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:
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.
- 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:
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:
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).
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.