Stable winner set: Difference between revisions
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In proportional representation, a '''stable winner set'''
{{Definition| Given a winner set <math>S</math> of <math>k</math> winners, another winner set <math>S^\prime</math> containing <math>k^\prime</math> winners blocks <math>S</math> iff <math>\frac{V(S,S^\prime)}{n} \geq \frac{K^\prime}{K}</math>, where <math>V(S,S^\prime)</math> is the number of voters who strictly prefer <math>S^\prime</math> to <math>S</math>, and <math>n</math> is the number of voters.
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There are a few points which are important to note:
# In most cases K′ < K. This means that the definition of a stable winner set is that a subgroup of the population cannot be more happy with fewer winners given the relevant size comparison of the Ks and the group. This relates to how PR methods attempt to maximize voters' representation by electing those who sizeable subgroups each strictly prefer, rather than only those who only majorities or pluralities can agree on.
# There can be more than one stable winner set. The group of all stable winner sets is referred to as '''the''' '''core'''.'''''<ref>{{Cite journal|title=|url=https://dl.acm.org/doi/abs/10.1145/3357713.3384238|journal=}}</ref><ref>{{Cite journal|title=|url=https://arxiv.org/pdf/1911.11747.pdf|journal=}}</ref>'''''
#The definition of V(X,Y) must be such that, if Y is a subset of X, V(X, Y) must be 0. That is to say: it does not make sense for voters to want to "block" a winner set because of how much they like a set of candidates who all won.
#The term "strictly prefer" can have various meanings:
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