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

Wed Nov 5 14:05:30 PST 2014

The minmax philosophy in that example says roughly that one should elect a consensus candidate that is most probably widely accepted and least probable to cause problems like "we want X instead of D".

I already gave a quite detailed analysis of the Smith set in the previous mail below. The minmax philosophy says something like: there is no need to force the group opinion to be a linear ordering of the candidates. Actually in all the examples that we are discussiong here about, the group opinion is cyclic. If we want to form a linear order, then it is hard to argue against the fact that all members of the Smith set beat all the others, and it looks natural to make the order such that the Smith set members are ranked before the rest. But if we do not see the need for such linearization of the group opinions, then we are free to compare the weknesses and strengths of all the candidates (in other ways that may be more meaningful in the cyclic environment), and in this example all Smith set canidates are "bad" in the sense that some other candidates beat them badly (altough only slighly more than D is beaten in this particular example).

Then to the clone problem that I didn't cover yet. A, B and C could be clones in the sense that they are from the same party, and in fact all voters who vote e.g. A>B>C>D actually want to vote A>B>C>>>>D since all voters consider A, B and C about equally good. In this case it would make sense to think that party "ABC" is more popular than party "D" and therefore one of its members should win. But this we don't know. It is quite as possible that we have four parties that are fierce competitors, and all of them have just one candidate. B is beaten badly by "enemy and competitor" A, not by "friend and supporter" A.

In the given example candidates A, B and C are clones also in the (EM clone criterion) sense that A, B and C are ranked next to each others in every ballot. This could mean that they could be clones in the sense of the previous paragraph. On the other hand a similar pairwise preference matrix could be generated also using other kind of votes. It could as well be so that large majority of the ballots do not rank A, B and C next to each others. All typical Condorcet methods use the matrix to count the results, and therefore the method has no idea if the voters ranked A, B and C next to each others not. If some voter ranks them next to each others, they could or could not be clones in the eyes of this voter. If they are not next to each others, they can not be a three member clone set in the eyes of this voter.

The point thus is that although following clone criterion strictly may make sometimes improve the outcome of the election (when there are indeed three very closely related clones that all happen to lose to each others badly), there are also many situations where such a requirement forces some non-clones to be treated as if they were clones. In the matrix of this example A, B and C could be bitter rivals as well. And in this case D would be a less controversial compromise winner. In this case not electing D would be a violation of some other principles - like the idea that the least controversial candidate / the candidates with smallest opposition in favour of the other candidates should be elected. You could call it the LAV criterion (Least Additional Votes required to beat all others).

I note that there are also methods where the clones are explicitly marked as clones, e.g. if they come from the same party. That approach allows the method to treat declared clones in such a way that their losses to each other will not be counted against them when the winner is determined. That would solve the clone problem in a way that makes it possible to separate treament of actual clones from rules that force us to always elect such candidates whose results in the pairwise matrix could be a result of a vote set in which certain groups of candidates could be next to each others in every ballot. The former would be a more accurate technique. The latter could be called an overkill.

I hope this clarified why I don't see the clone criterion to be an absolute recuiremet. It is pretty inaccurate and can lead to bad results too. In the minmax(margins) case it forbids electing some candidates that clearly are good winners from one point of view, i.e. if one wants to respect the LAV criterion. It is in theory possible that reaspecting the LAW criterion would sometimes work against the clone criterion (when all voter rank A, B and C together), or lead lead to some party losing because of having nominated three candidates that end up in a strong loop with each others. But this is not a likely scenario, and we can not tell if it happened, or if the candidates of that one party were amicable enough to ignore their defeats to each others in the counting process as insignificant. It may be that the four competing parties scenario with mixed votes of all kind (not only votes that rank A, B and C together) is a more probable real life election sceario - also when the election produces a similar matrix to this example.

> I, like you, don't have a clear notion of how to define the best winner in a ranked-ballot election

Note that I do have a clear definition of who the best winner is for some methods like minmax(margins). It clearly elects exaxtly the candidate that needs least additional votes to beat all others. The LAV criterion may not be the preferred choice for all needs, but it certainly is one possible target for many possible uses.

Note also that I didn't have a clear defnition for the _path_based_ Condorcet methods, and I'd like to hear some argumentation that would defend the choices that they make, hopefully in the form of a complete and exact dfinition of the ideal winner for some such method from some chosen point of view.

Juho

On 05 Nov 2014, at 22:43, Toby Pereira <tdp201b at yahoo.co.uk> wrote:

> On minimax, because it doesn't always elect from the Smith set and isn't cloneproof, it can give some weird results. For example:
> 
> 10: A>B>C>D
> 10: B>C>A>D
> 10: C>A>B>D
> 6: D>A>B>C
> 6: D>B>C>A
> 6: D>C>A>B
> 
> With these ballots, A beats B 32:16, B beats C 32:16 and C beats A 32:16. All of A, B, C beat D 30:18. But that's the smallest defeat so D is the minimax winner. I, like you, don't have a clear notion of how to define the best winner in a ranked-ballot election, but one definition I wouldn't use is the minimax definition.
> 
> From: Juho Laatu <juho4880 at yahoo.co.uk>
> To: em <election-methods at electorama.com> 
> Sent: Tuesday, 4 November 2014, 23:12
> Subject: Re: [EM] Condorcet methods - should the cycle order	always	determine	the result order?
> 
> 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.
> 
> 
> 
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