Stable winner set: Difference between revisions
→Example in which definition of "strictly prefers" matters: write out "polarizing" example.
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=== Example in which definition of "strictly prefers" matters===
Consider the following example (where each candidate letter stands for an unlimited group of clones):
61% vote A5 B4 C0
39% vote A0 B4 C5
Under the above definition 4.1. of V(S,S'), the winner set {B, B, B, B, B} is strictly preferred by all voters over any other set of size 5 or less, so it is the unique stable winner set.
Under the above definition 4.2. of V(S,S'), the winner set {B, B, B, B, B} is not stable. For instance, the set {A, A, A} blocks it, because the first group of voters — over 3/5 of all voters — prefers {A, A, A} over any 3 candidates from {B, B, B, B, B}. Thus, the only stable sets under this definition are {A, A, A, A, C}, {A, A, A, B, C}, and {A, A, A, C, C}.
Under the above definition 4.3. of V(S,S'), the situation is the same as for definition 4.2. (These definitions might still differ in more-complex situations. Since definition 4.3. has stronger criteria for "strictly prefer", the set of stable winner sets under 4.2. will be a non-strict subset of that for 4.3.)
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== Droop Version==
If the formula V(S,S′)/n >= K′/K is modified to instead be V(S,S′)/n >= K′/'''(K+1),''' (it may be appropriate to make it only a > rather than an >=, for reasons to be explained below), then this makes stable sets' definition of proportionality become more similar to other definitions of PR that use Droop [[Quota]]s (or more specifically, Hagenbach-Bischoff Quotas) rather than Hare [[Quota]]s, and begins to resemble a Condorcet PR method. <ref>{{Cite web|url=https://arxiv.org/abs/1701.08023|title=The Condorcet Principle for Multiwinner Elections: From Shortlisting to Proportionality|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=|quote=A size-k committee is locally stable in an election with n voters if there is no candidate c and no group of more than n/(k+1) voters such that each voter in this group prefers c to each committee member.}}</ref>
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