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

[James Green-Armytage]

Hi Kevin, thanks for your reply! My answers follow


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?

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.

James:
>> 	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.

Kevin:
>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.

	Yes, exactly.

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.

James:
>> 	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. 

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). 

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.

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).

James:
>> 	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.

Kevin:
>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.

	If all of the members of the Smith set happen to be candidates who “must
not win”, for whatever reason (approval score?), then I argue that it
would indeed be a violation of majority rule.

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).

my best,
James