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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' containing K’ winners blocks S ififf V(S,S’)/n >= K’/K.
Where V(S,S’) is the number of voters who strictly prefer 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. 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''. Under certain variations of the stable set definition, this is equivalent to the Smith Set.
#The term "strictly prefer" canis begenerally thoughtconsidered ofto either asmean "receive more utility from the overall winner set". orOne possible variation is "receive more of their highest-preferred candidates." Also, the formula V(S,S’)/n >= K’/K is analogous to a quota and can be modified accordingly.
#The formula V(S,S’)/n >= 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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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]].
 
(It is possible to create various hybrids of the two definitions i.e. One example is a voter prefersbeing considered to prefer a set which offers them slightly less utility so long as it has a certain additional number of more-preferred candidates in it. So between one set where a voter gets their favorite candidate and 10 utility and another set where the voter gets their 2nd and 3rd favorite candidates and 11 utility, the former set could be considered preferred if the definition of "strictly prefer" added at least 1 or more points of utility to the voter's preference for the former set because it had a more-preferred candidate, the favorite, in it.)
 
== Droop Version==
If the formula V(S,S’)/n >= K’/K is modified to instead be V(S,S’)/n >= 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. The(However, mostif 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. problematicThis is analogous to the singleSchwartz Set, and always results in a non-winnerempty casecore. In this example, whichboth is{A} thatand {B} are in the Scorecore winnerunder maythis nodefinition longerbecause bethey stableare not blocked by any other sets, "any other" being an empty set of sets. ConsiderThis core produced by this definition seems to always reduce to the followingcore produced by the original definition, when one exists.) example:
 
49% vote A5 B3 C0▼
The biggest issue when using the Droop stability definition arises in the single-winner case, which is that the Score winner may no longer be stable. Consider the following example:
2% vote A5 B3 C5▼
 
▲49% vote A5 B3 C0
 
▲2% vote A5 B3 C5
 
49% vote A0 B3 C5
 
B is the Score/Utilitarian winner and is stable under Hare but not Droop.
 
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. This is analogous to the Schwartz Set.
 
Also, stable sets can have this "quota" computed based solely on voters who have preferences between any pair of sets that are being compared, so that in a 2-winner Approval Voting election with 67 A 33 B 10 C, the quota when looking at matchups between sets including either or both A and B is only computed off of at most the 100 voters that have preferences between them, rather than all 110. This would fix some but not all of the issues with this definition.
 
Using Droop Quotas and this "only voters with preferences between the relevant sets are used to compute the quota" trick makes stable sets become a Smith-efficient Condorcet method in the single-winner case. However, useand ofappear to become a Condorcet PR method in the [[KPmultiwinner transform]]case (mostwith simplythe thoughtcore ofbeing as:analagous convertto scoresa intoSmith fractionalSet approvalof ballotswinner sets). However, use of the [[KP transform]] appears to prevent this, and makemakes stable sets always include degree of preference and remain a cardinal method so long as it is done on rated ballots.<ref>[https://forum.electionscience.org/t/the-concept-of-a-stable-winner-set/553/26?u=assetvotingadvocacy https://forum.electionscience.org/t/the-concept-of-a-stable-winner-set/553/26]</ref>
 
Interestingly, the two varying modes of deciding which set a voter prefers in each pairwise matchup (evaluate a voter's preference between sets as based either on, either: first whether they have a more-preferred candidate in one set, and then second more of their more-preferred candidates in that set, or as being based on: which set gives them more utility), as well as the discussion over whether to use Droop Quotas vs. Hare Quotas within the formula, has already been discussed before for Condorcet PR methods:<blockquote>We deferred the question of how to decide whether a voter prefers one set of ''f'' candidates over another, where a set of candidates is a subset of a committee. In [[Proportional representation]] mode, there is only one difference from the voter's perspective. The voting algorithm decides which of two committees would be preferred by a candidate using one of two criteria, ''combined weights'' or ''best candidate''.</blockquote><blockquote>The factor (''k''+1) may be surprising in the condition for proportional validity, but it actually agrees with proportional representation election methods developed elsewhere; it is analogous to the Droop quota used by many STV election methods.<ref name=":0" /></blockquote>
 
==Further Reading==
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