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

Toby Pereira tdp201b at yahoo.co.uk
Wed Nov 5 12:43:32 PST 2014

```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.

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> 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?
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>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".
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>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.
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>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.
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>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.
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>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.
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>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.
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>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.
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>Juho
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>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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