Most of my previous articles have been about methods (voting systems) that I recommend for some particular application and set of conditions. I’d now like to discuss a popular method that I don’t recommend: Schulze. I’ll call it by that name, because that’s how it’s usually known. A descriptive name would be “beatpath”, because Schulze defines it in terms of beatpaths.
Since I don’t recommend it, there’s no point in defining it. On the other hand, however, if you’re curious, I don’t want to make you have to look up the definition, and so below I’ll include Schulze’s beatpath definition of his method. The inclusion of this definition doesn’t mean that I suggest that you read it. I suggest that you skip it. Only read it if you’d otherwise take the trouble to look it up elsewhere.
I’ve already defined “beats”:
X beats Y if more voters rank X over Y than rank Y over X.
A pairwise defeat is what Y has if X beats Y.
Definition of a beatpath:
There is a beatpath from X to Y if either X beats Y, or if X beats something that has a beatpath to Y. The strength of a beatpath is measured by the strength of the weakest defeat that is part of that beatpath. If X beats Y, then the strength of that pairwise defeat is measured by the number of voters ranking X over Y. If there is a stronger beatpath from X to Y, than from Y to X, then X has a beatpath-win over X. The winner is the alternative that has a beatpath-win over each of the other alternatives.
As I said, Schulze is popular. One website lists 72 organizations that use Schulze for their voting. Evidently someone has been very busily promoting Schulze. Schulze is probably the most popular rank method other than IRV.
What’s good about Schulze Ranked Voting System?
I’ll start by saying that, if everyone ranked sincerely, then Schulze would be an excellent method. It’s often been assumed, in academic discussion, that everyone ranks sincerely. Academic assumptions and premises tend to be fashionably copied by many outside of academia. Maybe that’s a reason why some consider Schulze to be a fine method.
But Schulze isn’t so good when we drop the assumption of universally sincere ranking, and consider strategy incentives and strategy needs. I’ll return to that later, when I discuss some strategy criteria and properties.
First a word about Schulze-promotion by criteria:
Well, its first-proposer provides a list of criteria that it meets. It’s a long list, and one could easily get the impression that it must be very good, meeting so many criteria. But anyone can make up a criterion. When citing a criterion-compliance, one should tell why that criterion is important, or why one claims that it’s necessary. Without that, a list of criterion compliances is quite meaningless.
Obviously it’s the task of the proponent to tell what _important or necessary_ criteria his proposed method meets, and why those criteria are important or necessary. What are some of Schulze’s criterion compliances that I acknowledge as important? I defined the Mutual Majority Criterion (MMC) in my two most recent articles, but let me define it again here, because I consider it an important criterion for a rank method.
- The Mutual Majority:
- A mutual majority (MM) is a set of voters, comprising a majority of the voters, who all prefer some same set of candidates to all of the other candidates. That set of candidates is that MM’s “MM-preferred set”.
- Mutual Majority Criterion (MMC):
- If a MM vote sincerely, then the winner should come from their MM-preferred set.
- As a supporting definition, it’s necessary to define sincere voting:
- A voter votes sincerely if s/he doesn’t vote an unfelt preference, or fail to vote a felt preference that the balloting system in use would have allowed hir to vote in addition to the preferences that s/he actually did vote.
- To vote a preference for X over Y is to vote X over Y.
- To vote a felt preference is to vote X over Y, if preferring X to Y.
- To vote an unfelt preference is to vote X over Y if not preferring X to Y.
MMC is very important and valuable. It’s what guarantees majority rule for a MM, without requiring anything more than sincere ranking. The problem is that when there’s a chicken dilemma, that chicken dilemma gives dis-incentive for sincere ranking. The chicken dilemma can and will (when it occurs) spoil a mutual majority. When that happens, that chicken dilemma makes MMC compliance meaningless and useless. Therefore, Schulze’s MMC compliance is meaningless and useless. I’ll give a brief definition of the Condorcet Criterion, for actual votes, for rank methods:
Condorcet Criterion (CC):
If there’s an alternative that beats each one of the others, then it should win.
But, as with MMC, CC compliance is made meaningless and worthless when there’s a chicken dilemma. Therefore, Beatpath’s CC compliance doesn’t mean much. As I’ve said in my previous two articles, MMC compliance, and freedom from chicken dilemma, are a very powerful combination. That combination of properties is possessed by IRV, Benham, Woodall, and Schwartz Woodall (defined in my 2 most recent articles).
Additionally, Benham, Woodall, and Schwartz Woodall comply with the Condorcet Criterion, making them, in comparison to plain IRV, less vulnerable to replacement by a dis-satisfied majority. I’ve discussed the relative merits, for various applications, of IRV, Benham, Woodall and Schwartz Woodall. They’re all excellent methods. I’ve often pointed out that there’s no excuse for a rank method to have chicken dilemma, and that there’s no reason to propose, consider, or use a rank method that has a chicken dilemma. That disqualifies Schulze.
As I’ve been saying, I suggest that, for “current conditions” (dishonest, disinformational media and a public who believe everything that those media say, such as who is a viable candidate, and which candidates and policies are acceptable), it’s necessary for a voting system to be favorite-burial foolproof, and never, under any circumstances, cause a worsening or your outcome because you top-rated or top-ranked your favorite. That’s the Favorite-Betrayal Criterion, precisely defined in previous articles. Without that complete freedom from favorite-burial incentive, disinformational media can easily have millions of voters burying their favorites in order to instead favor some “lesser-evil”, …as happens now, of course.
Schulze’s FBC failure, its possibility of favorite-burial incentive, is well-known among voting system reform advocates, and not denied. That disqualifies Schulze for “current conditions”. Schulze’s chicken dilemma disqualifies Schulze under any and all conditions.
Now, lastly, let me give 3 examples of Schulze’s chicken dilemma. But first, let me define the chicken dilemma, though I’ve defined it in previous articles:
Say there are 3 candidates, A, B, and C. The A voters + the B voters add up to a majority. The size-relation of the candidates’ support-factions is:
C>A>B
The A voters and the B voters greatly prefer both A and B to C. In fact, the A voters and the B voters “despise and detest” C. For the sake of simplicity, let’s say that the C voters are indifferent between A and B.
Below, I’ll show examples of what can happen, but first I’ll just verbally summarize what can happen: First of all, of course A is the CW. (the “Condorcet Winner—the candidate who’d pairwise-beat each of the others, under sincere voting). Candidate A is the “sincere CW”. In comparison to each of the other candidates, more people prefer A to the other candidate than vice-versa. A should win, and would win in Schulze, Benham, Woodall, or Schwartz Woodall, or any Condorcet method, under sincere voting.
Obviously, if each faction just rank only their favorite, then C will win. To defeat C, it’s necessary that at least one faction rank both A and B over C. So the A voters, being co-operative, and wanting to defeat C, rank B in 2nd place. But the B voters, knowing that the A voters are co-operative and responsible, decide to take advantage of the A voters’ co-operativenes and responsibility: The B voters refuse to rank A. The result? B wins, by defection, by taking advantage of the A voters’ co-operativeness. By taking advantage of the fact that the A voters wanted to help B.
The message that Schulze is sending is: “You help, you lose.”
That isn’t good. That’s the chicken dilemma. Schulze is always susceptible to the chicken dilemma, whenever the above-stated conditions obtain. The chicken dilemma is well known and much discussed in game theory.
Some like Schulze because it does better than some other Condorcet methods, in regards to Condorcet offensive strategy—strategy intended to take the win from a majority by burying (ranking insincerely low) or refusing to rank a candidate. But, with a MMC complying method, those strategies can’t take the win from a sincere-voting mutual majority. MMC holds. MMC trumps Condorcet offensive strategy.
But the chicken dilemma can disrupt and disband a mutual majority so that its members won’t mutually help each other. Chicken dilemma, not Condorcet offensive strategy, threatens mutual majorities. Therefore, I claim that chicken dilemma is worse than Condorcet offensive strategy, and that therefore, Schulze’s chicken dilemma outweighs its slightly better defense against Condorcet offensive strategy.
Now, let me show a few examples of Schulze’s chicken dilemma:
In the example-tables below, the number on the left, on each line, is the number of voters who have the preferences stated on that line, or who vote the rankings stated on that line. “A>B>C” indicates preference for A over B, and for B over C. …or a ranking of A over B, and B over C. “>>” indicates a much stronger preference. I’ll give 3 examples.
Example 1:
- Sincere preferences:
- 99: A>B>>C
- 2: B>A>>C
- 100: C>>(A=B)
- The A voters rank sincerely, and the B voters defect:
- 99: A>B
- 2: B
- 100: C
Schulze elects B. The B voters’ defection has stolen the election from A, the CW. …even though there are only 2 voters to whom B is favorite. 2 defecting B voters have stolen the election from 99 co-operative A voters.
Example 2: Here’s an example in which the 3 factions are nearly equal in size
- Sincere preferences:
- 33: A>B>>C
- 32: B>A>>C
- 34: C>>(A=B)
- Actual votes, when A voters co-operate and B voters defect:
- 33: A>B
- 32: B
- 34: C
Again, though A is CW, B wins by defection.
Example 3: Here’s an example in which the A and B voters barely add up to a majority, and are nearly equal to each other
- Sincere preferences:
- 26: A>B>>C
- 25: B>A>>C
- 49: C>>(A=B)
- Actual rankings, when the A voters rank sincerely and the B voters defect:
- 26: A>B
- 25: B
- 49: C
Again, the B voters’ defection elects B, stealing the election from A, the CW.
If the chicken dilemma were unavoidable, then we could just say, “Oh well”, and hope for the best. But the chicken dilemma is easily avoided. There are rank methods that avoid it, while retaining other important properties. I’ll list a few below, with their desirable properties, under two conditions-headings:
For current conditions:
Symmetrical ICT
It meets FBC, has no chicken dilemma, and meets 0-info LNHe (defined in previous articles). It doesn’t strictly meet MMC, but meets it unless there’s a top-cycle among the MM-preferred set of candidates.
For the Green scenario:
IRV:
Meets MMC, has no chicken dilemma. Meets LNHa and LNHe, and is clone-independent.
Benham, Woodall, and Schwartz Woodall:
They meet MMC, have no chicken dilemma, and meet the Condorcet Criterion, and the Smith Criterion. (defined in the article with the criterion-compliance chart).
Schulze Conclusions
There’s no reason to advocate, consider, or use Schulze, because there are rank methods with better properties, for any and all applications and conditions.
Richard Fobes says
The opinions expressed in this article are not shared by most election-method experts.
The author of this article dismisses attempts to point out why his opinions are not widely shared, so if you want peer-reviewed opinions about voting methods, please ask questions at the election-methods forum (at https://lists.electorama.com/pipermail/election-methods-electorama.com/) where multiple election-method experts can answer your voting questions without the unusual bias presented here.
Or you can read the “Declaration of Election-Method Reform Advocates” at http://www.BanSingleMarkBallots.org to verify that many experts agree that the Condorcet-Schulze method is better than instant-runoff voting (IRV).
It’s not the method I advocate, but it is far better than Runoff voting methods and, in my opinion, much better than Approval voting.
Besides offering significant advantages over plurality and IRV, a major reason why the Condorcet-Schulze method is often used is that software for this method has been available for many years (because it is easy to code), and organizations naturally tend to choose what is already used in other organizations.
Michael Ossipoff says
One more reply-comment:
Richard says:
[quote]
so if you want peer-reviewed opinions about voting methods, please ask questions at the election-methods forum where multiple election-method experts can answer your voting questions without the unusual bias presented here.
[/quote]
It wasn’t my purpose or intent to criticize the election-methods forum. But Richard brought it up, and so it isn’t unfair for me to reply.
Richard’s amusing reference to EM as “peer-reviewed”, exemplifies a pseudoscientific pretense that is unproductive in voting-system discussion.
Regrettably, it’s common in Internet discussioin for people to do a poor job of saying what they mean, and an even poorer job of justifying what they say. Richard has been exemplifying that tendency as well.
Michael Ossipoff
Michael Ossipoff says
Richard says:
[quote]
The opinions expressed in this article are not shared by most election-method experts.
[/quote]
Yes, that’s what Richard always says. …as if polling were the ultimate source of truth on these matters.
As I explained last time Richard repeated that refrain, the experts don’t agree on what voting system is best, or what criteria are the most important.
Richard, wanting to disagree with what I’ve said, can’t support his disagreement other than by invoking some sort of vague and nonexistent consensus among unspecified experts.
Is it too much to ask that we support our claims?
[quote]
The author of this article dismisses attempts to point out why his opinions are not widely shared
[/quote]
Not many opinions are widely-shared, on voting systems.
But that’s another thing that Richard has claimed before. The last time he said it, at EM, I asked him to cite an instance in which I’d dismissed, without adequate reply, someone’s arguments. He couldn’t cite an instance.
I don’t want to miss replying to an argument regarding something that I’ve said. As I said on EM, if I’ve missed something, if I’ve missed answering some criticism or argument, then please tell me what I missed, because I want to answer it.
What I _do_ dismiss is vague, unssupported expression of personal opinion, which is what Richard is indulging in now.
[quote]
, so if you want peer-reviewed opinions about voting methods, please ask questions at the election-methods forum
[/quote]
If, by “peer-reviewed” means that anyone can commment on what anyone else has posted, then sure the EM forum is peer-reviewed :-)
Similarly, anyone, including Richard, can comment on articles at Democracy Chronicles, just as Richard is doing. And if Richard wants to say that I’ve said something that is incorrect, then I invite him to say, specificallly, what he thinks is incorrect, and then share with us why he thinks so.
But that isn’t what Richard does. Instead, he makes angry noises, and vaguely refers to some expert consensus that disagrees with me. Every time he doesn’t like one of my articles, he trots out his repetition of that vagueness.
How can I answer someone who doesn’t cite a specific incorrect fact or claim, or support anything that he says?
Forgive me if I sound dismissive :-)
[quote]
…where multiple election-method experts can answer your voting questions without the unusual bias presented here.
[/quote]
Does Richard even know what he means by “bias”? Does “bias” mean any position that differs from that of the current few frequent posters at EM? What about other forums, such as electology? Are they, too, biased, because their position (advocacy of Approval and Score as superior to the Condorcet methods) differs from that of the most frequent posters at EM (who don’t really agree with eachother anyway)?
Bias is a prejudice, a pre-judgment, that makes someone repeat their claims in complete disregard for arguments to the ocntrary.
A sign of bias is a failure to support one’s statements and claims–like Richards consistent complete failure to support his claims.
As I said last time Richard did this, I would be glad to answer Richard’s arguments regarding my article. If Richard would tell me what, specifically, he disagrees with in my article, share with us why he thinks it’s wrong, then I’d be glad to answer his arguments.
But that isn’t what Richard is all about.
[quote]
Or you can read the “Declaration of Election-Method Reform Advocates” at https://www.BanSingleMarkBallots.org to verify that many experts agree that the Condorcet-Schulze method is better than instant-runoff voting (IRV).
[/quote]
Actually, the declaration expressed reservation about IRV, but didn’t take a position on it.
I’ve always made it clear, at Democracy Chronicles, at EM, and at electology, that IRV would be a fine method in the Green scenario, but is inadequate for current conditions in the U.S. I don’t advocate IRV for current conditions in the U.S.
A few people (Richard and one other person?) wrote the “declaration”. Others signed it. I signed it too, because I assumed that the declaration’s reservations about IRV referred to current conditions. As I said, I oppose the use of IRV under current conditions, and I’ve expressed that opposition here, at EM, and at Electology.
By the way, the methods Benham, Woodall, and Schwartz Woodall, which I’ve recommended for the Green scenario, are Condorcet methods.
[quote]
It’s [referring to Schulze] not the method I advocate, but it is far better than Runoff voting methods…
[/quote]
Richard is statig his opinion as fact. Richard doesn’t seem to understand the need to tell _why_ he believes what he believes. Asserting his beliefs won’t do.
I told why Schulze is inferior to IRV, for the Green scenario, and for organizational voting. I told of the disadvantages that disqualify Schulze for consideration. I supported my claims, and I gave examples.
If Richard wants to be more forthcoming wit us, more specific with us about why he thinks that Schulze is a good method, maybe even better than those that I’ve recommended for current conditions, &/or for the Green scenario, &/or for organizational voting, the’d welcome to.
If Richard wants to tell us specifically what statement(s) in my article he disgrees with, and give supporting reasons, then I cordiallly invite him to do so.
[quote]
…and, in my opinion, much better than Approval voting.
[/quote]
…in Richard’s unsupported opinion.
For current conditions, Schulze is inferior to Approval, Score, and Symmetrical ICT, because of Sculze’s failure of FBC, the Favorite-Betrayal Criterion. Given voters’ beliefs, under current conditions, a progressive’s perceived optimal strategy, in Schulze, is to rank the Democrat alone in 1st place, even though s/he prefers the Green to the Democrat.
Such a voting system, under current conditions, would llikely just perpetuate the old lesser-of-2-evils voting that now keeps two unliked parties in office.
For the Green scenario, and for organizationl voting, Schulze is inferior to IRV, Benham, Woodall, and Schwartz Woodall, because, unlike those methods, Schulze has the chicken dilemma, which will render meaningless Schulze’s compliance with the mutual majority crierion and the Condorcet Criterion.
But I already covered that in greater detail in my Schulze article, with examples.
[quote]
Besides offering significant advantages over plurality and IRV
[/quote]
See above.
[quote]
, a major reason why the Condorcet-Schulze method is often used is that software for this method has been available for many years (because it is easy to code), and organizations naturally tend to choose what is already used in other organizations.
[/quote]
Inertia is common. Not just in organizations, but more-so in public elections. In my article, and in my article’s title, I acknowledged that Schulze is popular. That was why the article was needed.
Michael Ossipoff
Raphael Wimmer says
Thank you for this interesting explanation.
Just to make sure: the Schulze method is perfectly acceptable if I can rule out tactical voting, right?
We have used Schulze ranking for ergonomics research: study participants were asked to rank different shapes of an input device. We then calculated an overall winner using the Schulze method.
As they had no incentive to rank them insincerely, I would assume that your valid objections against the Schulze method do not apply to our use case.
michaelossipoff says
Yes, Schulze (also called “Beatpath”) works fine 1) if there’s no chicken-dilemma; and 2) if there’s no deception/gullibility combination that would make people mistakenly believe that they need to bury their favorite to help a lesser-evil.
Poll-results at the Condorcet Internet Voting Service (CIVS) show that successful chicken-dillema defection isn’t taking place, and that people are voting sincerely. The Debian organization hasn’t had a chicken-dilemma problem either, and is satisfied with Schulze.
I don’t think there’ll be chicken dilemma in polls (unless maybe they’re orricial or semi-official polls just before a major election). Amicable organizations aren’t likely to have chicken-dilemma.
But political parties in official public political elections are subject to chicken dilemma, among parties, and maybe within parties. That’s why I propose Benham or Woodall for public elections in the Green scenario.
And, in current conditions, we have disinformational, thoroughly-dishonest, media, and a public who believe them. The media own the public. Under current conditions, therefore, there’s a serious lesser-of-2-evils problem, and a serious giveaway compromise problem. I claim that we have the situation, described by Myerson & Webser, in which two disliked parties will keep winning forever, at Myerson-Weber voting equilibrium–where people vote only for a certain two disliked parties in the belief that only they can win–with the result that, of course, only they ever win.
The conditions in your organization, and at Debian, and in CIVS polls, I call “Ideal Majoritarian Conditions”. Under such conditions, Schulze is fine. But I prefer MAM, because 1) It has a simpler, briefer, and more obviously and naturally justified definition and rule; and 2) Simulations show that, when Schulze and MAM choose differently, then in the vast majority of cases, the MAM winner pairwise-beats the Schulze winner.
The Condircet Internet Voting Service (CIVS) automated polling website can be reached at:
https://www.cs.cornell.edu/andru/civs.html
Michael Ossipoff
Raphael Wimmer says
Thank you for the elaborate response!
Blake Hindrichs says
Thanks for the concise rebuttal! I’m only very recently engaging in the voting methods debate, and I came across this article today while researching the Schulze method. This may therefore be a deeply naive question, but is possible that the problem of the chicken dilemma might be mitigated by the outcome of the iterated prisoner’s dilemma?
https://youtu.be/BOvAbjfJ0x0