Schulze method: Difference between revisions
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The Schulze ranking is a [[Smith set ranking]]. This is because every candidate in the n-th Smith set will have a beatpath to all candidates in lower Smith sets (because they directly pairwise beat them), but all candidates in lower Smith sets will have no beatpath back to the candidates in the n-th Smith set, because by definition the candidates in the lower Smith sets are pairwise beaten by all candidates in higher Smith sets, and can thus only pairwise beat fellow members of lower Smith sets, who are also all pairwise beaten by all candidates in the n-th Smith set. Therefore, the strength of the path for candidates in the n-th Smith set to candidates in lower Smith sets is always stronger than the other way around. The same logic demonstrates why all candidates in the n-th Smith set will be ranked lower than all candidates in higher Smith sets.
[[File:Smith based Schulze example.png|thumb|An example of the Smith set-based variation of the Schulze method.]]
A possible variation of Schulze (caution; not proposed or seriously analyzed by Markus Schulze) which is only Smith-efficient and not Schwartz-efficient (see the image to the right for an example) can be described as "Iteratively repeat the following two steps until there are no more pairwise defeats, at which point all of the remaining candidates are tied to win: Eliminate all candidates not in the Smith set, and then turn the weakest pairwise defeat into a pairwise victory for both candidates in the matchup." This can be argued to be simpler than regular Schulze, since the Smith set is easier to understand than the Schwartz set. It will return the same result as regular Schulze when there are no pairwise ties between any members of the Smith set. (Unverified) It may be the case that this variation of Schulze can be described by changing the definition of a path in the Schulze description from a [[beatpath]] to a [[beat-or-tie path]] (i.e. changing the third property of a path from <blockquote>For all i = 1,...,(n-1): d[C(i),C(i+1)] > d[C(i+1),C(i)] </blockquote>to <blockquote>For all i = 1,...,(n-1): d[C(i),C(i+1)] '''>=''' d[C(i+1),C(i)] </blockquote>in which case this variation could be called the '''beat-or-tie path method''' or '''cloneproof Smith sequential dropping''' (though instead of dropping defeats, they are "flipped" to victories for both candidates in the matchup). It may be possible when using this variation to pretend a particular pairwise matchup simply didn't happen, rather than to say that both candidates in the matchup got a pairwise victory, when flipping or dropping defeats.
== Use of the Schulze method ==
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