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 number B > 1 and then divided equally among all players in the game (including those who do not contribute.) Thus if all N players contribute $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 contributed 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? _________.
(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 contributors? _________. What is the payoff of the player who keeps his $10? $10+$60×4/ = $58. 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 _________, what should the fifth player to maximize his absolute payoff? Contribute. 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? _________.