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PS 30:
Sections 1I, 1J, & 1K
Practice Problems for Midterm
This is designed to give you more problem sets, which will help prepare for final and,
more importantly, understand basic game theory. In short, these are “practice problems
for midterm,” rather than “practice midterm.”
1. Two candidates, Incumbent and Challenger, are competing for the same office. Each
faces
a
choice
of
whether
to
run
a
positive
campaign
emphasizing
their
own
accomplishments or a negative campaign that emphasizes their opponent’s shortcomings.
If one runs a negative campaign and the other runs a positive campaign, the one that ran
the negative campaign will win. If they both run the same kind of campaign, Incumbent
will win.
Winning the election is worth 5 units of payoff to either candidate.
Neither
candidates likes running a negative campaign, however. Each candidate loses 2 units of
payoff if they run a negative campaign.
(a) Suppose that Challenger decides her campaign strategy first. Draw a game tree and
solve it using backwards induction.
What do you predict to happen?
Transform the
game tree into a strategic form.
Find PSNE and SPE.
(b) Now suppose that Incumbent decides her strategy before Challenger does.
Draw a
game tree and solve it. What do you predict to happen? Transform the game tree into a
strategic form.
Find PSNE and SPE.
(c) Now suppose that Challenger and Incumbent decide their strategies simultaneously.
Draw a matrix and find all Nash equilibria (PSNE and MSNE if they exist). What is the
probability of Challenger winning?
(d) Suppose again that Challenger moves first, but now Incumbent is “extra ethical” and
loses six (rather than two) units of payoff from running a negative campaign.
Draw a
game tree that incorporates this change.
What will Challenger choose?
Transform the
game tree into a strategic form.
Find PSNE and SPE.
(e) Suppose that Challenger and Incumbent decide their strategies simultaneously and
Incumbent is “extra ethical.”
Draw a matrix and find all Nash equilibria (PSNE and
MSNE if they exist).
1
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 Fall '07
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