What is a Majority

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This is a draft. This is a work in progress. wegerje 09:43, 12 Jan 2005 (PST)


[Editor's Note: This is an exchange between James Green-Armytage and members of the Election-methods mailing list - see http://electorama.com/em for list info. The exchange will at times use some unlinked technical language.]

Dear M___,

Yes, it's true that "majority rule" is an ambiguous term, and so there is no one 'correct' definition. However, I find some definitions more useful than others. Let me suggest 4 general categories of majority rule methods, based on 4 basic criteria.

  • Criterion 1: If a majority of the electorate coordinates their efforts, they can assure that a given candidate is elected, or that another given candidate is not elected.
  • Criterion 2: Mutual majority criterion
  • Criterion 3: Condorcet efficiency
  • Criterion 4: Minimal dominant set (Smith, GeTChA) efficiency
  • Criterion 1 only: Pseudomajority methods.
  • Criteria 1 and 2 only: Weak majority rule methods.
  • Criteria 1, 2, and 3: Intermediate majority rule methods.
  • Criteria 1, 2, 3, and 4: Strong majority rule methods.
  • Pseudomajority methods: Plurality, approval, range voting, Borda
  • Weak majority rule methods: single-winner STV
  • Intermediate majority rule methods: Minimax (aka SD, PC, etc.), Nanson, Black, etc.
  • Strong majority rule methods: ranked pairs, beatpath, river, cardinal pairwise (assuming that a strong-majority base method is used)

You are correct in saying that in pseudomajority methods (like plurality and range voting), a given majority of the electorate *can* coordinate their intentions and decide the winner, but this merely postpones the question of how they do this. The stronger majority methods not only enable firmly coordinated majorities to assert themselves, but they allow majorities to *reveal* themselves, with out any need for prior coordination. I think that voting methods that facilitate this process of revelation are superior to those that do not.

The remaining three categories allow mutual majorities to reveal themselves (in the absence of a self-defeating strategy by supporters of this majority). Strong majority rule methods not only reveal mutual majorities, but they reveal minimal dominant sets and Condorcet winners (in the absence of a severe burying strategy). This is especially valuable because it means revealing possible compromises on divisive issues, thus avoiding a lot of political polarization and strife.

As you have said, many progressive policies may be supported by a majority, but they have not been brought to a fair vote. A major goal of a good voting system is to *reveal* these majority-supported policies, to avoid their getting lost in the noise of the two party system. Can they be revealed if we use a pseudomajority method like range voting? Yes, it's quite possible, but it's more difficult. In many cases it means that the intention-revealing part of the process, which is ideally the purpose of the voting system, will have to be deferred to some other mechanism or institution.

Sincerely, James Green-Armytage


Dear R___,

You disagreed with my use of the derogatory term "pseudomajority" for approval voting, and I can sympathize with you there, but I'm disappointed to note that your response doesn't touch on the major points that I made in the post that you are replying to, and therefore I wonder whether you read it with an open mind.

Anyway, is "pseudomajority" unnecessarily harsh? Maybe. I guess it doesn't matter too much what name you use. The point is that the methods in question fail the majority and mutual majority criteria, which I think should be a bare-bottom minimum requirement for a method to be called a majority rule method, and therefore it also fails the Condorcet and minimal dominant set criteria, which I think provide the most authentic definition of majority rule. And, in the portion of the e-mail that you seem to have ignored, I go on to talk about why this is a problem. So, a more proper classification would be "non-majoritarian single winner methods", but "pseudomajority methods" just seemed a bit shorter.

The rest of your e-mail goes on to detail reasons why approval voting is a good method. I agree with some of your points. I have always said that I prefer approval to plurality, and I can also say that the marginal benefit of using approval rather than plurality is non-trivial. In my classification, approval and plurality are both pseudomajority methods, and approval is superior to plurality.

In actual practice, approval is far more likely to produce a majority winner than either plurality or Borda. (I'm not sure about range.)


Okay, but which definition of a "majority winner" are you using? I think that "a member of the smallest non-empty mutual majority set" would be one possible definition, but I think that it is still weak. My preferred operational definition of a majority winner is "a member of the minimal dominant set". Do we agree on this?

For this reason, even though approval may not be the best method for conducting elections of public officials, it is such a good method for making quick decisions at meetings that it should be an integral part of all "rules of order."


I don't think that I agree with you here. I think that pairwise counts should be part of the rules of order of legislatures, but why approval voting? Pairwise counts can reveal majority-supported compromises; what can approval counts reveal? The result of approval tallies constitute ambiguous information, because the definition of "approval" is ambiguous.

Given the widespread problems affecting elections in the U.S., including the current near universal lack of election administration and voting equipment adequate to conduct elections using more sophisticated voting methods, approval probably is -- at this time in the U.S. -- the method most likely to produce a majority winner. Only after election problems are fixed and better voting equipment is obtained will the methods you prefer be viable for U.S. elections. So in the U.S. at this time, approval is less deserving of being called a "pseudomajority" method than ANY other method.


Less deserving? ... or you just don't think that it's politically wise to call it that? I think it is deserving of the title, but I can see how it might be politically counterproductive. Although, in going from plurality to approval we'd just be going from one pseudomajority method to another, so the term wouldn't really be part of the discussion... unless switching to approval means that we have to pretend that approval is the best possible single-winner method, in which case I'm not interested.

Approval voting would be a good way of introducing the problem of single winner election methods to students and other election method novices, because it is very easy to explain and to compare with plurality. If the Center for Voting and Democracy had developed an election methods instruction program directed at schools and included approval voting in the program, chances are CVD would have had far more success than it has had in building popular support for alternatives to plurality.


Yeah, maybe, but we shouldn't put everything on CVD's shoulders. Sure, it does soak up some grant money, but it doesn't really run on a huge budget. It is quite possible for more than one national voting methods NGO to exist; I suspect that you could start the sort of organization you're describing if you had the right nonprofit management skills and you were willing to invest the necessary time.

Election method educational programs could even be developed for elementary schools.


Oh, I definitely agree. The math involved is so simple... even the math in a lot of the STV methods. And it's a great way of applying math to actual situations. Actually, voting methods are what got me interested in math again; I had given it up at the end of high school and not touched it for years, and now I'm trying to become an economist. I wonder who you'd have to fight with to get that into the elementary school curriculum...

Sincerely, James Green-Armytage http://fc.antioch.edu/~james_green-armytage/voting.htm

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