question 4 economics
4. (30 pts') Suppose two players, Player A and Player B, are playing a game called, \"Split the Pot.\" It works like this: in front of them is a pot with $1500. Player A proposes a division of the pot in the form of (TIA, are), where m is the amount Player A gets to keep, ms is the amount Player B gets to keep, and TIA + TIE = 1500. Player B then decides to accept or reject Player A's proposal. If Player B accepts the proposed division of the pot, then Player A gets HA, and Player B gets mg. If Player B rejects the proposed division of the pot, then they both get 0. Suppose that Player A's utility from an amount of money is equal to the amount of money: U(m) = 11]. Suppose the same is true for Player B. This means that if Player B accepts Player A's proposal, Player A's utility will be U(rm) = 1m, and Player B' 3 utility will be U(ma) = 1m. Suppose for now that there is no other source of utility in this game for either player. Each player M cares about how much money he or she gets and absolutely nothing else. Also suppose that if Player B is indifferent between accepting a proposal and rejecting it that is, if Player B's utility from accepting a proposal is equal to her utility from rejecting it then she will accept it. a. Draw the game tree for this sequential game. What will Player B's utility be in Nash Equilibrium? b. Now suppose that Player B has a different utility function from what was described earlier. She has two components to her utility: money and comparison. Let (1 = HA mg be the difference between the two players' payoffs in the game. Then assume Player B's utility function is U(m,d) = m d. The game still functions exactly as it did before, with Player A proposing a split of the pot and Player B deciding to accept the proposal or reject it. Player B still gets 0 utility if she rejects Player A's proposal, and she still accepts a proposal if she is indifferent between accepting and rejecting. Draw a new game tree reecting Player B's different utility function. What rule will Player B follow in deciding whether to accept an offer? Knowing this, how much will Player A offer Player B in Nash equilibrium? c. Now suppose that there are two types of people: Type I, those whose utility functions are U(m) = m, and Type 11, those whose utility functions are U(m,d) = m d. Suppose everyone knows that half the population is the rst type, and half is the second type. Suppose Player A knows that he is Type I, but he does not know Player B's type. What is Player A's expected utility (that is, the expected value of his utility) in this case from making his proposal as if Player B were Type I and didn't care about comparison? What is his expected utility from making his proposal as if Player B were Type II and did care about comparison? Which proposal of those two should he make? d. How much would Player A be willing to spend on a screening system that would allow him to know Player B's type before he made the proposal
