Stable winner set: Difference between revisions
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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]]).
== Droop version==
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