QUESTION 5 A social scientist (let's call him Bob) walks up to you on a sidewalk and pulls out a wad of 100 one dollar bills. He stops a second pedestrian and proposes the following game. You are to propose a division of the wad of bills (between you and the second pedestrian) and then the second pedestrian either accepts the division or does not accept. If she accepts, you split the money according to your proposal. If she does not accept, Bob keeps the money. a) What does the game tree for this game look like? b) Find a backward induction solution to the game. c) How well does this solution describe outcomes of ultimatum game experiments? For the toolbar press ALT+F10 (PC).or ALT+EN+F10 (Mac) QUESTION 3 Consider the following game in which it is your interest to be in the minority of the population. Suppose that there are three traders of a financial security each of whom must independently chose whether to place a buy order or a sell order this morning. These orders are large, and so will influence the market price. Consequently, for each player, the payoff to Buy is 3 if the other two Sell and -1 if at least one other Buys. Similarly the payoff to Sell is 3 if the other two Buy and -1 if at least one other Sells. I.e., it pays to be in the minority. a) Are there any symmetric pure strategy NE in this game? b) Is there a symmetric NE in mixed strategies? For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). QUESTION 4 1. Suppose that you are one of 2 students who witness another student, Biff, cheating on an exam. Each of you is angry about the cheating, and each believes that he or she will gain utility of 100 if the student is reported, regardless of who reports him. However, there is also a cost to any student who gets personally involved, which each assesses at 80. Note that, if you knew you were the only student that witnessed the cheating, you would report Biff, as your benefit would outweigh your cost (you would get net utility of 20). However, you also know that either you or the other student who witnessed the cheating could do so, and you would prefer that the other student bears the cost. You also know that the other student feels the same way. Suppose that, after leaving the exam, both you and the other student who witnessed the cheating decide independently whether to report the cheater (R) or not (N). Further, suppose that each of you plays a mixed strategy. What is the probability that neither reports the cheating episode at the Nash equilibrium? If the number of witnesses was 10 instead of 2, would the probability that no one reports the incident be higher or lower? Provide intuition for this result. QUESTION 1 Save Answer Consider a subtraction game in which a player can remove from 0 to 4 chips, but removing 0 chips is only allowed if the previous player removed at least 1 chip. We can model this game as the following chip-and-cup game: There is a pile of n 0 chips and a cup that can hold either 0 or 1 chip. A legal move is to either remove from 1 to 4 chips from the pile, placing one of them in the cup if it is empty, or to remove a chip from the cup. (a) Show that the game is progressively bounded. (b) Find the sets N and P. (c) Describe the winning strategy. For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). QUESTION 6 Two players (Lisa and Bart) take turns choosing a number between 1 and 10 (inclusive). Lisa goes first. The player who takes the total to exactly 100 is the winner. Who should win this game? For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). QUESTION 2 Save Answer Consider a game of Nim in which the players have some number of chips in their pockets. A legal move is either one of the usual legal moves of Nim, or involves moving some positive number of chips from the player's pocket on to one of the piles. (Notice that no chip is ever returned to a pocket.) (a) Show that the game is progressively bounded. (b) Find the sets N and P. (c) Describe the winning strategy. For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac).