[EM] a majority rule definition based on the Smith set

Sat Apr 9 10:16:01 PDT 2005

Hi James,

 --- James Green-Armytage <jarmyta at antioch-college.edu> a écrit : 
> James:
> >> 	Okay then, how do you define majority rule? The question I'm interested
> >> in is not whether we can invent an interesting new concept; the question
> >> is what is the most appropriate criterion to be identified as "majority
> >> rule". When we say that a given method is a majority rule method, what
> >> should we mean by this?
> 
> Kevin:
> >I think the most intuitive is Steve Eppley's criterion that Markus quoted.
> >When v(i,j)>50% and there is no beatpath of strength >50% from J to I,
> >then
> >J mustn't win. Basically, when more than half prefer I to J, in the
> >"normal"
> >case you mustn't elect J.
> 
> 	We disagree about whether "majority" should refer to the a majority of
> ballots cast, or a majority of ballots that express a preference between a
> given pair of candidates. If you remove the >50% stipulation from your
> definition above, you are left with what? The Schwartz set? Anyway,
> something close to the Smith set, right?

Right, you're left with "when I beats J and J has no beatpath I, then J
can't win."

> Kevin:
> >If this criterion is too strong (although I doubt you think so), then I'd
> >suggest Minimal Defense: When a majority rank X>Y and Y over no one, then
> >Y mustn't win.
> 
> 	Sounds like a useful criterion, but it strikes me as being a bit too
> specific to be regarded as the technical definition of majority rule. It
> seems like there are plenty of scenarios where one could fairly say that
> majority rule has not been satisfied although the above criterion has not
> been violated.

Here is a weaker quick test:
If for some A and B, v[a,b]>50%, and there is no pair of candidates C and D
(where C is not A) for whom v[c,d]>50%, then B must not win.

IRV still fails this.

> Kevin:
> >But what you write above doesn't seem true. Choosing just anyone from the
> >Smith set can "unnecessarily overrule" defeats, unless I don't understand
> >what you mean by "unnecessarily."
> 
> 	I don’t know whether you understand what I mean or not. I mean that if
> there is a CW, there is no need to overrule any defeats in order to choose
> a winner. If there is a majority rule cycle, then it is necessary to
> overrule at least one defeat, but it is not necessary to overrule a defeat
> outside of the Schwartz set (= Smith set in large electorates). Hence,
> choosing a candidate outside of the Smith set always involves
> unnecessarily overruling a pairwise defeat. If you disagree, please state
> your reasoning in more detail.

Well, I believe the >50% interpretation is useful. So given that, and the
following two elections:

49 A
24 B
27 C>B

9 A>B
5 B
8 C

In my mind it's a majority rule violation (and a complete waste of an election)
to elect A in the first scenario, but nothing near that to elect B in the
second scenario.

Neither the Smith set, nor the set of candidates I consider electable according
to majority rule, is a subset of the other.

> Kevin:
> >IRV guarantees majority rule by a solid coalition. But in general I don't
> >consider it to be a "majority rule method."
> 
> 	Nor do I, really. Or perhaps I consider it a majority rule method in a
> “weaker” sense than a Smith-efficient method. Here are some key criteria
> that I consider to be indicative of the “majoritarian-ness” of a given
> method: 
> (1) Mutual majority criterion, 
> (2) Condorcet criterion
> (3) Condorcet loser criterion
> (4) Smith criterion
> 	IRV only meets (1) and (3). Beatpath, ranked pairs, and river meet all 4.
> Minimax only meets (2). Approval and plurality meet none of the above.
> Borda, I believe, only meets (3). 

I've criticized before that this list lacks a criterion dealing with majority
(my sense) pairwise wins.

The CDTT methods I like only meet (1), but I consider them more majoritarian than
e.g. Margins.

> Kevin:
> >I don't agree that we should use Smith as the reason, since Smith is too
> >weak to satisfy "majority rule" in the >50% sense.
> 
> 	We disagree on the importance of the >50% of cast ballots definition.

I consider it a fundamental component of IRV's badness that it can't see
>50% opinions. If you don't insist on that kind of capability, I wonder if
you won't end up with a method just as poor.

> Kevin:
> >In my mind the problem is IRV's failure of Minimal Defense slash SDSC:
> >In a race primarily between A and B, even when a majority prefer A to B,
> >they can "confuse" the method into electing B just by ranking weaker
> >candidates above A.
> 
> 	Wouldn’t they have to rank the weaker candidate above B as well?
> Otherwise, the insincere ranking wouldn’t have an impact until B was
> already eliminated. I’m not saying that IRV passes minimal defense, I’m
> just saying that what you’re talking about sounds like the push-over
> strategy, which I suggest is relatively hard to use in practice (low
> reward/risk ratio).

No, not push-over. I could have been clearer. When I say "they" can confuse
the method, I mean that the A>B voters can *unintentionally* confuse the
method. It's just a compromising problem. IRV and FPP both have it.

> Kevin:
> >I agree that MMC is *far* too "broad," but in some cases I think Condorcet
> >is too narrow:
> >9 A>B
> >5 B
> >8 C
> >I don't agree that a method "fails majority rule" if it elects B here.
> 
> 	A is a Condorcet winner. A 9-5 majority (a majority as I define it, not
> as you define it) prefers A to B, and a 9-8 majority prefers A to C. Hence
> I assert that electing A is the only way to satisfy majority rule (as I
> define it).

I wouldn't feel too bad insisting on Condorcet if it were more compatible
with Later-no-harm.

Kevin Venzke

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