# Stable Winner Set

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 if 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:

1. 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.
2. There can be more than one stable winner set. The group of all stable winner sets is referred to as the core.

## Relation to Proportional Representation

Each group of voters should feel that their preferences are sufficiently respected, so that they are not incentivized to deviate and choose an alternative winner set of smaller weight. In the common scenario that we do not know beforehand the exact nature of the demographic coalitions, we adopt the robust solution concept which requires the winner set to be agnostic to any potential subset of voters deviating. This means that the requirement of a stable winner set is equivalent to but more robust than the concept of Proportional representation.

This is a more strict definition than the Hare Quota Criterion which is typically what used as a stand in for Proportional Representation in non-partisan systems since there is no universally accepted definition. The existing definitions of Proportional Representation are unclear and conflicting.

## Example

Let's look at a common example lets say we have two voting blocks group A and B. B makes up 79% of the population and A 21%. In 5 winner election with max score of 5 and 100 voters, Group A will score all the A candidates 5 and the B candidates 0. Group B will do the opposite.

The best winner set for group A is {A1,A2,A3,A4,A5}. This is the bloc voting answer and is not the proportional answer. So lets prove it is not stable

S = {A1,A2,A3,A4,A5}

A blocking set is S’ = {B1}

V(S,S’) = 21 since their total utility from S is 0 and S’ is 5.

V(S,S’)/n = 21/100 = 0.21 K’/K = 1/5 = 0.2

0.21≥ 0.2 so 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

2 to elect, scores out of 10

10 voters: A=10, B=0 C=9, D=9

10 voters: A=0, B=10, C=9, D=9

S = {A,B}

S’ = {C,D}

K=K’=2

n=20

V(S,S’) = 20 since everybody has a Utility of 10 from S and 18 from S’

V(S,S’)/n=20/20=1

K’/K = 2/2 = 1

1≥ 1 so S’ block S. Since S’ is not also blocked by S then 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:
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.[1]
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.

## Notes

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 Quotas (or more specifically, Hagenbach-Bischoff Quotas) rather than Hare Quotas.

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.

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, use of the KP transform (most simply thought of as: convert scores into fractional approval ballots) appears to prevent this, and make stable sets include degree of preference.[2]

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, 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 more utility), as well as the discussion over whether to use Droop Quotas vs. Hare Quotas within the formula, has already been used before for Condorcet PR methods:
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.
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.[1]