[EM] Majority Criterion, hidden contradictions

[David Cary]

As with the Independence of Irrelevant Alternatives (IIA) Criterion,
it is perfectly acceptable to interpret the Majority Criterion (MC)
as being a theoretical assertion about a certain relationship between
voter mental preferences and the outcome of an election.  Just
because mental preferences are the subject of mathematical modeling,
does not make the criterion as a theoretical assertion any less
objective.

Unlike IIA, however, the quoted version of MC can not be legitimately
construed to only refer to voted preferences.  It would make the
restriction to sincere voting irrelevant.  The clause about sincere
voting is the clue that the first preferences, the strictly
preferring, are intended to mean mental preferences and that sincere
voting is the connection between the mental preferences and the votes
cast.

As a result, it is also not a legitimate argument that Approval
Voting satisfies MC because first preference means first voted
preference and the only time the premise holds is when a majority
have bullet voted for a candidate.

That leaves the satisfaction of MC by Approval Voting depending on
the interpretation of sincere Approval Voting.  There are several
approaches.

1. If the model of voter preference is one that only allows a binary
scoring of candidates and nothing more, then every individual
preference would produce exactly one sincere approval vote, and the
premise of MC would guarantee that a majority would bullet vote for a
candidate.

But that is an extremely limited and unappealing model of human
preference.  This points out that part of a good full statement of MC
is a guarantee of a minimum variety of mental preferences.

2. Voter preferences allow at least any complete ordering of the
alternatives, and there is also a binary scoring function that is
consistent with (does not contradict) the ordinal preferences. 
Sincere voting is based on that binary scoring function.  In this
case, first preferences would be based on the more discriminatory
ordinal preferences, allowing a violation of MC.

3. Voter preferences allow at least any complete ordering of the
alternatives.  Any approval vote that is consistent with (does not
contradict) the ordinal preferences is considered sincere.  This
means that even with a linear ordering of preferences, there can be
more than one way to sincerely vote.  So a violation of MC is
possible.

4.  Voter preferences allow at least any complete ordering of the
alternatives.  No votes are considered sincere.  Strictly speaking,
Approval Voting then vacuously satisfies the quoted MC because the
premise is never satisfied.  I'd rather wordsmith the criterion so
that MC does not apply to Approval Voting in this interpretation of
sincere Approval Voting.

It would make sense to me to take the sincere voting clause out of
MC.  That would leave it up to the individual application of MC to
specify what restrictions are imposed in converting personal mental
preferences into votes.  It would also allow a version of MC that was
based solely on voted preferences.  
 
-- David Cary



--- Abd ul-Rahman Lomax <abd at lomaxdesign.com>, on Wed, 01 Nov 2006
23:18:32 -0500, wrote:

> I've been realizing just how defective the Majority Criterion is. 
> People tend to assume that the Majority Criterion is an important 
> characteristic of any proper democratic election system. Yet the 
> Criterion itself suffers from a number of serious problems.
> 
...
> (2) There is a serious semantic problem in how the Criterion is 
> expressed and applied. It is commonly stated that the Majority 
> Criterion is satisfied by Plurality and not by Approval. In order
to 
> state this about Approval, one must presume that there are 
> unexpressed preferences on the part of voters.
> 
> Approval, just like Plurality -- Approval *is* Plurality plus an 
> additional freedom of the voter -- allows voters to express a 
> preference between two candidates. Just vote for only one of them.
If 
> a majority express this preference for one candidate, then, under 
> Approval as well as under Plurality, then that candidate cannot
lose 
> to the less-preferred candidate. (And if this preference is
expressed 
> over all other candidates, the candidate cannot lose the election.)
> 
> Looks to me like Approval *does* satisfy the Majority Criterion. 
> Range, generally, does not.
> 
> However, Approval allows the voter to abstain from a pairwise 
> election. The voter does so by giving both candidates the same
vote. 
> By voting for both candidates, the voter has expressed a preference

> for both of them over all other candidates not so marked, and has 
> abstained from the pairwise election between them. Only by assuming

> that the voter has a preference between the two even though this 
> preference has not been expressed, and then by using this
unexpressed 
> preference to determine if the method satisfies the Majority 
> Criterion, do we come up with the answer that Approval does not
satisfy
> it.
> 
> The Majority Criterion has been crafted, though, to create a 
> technical correctness of the common analysis:
> 
> >The majority criterion is a 
> ><http://en.wikipedia.org/wiki/Voting_system_criterion>voting
system 
> >criterion, used to objectively compare 
> ><http://en.wikipedia.org/wiki/Voting_system>voting systems. The 
> >criterion states that 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.
> 
> No method can determine a winner based on unexpressed preferences. 
> The definition glosses over this. Approval allows voters to express

> that they "strictly prefer a given candidate to every other 
> candidate," in exactly the same manner as Plurality. If they choose

> not to do this, then, within the definition of preference as it can

> be expressed in the method, *they have not expressed a preference
and 
> we cannot claim that they have one.*
> 
> The only difference between Approval and Plurality is that one may 
> express preference for a group of candidates over all others,
instead 
> of just for one. But expressing preference for a group is outside
the 
> purview of the Majority Criterion, it is irrelevant to it. So we
can 
> only consider the expressed preferences, and Approval satisfies the

> Majority Criterion with respect to them. In Approval, one has 
> expressed a preference for every candidate for whom one has voted, 
> over all other candidates. If a candidate enjoys such an expression

> from a majority of voters, that candidate cannot lose unless a 
> majority has *also* expressed that same preference for another 
> candidate. There is more than one possible winner who satisfies the

> Majority Criterion, just as there can be more than one winner who 
> satisfies the Condorcet Criterion.
> 
> This is important because many writers assume that the Majority 
> Criterion is some kind of gold standard for elections, and when it
is 
> asserted that Approval fails to satisfy it, this can be and is 
> considered a fatal argument, or at least a serious defect of
Approval.
> 
> I came to write this from reading the following:
> 
> http://www.reformthelp.org/issues/voting/glitch.php
> 
> Thanks to Jan Kok for suggesting I look at this article.
> 






 
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