[EM] Majority Criterion, hidden contradictions

[Michael Poole]

Abd ul-Rahman Lomax writes:

> At 08:35 AM 11/7/2006, Michael Poole wrote:
>>Abd ul-Rahman Lomax writes:
>>
>> > No voting method can use preferences that are not expressed.
>> >
>> > Linguistically, the Criterion contains a lost performative -- or
>> > something like that. *How* do the voters answer affirmatively. It
>> > could only mean that they so answer on the ballot. Which in Approval
>> > *requires* that they vote for  X and not for any other candidate. And
>> > if a majority of voters do this, that candidate cannot lose. So why is
>> > it said that Approval fails the Majority Criterion?
>>
>>Nothing in the MC talks about what the ballot contains, only about how
>>voters answer a specific yes/no question.
>
> A question which is not on the ballot. If the Majority Criterion is
> held to apply to Approval Voting, and it is held that Approval Voting
> fails the criterion, it must be that the "question" asked the voters
> is *not* what is on the ballot, it is some theoretical question that
> seeks to discover any preference at all, no matter how weak. *Where is
> this question?* Is it on the ballot?

The original claim is "Approval does not satisfy the Majority
Criterion".  If you redefine either Approval or Majority Criterion
such that you can accurately say "Approval-prime satisfies the
Majority Criterion" or "Approval satisfies the Majority-prime
Criterion" it is distinct from the original claim.

> Now, it is possible for voters to answer the question posed in the MC,
> using an Approval ballot. Same as a plurality ballot. Simply vote for
> the candidate preferred and for no others.

Yes, but that is not how Approval is defined to work.  "Approval voted
according to Plurality satisfies Majority Criterion" is true but not
useful because it is the same method as Plurality.  "Approval
satisfies Majority Criterion" is false because Approval permits voters
to approve of more than one candidate.

> Here is another statement of the Criterion, from Wikipedia. The one I
> quoted before is from the same article, but is not, formally, the
> definition. The "question" was mentioned in an explanation. Mr. Poole
> fell into the same trap, he also referred to "how voters answer a
> specific yes/no question."
>
>> if a majority of voters strictly prefers a given candidate to every
>> other candidate (i.e. the given candidate is the first preference of
>> more than half the voters) and they vote sincerely, then that
>> candidate should win.
>
> What does it mean to "strictly prefer?" And which is the Criterion, "a
> majority strictly prefers" or "the given candidate is the first
> preference"? The latter is clearer, but the former is the language
> I'll use.

Strict preference means that equivalence is excluded but it does not
address magnitude.  "Strictly prefer X to Y" means, in preference
terms, "X > Y" rather than "X >= Y".  This is basic set theory.

"Strictly prefers X to every other candidate" is the same as "for all
Y != X, X > Y" -- this is the same as "first preference".  It does not
mean, as you assume later, "for all Y != X, Y = 0".

It is not accurate or honest to ignore voter preference or treat it as
irrelevant when the ballot does not allow it to be expressed!

Translating the logical flaw to a different metric: Strength of
preference is not expressed on a simply ranked ballot.  No one claims,
on that basis, that preference strength should be considered evenly
distributed for systems using such ballots.  Yet given that bogus
lemma, one could claim that Condorcet methods optimize for the same
social utility function as Range voting, which is plainly wrong.

Also, could you please try to edit your posts down?  It is a painful
waste of time to read through five paragraphs that boil down to "I
want to treat information as irrelevant that the voting system
ignores", or through another five paragraphs that boil down to "I want
to add conditions to Range Voting so that it satisfies additional
criteria -- and treat the result as general Range Voting".

[snip]
> In my view, *whatever* election method or methods are used, there is
> not full democratic process if the election is not explicitly accepted
> by a majority. The identity of a Condorcet winner is certainly of
> interest, but the Condorcet winner could fall far short of being
> accepted by a majority. Approval and Range can as well, but are much
> more likely to find the candidate who will be most broadly accepted
> and for whom, thus, the "shall we elect this candidate" would prevail
> by a majority.

The Condorcet Winner is a candidate who, for every other candidate, is
(strictly) preferred by a majority of voters over the other.  If the
CW is not acceptable to a majority of voters, then there *is no*
candidate that is acceptable to a majority -- otherwise that same
majority would prefer that candidate over the CW, which violates the
definition of CW.

Michael Poole