Problems for the sixth seminar: Static games
ECON3200 Microeconomics and game theory — Fall semester 2011
Solutions to the problems will be presented 20–26 October 2011.
Please direct any questions to Geir B. Asheim (Room ES1152, Tel: 228 55498,
e-mail: g.b.asheim@econ.uio.no)
Problem 1 (Best response sets)
Watson Exercise 6.4
Problem 2 (Rationalizability)
Watson Exercise 7.3
Problem 3 (Best response property / best response complete)
Watson Exercise 9.6
Problem 4 (Choosing how to dress)
You are going to meet a friend at a party and you care about the colors that
you wear and your friend wears. Provided that your friend wears a different
color from you, your payoff is 4 for wearing blue, 3 for wearing green, 2 for
wearing red, and 1 for wearing yellow. However, if your friend wears the same
color as you, your payoff is 0, independently of what color you and your friend
wear. The payoffs for your friend are as follows: Provided that you wear a
different color from your friend, your friend’s payoff is 4 for wearing red, 3 for
wearing yellow, 2 for wearing blue, and 1 for wearing green. However, if you
wear the same color as your friend, your friend’s payoff is 0, independently of
what color you and your friend wear.
(a) Represent this strategic situation as a normal form game.
(b) Is there any belief about the behavior of your friend for which yellow is
a best response for you?
(c) Specify a belief about the behavior of your friend for which red is a best
response for you.
(d) Which of your strategies is/are rationalizable?
(e) Determine the Nash equilibrium/a of this game.
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Problem 5 (Choosing a route)
Four people must drive from A to B at the same time. Two routes are
available, one via X (A → X → B) and one via Y (A → Y → B).
The driving time for each car on each road is a function of how many cars
are using this road. The driving times are as follows:
• A → X: 6 minutes if 1 car, 9 minutes if 2 cars, 12 minutes if 3 cars, 15
minutes if 4 cars.
• X → B: 20 minutes if 1 car, 20.9 minutes if 2 cars, 21.8 minutes if 3 cars,
22,7 minutes if 4 cars.
• A → Y: 20 minutes if 1 car, 21 minutes if 2 cars, 22 minutes if 3 cars, 23
minutes if 4 cars.
• Y → B: 6 minutes if 1 car, 9 minutes if 2 cars, 12 minutes if 3 cars, 15
minutes if 4 cars.
(a) Formulate this as a normal form game and find the Nash equilibria.
Suppose now that a new short and wide road is built between X and Y,
implying that there is now a third route between A and B, namely A → X →
Y → B. The driving times on the new road are as follows:
• X → Y: 7 minutes if 1 car, 8 minutes if 2 cars, 9 minutes if 3 cars, 10
minutes if 4 cars.
(b) Find the Nash equilibria after the new road has been built.
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