PROBLEM SET Explain the difference between mutual knowledge and common knowledge.

Transcription

1 PROBLEM SET 1 1. Define Pareto Optimality. 2. Explain the difference between mutual knowledge and common knowledge. 3. Define strategy. Why is it possible for a player in a sequential game to have more than n strategies even if he has only n available actions? 4. (Geanokoplos, 1992) An honest but mischievous father tells his two sons that he has placed dollars in one envelope and dollars in the other envelope, where n is chosen with equal probability among the integers between 1 and 6. The sons completely believe their father. He randomly hands each son an envelope. The first son looks inside his envelope and finds $10,000. Disappointed at the meager amount, he calculates that the odds are fifty-fifty that he has the smaller amount in his envelope. Since the other envelope contains either $1,000 or $100,000 with equal probability, the first son realizes that the expected amount in the other envelope is $50,500. The second son finds only $1,000 in his envelope. Based on his information, he expects to find either $100 or $10,000 in the first son s envelope, which at equal odds come to an expectation of $5,050. The father privately asks each son whether he would be willing to pay $1 to switch envelopes, in effect betting that the other envelope has more money. Both sons say yes. The father then tells each son what his brother said and repeats the question. Again both say yes. The father relays the brothers answers and asks each a third time whether he is willing to pay $1 to switch envelopes. Again both say yes. But if the father relays their answers and asks each a fourth time, the son with $1,000 will say yes, but the son with $10,000 will say no. Why? 5. There once was a small island whose inhabitants had a strange tradition: anyone who discovered his or her own eye color was obligated to commit ritual suicide by jumping off a cliff at midnight on the day the discovery was made. Due to this tradition, the topic of eye color on the island was taboo. Everyone on the island knew everyone else s eye color, and they all knew that only three possible colors are possible: blue, green, and brown. But no one knew his or her own eye color, and so the island s seven blue-eyed inhabitants lived in peace and happiness with their fifteen green-eyed compatriots and their three hundred brown-eyed compatriots. All went smoothly until one visiting stranger, unfamiliar with the taboos of the island, made a shocking declaration: Some of you have blue eyes. The stranger spoke loudly, in the middle of the town square, where it was clear everyone could hear. What the stranger said wasn t news: even before he spoke, everyone knew that some of the island s inhabitants had blue eyes. Describe what happened during the first thirty days that followed the stranger s announcement. (You may assume that strangers are commonly known to be truthful.) 6. Find the Nash Equilibrium of the game below. Is the equilibrium Pareto Optimal?

2 7. Two distinct proposals, A and B, are being debated in Washington. The Congress likes A and the President likes B. The proposals are not mutually exclusive; either or both or neither may become law. Congress and the President receive the following payoffs from the four possible alternatives: Congress President A becomes law (4 for congress) (1 for president) B becomes law (1 for congress) (4 for president) Both A and B become law (3 for congress) (3 for president) Neither A nor B become law (2 for congress) (2 for president) (a) Suppose the game has the following structure. First, Congress decides whether to pass a bill and whether it contains just A, just B, or both. Then the President decides whether to sign the bill into law or to veto the bill. Congress does not have enough votes to override a veto. Draw the game tree. Find the Nash equilibrium in this game. (b) Now suppose the President is given the extra power of a line item veto. Thus the President can sign an entire bill into law, veto an entire bill, or veto just part of a bill and sign the other part into law. As before, Congress does not have enough votes to override a veto. Draw the game tree. Find the Nash equilibrium in this game. 8. (Alternating-Offer Bargaining) In this game, Players 1 and 2 take turns. Similarly to what happens in the Ultimatum Game, Player 1 receives a certain endowment, and has to make an offer to Player 2. The difference now is that, if Player 2 rejects the offer, instead of both getting zero, the size of the pie being divided shrinks, and Player 2 makes a counteroffer to Player 1. If

3 Player 1 rejects the counteroffer, then the game is over and neither player gets anything. If the payoffs in each round of this two-round game are $5.00 and $2.75, what is the equilibrium of this game, assuming that both players are selfish? What if we had three rounds (player 1, player 2, player 1), with payoffs $5.00, $2.50, and $1.25? And what if we had five rounds (player 1, player 2, player 1, player 2, player 1) with payoffs of $5.00, $1.70, $0.58, $0.20, and $0.07? What if the initial endowment is 1, the discount factor in each period is δ, and the game is played for n rounds? 9. (Strategic Voting) Three legislators are voting to give themselves a pay raise. The raise is worth b, but each legislator who votes for the raise incurs a cost of voter resentment equal to c < b. The outcome is decided by majority rule. a) Draw a game tree for the problem, assuming the legislators vote sequentially and publicly. b) Find the equilibrium of this game using backward induction. Is it better to go first? 10. Three gangsters armed with pistols, Al, Bob, and Curly, are in a room with a suitcase containing 120,000 dollars. Al is the least accurate, with a 20% chance of killing his target. Bob has a 40% probability. Curly is slow but sure; he kills his target with 70% probability. For each, the value of his own life outweighs the value of any amount of money. Survivors split the money. a) Suppose each gangster has one bullet and the order of shooting is first Al, then Bob, then Curly. Assume also that each gangster must try to kill another gangster when his turn comes. What is an equilibrium strategy profile and what is the probability that each of them dies in equilibrium? b) Suppose now that each gangster has the additional option of shooting his gun at the ceiling, which may kill somebody upstairs but has no direct effect on his payoff. Does the strategy profile that you found was an equilibrium in part (a) remain an equilibrium? Why? Why not? c) Now generalize this game. Assume that Al kills his target with probability p, Bob kills his target with probability q, and Curly kills his target with probability r. The option of shooting the ceiling is available to all players. Find all the possible equilibria for this game. d) In the generalized game, what is the probability that each player dies in equilibrium? If you had to choose between being the first-mover, the second-mover, or the thirdmover in this game, what would you choose? e) What is the equilibrium of the generalized game if you add a fourth player, Dan, who moves after Curly if he is still alive and kills his target with probability s? What is the equilibrium of this game? What if you keep adding more players? f) In the generalized game with three players (Al, Bob, and Curly), what is the equilibrium if a second round is allowed for those who are still alive? And what is the equilibrium if the game continues until there is only one man standing? 11. The five Dukes of Earl are scheduled to arrive at the royal palace on each of the first five days of May. Duke One is scheduled to arrive on the first day of May, Duke Two on the second, etc. Each Duke, upon arrival, can either kill the king or support the king. If he kills the king, he

4 takes the king s place, becomes the new king, and awaits the next Duke s arrival. If he supports the king, all subsequent Dukes cancel their visits. A Duke s first priority is to remain alive, and his second priority is to become king. Who is king on May 6? 12. Three players have to guess the price x of an object. It is common knowledge that x is a random variable that is distributed uniformly over the integers 1, 2,, 9. The game runs as follows: First, player 1 chooses a number n1, which the other players observe. Then player 2 chooses a number n2 that is observed by the other players. Player 2 is not allowed to select the number already chosen by player 1. Player 3 then selects a number n3, where n3 must be different than n1 and n2. Finally, the number x is drawn and the player whose guess is closest to x without going over wins $1000; the other players get 0. Find the Nash equilibrium of this game. 13. Suppose Alice contemplates suing Bob over some purported ill Bob perpetrated upon her. Suppose Alice s court cost for initiating a suit is c, her legal costs for going to trial are p, and Bob s cost of defending himself is d. Suppose both sides know these costs and also share the knowledge that the probability that Alice will win the suit is γ and the expected amount of the settlement is x. We assume γ x < p, so the suit is a frivolous nuisance suit that Alice would not pursue if her goal were to win the case. Finally, suppose that before the case goes to trial (but after the suit is initiated), the parties can settle out of court for an amount s. We assume s is given. a) Draw the game tree and find the Nash Equilibrium using backward induction. b) Now suppose the plaintiff puts his lawyer on retainer, meaning that he pays the amount p in advance, whether or not the suit is taken to trial. What is the equilibrium? Explain the intuition. 14. (Brams and Straffin, 1979) In the draft system used in football, basketball, and other professional sports in the United States, the teams having the worst win-loss records in the previous season get first pick of the new players in the draft. The worst team gets first choice, the next-worst team second choice, and so on. After each team has made a draft choice, the procedure is repeated, round by round, until every team has exhausted its choices. Presumably, this system, by giving the worst teams priority in the selection process, makes the teams more competitive the next season. Consider a game with only two teams (A and B) and four players in the draft (1,2,3,4), with preference ordering as follows: A = {1,2,3,4}and B = {2,3,4,1}. Assume that the both sets of preferences are common knowledge. Team A is the first-mover. Given this information, answer the following questions: a) If both teams choose sincerely, what is the outcome of the game? Is it Pareto Optimal? b) To choose sincerely, however, is not an equilibrium in this game. If both teams act rationally, what is the equilibrium of the game? Is it Pareto Optimal? c) Can you find an algorithm to find the equilibrium of the game when the number of teams is 2? d) Now consider the same game with three teams (A,B,C) and six players (1,2,3,4,5,6). Each team preference ordering is given by: A = {1,2,3,4,5,6}, B = {5,6,2,1,4,3}, and C =

5 {3,6,5,4,1,2}. Team A moves first, team B moves second, and team C moves last. If the teams choose sincerely, what is the outcome of the game? Is it Pareto Optimal? Why or why not? e) If the teams act strategically, what is the equilibrium of the game? Is it Pareto Optimal? f) Now assume that B moves first, C moves second, A moves last, and they behave rationally. What is the equilibrium of the game? Compare this result with your findings from part (e). What do you conclude? 15. (A very simple model of team production) Two workers have to choose sequentially whether to exert a high level or a low level of effort. The benefit that each worker gets is equal to, where n is the number of workers choosing a high level of effort, and x > 1. The cost of providing a high level of effort is equal to and the cost of choosing a low level of effort is equal to, where. a) What is the variable x capturing? Why is it greater than 1? b) Check for all possible Nash Equilibria. c) What happens to the probability of both workers choosing high effort when x increases? Explain the intuition. d) Assume that the manager of the firm moves last (after the two workers chose their respective levels of effort) and chooses to implement a fine f on those workers who chose a low level of effort. What is the minimum fine that guarantees high effort from both workers? How does the size of the fine change with, and x? e) Now add a third worker (without a manager). Which situation are we more likely to see all workers exerting high effort? With 2 workers or with 3? Explain the intuition. 16. Two staff managers in a sorority, the house manager (player 1) and kitchen manager (player 2), must select a resident assistant from a pool of three candidates: {a, b, c}. Player 1 prefers a to b, and b to c. Player 2 prefers b to a, and a to c. The process that is imposed on them is as follows: First, the house manager vetoes one of the candidates and announces the veto to the central office for staff selection and to the kitchen manager. Next the kitchen manager vetoes one of the remaining two candidates and announces it to the central office. Finally the director of the central office assigns the remaining candidate to be a resident assistant at the sorority. a) Find the Nash equilibrium. Is it unique? b) Now assume that before the two players play the game, player 2 can send an alienating e- mail to one of the candidates, which would result in that candidate withdrawing her application. Would player 2 choose to do this, and if so, with which candidate? 17. (Divide and Choose) Two players use the following procedure to divide a cake. Player 1 divides the cake into two pieces, and then player 2 chooses one of the pieces; player 1 obtains the remaining piece. The cake is continuously divisible (no lumps!), each player likes all parts of it, and each player cares only about the size of the piece of cake she obtains. a) Find the equilibrium (assuming that they only care about the amount of cake they receive);

6 b) Come up with two different utility functions (one for player 1 and one for player 2) so that the equilibrium is different than the one from part a; c) Come up with one utility function (the same one for each player) so that the equilibrium is different than the one from part a. 18. (Brams and Straffin, 1979) In the draft system used in football, basketball, and other professional sports in the United States, the teams having the worst win-loss records in the previous season get first pick of the new players in the draft. The worst team gets first choice, the next-worst team second choice, and so on. After each team has made a draft choice, the procedure is repeated, round by round, until every team has exhausted its choices. Presumably, this system, by giving the worst teams priority in the selection process, makes the teams more competitive the next season. Consider a game with only two teams (A and B) and four players in the draft (1,2,3,4), with preference ordering as follows: A = {1,2,3,4}and B = {2,3,4,1}. Assume that the both sets of preferences are common knowledge. Team A is the first-mover. Given this information, answer the following questions: g) If both teams choose sincerely, what is the outcome of the game? Is it Pareto Optimal? h) To choose sincerely, however, is not an equilibrium in this game. If both teams act rationally, what is the equilibrium of the game? Is it Pareto Optimal? i) Can you find an algorithm to find the equilibrium of the game when the number of teams is 2? j) Now consider the same game with three teams (A,B,C) and six players (1,2,3,4,5,6). Each team preference ordering is given by: A = {1,2,3,4,5,6}, B = {5,6,2,1,4,3}, and C = {3,6,5,4,1,2}. Team A moves first, team B moves second, and team C moves last. If the teams choose sincerely, what is the outcome of the game? Is it Pareto Optimal? Why or why not? k) If the teams act strategically, what is the equilibrium of the game? Is it Pareto Optimal? l) Now assume that B moves first, C moves second, A moves last, and they behave rationally. What is the equilibrium of the game? Compare this result with your findings from part (e). What do you conclude?