1. (4 pts + 1 bonus pt) A committee consisting of three members is supposed to determine the budget for a student activi

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1 4 Pts 1 Bonus Pt A Committee Consisting Of Three Members Is Supposed To Determine The Budget For A Student Activi 1 (217.35 KiB) Viewed 55 times

1. (4 pts + 1 bonus pt) A committee consisting of three members is supposed to determine the budget for a student activity. Each member of the committee (faithfully) represents a student group; since these groups differ, the committee members also have different preferences over the budget. We can represent committee member i's preferences by a utility function Uj(x) = -| x - x 1 where | . | denotes absolute value, x" is i's most preferred budget, and x is the actual budget. Let x; < x; < x;. Assume that committee members care only about this budget, and this is the only issue facing the committee. (No side deals, outside threats, etc.) The committee uses majority voting to choose the budget, successively pitting alterna- tive proposals against each other in a pairwise vote. Any member may make a proposal (even several proposals over the course of the voting process). (a) (1 pt.) Is there a proposal that would survive this process, in the sense that it cannot be beaten by any other proposal in a majority vote? Prove your answer. (b) (1 pt.) Suppose that committee member 2's preferences are given by uz(x) = {- \ *; - *.* +(x3 - x) if x<* where ß > 0. This preference implies that member 2 either wants the budget close to x; when the budget is moderate, or wants to minimize the budget when the budget falls below x1. The other members' preferences are the same as before. Draw a graph with the three utility functions. (c) (1 pt.) Given this new configuration of preferences, is there any proposal that would survive under simple majority voting when B < 1? (Hint: First you want to show that if there is a proposal that would survive, it must be xj.) (d) (1 pt.) Given this new configuration of preferences, is there any proposal that would survive under simple majority voting when ß > 1? (e) (Bonus 1 pt.) Suppose that committee member 2's preferences are given by uz(x) = ax where a > 0. The other members' preferences are given by (1) above. Suppose also that member 2 is the chair of the committee, and she is the only committee member who may make a proposal. There is only one vote. If a majority approves the chair's proposal, that becomes the budget. Otherwise, the budget is xo < x What is the best proposal, from the chair's point of view, that can pass? (Again, no side deals, outside threats, etc.) How does your answer change if xo 2 x ?