Stable winner set: Difference between revisions
→Droop version
mNo edit summary |
|||
Line 12:
##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 utility 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."{{Clarify|reason=What does "counting from the top" mean? A solid coalition?|date=May 2024}}
#The formula V(S,S′)/n >= K′/K is analogous to a (Hare) quota; formulas analogous to other quotas may be used instead.
Line 19:
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
==Example==
Let's look at a common example. Let's say we have two voting blocs: group A and B. A makes up 79% of the population and B 21%. In a
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.
S = {A1,A2,A3,A4,A5}
Line 85:
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 <math>L</math> be some integer and consider a multi-winner election with <math>n=kL</math> voters and <math>k</math> seats
* Voters <math>1
* <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>
*
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.
Line 102:
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==
If the formula V(S,S′)/n >= K′/K is modified to instead be V(S,S′)/n >
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. (This is not the case if the formula is done only with an >, however. Consider that the above formula is really analogous to a Hagenbach-Bischoff quota; it has been [https://en.m.wikipedia.org/wiki/Hagenbach-Bischoff_quota#Disadvantage_of_Hagenbach-Bischoff_quota proposed] in STV that when using the HB quota, the rules should be modified such that a candidate must not only meet but exceed the quota to win, to avoid there being more winners than seats. Such a suggestion applies here in the form described
However, 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 instead. This is analogous to the Schwartz Set, and always results in a non-empty core. In this example, both {A} and {B} are in the core under this definition because they are not blocked by any other sets, "any other" being an empty set of sets. This core produced by this definition seems to always reduce to the core produced by the original definition, when one exists.
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:
Line 119 ⟶ 121:
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
==Further reading==
| |||