Smith has been injured during an operation. with probability .5 the doctor is guilty of malpractice and with probability .5 he is innocent. the doctor knows whether or not he is actually guilty. smith threatens to sue the doctor.

before the case goes to trial, the doctor can offer to settle out of court by paying smith 1,000.

if the doctor offers to settle and smith accepts, then smith would have to give 1/3 of this 1,000 to her lawyer and the doctor would not have to pay any legal fees.

if the doctor does not offer to settle then smith can either drop the suit or let the case go to trial.

if smith drops the suit smith and the doctor each receive a payoff of 0.

if the case goes to trial, the doctor's guilt or innocence will be discovered.

if smith wins the trial, then (i) she will win 1,000 but will have to give 1/3 of this 1,000 to her lawyer, and (ii) the doctor will lose the 1,000 he pays smith and an additional 100 in legal fees.

if smith loses the trial then she must pay the doctor's legal fees of 100.

(a) show that this game has all of the basic elements of a signaling game.

(b) show that this game does not have a separating equilibrium.

(c) show that this game does not have a pooling equilibrium.

(d) find the mixed strategy perfect bayes nash equilibrium in this game.