[EM] Majority Criterion, hidden contradictions

[Abd ul-Rahman Lomax]

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.

(1) It is clear that any method which satisfies the Majority 
Criterion cannot maximize the expected value of the election. Range 
is the method which directly does the latter, and this is directly 
connected with its non-satisfaction of the Majority Criterion.

This is because the Majority Criterion neglects preference strength, 
and thus a minor, practically inconsequential preference is given the 
same weight as a clear, stark preference. So if Preference Strength 
is considered in determining the winner, in the presence of expressed 
first-preference votes of a majority, it must be true that under some 
circumstances preference would alter the winner and thus the Majority 
Criterion would not be satisfied.

(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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