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

Juho Laatu juho4880 at yahoo.co.uk
Tue Nov 4 15:12:30 PST 2014


Condorcet methods are usually designed using two different kind of criteria. One is "who is the best winner" and the other one is "is the method strong enough against strategic voting".

It is thus possible to use a method that does not always elect the best winner with sincere votes but that is tailored to be stronger against some kind of strategic voting attempts. The end result may be good enough if the anti strategic nature of the method efficiently makes the voters more sincere, and thereby improves the chances of electing a good candidate, although the method may not elect the best winner (whatever the criterion is) with sincere votes. In principle you could also have methods that try to cancel the effects of strategic voting, but that is very difficult since it is almost impossible to tell which votes are sincere and which ones are not.

Let's now forget the strategic concerns for a while, and focus on who is the best winner. Most Condorcet methods do not have a complete definition of what kind of a winner they want to elect. They may often be designed just as different technical algorithms with differen characteristics (i.e. with no clear plan on "who is the best winner"). Of course all of them think that Condorcet winners are good when they exist.

Minmax(margins) is one of the few that has a complete definition of the best candidate: elect the one that needs least number of additional votes to beat all the others / become a Condorcet winner. That definition is one approach to minimizing the level of opposition (in favour of any of the other candidates) after the election.

One partial definition (of who is the best winner) is the Smith set. Many Condorcet methods are designed to elect the winner always from the Smith set. That approach conflicts with the minmax(margins) definition of the best winner. As I already noted, this is related to the question whether one should force the group opinion to be presented as a linear order or not. Beatpaths are closely related to establishing this kind of linear order and guaranteeing that the winner comes from the Smith set. Personally I don't see any obvious need to establish linear orders in group preferences and to respect the Smith set, but many others probably do. The alternative philosophy is that defeats within the Smith set may well be worse (according to some criterion) than the defeats of those candidates that are outside the Smith set. In practce the winner comes with about 99.9% probability from the Smith set also in minmax(margins), and the question is whether the Smith set candidates can be too bad in some extreme cases.

My ability to analyze the differences of different defeat chain based and Smith set based methods ends here. I have no clear definition on how they differ in answering the question "who is the best winner". If any reader is able to give some clear definitions on how they differ from this point of view, please tell.

Within the group of Condorcet methods there are thus at least two main categories with respect to "who is the best winner". Some try to establish a complete preference ordering among the candidates, and some try to analyze the quality of the candidates without considering the complete chains of defeats.

Juho


P.S. I note that although most Condorcet methods use ranked votes, some methods can use also additional information like ratings or approvals (explicit cutoff or implicit cutoff after the last ranked candidate). This kind of additional information of course makes the answer to question "who is the best winner" somewhat different when compared to the plain rankings case.



On 04 Nov 2014, at 17:38, Toby Pereira <tdp201b at yahoo.co.uk> wrote:

> Thank you Juho and Kevin for your replies. I agree with you both that the main point of a Condorcet method is to find a winner. Anything else is relatively unimportant. If, for example, we chose the Minimax method because we decided that it gave a better overall order, we might find cases where we'd disagree with the winner. So using something like Schulze or Ranked Pairs would still be better if we thought they were better at finding a winner regardless of what they said about the other candidates.
> 
> With Schulze, I thought its beatpath method was transitive. I wasn't thinking of removing the winner and recalculating. Maybe I'm wrong about that. But yes, River obviously only affirms the pairwise results that it needs to in order to get a winner, so certainly can't be said to produce an overall ranking in all cases.
> 
> But I was thinking that maybe a Condorcet method might be used to generate an ordering in a sporting competition, where you have head-to-head matches, and where you want to find more than just first place. I'm not sure many sports would want the added complication of Condorcet though.
> 
> Toby
> 
> 
> 
> 
> From: Kevin Venzke <stepjak at yahoo.fr>
> To: em <election-methods at electorama.com> 
> Sent: Tuesday, 4 November 2014, 13:47
> Subject: Re: [EM] Condorcet methods - should the cycle order always	determine	the result order?
> 
> Hi Toby,
> 
> Especially with more than three candidates I think it is unclear how to define second or third place under Schulze. Are you removing the winner and recalculating?
> 
> River's "ranking" can fail to order two candidates, despite there being a pairwise winner between them.
> 
> I would say the only reason Ranked Pairs appears to indicate a ranking is that the only way we know how to find the RP winner is to construct this ranking. In other words, if you somehow find another method of calculating the RP winner, which doesn't involve constructing the full ranking, then it would be unclear to me that RP says anything about second or third place.
> 
> Kevin Venzke
> 
> De : Toby Pereira <tdp201b at yahoo.co.uk>
> À : "election-methods at electorama.com" <election-methods at electorama.com> 
> Envoyé le : Lundi 3 novembre 2014 19h00
> Objet : [EM] Condorcet methods - should the cycle order always determine	the result order?
> 
> 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
> 2. The intuition that C should finish ahead of B is poorly thought out.
> 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.
> 4. Other?
> 
> 
> 
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