On 10/02/2012 12:50 AM, Juho Laatu wrote:
I just note that there are many approaches to making the pairwise
comparisons.
- One could use proportions instead of margins => A/B isntead of
A-B.
- If one measures the number of poeple who took position, one would
have to know which ones voted for a tie intentionally, and which ones
voted for a tie because they thought those candidates were already
irrelevat, or because they didn't know the candidates, or were just
too lazy to mark all the details in the ballot. An wlternative would
be to assume that any tie is interpreted as an intentionally marked
tie. A candidate taht is not known by many voters probably will not
be ranked high anyway, so there may be no need for adjustments.
- Winning votes counts the amount of opposition, but doesn't care
about the amount of support.
- Also other more fine-tuned approaches to making the pairwise
comparisons could be developed. Or maybe rough and simple rules are
easier to justify.
- Truncation as a way to make the results of the truncated candidates
worse is not a nice option because it may lead to people not ranking
the candidates, which is contrary to the targets of ranked voting (=
collect all preference opinions). The worst case would be bullet
voting.
My earlier voting software has a number of ways of doing Condorcet
comparisons, although most are pretty obscure. These are:
- wv: winning votes, number of voters on the victorious side, 0 if losing
- lv: losing votes, number of voters in total minus number of voters on
the losing side, or 0 if this is the losing side
- margins: maximum of A>B - B>A and 0.
- lmargins: A>B - B>A, so negative numbers are permitted.
- pairwise opposition: number of voters on this side (even if this is
the losing side).
- wtv: same as wv, but ties also count (on both sides).
- tourn_wv: 1 if this is the winning side, otherwise 0.
- tourn_sym: 1 if this is the winning side, 0 for a tie, otherwise -1.
- fractional_wv: (A>B) / (A>B + B>A) if on the winning side, otherwise 0.
- relative_margins: (A>B - B>A) / (A>B + B/A)
- keener_margins: h((A>B + 1) / (A>B + B>A + 2)) where h(x) = 0.5 + 0.5
sign(x - 0.5) * sqrt(|2x - 1), as per
meyer.math.ncsu.edu/Meyer/Talks/OD_RankingCharleston.pdf .
It's not that hard to find different ways to compare Condorcet. I think
someone on the list had an idea of using a statistical comparison, i.e.
to say A>B if A beats B with a certain level of confidence (as one would
reason with polls), B>A if B beats a within the same level, and unknown
otherwise.
Perhaps the important part is not really what kind of interpretation one
uses as how well it goes with the three categories I have talked about
earlier. Well, both might be important. Say you had an interpretation
that gave second place votes much more weight (e.g. A>B plus two times A
votes in second place) than others. Even if this interpretation had some
criterion-failure avoiding properties, it could easily lead to people
doubting the legitimacy of the method with such a seemingly arbitrary
component to it.
And even in the three-categories classification, it's hard to find any
objectively "best" method. You can find Pareto-dominating and
Pareto-dominated methods. For instance, unless the societal value under
sincerity of Black (Condorcet/Borda) is better than, say, Ranked Pairs,
Ranked Pairs would Pareto-dominate Black and so we wouldn't have to
consider Black. This helps remove methods where you can get "something
for nothing" by switching to another method, but it still leaves the
frontier intact. It still leaves EM members free to argue about whether
Mono-Add-Top is more important than Plurality in methods passing Smith,
for example. Finally, some methods are pretty much on their own in their
area of the frontier: if you have a society that insists on mutual
majority, LNHelp, and LNHarm, you pretty much have to pick IRV (and lose
monotonicity in the process). It might be so with Condorcet
interpretations, too.
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