1. Two women 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 impostor. The king knows that one of the two women is the true mother of the child, and he knows the "values" that the true mother and the impostor ascribe 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 kind explains to both women that he will implement the following steps:

(a)He will ask the first woman whether the child is hers. If she

answers negatively, the child will be given to the second woman.

If she answers affirmatively, the king will continue to the next

step.

(b)He will ask the second woman if the child is hers. If she answers

negatively, the child will be given to the first woman. If she

answers affirmatively, the second woman will pay the king 75

dinars, and receive the child, and first woman will pay the king

10 dinars.

i. Assume that each woman has a payoff of

0 dinar if she does not

receive the child. Draw the game tree when (a) The first woman

is the true mother and when (b) The second woman is the true

mother.

ii. Find and report the subgame perfect equilibrium under (a) and

(b), and describe what happens (who gets the child and how

much is paid by each woman) in each equilibrium.

2. From the movie A Beautiful Mind :

there are 4 men and 5 women at the bar. There are two types of women γ and ψ . Out of 5 women one is type γ and 4 are

type ψ. Each man decides awoman (γ or ψ) to hit on, simultaneously. Suppose the ψ’s are equally attractive to the men, with appeal b. The γ is more attractive, with appeal a > b. If more than one man attempts to hit on the γ, they succeed neither with the γ(they block each other), nor with any ψ(the latter feel slighted). Any man choosing not to hit on the γ succeeds with a ψ, for a payoff of b. If a man is the only one who hits on the γ, he succeeds, with a payoff of a. Find all Nash equilibria of this game(both pure and mixed).

3. Suppose there are n firms in the Cournot oligopoly model Let q i denote the quantity produced by firm i, and let Q =q1+...+qn denote the aggregate quantity on the market. Let P denote the market- clearing price and assume that inverse demand is given by P(Q) =γ−Q(assumingQ<γ, elseP=0). Assume that the total cost offirmifrom producing quantityqiisCi(qi) =αq2i+βqi. That is, there are no fixed costs and β<γ.

(a)Following Cournot, suppose that the firms choose their quantitiessimultaneously. What is the Nash equilibrium? What happens asn approaches infinity?

(b)Now assumen=2. Further, assume that the total cost of firm1 isC1(q1) =αq21+βq1and firm 2 isC2(q2) =cq2. Followingthe Cournot model, what is the Nash equilibrium? Under whatconditionq∗1>0?