You should answer all four questions. The weight of each question is indicated below.

1. [30 marks] Consider the two player game described by the following matrix:

where the payoff of the two players for the action profile (B,L) is given by x, left unspecified.

• [10 marks] Find for which values of x action T is strictly Can you find values of x for which action B is strictly dominated?
• [10 marks] Characterize the set of pure strategy Nash equilibria as a function of the parameter
• [10 marks] Find the values of x for which a mixed strategy Nash equilibrium

2. [25 marks] Consider the following game with simultaneous moves:

Player 2

Player 1

The game is repeated twice (that is, played twice in total). Players have common

discount factor δ>0.

• [10 marks] Construct a subgame-perfect Nash equilibrium in which players play the same action profile in each period and after any
• [15 marks] Find the values of δ for which there exists a subgame-perfect Nash equilibrium in which:
  • players play (U,R) in the first period
  • In the second period, players play (U,L) if (U,R) was played in the first period, otherwise they play (D,R).

3. [25 marks] Consider the following game: two firms have to choose simultaneously where to produce a high H) or a low (L) quantity of The payoffs are described by the following matrix:

Firm 1

The game is played repeatedly T times. The two firms have common discount factor 1>δ>0.

• [5 marks] Describe the unique subgame perfect equilibrium when T is
• [10 marks] Suppose the game is played infinitely often (T is infinite). Derive the values of δ for which there exists a subgame perfect equilibrium where each firm obtains a payoff of 5 at each Describe the strategies of the players at such equilibrium.
• [10 marks] Suppose the game is played infinitely often (T is infinite). Construct a subgame perfect equilibrium such that in the limit as δ→1, each firm’s equilibrium average payoff equals

4. [20 marks] Consider the following sequential bargaining game between two players for the division of a pie of size Player 1 makes an offer to player 2, that is proposes an amount x to player 2. If player 2 accepts the game ends and player 1 gets 10-x and player 2 x. If player 2 rejects, player 2 makes an offer to player 1, proposing an amount y to player 1. If player 1 accepts player 1 gets δy and player 2 gets δ(10-y). If player 1 rejects, the game ends.
   • [10 marks] Find a subgame perfect equilibrium when δ=0.6. Is this the unique subgame perfect equilibrium of the game?
   • [10 marks] Can you find a Nash equilibrium where both players get a payoff of 5?