ECOM010 Microeconomics B

Question 1 [50 marks]

Consider the following game:

(a) Find the strictly and weakly dominant strategy equilibria (if any). [10 marks]

(b) Solve the game by iterative elimination of strictly dominated strategies. [10 marks]

(c) Find all pure strategy Nash equilibria as well as all pure strategy subgame perfect Nash equilibria. [10 marks]

(d) Now suppose that there is a small probability, p ∈ (0, 1), that 2 does not observe 1’s move correctly. Suppose we model this by a move by Nature as shown in the figure below. In particular, if 1 chooses a, Nature “tells” player 2 the correct action (i.e., N chooses A) with probability 1 − p and “lies” (i.e., N chooses B) otherwise. Analogously, 2 sees B with probability 1 − p if 1 chooses b and sees A otherwise. Payoffs depend on 1’s actual action and 2’s action, not on the observation by 2. Find all the pure strategy weak perfect Bayesian equilibria of the game. [20 marks]

Question 2 [30 marks]

Consider a repeated game where the stage game is represented by

where both players discount future payoffs by δ ∈ (0, 1).

(a) Suppose that the stage game is repeated twice (T = 2). Can (A, A) be sustained in the first stage as a subgame perfect Nash equilibrium? [15 marks]

(b) Suppose now that the game is infinitely repeated. Find the minimum value for δ such that the outcome (A, A) can be supported in each period by a trigger strategy in a subgame perfect equilibrium. [15 marks]

Question 3 [20 marks]

(a) Show an example of a game in which the minimax value of player 1 is equal to 2 and the minimax value of player 2 is equal to 3. [10 marks]

(b) Show an example of imperfect recall that we have not seen in the module, in which the outcome of a particular behavioural strategy cannot be reproduced under a mixed strategy. [10 marks]