Uncovered set: Difference between revisions
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==Notes== |
==Notes== |
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The uncovered set can be thought of as requiring its candidates to have a beatpath |
The uncovered set can be thought of as requiring its candidates to have a two-step beatpath to every candidate not in the uncovered set. The Smith set requires a one-step beatpath (i.e. of at most two candidates, a direct pairwise victory). |
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'''Independence of covered alternatives''' says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the [[uncovered set]]. Independence of covered alternatives implies [[Independence of Smith-dominated Alternatives]] (since independence of covered alternatives implies that one can eliminate everyone outside of the uncovered set without changing the winner, and the uncovered set is a subset of the Smith set, therefore eliminating everyone outside of the Smith set also can't change the winner), which further implies [[Smith criterion|Smith]] and thus [[Condorcet criterion|Condorcet]]. If a method is independent of covered alternatives, then the method fails monotonicity if perfect ties can always be broken in favor of a choice W by using ballots ranking W first. |
'''Independence of covered alternatives''' says that if one option (X) wins an election, and a new alternative (Y) is added, X will win the election if Y is not in the [[uncovered set]]. Independence of covered alternatives implies [[Independence of Smith-dominated Alternatives]] (since independence of covered alternatives implies that one can eliminate everyone outside of the uncovered set without changing the winner, and the uncovered set is a subset of the Smith set, therefore eliminating everyone outside of the Smith set also can't change the winner), which further implies [[Smith criterion|Smith]] and thus [[Condorcet criterion|Condorcet]]. If a method is independent of covered alternatives, then the method fails monotonicity if perfect ties can always be broken in favor of a choice W by using ballots ranking W first. |
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