Not an Essay, Game theory based Micro Questions
EC322 MICROECONOMICS (ADVANCED) Takehome assignment: You should answer all four questions. The weight of each question is indicated below. You should upload your answers via Faser by the deadline. 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. (a) [10 marks] Find for which values of x action T is strictly dominated. Can you find values of x for which action B is strictly dominated? (b) [10 marks] Characterize the set of pure strategy Nash equilibria as a function of the parameter x. (c) [10 marks] Find the values of x for which a mixed strategy Nash equilibrium exists. 2. [25 marks] Consider the following game: L R U 4,4 0,2 D 2,0 1,1 L R T 2,2 0,0 B x,x 2,2 Player 2 Player 1 The game is repeated twice. Players have common discount factor δ>0. (a) [10 marks] Construct a subgame-perfect Nash equilibrium in which players play the same action profile in each period and after any history. (b) [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 output. The payoffs are described by the following matrix: L H L 3,3 6,2 H 2,6 5,5 The game is played repeatedly T times. The two firms have common discount factor 1>δ>0. (a) [5 marks] Describe the unique subgame perfect equilibrium when T is finite. (b) [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 date. Describe the strategies of the players at such equilibrium. (c) [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. 4. [20 marks] Consider the following sequential bargaining game between two players for the division of a pie of size 10. 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. (a) [10 marks] Find a subgame perfect equilibrium when δ=0.6. Is this the unique subgame perfect equilibrium of the game? (b) [10 marks] Can you find a Nash equilibrium where both players get a payoff of 5? Firm 1 Firm 2