[EM] Condorcet methods - should the cycle order always determine the result order?

[Juho Laatu]

On 04 Nov 2014, at 03:00, Toby Pereira <tdp201b at yahoo.co.uk> wrote:

> In a lot of the more preferred Condorcet methods (e.g. I think all of Schulze, Ranked Pairs, River, Kemeny), if you have, for example, an A>B>C>A cycle, then if, say, A wins then B will automatically finish second and C third (if B wins, C will be second etc.). But you could have a similar number of A>B>C, B>C>A and C>A>B ballots but then also a lot of A>C>B ballots, meaning that in some sense C looks better than B. But as long as this doesn't break the cycle and A wins, then B will still finish second. I think the following example does it:
> 
> 11: A>B>C
> 10: B>C>A
> 10: C>A>B
> 8: A>C>B
> 
> A beats B 29:10
> C beats A 20:19
> B beats C 21:18
> 
> I'm not saying these methods are wrong for doing this, but there is an intuitive sense in which C is arguably a better choice than B. So is it:
> 
> 1. There is a reasonable Condorcet method that would rank them A>C>B

Yes, I think e.g. minmax(margins) (= least additional votes) is a reasonable method, and it elects A (1 additional vote required to not lose to anyone, and one more to become a Condorcet winner) and considers C second best (3 additional votes required) and B last (19 additional votes required).

> 2. The intuition that C should finish ahead of B is poorly thought out.

That reasoning makes sense to me. There could be also other viewpoints since in a single winner election second place is not really well defined. It depends on what we intend to do with the second position. All we care is to elect the winner. In principle we don't care about the position of the remaining candidates.

Let's say that in this example the elected person (A) has to resign for some reason, and we don't want to arrange a new election, but we want to use the results of the old election to determine the alternative winner. In this case B looks like a natural second best winner since she was 4 votes short of being a Condorcet winner. So we elect B. But we could also think that although B would need 20 additional votes to become a Condorcet winner (very bad when compared to the others), we now note that A is no longer an electable candidate, and therefore B actually loses to no electable candidate, and therefore B should be elected. I thus mean that different elections may have different needs, and therefore different rules on who is "second" may apply.

Here's one more argument why the results of an election should not be a total ordering of the candidates. We can assume that it is easy for the voters to order the candidates in their personal order of preference. We may assume that within their heads the voters can rate the candidates using numbers or whatever internal feelings they may have. That is the common assumpton anyway. We don't expect any individual to have circular preferences. On the other hand we know that group opinions do not follow this rule. It is quite possible that a group would prefer A over B, B over C, and C over A. Group opinions can be circular. We should accept that. There is no point in trying to foce the group opinion (= declared outcome of the election) to be linear. We should basically tell that there was an opinion cycle but still A was found to be the best winner. We should not tell that the group opinon was A>B>C or A>C>B. I.e. there is really no point in forcing the group opinion to be a complete ordering of the candidates when it is not. We can discuss on who was second though, if we define what that "second" means (i.e. not second in the sense of having a linear group preference order). (Note that this point is also related to minmax failing the Smith set criterion. Is there a need to establish a linear ordering where the Smith set candidates must be ahead of the others, or can we elect a candidate that is almost a Condorcet winner (few votes short) but outside the Smith set?)

> 3. It is in a sense reasonable to think that C should finish ahead of B, but doing so would cause a method to fail certain criteria and end up worse as a result.

Yes. All methods fail some criteria, so it is not a big problem to add another one. There are not many criteria that would be absolute requirements, i.e. agreed by all and valid in all possible kind of elections. Often one can also think that violating multiple criteria, but violating all of them only a little bit, is a better situation that meeting all of those criteria except one that is violated a lot. That method could be very vulnerable to attacks on this one point, while the method with "more balanced" approach to vulnerabilities could be safe. At least in principle that's how you could spread the risks to multiple vulnerabilities that are small enough not to cause any problems.

Another approach to this question is to focus on determining the ideal winer instead of focusing on determining the full linear order. That way you can skip all the complete ordering related criteria :-).

Juho


> 4. Other?
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