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.
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
|