Majority Criterion refutation without invoking IoIA

[Clay Shentrup]

Imagine a majority of voters want to build a bridge to nowhere. And a majority of voters want to take a hostile stance toward Petrolistan.

But a majority of voters opposes a ballot initiative which enacts the bridge to nowhere and asserts a hostile stance toward Petrolistan (and has no other stipulations).

A majority is thwarted whether or not the initiative passes.

[William Waugh]

Are you arguing that a definition of Majority {constraint or criterion} is incoherent?  If so, which definition?

Wikipedia says "if one candidate is preferred by a majority (more than 50%) of voters, then that candidate must win". The use of the word "must" suggests that the intent is to define a constraint. My next step in analyzing the definition is to examine "preferred by" and ask whether there is an obvious unambiguous definition for it. I suppose it means "preferred over each other candidate running in the election by". Under this definition, your example has only two candidates. One is a conjunction of two measures, and the other is the negation of that conjunction. Whether a voter prefers one of the measures by itself is irrelevant, because that is not a candidate in the election you describe. If the measure fails, no majority is thwarted.

On Tuesday, May 12, 2015 at 10:55:11 AM UTC-4, Clay Shentrup wrote https://groups.google.com/forum/?fromgroups#!topic/electionscience/gcGymYvu434

[Warren D Smith]

I think what Clay had in mind, to make things more concrete, is this.
There are two independent and unrelated possible moves, A & B.

#voters......what they want
40%...........A and B
30%...........A but not B (the latter being more important)
30%...........B but not A (the latter being more important)

Under those circumstances:
* a ballot question "want A?" would pass 70-30.
* a ballot question "want B?" would pass 70-30.
But the ballot question "want both A and B?"
would fail by 60-40 margin.

You could also change all the signs to get the opposite
effect, where they'd pass "want A & B?" but neither A nor B
alone would pass.

Even though in all these examples A and B are unrelated independent things.

Still further dysfunctions can happen if A and B actually are related, such
as building a highway, and building a bridge so that highway to cross over
a river.



--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)

[Clay Shentrup]

I should say that this refutes the majority *axiom*. The idea that if a majority favors X to Y, then the group favors X to Y.

[William Waugh]

What is your operational definition of "the group favors X to Y"?

[Clay Shentrup]

On Wednesday, May 13, 2015 at 7:48:18 AM UTC-7, William Waugh wrote:

What is your operational definition of "the group favors X to Y"?

That's the whole point of the debate. Majority Criterion zealots say that the group favors X to Y if a majority of its members favor X to Y. Math says that's not necessarily true.

[William Waugh]

How can math say anything one way or the other about whether there is a logical connective that applies between "the group favors X to Y" and "a majority of its members favor X to Y" absent a concrete definition of "the group favors X to Y"? Do you have a clear enough definition of it in your mind that you could describe how to falsify or verify such a proposition? Math only works on concepts that can be defined to a certain level of precision.

[Clay Shentrup]

On Wednesday, May 13, 2015 at 5:51:15 PM UTC-7, William Waugh wrote:

How can math say anything one way or the other about whether there is a logical connective that applies between "the group favors X to Y" and "a majority of its members favor X to Y"

Because if that axiom contradicts itself, then it must not be correct.

[Warren D Smith]

More stuff, some of it by Clay himself, on how "majority axiom
contradicts itself"

http://rangevoting.org/CondorcetCycles.html

[William Waugh]

I still don't see what the observation about a conjunction brings to the party. I don't see how it applies.

The pizza argument, on the other hand, convinces me that following the majority constraint is not necessarily good, and so I can see citing it to people who contend that the majority constraint should count. http://rangevoting.org/MajCrit.html

From Wikipedia, I gather that the majority "criterion"[1] is defined to be met by those voting systems and only those voting systems in which if a candidate is preferred to all others by a majority, that candidate will win. I go to Wikipedia because our primary satan[2], Fairvote, doesn't define the majority constraint on their main web site. The W'pedia editors go into the ambiguity of whether by "preferred" we are to take it to refer to the voters' states of mind or only to their concrete manifestations on their ballots. I consider a reference to state of mind as an unnecessary complication in this context, so my first try in seeking a clear definition is to leave that out and ask whether a reasonable attempt to make the definition precise would depend only on the markings on the ballots. The W'pedia editors say that there is still some question about how the majority "criterion" would be evaluated with Range, but to me it seems straightforward to interpret that if a voter rates candidate A strictly higher than candidate B on her ballot, she prefers A to B.

I feel as though I had in mind some further point to mention once I should get a precise definition of the majority constraint in place, but now I can't remember what that point might have been.  I'll let my interpretation of the definition stand anyway for what it might be worth.

[1] The term "criterion" is consistently abused by almost everyone I've seen use it outside the field of engineering. What they almost always mean when they say "criterion", is actually "constraint". If the output of applying the criterion or constraint to an entity under test (e. g. a voting system) is an up-or-down result as opposed to a measure of degree of goodness that has more than two possible outcomes, it's a constraint.  The discipline of engineering teaches the distinction between these terms.

[2] People know the name Satan as a name for the character Lucifer from some of the Abrahamic religions, but I understand that originally "satan" was a generic term meaning an opponent in argument.  Etymonline traces it to a Hebrew root  << s-t-n "one who opposes, obstructs, or acts as an adversary."  >>.  And that seems to describe Fairvote perfectly.

[Clay Shentrup]

On Friday, May 15, 2015 at 4:50:26 PM UTC-7, William Waugh wrote:

I still don't see what the observation about a conjunction brings to the party. I don't see how it applies.

It just proves that "Naive Majoritarianism" is wrong via reductio ad absurdum.

The pizza argument, on the other hand, convinces me that following the majority constraint is not necessarily good

Not that it is "not necessarily good", but that the majoritarian axiom is objectively false.

[William Waugh]

<< It just proves that "Naive Majoritarianism" is wrong via reductio ad absurdum. >>

Please state naïve majoritarianism so I can see whether it's a logical proposition possibly vulnerable to reductio ad absurdum.

[Clay Shentrup]

A group favors X to Y if a majority of its members favor X to Y.

[Abd ul-Rahman Lomax]

At 12:36 AM 5/13/2015, Clay Shentrup wrote:
>I should say that this refutes the majority *axiom*. The idea that
>if a majority favors X to Y, then the group favors X to Y.

But in the example, there are two candidates, A+B or not(A+B). A
majority prefers not(A+B) to A+B, therefore not(A+B) prevails. I'm
surprised to see this naive example presented.

If this were presented as two measures on the ballot, A and B, both would pass.

Usually laws require that referenda submit a *single issue* to the
voters. In parliamentary process, as well, a motion should have a single topic.

Sometimes topics get linked. Indeed, an amendment to a proposal may
be suggested as a "poison pill."

The basic flaw here is using public elections in an attempt to
resolve issues that may require negotiation. The example given would
probably not be a legal referendum or ballot question.

The majority criterion assumes independent candidates for a single
result. It has been misapplied here, so this is not a "refutation" of
the majority criterion.

We know well that the majority criterion is defective, because of the
situation of the majority with a weak preference for A>B and a
minority with a strong preference for B>A. Score voting would reveal
this, if the candidate field is large enough. One solution is fairly
simple: if score analysis shows that B>A even though the majority
prefers A>B, the result would be submitted for majority approval. (Or
the voting system would have an approval cutoff, such that a majority
has actually approved the result).

In real multiple-round elections, the turnout in a runoff varies from
that in a primary. If the majority preference for A>B is weak, there
will be preferential turnout, which can readily flip the result to B.
Years ago, analysis showed that this actually happened in nonpartisan
top-two runoff elections.

[Abd ul-Rahman Lomax]

At 01:06 AM 5/17/2015, Clay Shentrup wrote:
>A group favors X to Y if a majority of its members favor X to Y.

Undefined: what it means for a "group" to "favor" X to Y.

Wikipedia has this definition:
The criterion states that "if one candidate is preferred by a

majority (more than 50%) of voters, then that candidate must

win".<https://en.wikipedia.org/wiki/Majority_criterion#cite_note-1>[

For this to make any sense, the

Okay, what is the history of this question? Clay posted

>Subject: [CES #11998] Majority Criterion refutation without invoking IoIA

This is not a "refutation" of the Majority Criterion, which is not a
proposition, even though some use it that way. It is a criterion, a measure.

The word "must" could have normative value, but, in fact, in this
context it is only describing the mechanics of the voting system,
that should voters *express* that preference, the candidate *will*
win. In considering whether or not Approval Voting satisfies the
Majority Criterion, the concept of hidden preferences was introduced.
Obviously, if the voters do not express hidden preferences, they
cannot affect the outcome.


The Majority Criterion is not that "it is good" or "it is necessary,"
for the Criterion to be satisfied. However, Clay must be "refuting"
not the Criterion itself but the idea of its necessity.

And the example doesn't show that with any clarity.

The circumstances, as I pointed out, are arbitrarily set up to
combine two different issues.

The "initiative" is (A and B). That a majority favors A and a
majority favors B does not imply that a majority favors A and B. The
separate issues were not presented to the voters. Only the combination.

If we are choosing between A and B, the majority criterion applies.

It is stated in Clay's setup that, choosing between A and nothing, a
majority favor A, and choosing between B and nothing, a majority
favor B. However, choosing (A+B) and nothing, a majority favor
nothing. There is no contradiction.

For example, an electorate of one, me, choosing between buying a
Subaru and nothing, may prefer the Subaru. Choosing between buying a
Toyota and nothing, I may prefer the Toyota. But choosing between
buying a Subaru *and* a Toyota, or nothing, I may choose nothing. I
can't afford two cars.

The electorate cannot afford the two measures both winning.

The majority criterion is best applied to actual votes, rather than
"preferences," which confuses it all to hell, because people may vote
against preference. The majority criterion doesn't apply to Score
Voting, unless it's been redefined to make it applicable. It only
applies to strict preference ordering, and cannot assess hidden
preferences (which is an error that James Armytage-Green made, in
applying the Criterion to Approval Voting.)

Warren gave an example showing the setup more precisely (as an example)

>I think what Clay had in mind, to make things more concrete, is this.
>There are two independent and unrelated possible moves, A & B.
>
>#voters......what they want
>40%...........A and B
>30%...........A but not B (the latter being more important)
>30%...........B but not A (the latter being more important)
>
>Under those circumstances:
>* a ballot question "want A?" would pass 70-30.
>* a ballot question "want B?" would pass 70-30.
>But the ballot question "want both A and B?"
>would fail by 60-40 margin.

Notice that a majority of the voters consider rejecting the other
measure more important than getting the ersult they want. Now,
suppose this goes for individual initiatives. If they are actually
unrelated, we can largely assume that if both initiatives were on the
ballot *separately*, they would pass.

The setup is odd in that 100% of the electorate wants A or B or both.
Normally, some will want neither. But the example still works.

William Waugh's analysis was clear:

>Under this definition, your example has only two candidates. One is
>a conjunction of two measures, and the other is the negation of that
>conjunction. Whether a voter prefers one of the measures by itself
>is irrelevant, because that is not a candidate in the election you
>describe. If the measure fails, no majority is thwarted.

Another way to see this is that, by a 60/40 majority, the electorate
does not want two measures to pass, only one.

The example is interesting because if the measures are presented
separately on a ballot, and they are *not* linked, both will pass. In
that case, we could claim that a majority is thwarted! But that
question was not asked. To truly resolve these issues takes much more
complex deliberative process.

Notice that no voting system can handle this problem, as presented.
This is not about the Majority Criterion. If there are *fixed
preferences*, the example shows, there may be no outcome which will
satisfy a majority without dissatisfying a different majority.
However, preferences are not fixed. We make choices in the presence
of conflicts all the time. Not Subaru vs. Toyota, but Subaru vs.
Vacation in Tahiti.

What Clay said above, he had already written:

>I should say that this refutes the majority *axiom*. The idea that
>if a majority favors X to Y, then the group favors X to Y.

There is no such axiom, the basic problem is the definition of what
it means that a "group favors" X to Y. The example doesn't really address this.

Score voting "fails" the majority criterion, unless we define it
differently, say, considering each voter to be 10 voters in Approval Voting.

"Favors" only refers to relative preference. If the voters do not
normalize, that is, express full preference, if they have "weak
preference" so that is what they express, then, yes, the majority
criterion fails because it only looks at all-or-nothing positions of
the voters. Classic example, from Saari, 99% of voters vote A 100,
B, 99, and one voter votes A 0, B 100. B wins. Somehow this is
supposed to be Bad. In fact, all voters will be happy with the
outcome, assuming the votes were sincere. The preference for A was miniscule.

It's just obvious: the axiom stated is defective. But behind the
Majority Criterion is a fundamental principle, majority rule.
Majority rule is not about voting systems. It is only about Yes/No questions.

It is easy to show that any voting system that, with a single ballot,
guarantees the Majority Criterion is satisfied, must be a defective
system, that is, it will produce results that a strong majority will
agree are deficient. Systems that handle the problem do so
interactively, with multiple polls.

Social utility analysis can easily show that a majority-approved can
generate inferior social utility. However, it is quite unlikely that
a group informed about such a situation would still approve it by a
majority, for the conditions require weak majority preference vs
strong minority preference, and human societies do respect preference
strength. The voting systems in common use for public elections are
not good at this!

Score is much better, but, then, might fail the majority criterion,
which can indicate one of two things: a situation where such failure
is a positive benefit overall, and a situation where the voters were
not aware of the balance of preference. How to test this? Submit it to vote.

Score voting systems should, first of all, incorporate approval
cutoff and should never produce a final result which is not approved
by a majority. Secondly, if there is a score winner who is not
approved by a majority, this should be presented again to the voters.
Whether or not to submit that to the voters if the situation is one
of multiple majorities is another matter.

Multiple round systems are intrinsically more powerful than systems
limited to a single ballot.

William Waugh wrote:
>The W'pedia editors say that there is still some question about how
>the majority "criterion" would be evaluated with Range, but to me it
>seems straightforward to interpret that if a voter rates candidate A
>strictly higher than candidate B on her ballot, she prefers A to B.

Right. So Score, analyzed this way, fails MC. However, if we consider
that the voter becomes 10 voters as the voter's collective
expression, MC holds. Still, it *is* straightforward to conclude that
Score/Range fails the Criterion, and it is merely that this is
*disconcerting*, so accustomed are we to what Clay is apparently
calling the "Majority Axiom," which is really about it being "better"
that the majority preference win.

William pointed to the Pizza election. My favorite.
http://www.rangevoting.org/LomaxRLame.html This was written when
FairVote was still the Center for Voting and Democracy.

The resolution of the apparent contradictions is in requiring
majority approval of every result. Ultimately it will be found in
using Asset systems to create an electoral body that can elect a
fully representative Assembly that can then elect officers as in a
parliamentary system.

Short of Asset, there are hybrid systems that would settle within two
ballots to very good results.

And my generic path to implementation: if an organization uses Asset,
for its own decisions, it is likely to be more successful than with
any other system, other things being equal. It's not magic pixie
dust. Just a way of creating a network of trust.