Simple reductio ad absurdum proof that "majoritarian axiom" is false

[Clay Shentrup]

Suppose a majority of voters want Ballot Initiative X to pass, regardless of whether Y passes.

And a majority of voters want Ballot Initiative Y to pass, regardless of whether X passes.

And a majority of voters want Ballot Initiative Z to fail, where Z is just X and Y combined.

No matter what, a majority is wrong. Even on binary issues.

This seems to me to be directly related to Anscombe's paradox.

[Warren D Smith]

On 8/24/15, Clay Shentrup <cl...@electology.org> wrote:
> Suppose a majority of voters want Ballot Initiative X to pass, *regardless
> of whether Y passes.*
>
> And a majority of voters want Ballot Initiative Y to pass, *regardless of
> whether X passes.*

>
> And a majority of voters want Ballot Initiative Z to fail, where Z is just
> X and Y combined.

--well, no. Your suppositions are contradictory.
Calling them A, B, and C, it seems to me that A contradicts C, and
also that B contradicts C.

The problem here is not the voting system or some majoritarian desire. The
problem here is that your voters are insane.

[Warren D Smith]

Perhaps this can be saved by saying what is truly meant, i.e. better wording?

We could, say, make X be "raise taxes 10%"
and Y also be raise taxes 10%.
If both X and Y pass, that raises taxes 21%.
The voters want a 10% rise but not a 21% rise.

So, if voters were to vote on X by itself: yes-majority.
If vote on Y by itself: yes-majority.
If vote on X&Y: no-majority.

I do not see any problem with majoritarianism in this example, though, and
even if there were only a single voter, i.e. me, I see no logical problem.

In Clay's wording, he claimed the voters wanted X, even if Y was true,
then also claimed the voters did not want X&Y (forcing the existence
of self-contradictory voters) than ascribed the problem not to his
voters, but to majoritarianism.
Well, no.

[Clay Shentrup]

On Monday, August 24, 2015 at 6:38:43 AM UTC-7, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

--well, no.   Your suppositions are contradictory.

Not one bit.

Calling them A, B, and C,   it seems to me that A contradicts C, and
also that B contradicts C.

You're confused.

The problem here is not the voting system or some majoritarian desire.  The problem here is that your voters are insane.

Not true.

Look. It is simple. You want X, but you REALLY do not want Y. You would oppose a bundled X+Y initiative, but even if Y passes, you still want X to pass. 

[William Waugh]

Dear Mr. Shentrup, I think that to make your argument cohere, you have to say how many voters place what valuation on X without Y, Y without X, X with Y, and neither X nor Y. And remind me what "axiom" you are trying to refute, not just by name, but by specification.

On Monday, August 24, 2015 at 1:28:43 AM UTC-4, Clay Shentrup wrote https://groups.google.com/d/msg/electionscience/UP67dBpeTEk/ZKLSDbYBAAAJ

[Warren D Smith]

> Look. It is simple. You want X, but you REALLY do not want Y. You would

> oppose a bundled X+Y initiative, *but* even if Y passes, you still want X
> to pass.

--I'm trying to translate this into some concrete form. Given this latest
"clarification" from Clay S, perhaps he has in mind these utilities:

Util(X and not Y) = 9
Util(Y and not X) = 0
Util(X and Y) = 4
Util(not X and not Y) = 3

in which case this voter would prefer X over not X, regardless of Y, and
would prefer not Y over Y, regardless of X?

Who knows what he meant. I suggest that he actually say.

[Nevin Brackett-Rozinsky]

Clay, does this minimal example fit your scenario?

Alice wants X only, not Y or Z

Bob wants Y only, not X or Z

Carol wants each of X and Y, hence also Z

Now:

If X alone passes, Bob and Carol are upset Y did not pass.

If Y alone passes, Alice and Carol are upset X did not pass.

If Z (which is X and Y) passes, Alice and Bob are upset.

If nothing passes, everyone is upset.

Every outcome leaves a majority dissatisfied. This is, in a sense, similar to what happens in a Condorcet cycle: no matter who wins, a majority would have preferred someone else.

It is certainly interesting and worth analyzing, but it does not say anything about the situation where there is an outcome with majority support.

Nevin

[Clay Shentrup]

On Monday, August 24, 2015 at 8:19:49 AM UTC-7, William Waugh wrote:

Dear Mr. Shentrup, I think that to make your argument cohere, you have to say how many voters place what valuation on X without Y, Y without X, X with Y, and neither X nor Y.

No I do not.

And remind me what "axiom" you are trying to refute, not just by name, but by specification.

The majoritarian axiom: that if a majority of a group's members favor X, the group also favors X. E.g. IRV advocates often ignorantly criticize Score/Approval Voting for failing the Majority Criterion, because they don't understand that the majority criterion is actually wrong.

[Clay Shentrup]

On Monday, August 24, 2015 at 8:50:11 AM UTC-7, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

Who knows what he meant.  I suggest that he actually say.

I meant exactly what I said. It was perfectly clear and unambiguous. Your utilities are a perfectly fine example.

[Clay Shentrup]

On Monday, August 24, 2015 at 12:48:34 PM UTC-7, Nevin Brackett-Rozinsky wrote:

Clay, does this minimal example fit your scenario?

Absolutely not. It's not sufficient for some segment to want X and not Y. They have to want X *regardless of* whether Y passes, but also some of them have to dislike Y so much that they'd prefer ~X~Y to XY.

[Clay Shentrup]

On Monday, August 24, 2015 at 8:50:11 AM UTC-7, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

--I'm trying to translate this into some concrete form.  Given this latest
"clarification" from Clay S, perhaps he has in mind these utilities:

Util(X and not Y) = 9
Util(Y and not X) = 0
Util(X and Y) = 4
Util(not X and not Y) = 3

That would be one of many examples, yes.

in which case this voter would prefer X over not X, regardless of Y, and would prefer not Y over Y, regardless of X?

Who knows what he meant.  I suggest that he actually say.

I meant exactly what I said. I don't see how you can think I was the least bit ambiguous. 

[Clay Shentrup]

On Monday, August 24, 2015 at 12:48:34 PM UTC-7, Nevin Brackett-Rozinsky wrote:

Clay, does this minimal example fit your scenario?

Here's the most minimal example I can readily think of.

35% X~Y > ~X~Y > XY > ~XY

33% ~XY > ~X~Y > XY > X~Y

32% XY > X~Y > ~XY > ~X~Y

A 67% majority prefers to have X pass, regardless of whether Y passes.

A 65% majority prefers to have Y pass, regardless of whether X passes.

A 68% majority prefers to have X and Y both fail to having them both pass.

[William Waugh]

In the premise, "a majority of a group's members favor X", does X denote a class of possible outcomes of the election, or just a single possibility? For example, if the election could enact A without B, B without A, A and B, or neither, and if there were no other possible outcomes, does X denote something like A (which I see here as a class of two of the four possible outcomes), or does it denote something like A and B together?

On Monday, August 24, 2015 at 11:41:32 PM UTC-4, Clay Shentrup wrote:

...

[Clay Shentrup]

X and Y are ballot initiatives. They could do anything, such as tax sugary beverages or sell bonds to fund subsidized housing.

[Warren D Smith]

On 8/24/15, Clay Shentrup <cl...@electology.org> wrote:

> On Monday, August 24, 2015 at 8:50:11 AM UTC-7, Warren D. Smith (CRV
> cofounder, http://RangeVoting.org) wrote:
>>
>> Who knows what he meant. I suggest that he actually say.
>
>
> I meant exactly what I said. It was perfectly clear and unambiguous

--well... so far, every reader has failed to understand your "perfectly clear"
example. Amazing, huh? Could that possibly mean it wasn't perfectly clear?

>. Your
> utilities are a perfectly fine example.

--no they weren't (which I think I'm allowed to say, since I devised them).

Again. You need to state a set of voters, a set of candidates (yours were X and
Y), and the utilities of each voter for the 4 possible outcomes XY, Xy, xY, xy
(where "x" means "not X").

Once you have actually stated this, which you seem to have an allergy
to doing, then the example could be evaluated.



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

[Warren D Smith]

I worked out what (perhaps) Clay Shentrup had in mind.
(I hope he never tries to convince anybody of anything using his wording.)
Two events named X and Y are considered.
Notation:
XY means "X and Y,"
xY means "Y and not X,"
Xy means "X and not Y,"
xy means "not X and not Y."

Voter their honest valuation of the 4 possible outcomes
voter #1: xy=6, Xy=9, xY=0, XY=3
voter #2: xy=6, Xy=0, xY=9, XY=3
voter #3: xy=0, Xy=5, xY=5, XY=9

Hence:
Using simple majority vote:
(1) X would be enacted by 2:1 majority (voters 1 & 3) whether or not Y was.
(2) Y would be enacted by 2:1 majority (voters 2 & 3) whether or not X was.
(3) "X and Y" would be defeated by 2:1 majority (voters 1 & 2) versus "neither."

Honest score voting: would prefer "X and Y."

[Warren D Smith]

Sorry, my attempted numerical instantiation of Clay S's claim last post,
actually was wrong.

[Warren D Smith]

Sorry, it looks ok again. Here is the example again with more annotations:

I worked out what (perhaps) Clay Shentrup had in mind.

(I hope he never tries to convince anybody of anything using his wording...)

Two events named X and Y are considered.
Notation:
XY means "X and Y,"
xY means "Y and not X,"
Xy means "X and not Y,"
xy means "not X and not Y."

Voter their honest valuation comment
===== of the 4 possible outcomes =======
voter #1: xy=6, Xy=9, xY=0, XY=3 any x-situation gains 3 points with X
any y-situation loses 6
points with Y

voter #2: xy=6, Xy=0, xY=9, XY=3 gain 3 with y->Y, lose
6 with x->X

voter #3: xy=0, Xy=4, xY=4, XY=8 gain 4 with x->X, gain
4 with y->Y

Hence:
Using simple majority votes:

(1) X would be enacted by 2:1 majority (voters 1 & 3) whether or not Y was.
(2) Y would be enacted by 2:1 majority (voters 2 & 3) whether or not X was.

(3) "X and Y" (XY) would be defeated by 2:1 majority (voters 1 & 2)
versus "neither" (xy).

Honest 4-option score voting: would prefer "X and Y" with total score
14, versus others
(scoring 12 or 13). Also XY would win using normalized scores.

[Clay Shentrup]

On Tuesday, August 25, 2015 at 10:03:37 AM UTC-7, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

--well... so far, every reader has failed to understand your "perfectly clear"
example.  Amazing, huh?   Could that possibly mean it wasn't perfectly clear?

I repeat:

Suppose a majority of voters want Ballot Initiative X to pass, regardless of whether Y passes.

And a majority of voters want Ballot Initiative Y to pass, regardless of whether X passes.

And a majority of voters want Ballot Initiative Z to fail, where Z is just X and Y combined. 

Which of those sentences do you not understand? Is it the word "majority"? Is the letter X?

Again. You need to state a set of voters, a set of candidates (yours were X and
Y), and the utilities of each voter for the 4 possible outcomes XY, Xy, xY, xy
(where "x" means "not X").

A) I do not need to list utilities, just ordered preferences.

B) I do not need to actually list the preferences, I just need to describe the criteria that would satisfy them—which I did in my original post.

C) I already did cite a specific example, which I will repeat here:

35% X~Y > ~X~Y > XY > ~XY

33% ~XY > ~X~Y > XY > X~Y

32% XY > X~Y > ~XY > ~X~Y

X~Y = Xy (in case the tilde was not obvious)

[Warren D Smith]

Clay, it was not clear an example existed meeting your criteria.
I could state plenty of criteria which if satisfied would prove the
moon is made of green cheese, but it might not be possible to satisfy
the criteria.
Anyhow, I will now attempt to make yet Another Web Page on this (now) example.

[Warren D Smith]

RangeVoting.org/XYvote.html

[Clay Shentrup]

On Wednesday, August 26, 2015 at 5:31:09 AM UTC-7, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

Clay, it was not clear an example existed meeting your criteria.

Oh, well that's different. It sounded like you were complaining that you didn't even understand my criteria.

[Clay Shentrup]

On Wednesday, August 26, 2015 at 1:59:22 PM UTC-7, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

RangeVoting.org/XYvote.html 

Error:

> ..and they each will simply value the combined event X&Y as the sum of their X and Y values.

No, I did not state that constraint. Their U(XY) need not be the sum of X and Y.

[Toby Pereira]

For what it's worth after all this time has passed, I think the original post is actually quite clear. It's quite easy to see that you could have 51% of people want to raise some tax, 51% of people want to introduce some restrictions on cigarettes, but only 2% of people want to do both.

[Clay Shentrup]

On Sunday, April 17, 2016 at 1:39:19 PM UTC-7, Toby Pereira wrote:

For what it's worth after all this time has passed, I think the original post is actually quite clear. It's quite easy to see that you could have 51% of people want to raise some tax, 51% of people want to introduce some restrictions on cigarettes, but only 2% of people want to do both.

I didn't say it wasn't clear. My point is that that the present description is overly restrictive.