Player 1 is the plaintiff and player 2 is the defendant in a lawsuit for breach of contract. The expected in-court settlement, to be paid by player 2 to player 1 if they go to court, depends on the strength of both sides' cases as shown in Table 6.10. In addition, if they go to court, then each side must pay trial costs of $10,000. So, for example, if both cases are strong and they go to court, then the expected total cost to player 2 is $90,000, but the expected net benefit to player 1 is only $70,000. Each player knows whether his own case is weak or strong, and each side believes that the other side is equally likely to be weak or strong.
Assume that both players are risk neutral. The players could avoid these trial costs by settling out of court. However, in pretrial bargaining, there is nothing to prevent a player from lying about the strength of his case, if he has an incentive to do so. Going to court is the disagreement outcome in any pretrial bargaining.
a. Suppose that you have been asked to mediate the pretrial bargaining between these two players. Create an incentive-compatible mediation plan that has the following properties:
(1) if they settle out of court, then player 2 will pay player 1 an amount equal to the expected incourt settlement, given their reports to you about the strength or weakness of their cases;
(2) if both report to you that they are weak, then they settle out of court with probability 1; and
(3) if one side reports that it is weak and the other reports that it is strong, then they settle out of court with a probability q that does not depend on which side reported weakness. Make q as large as possible without violating any informational incentive constraints.
(HINT: When you compute expected payoffs, do not forget the net payoff from going to court. Player 1 expects to get money from player 2 even if they do not agree to settle out of court!)
b. Imagine that player 1, knowing that his case is actually strong, gets the following advice from his lawyer. "Given that our case is strong, our expected in-court settlement would be 0.5 x 80,000 + 0.5 x 144,000 = $112,000. So let us tell player 2 that our case is strong and offer to settle out of court for this amount."
Show that, even if player 2 were to believe player 1's claim about the strength of l's case, it would be a mistake for player 1 to make an offer to settle for $112,000 when his case is strong.