(Partial) Explanation of Schulze "beatpath voting," a complicated Condorcet-type voting system.

[Markus Schulze's papers on his method may be found here. His method was described in print in M.Schulze: A new monotonic and clone-independent single-winner election method, Voting Matters 17 (2003) 9-19. A short explanation of why Schulze invented it is here.]

Preface and disclaimer

In some sense the range versus Condorcet debate is a red herring since Condorcet methods have, I think, no chance of actual adoption by governments. And range does have a chance. So for practical purposes, forget Condorcet. Why do I say that?

Well, in our real-world-voter study of range & approval: USA voters by statistically clear margins, told us they wanted to stay with plurality and not switch to either range or approval voting. This makes it sound (correctly) like range will have a hard time getting adopted! Now as far as I and the other pollsters could imperfectly see from listening to them, the top reason the USA voters felt this way, was complexity. They felt range and approval were too complicated. But range voting is actually quite simple. From this my coauthor Doug Greene concluded that discussing methods significantly more complicated than range was just "mental masturbation" with no hope of actual political success.

That to me is a big reason to go with range: it seems to me to be by far the simplest method out there that much improves upon plurality voting. Approval voting is even simpler than range (though not as simple as plurality). So that may be a reason to go with approval. However, my feeling is that

  1. despite the simplicity advantage of approval, USA third parties would be foolish to support it because range experimentally gives them hugely more votes than approval.
  2. if we cannot get – at the very least – the USA third parties to support a method, there is no chance to get it adopted!
So it seems to me range is clearly more likely to be adopted by the USA because it, but not approval, has a good hope of getting at least a mildly large hard-core support base. In fact a key part of CRV's tactical plan is exactly to try to mobilize the third parties.

So it seems to me that from a practical point of view of maximizing our chances (as voting reform advocates) of success, there is only one reasonable choice: range voting. Not approval or IRV (not good enough to third parties) and not advanced Condorcet (too complicated for USA voters to accept).

So while I want the mental masturbation (that was a joke! I want the research and debate and education about all voting methods to continue, Ok!) to continue unabated, I think we have to unify as a practical matter behind range voting. And that is why I created both the range voting list and the CRV; I meanwhile recommend continuing the study and endless debate over all voting methods on Electorama. I encourage all Electoramers who feel similarly to also join the CRV. As some have already found, we also have interesting discussions here on CRV!

Now: Schulze's beatpaths method

To illustrate what I mean about advanced Condorcet methods being "complicated", let me now describe a Schulze-beatpaths method, which many favor, and in fact has been adopted by Debian for votes on Linux development. We will include the wv and equality-allowing enhancements, which both Tarr and I consider essential. Hopefully I will not make any severe errors in exposition.

(1) "votes": are orderings of the candidates of the form A>B>C=D>E (equalities are permitted). All candidates must be ordered and none omitted (if any are omitted, the system either refuses to accept your vote, or tells you it will assume all the unlisted candidates are ranked coequal last – either way you have no way to express ignorance about any candidate).

(2) For each pair of candidates (a,b) we compute the "winning vote count" for a over b, which is the number of voters whose vote said a>b, assuming this number exceeds the count for b>a. It would also be possible to base the method on "margins" but Schulze himself does not recommend that. The two are equivalent if there are no candidate-equalities in any votes. The result is a "directed graph" with N vertices where each vertex corresponds to one of the N candidates. For each pair (a,b) of candidates an arrow is drawn pointing from the loser (call it b) to the winner (here a) of that pairwise sub-election, and the arrow is labeled with the numerical value of the winning-vote count we just described.

(3) Now, in this directed graph, a "beatpath" is a directed path of edges, always walking in the direction of an arrow, which leads from some candidate L to some other W. The "strength" of this beatpath is the minimum value of the numerical labels on its arrows. Non-obviously, it is possible to compute all the beatpath strengths in polynomial time by a modification, invented by Schulze, of R.W.Floyd's dynamic programming algorithm for digraph all-shortest-paths.

(4) If the strongest path from L to W, is stronger than, or at least as strong as, the strongest path from W to L, and if this is simultaneously true for every L, then W is a "Schulze winner." Schulze proved the theorem that such a W always exists (at least using "margins"; I am confused re the "winning-votes" enhancement).

(5) Usually (i.e. in the absence of unlikely "ties"), W will be unique, but ties for winner are possible. In that case, Schulze advocates breaking the ties by a rather complicated procedure which I thoughtfully will spare you from having to know! (If I described it, the length of this description would roughly double.)

(6) In this procedure, it perhaps is possible or desirable to further enhance it by permitting "partial-order-type" votes so voters can express ignorance/no-opinion. Some such votes would be illegal since they contain a "preference cycle" and hence make no logical sense. They would have to be detected and rejected. That would add considerable additional complexity to the algorithm.

So. OK, what do you think? Does this method strike you as simple enough, compared to range voting, to allow its adoption by the public? Holy cow. I think it has some beautiful ideas, but... it's not going to fly with Joe Public.

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