Talk:Schulze method

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Local Independence of Irrelevant Alternatives

Since LIIA is one of the few criteria where Schulze and ranked pairs differ, should the article give an example where Schulze fails it or link to an external reference? WildGardener (talk) 01:22, 9 December 2020 (UTC)

In this example, the Schulze ranking is C > D > B > A. However, when candidate C is removed, then the Schulze ranking of the remaining candidates is D > A > B.

I had added this example in 2009 (diff). However, this example was removed by Daveagp in 2011 (diff). I didn't reinsert this example because I didn't want to be accused of starting an edit war. Markus Schulze 07:52, 9 December 2020 (UTC)

Thanks for the background! I didn't realize there was past history on this already. WildGardener (talk) 00:47, 10 December 2020 (UTC)

Simpler example

The 5-candidate pentagram example probably scares some people off. ;)

Maybe we could try simpler examples? 4 candidates is enough to show how Schulze differs from Minimax, without cluttering everything up with too many arrows (6 arrows total, and you can display them without intersections, vs. 10 arrows for a 5-candidate race). –Maximum Limelihood Estimator 17:02, 23 April 2024 (UTC)

Edits by expert / conflict of interest

MarkusSchulze, if you are the author / inventor of the Schulze Method ... first thank you for contributing to this article. Your expertise goes very far towards making this a great article. But, also, second, you have a potential conflict of interest when it comes to editing this article. The way I would balance these priorities ... please be extra deferential when it comes to disagreements with other editors. By all means, please argue strenuously (though politely) on the talk page if that is what it takes to make this article its best, but please stay very far away from edit warring (or edit skirmishes or edit stern looks ... you get the idea) when it comes to editing the article itself. It is the nature of authorship that you are both expert and potentially conflicted; so I ask you to adopt these measures to balance these priorities. (Actually, I'm not any sort of certified expert on the details of WP:COI; if you find that the formal advice is some other approach, please don't be shy about telling me.) Thank you —Quantling (talk | contribs) 20:38, 3 May 2024 (UTC)

Dear Quantling, it is a central aspect of the Schulze method that it can be proven that p[X,Y] > p[Y,X] and p[Y,Z] > p[Z,Y] together imply p[X,Z] > p[Z,X]. That's the whole point of why defeats are defined this way. If this aspect wasn't true then the Schulze method wasn't even well defined. But Closed Limelike Curves keeps removing this aspect. Markus Schulze 07:10, 4 May 2024 (UTC)

I'm confused why you think this is different from transitivity. If we define "X has a beatpath win over Y iff p[X,Y] > p[Y,X]", this just seems to be saying "if X has a beatpath-win over Y and Y has a beatpath-win over Z, X has a beatpath-win over Z"—i.e. beatpath-wins are a transitive relation. –Maximum Limelihood Estimator 21:18, 5 May 2024 (UTC)

Dear Closed Limelike Curves, this is not how you defined the Schulze method. You defined the Schulze method as follows [1]:

The idea behind Schulze's method is that if Alice defeats Bob, and Bob beats Charlie, then Alice "indirectly" defeats Charlie; this kind of indirect win is called a 'beatpath'.

Every beatpath is assigned a particular strength. The strength of a single-step beatpath is just the number of voters who rank A over B. The strength of a beatpath is equal to the strength of its weakest link, i.e. the victory with the smallest number of winning votes.

Alice is considered to have a "beatpath-win" over Charlie if their beatpath to Charlie is stronger than Charlie's beatpath to Alice. The winner is the candidate who has a beatpath-win over every other candidate.

Markus Schulze proved that this definition of a beatpath-win is transitive and free of cycles. Moreover, it will always produce a winner.

You are talking about beatpaths, about strengths of beatpaths, about transitivity, etc.. But you are not talking about strengths of strongest beatpaths. So the reader knows that he has to calculate all beatpaths and that he has to calculate the strengths of these beatpaths. But he doesn't know what to do with these values. Does he has to calculate the sum of the strengths of all beatpaths from Alice to Bob and the sum of the strengths of all beatpaths from Bob to Alice? Or do they have to remove beatpaths successively? Your description of the Schulze method is not a proper definition.

On the other side, this is a proper definition:

Let d[V,W] be the number of voters who prefer candidate V to candidate W.

A path from candidate X to candidate Y is a sequence of candidates C(1),...,C(n) with the following properties:

1. C(1) = X and C(n) = Y.
2. For all i = 1,...,(n-1): d[C(i),C(i+1)] > d[C(i+1),C(i)].

In other words, in a pairwise comparison, each candidate in the path will beat the following candidate.

The strength p of a path from candidate X to candidate Y is the smallest number of voters in the sequence of comparisons:

For all i = 1,...,(n-1): d[C(i),C(i+1)] ≥ p.

For a pair of candidates A and B that are connected by at least one path, the strength of the strongest path p[A,B] is the maximum strength of the paths connecting them. If there is no path from candidate A to candidate B at all, then p[A,B] = 0.

Candidate D is better than candidate E if and only if p[D,E] > p[E,D].

Candidate D is a potential winner if and only if p[D,E] ≥ p[E,D] for every other candidate E.

It can be proven that p[X,Y] > p[Y,X] and p[Y,Z] > p[Z,Y] together imply p[X,Z] > p[Z,X]. Therefore, it is guaranteed (1) that the above definition of "better" really defines a transitive relation and (2) that there is always at least one candidate D with p[D,E] ≥ p[E,D] for every other candidate E.

Markus Schulze 18:54, 6 May 2024 (UTC)

Ahh! You're right, I forgot to mention that it has to be the strongest beatpath from A to B. Are there any other issues with my description?

My real goal is to remove any LaTeX, because as beautiful as LaTeX typesetting is, it immediately terrifies even the average well-educated person. I know quite a few people whose first exposure to Condorcet methods is this article, and seeing math with dozens of single-letter variable names immediately puts them off the topic forever. –Maximum Limelihood Estimator 19:06, 6 May 2024 (UTC)