Question 1

Find the Nash equilibria of the following zero-sum games.

Player I/Player II 1 2 a) 1 96

2 110

Player I/Player II 1

2

1 2 3 4 10 6 10 13 8 -3 6 12

b)

Question 2

Find the Nash equilibria of the following nonzero-sum games.

3 10993

a)

b)

Player 1/Player 2 1

2

1 2 (7,6) (12,5) (6,-6) (-3,9)

Player 1/Player 2 1 2 3

1 2 3

(7,3) (4,1) (4,5) (3,7) (2,3) (3,4) (0,2) (2,3) (5,3)

Question 3

Two players play a sequential game, where A starts, then B plays, then A plays again, and finally B plays. They start with 12 units of resource between them. At each point, eitherr player can play Pass or Take. If a player plays Take, they take two units of resource, but then a further unit is lost from the total. If they play Pass, the total stays the same. At the end of the game, the remaining units are shared between the two players.

a) Formulate this game as an extensive form game, and then solve it.

b) How would the optimal strategy change if now the remaining units were not shared, but each player received the number remaining?

Question 4

Consider the Cournot game with price function P (q) = Γ − q − q2 and costs c1 = c2 = c. Find the Nash equilibrium of this game.

Question 5

Find all of the evolutionarily stable strategies of the following matrices. In matrix c) k is arbitrary, and you should find the solution for general k, but you may ignore the non-generic cases at the boundaries where the solutions change:

3602−1 k24 a) 7 5 b)−1 0 3 c)3 k 2.

3 −1 0 1 6 k

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