Stable winner set: Difference between revisions
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In proportional representation, a '''stable winner set''' (called the '''core''' in game theory parlance<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>) is a requirement on a winner set:
▲ 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 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 of how much they like a set of candidates who all 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.
==Relation to
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 is 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. A clear
==Example==
Let's look at a common example. Let's say we have two voting blocs: group A and B.
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 let's prove it is not stable
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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's 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 analogous to a Smith-efficient Condorcet PR method, such as [[Schulze STV]].
(It is possible to create various hybrids of the two definitions. One example is a voter being considered to prefer a set that 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.)
=== Example in which definition of "strictly prefers" matters===
== Droop Version==▼
In the above example, all the definitions of "strictly prefers" lead to the same winner sets being stable. But consider the following example (as above, each candidate letter stands for an unlimited group of clones):
61% vote A5 B4 C0
39% vote A0 B4 C5
Under the above definition 4.1. of V(S,S'), the winner set {B, B, B, B, B} is strictly preferred by all voters over any other set of size 5 or less, so it is the unique stable winner set.
Under the above definition 4.2. of V(S,S'), the winner set {B, B, B, B, B} is not stable. For instance, the set {A, A, A} blocks it (or any other set without at least three As), because the first group of voters — over 3/5 of all voters — prefers {A, A, A} over any 3 candidates from {B, B, B, B, B}. Similarly, any set without at least one C is blocked by {C} because of the second group of voters. Thus, the only stable sets under this definition are {A, A, A, A, C}, {A, A, A, B, C}, and {A, A, A, C, C}.
Under the above definition 4.3. of V(S,S'), the situation is the same as for definition 4.2. (These definitions might still differ in more-complex situations. Since definition 4.3. has stronger criteria for "strictly prefer", the set of stable winner sets under 4.2. will be a non-strict subset of that for 4.3.)
[[User:Jameson Quinn|Jameson Quinn]] has suggested the terms "sum-stable winner set" and "proportionally-stable winner set" for the stable winner sets under definitions 4.1 and 4.2 respectively.
==Limitations==
A set being in the core is not sufficient for "fairness", in the sense that core sets may exist where a quota of the voters gets all its approved candidates elected, even though nobody outside of that group of voters approved of them. This can be seen in the following example by Peters ''et al.'':<ref name="Peters Pierczyński Shah Skowron 2021 pp. 5656–5663">{{cite journal | last=Peters | first=Dominik | last2=Pierczyński | first2=Grzegorz | last3=Shah | first3=Nisarg | last4=Skowron | first4=Piotr | title=Market-Based Explanations of Collective Decisions | journal=Proceedings of the AAAI Conference on Artificial Intelligence | volume=35 | issue=6 | date=2021-05-18 | issn=2374-3468 | pages=5656–5663 | url=https://www.cs.toronto.edu/~nisarg/papers/priceability.pdf | access-date=2022-06-15}}</ref>
Let L be some integer and consider a multi-winner election with <math>n=kL</math> voters and <math>k</math> seats, and let the voters be split into factions who vote the following way:
* Voters 1 ... L approve candidates <math>c_1</math> ... <math>c_k</math>.
* <math>k-1</math> factions of <math>L-1</math> voters each approve a candidate nobody else approves: <math>c_{k+1},\ldots,c_{2k-1}</math>
* finally, <math>k-1</math> voters each approve a candidate nobody else approves: <math>c_{2k},\ldots,c_{3k-2}</math>.
Then the set <math>W_1 = \{c_1,\ldots,c_k\}</math> is in the core even though it denies representation to everybody but the first <math>L</math> voters. One can argue that the set <math>W_2 = \{c_1,c_{k+1},\ldots,c_{2k-1}\}</math> is a much fairer choice.
For instance, with k=3, L = 100:
{{ballots|1=
100: c1=c2=c3
99: c4
99: c5
1: c6
1: c7}}
The set <math>c_1, c_2, c_3</math> is in the core but <math>c_1, c_4, c_5</math> would arguably be more fair (and is the one elected by e.g [[proportional approval voting]]).
If the formula V(S,S′)/n >= K′/K is modified to instead be V(S,S′)/n >= K′/'''(K+1),''' (it may be appropriate to make it only a > rather than an >=, for reasons to be explained below), 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, and begins to resemble a Condorcet PR method. <ref>{{Cite web|url=https://arxiv.org/abs/1701.08023|title=The Condorcet Principle for Multiwinner Elections: From Shortlisting to Proportionality|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=|quote=A size-k committee is locally stable in an election with n voters if there is no candidate c and no group of more than n/(k+1) voters such that each voter in this group prefers c to each committee member.}}</ref>
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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: 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
* [https://arxiv.org/abs/1910.14008 Approximately Stable Committee Selection]
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