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29.3 (1) Let us consider a more general version of the voluntary public goods game described in the previous question. This game has N players, each of whom can contribute either $10 or nothing to the public fund. All money that is contributed to the public fund gets multiplied by some (including those who do not contribute.) Thus if all N players contribute number B > 1 and then divided equally among all players in the game $10 to the fund, the amount of money available to be divided among the N players will be $10BN and each player will get $10BN/N = $10B back from the public fund. (a) If B > 1, which of the following outcomes gives the higher payoff to each player? a) All players contribute their $10 or b) all players keep their $10. (b) Suppose that exactly K of the other players contribute. If you keep your $10, you will have this $10 plus your share of the public fund con- tributed by others. What will your payoff be in this case? If you contribute your $10, what will be the total number of contributors? What will be your payoff? (c) If B = 3 and N = 5, what is the dominant strategy equilibrium for this game? Explain your answer. NAME 349 (d) In general, what relationship between B and N must hold for "Keep" to be a dominant strategy? (e) Sometimes the action that maximizes a player's absolute payoff, does not maximize his relative payoff. Consider the example of a voluntary public goods game as described above, where B = 6 and N = 5. Suppose that four of the five players in the group contribute their $10, while the fifth player keeps his $10. What is the payoff of each of the four contrib- utors? What is the payoff of the player who keeps his $10? Who has the highest payoff in the group? What would be the payoff to the fifth player if instead of keeping his $10, he contributes, so that all five players contribute If the other four players contribute, what should the fifth player to maximize his absolute payoff? What should he do to maximize his payoff relative to that of the other players? (f) If B = 6 and N = 5, what is the dominant strategy equilibrium for this game? Explain your answer. 29.4 (1) The Stag Hunt game is based on a story told by Jean Jacques Nicourses on the Origin and Foundation of in- umething like this: "Two economists and other social scientists frequently conduct experiments in which subjects play games for money. One such game is known as the voluntary public goods game. This game is chosen to represent situations in which individuals can take actions that are costly to themselves but that are beneficial to an entire community. In this problem we will deal with a two-player version of the voluntary public goods game. Two players are put in separate rooms. Each player is given $10. The player can use this money in either of two ways. He can keep it or he can contribute it to a "public fund." Money that goes into the public fund gets multiplied by 1.6 and then divided equally between the two players. If both contribute their $10, then each gets back $20 X 1.6/2 = $16. If one contributes and the other does not, each gets back $10 x 1.6/2 = $8 from the public fund so that the contributor has $8 at the end of the game and the non-contributor has $18-his original $10 plus $8 back from the public fund. If neither contributes, both have their original $10. The payoff matrix for this game is: Voluntary Public Goods Game Keep Player B Contribute $16. $16 $8,918 $18.98 $10, $10 Player A Contribute Keep 10 8 (a) If the other player keeps, what is your payoff if you keep? If the other player keeps, what is your payoff if you contribute?- 29.3 (1) Let us consider a more general version of the voluntary public goods game described in the previous question. This game has N players, each of whom can contribute either $10 or nothing to the public fund. All money that is contributed to the public fund gets multiplied by some (including those who do not contribute.) Thus if all N players contribute number B > 1 and then divided equally among all players in the game $10 to the fund, the amount of money available to be divided among the N players will be $10BN and each player will get $10BN/N = $10B back from the public fund. (a) If B > 1, which of the following outcomes gives the higher payoff to each player? a) All players contribute their $10 or b) all players keep their $10. (b) Suppose that exactly K of the other players contribute. If you keep your $10, you will have this $10 plus your share of the public fund con- tributed by others. What will your payoff be in this case? If you contribute your $10, what will be the total number of contributors? What will be your payoff? (c) If B = 3 and N = 5, what is the dominant strategy equilibrium for this game? Explain your answer. NAME 349 (d) In general, what relationship between B and N must hold for "Keep" to be a dominant strategy? (e) Sometimes the action that maximizes a player's absolute payoff, does not maximize his relative payoff. Consider the example of a voluntary public goods game as described above, where B = 6 and N = 5. Suppose that four of the five players in the group contribute their $10, while the fifth player keeps his $10. What is the payoff of each of the four contrib- utors? What is the payoff of the player who keeps his $10? Who has the highest payoff in the group? What would be the payoff to the fifth player if instead of keeping his $10, he contributes, so that all five players contribute If the other four players contribute, what should the fifth player to maximize his absolute payoff? What should he do to maximize his payoff relative to that of the other players? (f) If B = 6 and N = 5, what is the dominant strategy equilibrium for this game? Explain your answer. 29.4 (1) The Stag Hunt game is based on a story told by Jean Jacques Nicourses on the Origin and Foundation of in- umething like this: "Two economists and other social scientists frequently conduct experiments in which subjects play games for money. One such game is known as the voluntary public goods game. This game is chosen to represent situations in which individuals can take actions that are costly to themselves but that are beneficial to an entire community. In this problem we will deal with a two-player version of the voluntary public goods game. Two players are put in separate rooms. Each player is given $10. The player can use this money in either of two ways. He can keep it or he can contribute it to a "public fund." Money that goes into the public fund gets multiplied by 1.6 and then divided equally between the two players. If both contribute their $10, then each gets back $20 X 1.6/2 = $16. If one contributes and the other does not, each gets back $10 x 1.6/2 = $8 from the public fund so that the contributor has $8 at the end of the game and the non-contributor has $18-his original $10 plus $8 back from the public fund. If neither contributes, both have their original $10. The payoff matrix for this game is: Voluntary Public Goods Game Keep Player B Contribute $16. $16 $8,918 $18.98 $10, $10 Player A Contribute Keep 10 8 (a) If the other player keeps, what is your payoff if you keep? If the other player keeps, what is your payoff if you contribute