One day, long time ago, King Solomon was faced with the following problem. Two women, Elizabeth and Mary were brought before him, along with an infant. Both women claimed to be the true mother of the infant (and indeed, one of them was the true mother). King Solomon did not know the identity of the true mother. If the true mother got the child she gets a payoff of 100. On the other hand, if the "fake" mother got the child she gets a pay-off of 50. Not getting the child would give each woman a pay-off of 0. All the payoffs are commonly known. What is not known is the identity of the true mother. Elizabeth and Mary knew who was the true mother, but King Solomon did not. King Solomon was a wise king, and in his wisdom, he decided to settle the dispute by setting up the following game.

Step 1: He will first 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 to continue to the next step.

Step 2: He will ask Mary whether the child is hers. If she answers negatively, the child will be given to Elizabeth. If she answers affirmatively, the king will ask Mary to pay 75 and Elizabeth to pay 10, and give the child to Mary. Utility from money is linear, i.e., paying p gives a utility of p. Suppose that Mary is indeed the true mother of the child. (King Solomon does not know this little fact.) Notice that the above is an extensive form game. We denote the extensive form game as GM (the subscript M signifies that Mary is the true mother indeed).

a) Describe the extensive form game GM, by representing the above problem in the form of a game tree.

b) Using Backward Induction, find an equilibrium in the game. Among Mary and Elizabeth, who gets the child in the equilibrium you identify?