Section A: Answer the ONE question

1.    Suppose that each of the games below is repeated for a finite number of T > 3 periods with discount factor 6.

a) b)

c)

Find all subgame perfect equilibria (SPEs) of game (a). (10 marks)

Describe a strategy profile that is a SPE of game (b) if the discount factor is high      enough and which gives the players payoffs of (3,3) in all but the last 2 periods. (20 marks)

For which discount factors does the subgame perfect equilibrium explained in part b) exist?  (20 marks)

Section B: Answer ONE question

2.

Consider the juror voting game as discussed in the lecture where there is a defendant     who is guilty with probability 几. Each juror knows 几 and, also each receive a private        signal about the defendant, s ∈ {g, i}, corresponding respectively to guilty and                innocent. The precision of the signal denotes the probability that a correct signal is          received; Pr(g|G) = Pr (i|G) = q where q is the precision. A defendant gets convicted if and only if all jurors vote to convict, while he gets acquitted otherwise. Each juror gets a payoff of 1 when a guilty defendant is convicted, or an innocent defendant is                 acquitted and get a payoff of 0 otherwise.

a)   Let there be 2 jurors who receive signals of precision respectively q1 and q2 . Find the conditions for a BNE in which both jurors vote to convict independently of their          signals. How does this condition change with respect to q1 and q2 ? (15 marks)

b)   Let there be 3 jurors who each receive signals of precision q. Find the conditions for a BNE in which two jurors vote truthfully (vote convict upon guilty signal and vote    acquit upon innocent signal) and one juror always votes to convict regardless of the signal. (15 marks)

c)   Let there be n jurors, each receiving a signal of precision q. Find the condition on q  for an equilibrium in which k jurors vote truthfully and n − k jurors vote to convict   regardless of their signal. What happens to this condition as k increases? (20 marks)