Talk:Schulze method

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MarkusSchulze (talk · contribs) / Schulze method This user has contributed to the article. This user has declared a connection. (User devised the method that is the subject of this article, assuming compliance with WP:REALNAME.)

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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 — $Q$ uantling (talk | contribs) 20:38, 3 May 2024 (UTC)[reply]

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)[reply]

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)[reply]

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:

C(1) = X and C(n) = Y.
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)[reply]

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)[reply]

"No" votes are only partially counted in this system.

@MarkusSchulze, I don't know if this should be considered a disadvantage, but when the strength of candidate A's preference over candidate B is taken into account, then candidate B's preference over candidate A should be taken into account equally. Let me show you what I meant: Let's say A→20B and B→25A. Your method will write B→25A in the graph, since 25>20. My idea suggests writing taking into account the "resistance", which in this case will look like A→5B, since 25-20=5. Otherwise, the method works similarly, and, if I haven't made any mistakes, using the example from the article, the final rating would look like A→E→B→D→C instead of E→A→C→B→D. Makuta Dionis (talk) 17:37, 21 June 2025 (UTC)[reply]

Treatment of incomplete ballots

The article indicates $d [A, B]$ represents either the number of voters who strictly prefer $A$ to $B$ ( $A > B$ ), or the margin of (voters with $A > B$ ) minus (voters with $B > A$ ). With complete ballots these are easily seen to be equivalent because they are linearly related:

margin = number - (N - number) = (2 \times number) - N

where $N$ is the number of ballots.

However, I believe that they give different election results if some ballot is incomplete and does not rate either of $A$ or $B$ as better than the other. Where incomplete ballots are tolerated, which system is usually employed? — $Q$ uantling (talk | contribs) 19:10, 24 June 2025 (UTC)[reply]

Beatpaths that include losses

The example code and my straightforward reading of the written description both set the initial $p [i, j] = d [i, j]$ only if $d [i, j] > d [j, i]$ ; otherwise they set $p [i, j] = 0$ . Does the election result change if we instead initially set $p [i, j] = d [i, j]$ regardless of whether this condition is satisfied? In other words, is it the case that the paths whose values are thus changed will prove to be irrelevant in determining the ultimate election results?

What I am trying to get at: is this mathematically important, or is it adopted because it makes the understanding of the Schulze method easier? Regardless of the answer, I'd like to see a quick mention of it in the article. — $Q$ uantling (talk | contribs) 19:25, 24 June 2025 (UTC)[reply]

Example where Schulze method and Ranked Pairs differ

Can we add to the article an example of ballots where the Schulze method and Ranked Pairs give a different winner? I believe that they give the same winner when the Smith set is only three candidates, so I believe that an example will have to have at least four candidates. — $Q$ uantling (talk | contribs) 20:47, 1 July 2025 (UTC)[reply]

The correct way would be to write a template (similar to the Tennessee voting example) and to include this template in every relevant article. Markus Schulze 10:22, 3 July 2025 (UTC)[reply]

If you have the set of ballots, I'd be willing to draft a section to discuss it. (Or you could draft it if you want.) And we could then decide template or no at that point. —

Q

uantling (talk | contribs) 16:35, 3 July 2025 (UTC)[reply]