Question 2 [20 marks] Princess Leia (player 1) has just started dating Han Solo. Each person can choose either to (C)ommit to the relationship, or (D)ump their partner and break up. Leia prefers a partner who will commit. Unfortunately, Leia knows there are two types of men in the galaxy: Gentlemen and Playboys. She doesn't yet know Han's type, but from experience, she assigns probability 3/4 to Han being a Playboy and probability 1/4 that he is a Gentleman. Han knows his own type. The payoffs for each action and type profile are given as follows: 112 Commit Dump 112 Commit Dump Commit 4,0 -4,4 Commit 4,4 -4,0 Dump 0,-4 0,4 Dump 0,-4 0,0 Han is a Gentleman [Probability 1/4] Han is a Playboy [Probability 3/4] (a) Write down the ex-ante normal form for this Bayesian game. (4 marks) (b) Solve for the unique Bayesian-Nash equilibrium of this game. (4 marks) Now imagine that Leia makes Han perform a grand public gesture of his affection for Leia in front of all the single ladies of the galaxy. The gesture has no effect on Han's payoffs if he is a Gentleman, but is costly to Han if he is a Playboy and chooses to Dump. Assume Playboy Han gets payoff -4c if he Dumps Leia after the public gesture, regardless of whether Leia chooses Commit or Dump. The payoffs now look as follows: Commit Dump 112 4,0 -4,-4a Commit 0,-4 Dump Han is a Playboy [Probability 3/4] 112 Commit Dump 0, -4a Commit 4,4 0,-4 Dump -4,0 0,0 Han is a Gentleman [Probability 1/4] (c) Solve for all pure Bayesian-Nash equilibria if a = 1/3. (6 marks) (d) For what values of a can Leia ensure that (C, CC) is the unique Bayesian-Nash equilibrium? (6 marks)