Consider majority voting over pairwise alternatives subject to agenda manipulation. Use thefollowing voting rules and preference profiles. There are three propositions to choose amongcandidates, A, B, and C. There are three voters, 1, 2, 3. The notation indicates strictpreference.Rules: There is a designer who sets the agenda, the order of voting. She announces twopropositions to choose between; the winner of that vote faces a runoff against the remainingalternative.Profile I:Voter 1: A B CVoter 2: B C AVoter 3: C A BProfile II:Voter 1: A B CVoter 2: B C, B A, (C vs. A preference is unspecified)Voter 3: C B AClaim: Under Profile I the designer can arrange that any one of the three propositions bethe winner by the chair’s choice of the order of voting.Under Profile II, the choice is independent of the order of the agenda.(a) Demonstrate the claim.(b) Show that Profile II fulfills the concept of single peaked preferences and that Profile Idoes not.
Question
Consider majority voting over pairwise alternatives subject to agenda manipulation. Use thefollowing voting rules and preference profiles. There are three propositions to choose amongcandidates, A, B, and C. There are three voters, 1, 2, 3. The notation indicates strictpreference.Rules: There is a designer who sets the agenda, the order of voting. She announces twopropositions to choose between; the winner of that vote faces a runoff against the remainingalternative.Profile I:Voter 1: A B CVoter 2: B C AVoter 3: C A BProfile II:Voter 1: A B CVoter 2: B C, B A, (C vs. A preference is unspecified)Voter 3: C B AClaim: Under Profile I the designer can arrange that any one of the three propositions bethe winner by the chair’s choice of the order of voting.Under Profile II, the choice is independent of the order of the agenda.(a) Demonstrate the claim.(b) Show that Profile II fulfills the concept of single peaked preferences and that Profile Idoes not.
Solution
(a) Demonstrate the claim.
For Profile I:
- If the designer wants A to win, she can set the agenda as B vs C first, then A vs (winner of B vs C). In the first round, B wins (voters 2 and 3 prefer B over C). In the second round, A wins (voters 1 and 3 prefer A over B).
- If the designer wants B to win, she can set the agenda as A vs C first, then B vs (winner of A vs C). In the first round, A wins (voters 1 and 3 prefer A over C). In the second round, B wins (voters 1 and 2 prefer B over A).
- If the designer wants C to win, she can set the agenda as A vs B first, then C vs (winner of A vs B). In the first round, A wins (voters 1 and 2 prefer A over B). In the second round, C wins (voters 2 and 3 prefer C over A).
For Profile II:
- Regardless of the order of the agenda, B will always win. This is because voter 2 prefers B over both A and C, and either voter 1 or voter 3 will also prefer B over the remaining option.
(b) Show that Profile II fulfills the concept of single peaked preferences and that Profile I does not.
Single-peaked preferences mean that there is one option (the "peak") that a voter prefers over all others, and their preference decreases the further an option is from their peak.
In Profile II, each voter has a clear peak: voter 1 prefers A, voter 2 prefers B, and voter 3 prefers C. No matter what other options are presented, they will always prefer their peak. This is why the outcome is independent of the order of the agenda.
In Profile I, however, the voters' preferences do not have a single peak. For example, voter 1 prefers A over B and B over C, but also prefers C over A. This cyclical preference means that there is no single peak, and the outcome can be manipulated by changing the order of the agenda.
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