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A stable winner set is a requirement on a winner set
 
Given a winner set S of K winners, another winner set S'S′ containing K’K′ winners blocks S iff V(S,S’S′)/n >= K’K′/K.
Where V(S,S’S′) is the number of voters who strictly prefer S’S′ to S and n is the number of voters.
A winner set is stable if no replacement set blocks it.
 
There are a few points which are important to note:
# In most cases K’K′ < K. This means that the definition of a stable winner set is that a subgroup of the population cannot be more happy with less winners given the relavant 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 term "strictly prefer" is generally considered to mean "receive more utility from the overall winner set". One possible variation is "receive more of their highest-preferred candidates."
#The formula V(S,S’S′)/n >= K’K′/K is analogous to a (Hare) quota; formulas analagous to other quotas may be used instead.
 
==Relation to Proportional Representation==
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S = {A1,A2,A3,A4,A5}
 
A blocking set is S’S′ = {B1}
 
V(S,S’S′) = 21 since their total utility from S is 0 and S’S′ is 5.
 
V(S,S’S′)/n = 21/100 = 0.21
K’K′/K = 1/5 = 0.2
 
0.21≥ 0.2 so S’S′ blocks S. Therefore, S is not stable.
 
Notably, stable winner sets focus on a voters' total utility from all candidates in a winner set, rather than the number of highest-preferred candidates they have in the set. The latter definition is closer to the traditional approximate definition of PR (voters receive as many of their highest-preferred candidates in proportion to their coalition sizes) which was meant to be used in conjunction with ordinal methods. The former definition is more relevant for cardinal methods; the difference can be seen with the example
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S = {A,B}
 
S’S′ = {C,D}
 
K=K’K′=2
 
n=20
 
V(S,S’S′) = 20 since everybody has a Utility of 10 from S and 18 from S’S′
 
V(S,S’S′)/n=20/20=1
 
K’K′/K = 2/2 = 1
 
1≥ 1 so S’S′ block S. Since S’S′ is not also blocked by S then S’S′ is the better solution.
 
It can be argued that even though {C, D} doesn't include any voter's 1st choice candidate, whereas {A,B} includes a 1st choice candidate of every voter, {C,D} is a better solution because it maximizes all voters' satisfaction with the overall set of winners. In essence, 1 point of utility is lost when comparing a voter's favorite candidate in each set but 9 points are gained for their 2nd-favorite candidate in each set. This form of analysis has been done with Condorcet PR methods before, though not while also discussing cardinal utility:<blockquote>Lifting preferences from candidates to committees is achieved through what we call ''f''-preferences. A given voter has an ''f''-preference for one possible committee ''A'' over another, ''B'', if the voter prefers ''A'' to ''B'' when considering in each committee '''only''' the ''f'' candidates most preferred by that voter. For example, a voter has a 1-preference for committee ''A'' over committee ''B'' if the voter' favorite candidate in committee ''A'' is preferred by that voter over the voter's favorite candidate in committee ''B''. The voter has a 2-preference for committee ''A'' over committee ''B'' if the two favorite candidates on committee 1 are preferred over the two favorite candidates on committee ''B''.<ref name=":0">https://civs.cs.cornell.edu/proportional.html</ref></blockquote>Note that under the "voter prefers the set with more of their highest-preferred candidates" definition, {A,B} would be the stable set here. This definition makes stable sets appear to become more analagous to a Smith-efficient Condorcet PR method, such as [[Schulze STV]].
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== Droop Version==
If the formula V(S,S’S′)/n >= K’K′/K is modified to instead be V(S,S’S′)/n >= K’K′/'''(K+1)''', 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.
 
This definition is more restrictive and as such has a number of undesirable situations where it eliminates all winner sets from being stable. This can happen even in super simple examples, e.g., two voters, one likes A, the other likes B, and one candidate to elect -- neither {A} nor {B} is stable. (However, if all sets are blocked by at least one other set, it may still be possible to come up with the smallest set of sets that aren't blocked by any other sets, and consider this the core instead. This is analogous to the Schwartz Set, and always results in a non-empty core. In this example, both {A} and {B} are in the core under this definition because they are not blocked by any other sets, "any other" being an empty set of sets. This core produced by this definition seems to always reduce to the core produced by the original definition, when one exists.)
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