Week 2 - Preference domain restrictions (single-peakedness & single crossing))

Question 1

Single peak preferences

Answer 1

• For each individual there is a peak
• That is a bliss point
• Such that desirability falls as we move away from the peak.
• It guarantees that a condorcet winner exists and the pairwaise majority relationship is transative.
• When preferences are single peaked, the majority voting rule escapes arrows theorem.
• This is because restriction on domain has been weakened.

Question 2

Reordering alternatives

Answer 2

• Define a function q such that in assigs each alternative with a unique number

Question 2

Reordering alternatives

Answer 2

• Define a function q such that in assigs each alternative with a unique number

Question 3

Single peakedness 2

Answer 3

We have shown that with single-peaked preferences there exists a SWF that satisfies:
- transitivity
- unanimity
- IIA
and is not a dictatorship

• We needed preference domain restriction (single-peaked preferences) to
  ensure transitivity. Didn’t need it for other properties.
• Preference of the median voter determines the condorcet winner.

Question 4

Medium Voter Theorem

Answer 4

Median Voter Theorem : If there is some way of lining up the alternatives on a graph so that all voters’ preferences are single–peaked, then the median of the voters’ most–preferred alternatives will defeat any other alternative in a pairwise vote.

The median of people’s preferred alternative is the alternative which has the number of people who prefer an alternative to the left equal to the number of people who prefer an alternative to the right

Question 5

What if every alternative is ranked worst by at least one indivual

Answer 5

• The preferences are not single peaked.