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.