The Judgment of Solomon Elizabeth and Mary appear before King Solomon at his palace, along with…

The Judgment of Solomon Elizabeth and Mary appear before King Solomon at his palace, along with an infant. Each woman claims that the infant is her child. The child is “worth” 100 dinars to his true mother, but he is only “worth” 50 dinars to the woman who is not his mother. The king knows that one of these two women is the true mother of the child, and he knows the “value” that the true mother ascribes to the child, and the “value” that the impostor ascribes to the child, but he does not know which woman is the true mother, and which the impostor. To determine which of the two women is the true mother, the king explains to Elizabeth and Mary that he will implement the following steps:

(i) He will ask Elizabeth whether the child is hers. If she answers negatively, the child will be given to Mary. If she answers affirmatively, the king will continue to the next step.

(ii) He will ask Mary if the child is hers. If she answers negatively, the child will be given to Elizabeth. If she answers affirmatively, Mary will pay the king 75 dinars, and receive the child, and Elizabeth will pay the king 10 dinars.

Answer the following questions:

(a) Describe the mechanism implemented by the king using two extensive-form games: in one extensive-form game Elizabeth is the true mother of the child, and in the second extensive-form game Mary is the true mother of the child.

(b) Prove that the mechanism implemented by the king guarantees that despite the fact that he does not know which of the above extensive-form games is being played, in each game the only subgame perfect equilibrium is the one under which the true mother gets her child and neither woman pays anything at all.

(c) Find another equilibrium of each game, which is not the subgame perfect equilibrium.