AMS 335/ECO 355
PROBLEM SET 3
Due Mon, June 6, 11:59 pm EST.
Solve the following exercises from the textbook:
• 5.5 Public Good Contribution
5.5 Public Good Contribution: Three players live in a town, and each can choose
to contribute to fund a streetlamp. The value of having the streetlamp is 3 for
each player, and the value of not having it is 0. The mayor asks each player to
contribute either 1 or nothing. If at least two players contribute then the lamp
will be erected. If one player or no players contribute then the lamp will not
be erected, in which case any person who contributed will not get his money
back.
a. Write out or graph each player’s best-response correspondence.
b. What outcomes can be supported as pure-strategy Nash equilibria?
• 5.10 Synergies
Synergies: Two division managers can invest time and effort in creating a
better working relationship. Each invests ei ≥ 0, and if both invest more then
both are better off, but it is costly for each manager to invest. In particular
the payoff function for player i from effort levels (ei, ej ) is vi(ei, ej ) =
(a + ej )ei − e2
i .
a. What is the best-response correspondence of each player?
b. In what way are the best-response correspondences different from
those in the Cournot game? Why?
c. Find the Nash equilibrium of this game and argue that it is unique.
• 5.18 Political Campaigning
Political Campaigning: Two candidates are competing in a political race.
Each candidate i can spend si ≥ 0 on ads that reach out to voters, which in
turn increases the probability that candidate i wins the race. Given a pair of
spending choices (s1, s2), the probability that candidate i wins is given by si
s1+s2
. If neither spends any resources then each wins with probability 1
2 . Each
candidate values winning at a payoff of v > 0, and the cost of spending si is
just si.
a. Given two spend levels (s1, s2), write down the expected payoff of a
candidate i.
b. What is the function that represents each player’s best-response function?
c. Find the unique Nash equilibrium.
d. What happens to the Nash equilibrium spending levels if v increases?
e. What happens to the Nash equilibrium levels if player 1 still values
winning at v but player 2 values winning at kv, where k > 1?
(http://assets.press.princeton.edu/releases/tadelis/game-theory-tadelis-part2.pdf)
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