cut his price so that Professor Ditt would loose $20. In so
doing, Professor Binmore would just break even over time.
c.
Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt’s
book so that Professor Ditt would loose $2. In so doing, Professor Binmore would still
make $190 over time.
d.
Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt’s
book so that Professor Ditt would only make $2. In so doing, Professor Binmore would
still make $100 over time.
e.
None of the above.
22. Professor Binmore has a monopoly in the market for undergraduate game theory textbooks. The
time-discounted value of Professor Binmore’s future earnings is $2,000. Professor Ditt is considering
writing a book to compete with Professor Binmore’s book. With two books amicably splitting the
market, the time-discounted value of each professor’s future earnings would be $200. If there is full
information (each professor knows the profits of the other), under what conditions could Professor
Binmore deter the entry of Professor Ditt into his market?
a.
Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt’s
book so that Professor Ditt would loose $8. In so doing, Professor Binmore would still
make $210 over time.
b.
Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt’s
book so that Professor Ditt would only make $8. In so doing, Professor Binmore would
still make $100 over time.
c.
Professor Binmore threatens to cut his price so that Professor Ditt would loose $800. In so
doing, Professor Binmore would loose $80 over time.
d.
Professor Binmore threatens to cut his price so that Professor Ditt would loose $80. In so
doing, Professor Binmore would just break even over time.
e.
None of the above.
23. Professor Binmore has a monopoly in the market for undergraduate game theory textbooks. The
time-discounted value of Professor Binmore’s future earnings is $4,000. Professor Ditt is considering
writing a book to compete with Professor Binmore’s book. With two books amicably splitting the
market, the time-discounted value of each professor’s future earnings would be $400. If there is full
information (each professor knows the profits of the other), under what conditions could Professor
Binmore deter the entry of Professor Ditt into his market?
a.
Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt’s
book so that Professor Ditt would loose $8. In so doing, Professor Binmore would still
make $410 over time.
b.
Professor Binmore threatens to cut his price so that Professor Ditt would loose $800. In so
doing, Professor Binmore would loose $80 over time.
c.
Professor Binmore threatens to cut his price so that Professor Ditt would loose $80. In so
doing, Professor Binmore would just break even over time.
d.
Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt’s
book so that Professor Ditt would only make $8. In so doing, Professor Binmore would
still make $200 over time.
e.
None of the above.
PROBLEM
1. The coach of the offensive football team has two options on the next play. He can run the ball or he
can pass. His rival can defend either against the run or against the pass. Suppose that the offense
passes. Then if the defense defends against the pass, the offense will make zero yards, and if the
defense defends against the run, the offense will make 25 yards. Suppose that the offense runs. If the
defense defends against the pass, the offense will make 10 yards, and if the defense defends against a
run, the offense will gain 2 yards.
a. Write down a payoff matrix for this game.
b. Is there a Nash equilibrium in pure strategies for this game? If so, what is it? If not, demonstrate
that there is none.
