[EM] Majority Criterion, hidden contradictions

[Abd ul-Rahman Lomax]

At 11:29 AM 11/6/2006, Michael Poole wrote:
>   By the majority criterion, a candidate X should win if a majority of
>   voters answers affirmatively to the question 'Do you prefer X to
>   every other candidate?'.
>
>A voter who cannot honestly or easily answer "yes" to some candidate
>is not a problem for this formulation of the majority criterion.

Note that Approval Voting provides a means of the voter expressing 
this preference. The claims that Approval does not satisfy the 
Criterion are based on a simultaneous assumption that voters "answer 
affirmatively to the question," but do not vote to express that 
answer. This is a contradiction.

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?

>   The
>existence of such a voter is an argument in favor of some other
>criterion, but does not make the MC subjective.  If you want to change
>the "prefer" to "strongly prefer", pick a new name for your modified
>criterion.
>
> > Further, it is alleged that Approval does not satisfy the Majority
> > Criterion. It seems to me that this requires a few assumptions that
> > have not been made explicit.
>
>It does not require any additional assumptions.  Approval can only
>capture preferences between members of the approved set and the
>not-approved set.

That's correct. However, voters are not required to approve more than 
one (indeed, they are not required to approve any at all, but that's 
another matter.) The ballot gives voters a means to do what the 
Criterion describes. If a majority of voters do that, their favorite 
cannot fail to win.

Approval satisfies the Majority Criterion. Range does not, because 
Range does allow the expression of a weak preference. Approval does 
not allow that. I've been arguing, it might be noted, that this very 
fact makes Range a superior method. That Range does not satisfy the 
Majority Criterion is not a defect. But that is another matter.

Here, the question is Approval. Does it satisfy the Majority 
Criterion *as stated?*

>When there are only two viable parties, Range Voting's use of strength
>of preference apparently encourages the factions to be strongly (even
>bitterly) divided: that will maximize the difference in scores between
>the two candidates among supporters of each party.

That claim is made. It's only true if the factions are already 
bitterly divided, have a strong preference for winning over 
maximizing the value of the election to the society. In other words, 
if the use of maximum range ratings is *real*.

The effect is here asserted as the cause.

We don't really know how Range will behave in actual elections. The 
only evidence we have comes from Warren's poll, and I consider that 
relatively weak. So confident assertions about how Range will behave 
are, quite often, premature.

I don't see how Range *creates* the use of maximum scores in any 
pathological way.

Range reduces to Plurality in the two-candidate case, unless voters 
follow non-standard strategy, that is, weaken their votes. It's 
possible in Range to cast a weak vote, a vote that does not have full 
effect in determining the outcome. In the pizza case, this would be 
like someone voting for Pepperoni, 90, for Mushroom, 80. That person 
does not have a strong preference, and so their vote is discounted, 
effectively.

Generally, the advice has been given to normalize your Range votes. 
That is, you would vote the maximum for your favorite, the minimum 
for the worst candidate, and then place the others either with these 
or in between. If there are only two candidates, this reduces to 
plurality. It is when there are more than two that it gets 
interesting. (Condorcet methods, of course, also reduce to plurality 
when there are only two candidates.)

But it is true that voters may decide to weaken their votes, in which 
case, we have something different than plurality. Call it a 
"half-abstention." I have an opinion, and I'm going to express it, 
but it is not strong, so I accept that it will not have full power in 
determining the winner. It will merely have a lesser influence.

>   I am not sure that
>more parties will necessarily resolve that; as Kevin Venzke showed, a
>voter is most likely to affect a Range Voting election by giving only
>extreme ratings.

Sure. "Most likely to affect by giving only extreme ratings" means 
that if you care, vote the extremes. So? If you don't care, you are 
not required in Range to vote the extremes.

It is absolutely true, it is not at all in controversy, that "a voter 
is most likely to affect a Range Voting election by giving only 
extreme ratings." And the decision of whether or not to do this is 
entirely up to the voter. It is not correct that therefore most 
voters will only give extreme ratings; there is some evidence, from 
Warren's poll, that they will not. But, I hasten to add, I consider 
that poll as interesting, because it is about all we have, but hardly 
conclusive.

I'll say it again. Range does not satisfy the Majority Criterion 
because it allows the expression of a weak preference, a preference 
that does not have full strength in determining the outcome. Approval 
does not have this trait, which is why, in my opinion, it does 
satisfy the Majority Criterion. It does allow the expression of 
preference as described in the Criterion, and if the voter so 
expresses the preference, and a Majority of voters do this for a 
candidate, that Candidate must win. QED.