Suppose that an odd number of individuals have single-peak

Question

Questions on Voting

1. Does the following family of preferences over the three alternatives X, Y, Z have the single-peak property? Explain. There are four preference orderings, each represented by a column, and a higher alternative is preferred to a lower one.

X	Z	Z	Y
Z	X	Y	Z
Y	Y	X	X

2. Suppose that an odd number of individuals have single-peak preferences. Explain why the most- preferred alternative of the median voter will defeat every other proposal by a majority votes.

3. Consider majority rule with six people. Because there is an even number of individuals there is a possibility of ties. We will say that α is a unique majority winner if there is no feasible alternative that defeats α by a majority, and for every other feasible alternative β there is at least one other feasible alternative that defeats β by some majority. Use Tables 1 and 2 to show that the rule that selects the unique majority winner can be manipulated by a single individual.

Table 1

Person 1	Person 2	Person 3	Person 4	Person 5	Person 6
X	X	W	Y	Z	Z
Y	W	Y	Z	X	W
Z	Y	X	W	W	Y
W	Z	Z	X	Y	X

Table 2

Person 1	Person 2	Person 3	Person 4	Person 5	Person 6
X	X	Y	Y	Z	Z
Y	W	Z	Z	X	W
Z	Y	W	W	W	Y
W	Z	X	X	Y	X