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

[Kevin Venzke]

James,

--- James Green-Armytage <jarmyta at antioch-college.edu> wrote:
> James G-A replying to Kevin Venzke...
> 
> Kevin:
> >You don't need to suppose that ballots are cast sincerely, since there is
> >no way to fill out ballots such that they couldn't possibly be sincere. If
> >a method fails Smith for any set of ballots, it must fail your criterion.
> 
> 	I just wanted to make sure that the criterion was failed by plurality,
> approval, and other methods that don't allow full rankings. Maybe there is
> a better way to do this.

I see. I forgot who you were talking to.

> Kevin:
> >
> >I doubt it is useful to identify the Smith set with "majority rule,"
> >unless
> >"majority" as in ">50% of all the votes" is not a useful idea.
> >
> 	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?

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.

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.

> 	I suggest that, essentially, we should mean that it is
> Smith-efficient.Why the Smith criterion? Because choosing from the Smith
> set is the way to ensure that no expressed majority preference (pairwise
> defeat) is unnecessarily overruled.

Smith is the way to do that if you already decided that "majority" refers
to the winners in all pairwise contests. If you don't define "majority" that
way then you don't have to use Smith.

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."

> 	In some ways, this question has more political significance than
> mathematical significance. Many of those who consider IRV to be the best
> single-winner method claim that it ensures majority rule. Is this true? If
> not, what criterion does IRV fail that makes it not a majority rule
> method. I suggest that it is the Smith criterion. 

IRV guarantees majority rule by a solid coalition. But in general I don't
consider it to be a "majority rule method."

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.

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.

> 	I suggest that narrower definitions, such as the one that Mike has
> formulated, are too narrow, in that it is necessary to choose one of
> several viable defeat strength definitions.

I also think Mike's definition is too narrow, but because it doesn't seem
to allow other methods (i.e. non-pairwise) to be used. I don't think it
would be a crime against majority rule for a method to rule out the 
candidates who must not win, and then pick the winner by e.g. the first
preferences.

>I suggest that broader
> definitions, such as the mutual majority criterion and Condorcet
> criterion, are unnecessarily broad.

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.

> I suggest that the Smith criterion is
> the majority rule definition that is broad enough to be generally
> acceptable, but narrow enough to be exclude methods that unnecessarily
> ignore the will of a majority, e.g. IRV, minimax, etc.

An advantage of using Minimal Defense rather than Smith is that you don't
have to use terms like "innermost," "minimal," "pairwise," or even "beats."

I begin to think that Minimal Defense slash SDSC would be more popular if
they had been named without reference to defensive strategy. Those criteria
are very useful even if no offensive strategy is possible.

Kevin Venzke



	

	
		
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