[EM] 3 or more choices - Condorcet

[Juho Laatu]

On 1.10.2012, at 19.16, Kristofer Munsterhjelm wrote:

> On 10/01/2012 12:13 AM, Juho Laatu wrote:
>> On 30.9.2012, at 15.41, Kristofer Munsterhjelm wrote:
> 
>>> As far as intrinsically Condorcet methods go, Ranked Pairs feels
>>> simple to me. The only tricky part is the indirect nature of the
>>> "unless it contradicts what you already affirmed" step.
>> 
>> To me the biggest problem of path based methods is that there is no very
>> good real life explanation to why chains of pairwise victories are so
>> important. In real life the idea of not electiong a candidate that would
>> lose to someone who would lose to someone etc. doesn't sound like an
>> important criterion (since it doesn't talk about what the candidate is
>> like or how strong the opposition would be, but about what the set of
>> candidates and its network of relations looks like). Probably there will
>> never be a long chain of changes from one winner to another in real life.
> 
> I don't think you need to go into path logic for Ranked Pairs. Rather, how about this?

Ranked Pairs is based on setting up a complete ranking where the result of one candidate may depend on pairwise comparisons of some distant candidates. If therere is a large top loop, changes in opinions between A and B may change the winner from C to D. In this sense some distant opinons along the paths somewhere may influence the "goodness" of a candidate.

> 
> "Because of the existence of cycles, it's obvious we need to discard some of the data. So, what data do we discard?

I wouldn't say that we have to discard some data but that we may violate some pairwise preference opinions in the sense that the winner may lose some pairwise comparisons.

The reason why I don't like word "discard" is actually related to the fact that this makes us too easily think of the end result as a complete ordeing of the candidates, where some facts had to be discarded because they did not fit in the picture. And here the problem is that group opinions may indeed be cyclic, and there is no need to "correct" them to a transitive order. The used words are not that important. But whatever the words, I do stick to the claim that group opinions are graphs, not linear orders, and we must decide who the winner is, in the presence of cyclic opinions (not by eliminating them, at least not in all methods).

(Same comments about terms like "breaking cycles".)

> If we have to discard a one-on-one victory, lets discard those that are as narrow, or involve as few voters, as possible.

Yes, it is in most cases better to violate some narrow victories rather than strong ones. (We can assume full rankings and skip the "few voters" criterion since it is not essential here and it would introduce new open questions.)

> Hence, we should go down the list of one-on-one contests and add the data they give to our order unless it would produce a cycle. That way, all the decisive contests get counted first and if we have to throw some away, it's the weaker ones."

I can see two approaches here. One is to measure the preference relations of each candidate seprately, e.g. how much and to which other candidates someone loses and how this influences this candidate's "goodness". The other approach is that also the pairwise preferences of other unrelated candidates may influence the "goodness" of this candidate. One special case of this second approach is to say that the best winner should be picked so that the group oinion is first forced into a linear opinion using some criteria, and then the first candidate of that order is the winner. Minmax is an example in the first category where only the "personal properties" of each candidate do count.

> 
> It's a little IRVish (justifying the method by the way it works rather than the outcome), but still...

I think the part that was "method oriented" was the formation of the linear ordering. The way Ranked Pairs arranges the candidates is however quite intuitive and natural (not as "heuristic" and "procedural" as IRV). But as already said, the intermediate result of a linear order of the candidates is not necessary, but just a method specfic trick.

Juho