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 mostpreferred
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 a is a unique majority winner if there is no feasible alternative
that defeats a by a majority, and for every other feasible alternative ß there is at least one other
feasible alternative that defeats ß 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