Game Theory Homework

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1. Consider a “divide four objects game” in which player 1 proposes a division scheme ((4,0),

(3,1), (2,2), (1,3) or (0,4)) and player 2 either accepts it or rejects. In the former case, the

proposed division takes place and in the latter both players get nothing.

a) Find the number of different pure strategy Nash equilibria of this game.

b) Find the number of SPE among these Nash equilibria.

2. Two players are engaged in a infinitely repeated game whose stage game is shown below.

a) Find all feasible individually rational payoffs.

b) Is payoff (5,5) possible in an SPE (assuming _ is sufficiently large)? Why?

c) Can you suggest a simple trigger strategy achieving payoff (5,5)? (Hint: find NE of

the stage game and recall the weak perfect folk theorem.)

d) What is the smallest value of _ making this simple trigger strategy possible?

3. Consider a modified public good game between four players in which for the public good

to be available at least two players out of the four have to contribute. If the public good

is available each player gets a payoff of 1 from it. The cost of contributing for player i is

equal to ci. Each player knows his or her own cost and has incomplete information about

the other players costs believing that the cost of each opponent is uniformly distributed on

the interval [0; 1] independently of each other. Find all symmetric PBE of this game.

4. In a multi-period chain store game with 30 cities, the payoffs of all players are just like

described in class with a = 2 and b = 0:5.

a) Suppose the local store in city 2 (counting from the end) was the first to enter in a

PBE. What is the possible range of the parameter pc in this problem?

b) What is the expected payoff of the chain store in this game?