Suppose n students are on a committee to decide the fraction of student fees to spend on student activities. The minimum fraction they can select is 0, and the maximum fraction is 1. Any amount they do not spend will go back into the university's general financial aid budget. Each committee member i has an ideal policy xi ∈ (0, 1) that represents the fraction member i wants to spend on student activities. Member i's utility is ui(xi , d) = - 0 d - xi 0 , where d ∈ [0, 1] is the fraction that the committee ultimately decides on. For example, if the committee chooses d = 0.5 and member i's ideal policy is xi = 0.7, then member i's utility is -0.2. The committee members have common knowledge of one another's preferred policies.
(a) Suppose that n = 3 and that the committee's choice is determined by a "median voter" rule. Simultaneously, each committee member i announces a policy position (i.e., "votes") yi ∈ [0, 1], and then the committee's decision d(y1 , y2 , y3) is the median of {y1 , y2 , y3}. Show that under the median voter rule, it is a Nash equilibrium for each committee member i to vote truthfully, so that yi = xi .
(b) Suppose that n = 2 and that the committee's choice is determined by a "phantom median voter" rule. The committee members simultaneously vote as in part (a), but now the committee's decision d(y1 , y2) is the median of {y1 , y2 , 0.5}. Show that under the phantom median voter rule, it is a Nash equilibrium for each committee member i to vote yi = xi .
(c) Suppose again that n = 3, but now assume the committee's decision is determined by an "average vote" rule, where d(y1 , y2 , y3) is the average of {y1 , y2 , y3}. Show that under the average vote rule, if the committee members have different ideal policies, then it is not a Nash equilibrium for the players to vote truthfully. Finally, find the unique Nash equilibrium of this game for the case in which x1 = 0.3, x2 = 0.6, and x3 = 0.9. Can you prove that it is the unique Nash equilibrium?