[EM] Modified Overall Preferences
Juho Laatu
juho.laatu at gmail.com
Sun Jun 16 06:51:05 PDT 2019
The point of this mail is to promote the idea of separating the strength of different pairwise preferences from the ordinary rankings (of ranked or Condorcet methods). Instead of giving voters the ability to adjust the strength of their own vote or pairwise preferences (the more common approach), they can influence the strength of the final preferences of the whole electorate. In this approach the pairwise preferences of the electorate will not be changed from A>B to B>A. Only the strength of that preference (A>B or B>A) will be modified.
You could make many complex tricks with this kind of tools, but I'll concentrate here on one example method that is very simple to the voters. It will consist of adding just one approval like threshold to a basic Condorcet method. I'll use Minmax(margins) as the base method to be modified. Instead of voting A >> B >> C >> D >> E (a normal ranked vote) voters can vote also A > B > C >> D >> E (i.e. one or more of the first preferences may be weakened/moderated). The philosophy of the latter vote is that A, B and C are promoted as "favoured candidates" or "near clones" or "protected candidates" (with the full strength of one vote).
In the counting process, first count the normal pairwise preferences (i.e. the matrix). After that, some of the pairwise preferences are made weaker. And then the final results are counted, (almost) as in Minmax(margins).
Our first approach is to count the number of votes that had preference A>B (moderated preference) (= Mab). Then the A over B preference count (of the whole electorate) will be weakened by multiplying it with 1-Mab/N (where N = number of votes). Moderated preferences are transitive in the sense that A > B > C increases also the Mac count. The idea is that if voters consider A and B to be "near clones", their defeats to each other should be seen and treated as "friendly defeats". Their strength can thus be weakened, although a large number of voters may have preferred one over another (in a friendly way).
(I note that some alternatives to the presented example method could be to make it symmetrical by using factor 1-(Mab+Mba)/N, or one could use the number of votes that preferred either A to B or B to A instead of N.)
I'll add one more trick to the example method. The "friendliness factor" can be stronger than presented above. Let's say that 50% of the voters (maybe one of two parties) think that A and B should be treated as "near clones". Half of those voters have voted A > B, and half B > A. This means that 25% of the voters have given (moderated) preference A > B. Maybe already this 25% is enough to convince us that A and B indeed are to be treated as full clones. If so, we can use factor max(1-4*Mab/N,0) instead of 1-Mab/N. Here reaching strength 0 means that those candidates will be declared as "clones" and treated as such.
The counting process may have some problems with ties if multiple preferences will be equal to 0. I will not discuss this question much more in this mail. Let's just say that also those preferences (although they all seem to be equal to 0) can be seen to have different strength, e.g. based on the initial margins. And that if there are two parties with 50 votes, and there are some "clones" in one of the parties, they could be seen as one alternative when flipping the coin. (Winner among the clones to be decided separately if the clone party wins.)
Here's one example set of votes.
45 A>B>>C --> A>>C>>B
15 B>A>>C
40 C>B>>A
B is the sincere Condorcet winner. Supporters of A will however strategically bury B under C. In Minmax(margins) the worst defeats are A:-10, B:-70, C:-20. The strategic voters will get what they wanted. There were however 40 voters that said Mcb. This means that in the example method the strength of B's defeat to C will be 0 (using the "4*" in the factor). There are also 15 voters that said Mba. This helps A a bit (-10 --> -4), but not too much. If all C supporters (or 17 or more of them) had voted C>>B>>A, A would win. C supporters may thus vote C>B>>A sincerely or as a (still quite sincere) defensive tactic (after hearing about A supporters' plans).
Another question where this ability to moderate the defeats of favourite near clones is interesting is whether to elect from the Smith set or outside of it.
17 A>>B>>C>>D
17 B>>C>>A>>D
17 C>>A>>B>>D
16 D>>A>>B>>C
16 D>>B>>C>>A
16 D>>C>>A>>B
A, B and C are not "clones" in the described sense (which means no weakening of their mutual defeats since not a single voter proposed that). D wins.
17 A>B>C>>D
17 B>C>A>>D
17 C>A>B>>D
16 D>>A>>B>>C
16 D>>B>>C>>A
16 D>>C>>A>>B
A, B and C are "clones" (weight of mutual defeats = 0, with the "4*" moderation). One of them wins.
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