Problems for the first seminar: Static games
ECON4240 Game Theory and Economics of Information — Spring semester 2011
Solutions to the problems will be presented Wed 26/Thu 27 January 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 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:
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• 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.
Problem 5 (Reporting a crime)
A crime is observed by a group of n people. Each person would like the
police to be informed but prefers that someone else make the phone call. Specifically, suppose that each person attaches the value v to the police being informed and bears the cost c if she makes the phone call, where v > c > 0.
(a) Model this situation as normal form game.
(b) Find the set of pure strategy Nash equilibria.
(c) Find the symmetric mixed strategy Nash equilibrium.
(d) In the symmetric mixed strategy Nash equilibrium, how does the probability that the police will be called vary with the number of people
n?
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