Perfect representation: Difference between revisions
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A winner set ''W'' provides '''perfect representation''' for a group of ''n'' voters and a total seat size ''k'' if ''n = ks'' for some positive integer ''s'' and the voters can be split into ''k'' pairwise disjoint groups ''N1, . . . , Nk'' of size ''s'' each in such a way that there is a one-to-one mapping ''μ : W → {N1, . . . , Nk}'' such that for each candidate ''a ∈ W'' all voters in ''μ(a)'' approve ''a''. |
A winner set ''W'' provides '''perfect representation''' for a group of ''n'' voters and a total seat size ''k'' if ''n = ks'' for some positive integer ''s'' and the voters can be split into ''k'' pairwise disjoint groups ''N1, . . . , Nk'' of size ''s'' each in such a way that there is a one-to-one mapping ''μ : W → {N1, . . . , Nk}'' such that for each candidate ''a ∈ W'' all voters in ''μ(a)'' approve ''a''. |
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In other words, in a multi-winner approval-voting election, for a given set of ballots cast, if there is a possible election result where candidates could each be assigned an equal number of voters where each voter has approved their assigned candidate and no voter is left without a candidate, then for a method to pass the perfect representation criterion, such a result must be the actual result. |
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| ⚫ | Every winner set that provides perfect representation also provides [[Justified representation | Proportional Justified Representation |
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Perfect representation is not compatible with strong monotonicity. Consider the following election with two winners, where A, B, C and D are candidates, and the number of voters approving each candidate are as follows: |
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100 voters: A, B, C |
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100 voters: A, B, D |
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1 voter: C |
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1 voter: D |
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A method passing the perfect representation criterion must elect candidates C and D despite near universal support for candidates A and B. This demonstrates that methods passing perfect representation can at best be only weakly monotonic, and could therefore be seen as an argument against perfect representation as a useful criterion. |
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| ⚫ | Every winner set that provides perfect representation also provides [[Justified representation | Proportional Justified Representation]] <ref> https://www.researchgate.net/publication/308022665_Some_Notes_on_Justified_Representation </ref>. In contrast, [[Justified representation | Extended Justified Representation]] may rule out all winner sets that provide perfect representation. <ref>https://arxiv.org/abs/1407.8269</ref> |
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=== Perfect Representation In the Limit === |
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While perfect representation arguably has flaws as a proportionality criterion, the Perfect Representation In the Limit criterion (PRIL) states that the result of a voting method must approach perfect representation as the number of elected candidates increases, as long as the ballots make this possible. This criterion does not demand that a method passes lower quota so does not disqualify the [[Sainte-Laguë method]] and is compatible with strong monotonicity. <ref>https://arxiv.org/abs/2305.08857</ref> |
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[[Category:Types of representation]] |
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Latest revision as of 22:48, 26 November 2023
A winner set W provides perfect representation for a group of n voters and a total seat size k if n = ks for some positive integer s and the voters can be split into k pairwise disjoint groups N1, . . . , Nk of size s each in such a way that there is a one-to-one mapping μ : W → {N1, . . . , Nk} such that for each candidate a ∈ W all voters in μ(a) approve a.
In other words, in a multi-winner approval-voting election, for a given set of ballots cast, if there is a possible election result where candidates could each be assigned an equal number of voters where each voter has approved their assigned candidate and no voter is left without a candidate, then for a method to pass the perfect representation criterion, such a result must be the actual result.
Perfect representation is not compatible with strong monotonicity. Consider the following election with two winners, where A, B, C and D are candidates, and the number of voters approving each candidate are as follows:
100 voters: A, B, C
100 voters: A, B, D
1 voter: C
1 voter: D
A method passing the perfect representation criterion must elect candidates C and D despite near universal support for candidates A and B. This demonstrates that methods passing perfect representation can at best be only weakly monotonic, and could therefore be seen as an argument against perfect representation as a useful criterion.
Every winner set that provides perfect representation also provides Proportional Justified Representation [1]. In contrast, Extended Justified Representation may rule out all winner sets that provide perfect representation. [2]
Perfect Representation In the Limit
While perfect representation arguably has flaws as a proportionality criterion, the Perfect Representation In the Limit criterion (PRIL) states that the result of a voting method must approach perfect representation as the number of elected candidates increases, as long as the ballots make this possible. This criterion does not demand that a method passes lower quota so does not disqualify the Sainte-Laguë method and is compatible with strong monotonicity. [3]