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:
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#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==
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==Limitations==
A set being in the core is not sufficient for "fairness",
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>.
== Droop Version==▼
* <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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