[EM] 3 or more choices - Condorcet

[Kristofer Munsterhjelm]

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.