Consider the following signaling game: Two players, a plaintiff (P) and a defendant (D), are negotiating over the possible settlement of their case. The defendant knows the strength of its own case, that it is either negligent and so would lose the case if it goes to trial or not, in which case it would win. The plaintiff is asking for damages of $5 million dollars, and the cost of a trial to the defendant is $6 million. The plaintiff pays nothing whether the case is settled or goes to trial. The game begins with the type of the defendant determined with probability 1/2 negligent and 1/2 not. Then the defendant can make a high offer of $5 million or a low offer of $3 million. The plaintiff then either accepts the offer or rejects it, leading to a trial.

(a) Write down an extensive form for this game from the chance move determining Ds type, to the offer and Ps response to both possible offers.

(b) Find two pooling equilibria of the game, including responses off the equilibrium path.

(c) Change the initial chance Ds is not negligent to 1/4 and negligent to 3/4. Is there still a pooling equilibrium? If so, what is it? (d) Find a semi-separating equilibrium of the game with these new probabilities.