1. A football team has the chance of scoring the game-winning touchdown on the last play of the game. It can either run
or pass. The defense can play for the run or play for the pass. The following normal form lists the payoffs from the game
made up by this last play. The payoffs are probabilities of winning the game.
Defense
Defend Pass
Defend Run
Offense
Pass
0, 1
1, 0
Run
1, 0
0, 1
What is the defense’s dominant strategy?
a.
Defend the run.
b.
Defend the pass.
c.
Defend both plays at random with equal probability.
d.
The defense does not have a dominant strategy.
2. Consider the same football situation as in the previous question, but now suppose the probabilities of winning are less
extreme than before, given by the following normal form:
Defense
Defend Pass
Defend Run
Offense
Pass
.2, .8
.8, .2
Run
.8, .2
.2, .8
What is the defense’s dominant strategy?
a.
Defend the run.
b.
Defend the pass.
c.
Defend both play at random with equal probability.
d.
The defense does not have a dominant strategy.
d
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3. Consider the same football situation as in the previous question, but now suppose the payoffs (probabilities of winning)
are as given in the following normal form:
Defense
Defend Pass
Defend Run
Offense
Pass
.2, .8
.3, .7
Run
.5, .5
.4, .6
Which team has a dominant strategy?
a.
Neither.
d
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b.
Both.
c.
Only the offense.
d.
Only the defense.
4. A football team has the chance of scoring the game-winning touchdown on the last play of the game. It can either run
or pass. The defense can play for the run or play for the pass. The following normal form lists the payoffs from the game
made up by this last play. The payoffs are probabilities of winning the game.
Defense
Defend Pass
Defend Run
Offense
Pass
0, 1
1, 0
Run
1, 0
0, 1
The Nash equilibrium is (equilibria are)
a.
the offense runs and the defense plays for the run.
b.
the offense passes and the defense plays for the pass.
c.
the offense runs and the defense plays for the pass.
d.
both pick their choice at random with equal probability.
d
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5. Consider the same football setting as in the previous question, but now suppose the payoffs (probabilities of winning)
are given by the following normal form:
Defense
Defend Pass
Defend Run
Offense
Pass
.2, .8
.9, .1
Run
.5, .5
.4, .6
The Nash equilibrium is (equilibria are)
a.
the offense runs and the defense plays for the run.
b.
both pick their choice at random with the offense focusing more on the run and the defense on defending the
pass.
c.
both pick their choice at random with the offense focusing more on the pass and the defense on defending the
run.
d.
both pick their choice at random with equal probability of either play.
b
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6. Teens A and B are smitten with each other but neither knows of the other’s feelings. Suppose the teachers at their
school organize a dance. The “payoff” is based on whether their advances are rebuffed or accepted. If they both Declare,
they get positive utility but if they are Rebuffed they face humiliation (significantly negative payoff). Rebuffing an
advance slightly elevates the teen’s standing with peers.
Teen B
Declare
Ignore/
Rebuff
c
1
Teen A
Declare
10, 10
−10, 1
Ignore/
Rebuff
1, −10
0, 0
What is a teen’s dominant strategy?
a.
Declare.
b.
choose at random.
c.
Rebuff/Ignore.
d.
No dominant strategy.
7. Consider the game between the teens from the previous question. The pure-strategy Nash equilibrium is (equilibria are)
a.
Both Declare.
b.
Both Ignore/Rebuff.
c.
There are two: in one, both Declare, and in the other, both Rebuff/Ignore.
d.
There are two: in both, the teens do the opposite of each other.
c
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8. Consider the game between the teens from the previous question. In addition to any pure-strategy Nash equilibrium,
there is another one in mixed strategies. In it, each teen chooses to declare with probability
a.
0.52
b.
0.5
c.
0.34
d.
0.1
a
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9. Consider the game between the teens from the previous question. Instead of being a simultaneous game, suppose it is
sequential, with teen A moving first. What is the subgame-perfect equilibrium of this new game?
a.
Both Declare.
b.
Both Ignore/Rebuff.
c.
It is a mixed strategy equilibrium.
d.
Teen A Declares and Teen B follows A’s action.
d
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10. A teacher curves the final exam such that the top half of students get an A and the bottom half an F (so their grade
depends only on relative and not absolute performance). Suppose that there are equal numbers of two groups, the
Brainiacs and the Numbskulls. If they both study or they both party, the Brainiacs will get the As but if the Brainiacs
party and the Numbskulls study, the Numbskulls will get the As. Suppose further that they both dislike studying and both
like good grades. Suppose all students of a type choose the same action (so we can view it as a two-player game). The
payoff matrix is
Numbskulls
Study
Party
Study
5, 0
5, 2
d
1
Brainiacs
Party
2, 5
7, 2
Characterize the Nash equilibrium or equilibria:
a.
There is only one equilibrium, in mixed strategies.
b.
The Brainiacs study and the Numbskulls party.
c.
There are two: in one, the Brainiacs study and the Numbskulls party, and in the other they do the reverse.
d.
Both types party.
11. Consider the same setup with the curved final exam as in the previous question. Now suppose the students move
sequentially. Which best describes the outcome in the subgame-perfect equilibrium?
a.
Brainiacs study and Numbskulls party regardless of who moves first.
b.
Brainiacs party and Numbskulls study regardless of who moves first.
c.
The party moving first studies.
d.
The party moving first parties.
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12. A lake supports a delicious variety of fish. The land around the lake is owned by two fisherman, so by state law both
are free to fish as much as they like. Let FA and FB be the number of fish each catches, respectively. Suppose the price of
fish is 100 – FA – FB. Given this demand curve, it turns out that marginal revenue is 100 – 2FA – FB for fisherman A and
100 – FA – 2FB for fisherman B. What is the total number of fish by both in the Nash equilibrium?
a.
30
b.
45
c.
60
d.
90
1
13. Consider the same game between the fishermen in the previous question. How does the total number of fish caught in
the Nash equilibrium compare to the number they would catch if they belonged to same company, so shared revenues and
costs equally?
a.
No change.
b.
They would catch more if they worked in the same company.
c.
They would catch less if they worked in the same company.
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14. Return to the version of the game between the fishermen in which they fish independently. If the marginal cost for
just fisherman A went up, what would be the likely effect on the Nash equilibrium?
a.
A would catch more fish, and B would catch fewer.
b.
A would catch fewer fish, and B would catch more.
c.
A would catch more fish, but B’s catch would not change.
d.
A would catch fewer fish, but B’s catch would not change.
b
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15. When a game has multiple equilibria, a useful method to sort out which one would be the “best” prediction is to
a.
find the one (if any) in which both players are better off than in any other equilibrium.
b.
find the equilibrium that is symmetric, if any.
c.
find the one which seems “focal”, if any.
d.
all of the above.
d
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16. Best-response functions
a.
can only be used to analyze games with continuous actions.
b.
are always downward sloping.
c.
always intersect at Nash equilibria.
d.
never intersect.
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17. The Prisoners’ Dilemma is so named because
a.
the Nash equilibrium is one of the worst outcomes for the players.
b.
the game has no Nash equilibrium.
c.
the game is zero-sum.
d.
players end up earning a lighter sentence than the prosecutor would like.
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18. If the Prisoners’ Dilemma is repeated over and over again with the same two players having an indefinite time
horizon,
a.
the unique equilibrium is to play the Nash equilibrium of Rat each period.
b.
players can cooperate on Silent if they are not too patient.
c.
players can cooperate on Silent if they are patient enough.
d.
players can only cooperate on Silent in the initial stages of the game.
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19. The game of Matching Pennies
a.
has no Nash equilibrium.
b.
has a pure-strategy Nash equilibrium.
c.
has a mixed strategy Nash equilibrium.
d.
has multiple Nash equilibria.
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20. The beauty of Nash’s equilibrium concept is that
a.
all games have one.
b.
all games have no more than one.
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c.
all games have a rich set to choose from.
d.
it is a Pareto optimum.
21. A subgame-perfect equilibrium is a Nash equilibrium that
a.
cannot persist through several periods.
b.
involves only credible threats.
c.
consists only of dominant strategies.
d.
is unique.
b
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22. In what way or ways can strategies more complicated than simple actions?
a.
They can be conditioned on a first mover’s action.
b.
They can involve random choices.
c.
Both a and b.
d.
Neither a or b.
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23. Two games that differ only in the timing of moves—one simultaneous, the other sequential move—can sometimes
have completely different subgame-perfect equilibria. Why?
a.
The second mover to choose non-credible threats.
b.
The first mover can choose an action that it would deviate from if its action were secret.
c.
Subgame-perfect equilibrium cannot be applied to simultaneous games.
d.
All of the above.
b
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