Two women came before King Solomon, disputing who was the true mother of a child. The Bible takes up the story in 1 Kings:

Then the king said, "Bring me a sword." So they brought a sword for the king. He then gave an order: "Cut the living child in two and give half toone [woman] and half to the other[woman]." The woman whose son was alive was filled with compassion for her son and said to the king, "Please, my lord, give her the living baby! Don't kill him!" But the other said, "Neither I nor you shall have him. Cut him in two!" Then the king gave his ruling: "Give the living baby to the first woman. Do not kill him; she is his mother."

a. Given that King Solomon does not initially know who the real mother is and assuming that the king prefers the true mother to have the kid, and also assuming that the true mother values the kid more than the other woman:

i.Is this a principal-agent problem? If so identify the principal(s) and the agent(s). Explain. ii.[3 points]According to the story, what kind of a strategy is King Solomon playing (mention the name of the class of such strategies)? Would the strategy still work if the second woman had understood

what was going on and simply repeated whatever the first woman said?

b. Consider a different version of this story in which now a game theory king is trying to persuade the two women tell the truth. Call the two women Anna and Bess. The king sets up the following game:

Move 1: The king decides on a fine, F.

Move 2: Anna is asked to either give up her claim, in which case Bess gets the child and the game ends (Anna gets

zero), or to assert her claim, in which case we go on to Move 3.

Move 3: Bess can either accept Anna's claim, in which case Anna gets the child and the game ends (Bess gets zero), or challenge Anna's claim. In the latter case, Bess must put in a bid, B, of her own choosing for the child and we go on toMove 4.

Move 4: Anna can either match Bess's bid, in which case Anna gets the child and pays B to the king, while Bess paysthe fine, F, to the king; or Anna does not match, in which case Bess gets the child and pays her bid to the king, Anna pays the fine, F.

Assume that the true mother values the child more than the false claimant. Denote the value of getting the child for Anna as vAand for Bess as vB. Note that the exact values are private information, that is, each woman only knows her valuation; however, theyboth know that the true mother's valuation is greater than that of the false claimant.

i. Draw the game tree. Consider only the payoffs to Anna and Bess.

ii.Consider the case in which Anna is the true mother, i.e. vA> vB, and find the subgame-

perfect equilibrium.

iii.Consider the case in which Bess is the true mother, i.e. vB> vA, and find the subgame-perfect

equilibrium.