[EM] 3 or more choices - Condorcet

[Juho Laatu]

> And even in the three-categories classification, it's hard to find any objectively "best" method.


The third category was "quality of the outcome under honesty". For this category only, finding the best method is straight forward in the sense that one can freely decide what the criterion of the best candidate is in each election.

The second category was "resistance to noise and strategy". It is difficult to estimate how much protection there should be against each threat scenario. It is easier to find the correct answers after the method has been in use for a while (in the given environment).

The first ctegory was "consistency with itself". Maybe this can be measured somehow, although opinions on what is good may be subjective.

Summing up all three properties to determine which system is best could be done in theory, but is of course quite complex.

Condorcet comparison methods are relevant in all three categories. Maybe the ability to separate comparison methods from the rest of the method makes discussion one step easeir / more structured.

I guess the most discussed topic around comparison methods has been strategy resistance (category two). Many of the comparison methods are so simple that category one doesn't cause major problems. In category three there might be something more to discuss. Also soft / heuristic approaches could be valid (in addition to the traditional simple and hard ones (that may be easier to define and agree)).

Juho


On 4.10.2012, at 22.44, Kristofer Munsterhjelm wrote:

> 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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