Independence of irrelevant alternatives: Difference between revisions
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(Import rock-paper-scissors example of IIA failure from Wikipedia. Add reference to River and Ranked Pairs in ISDA/IPDA section) |
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=== Ranked methods === |
=== Ranked methods === |
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[[Arrow's impossibility theorem]] states that no voting system can satisfy universal domain, non-imposition, non-dictatorship, unanimity, and independence of irrelevant alternatives. Since universal domain implies that the method is an ordinal method, the impossibility theorem only applies to [[ordinal voting]]. In practice, this means that no deterministic ranked ballot system can satisfy independence of irrelevant alternatives without either having a dictator (whose ballot decides who wins no matter the other ballots), failing to elect a candidate that the whole electorate ranks first, or rendering one or more outcomes impossible no matter the ballots. |
[[Arrow's impossibility theorem]] states that no voting system can satisfy universal domain, non-imposition, non-dictatorship, unanimity, and independence of irrelevant alternatives. Since universal domain implies that the method is an ordinal method, the impossibility theorem only applies to [[ordinal voting]]. In practice, this means that no deterministic ranked ballot system can satisfy independence of irrelevant alternatives without either having a dictator (whose ballot decides who wins no matter the other ballots), failing to elect a candidate that the whole electorate ranks first, or rendering one or more outcomes impossible no matter the ballots. |
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==== A simple example ==== |
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Let's say that we have a majoritarian ranked ballot method. With an election that's a Condorcet cycle (rock-paper-scissors situation), like this: |
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{{ballots| |
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25: A>B>C |
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40: B>C>A |
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35: C>A>B}} |
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at least one of A, B or C must be elected (or have a chance of winning the election if the method is nondeterministic). There are thus three cases: |
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*Case 1: ''A'' is elected. IIA is violated because the 75% who prefer ''C'' over ''A'' would elect ''C'' if ''B'' were not a candidate. |
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*Case 2: ''B'' is elected. IIA is violated because the 60% who prefer ''A'' over ''B'' would elect ''A'' if ''C'' were not a candidate. |
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*Case 3: ''C'' is elected. IIA is violated because the 65% who prefer ''B'' over ''C'' would elect ''B'' if ''A'' were not a candidate. |
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No matter who wins, the method can be made to fail IIA. |
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== Related criteria == |
== Related criteria == |
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To mitigate the reach of IIA failures, less strict properties have been proposed (some of which are incompatible with IIA): |
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* '''[[Independence of Smith-dominated Alternatives|Independence of Smith-dominated alternatives]]''' (ISDA) and '''[[Uncovered set|Independence of covered alternatives]]''' |
* '''[[Independence of Smith-dominated Alternatives|Independence of Smith-dominated alternatives]]''' (ISDA) and '''[[Uncovered set|Independence of covered alternatives]]''' |
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* Woodall's '''Weak IIA''': If x is elected, and one adds a new calternative y ahead of x on some of the ballots on which x was first preference (and nowhere else), then either x or y should be elected. |
* Woodall's '''Weak IIA''': If x is elected, and one adds a new calternative y ahead of x on some of the ballots on which x was first preference (and nowhere else), then either x or y should be elected. |
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Neither the [[Borda count]], [[Coombs' method]] nor [[Instant-runoff voting]] satisfies the less strict criteria above. |
Neither the [[Borda count]], [[Coombs' method]] nor [[Instant-runoff voting]] satisfies the less strict criteria above. [[Ranked Pairs]] does satisfy ISDA, and [[River]] satisfies IPDA. |
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== Anecdote == |
== Anecdote == |
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