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

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