Stable winner set: Difference between revisions
Example of better model for "strictly prefer".
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# 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''.
#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 they like a set of candidates, if all of that set of candidates won.
#The term "strictly prefer" can have various meanings:
##In the simplest model, voters have a certain quantity of "utility" for each candidate, and they strictly prefer set X over Y iff the sum of their utility for X is greater than the sum of their utilty for Y. However, this definition, while simple, is problematic, because it can hinge on comparisons between "utilities" for winner sets of different sizes.
##Another possible model is to restrict direct comparison to winner sets of the same size. Thus, when comparing sets of different sizes, we use the rule: a voter strictly prefers set X of size x over a set Y of size y, where x≤y, iff: there is no set Z of size y, where X⊆Z⊆X∪Y, such that they strictly prefer Z over Y. Using the same "utility sum" model as above, this would be equivalent to: they strictly prefer X over Y iff they strictly prefer X over any size-x subset of Y. For instance, they'd prefer the two "Greek" winners {Γ, Δ} over the three "Latin" winners {A, B, C} iff they prefer the two Greeks over any two of the Latins.
##Another possible variation is "voters strictly prefer a winner set iff they receive more of their preferred candidates, counting from the top."
#The formula V(S,S′)/n >= K′/K is analogous to a (Hare) quota; formulas analogous to other quotas may be used instead.
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