[EM] Chat on "A<B=C" & "=": to avoid infinitesimals & reject use of geometry

[Craig Carey]

Subject was: Re: [EM] Consistency, Truncation, etc. (was CR ballots, etc.)

At 01.09.26 11:19 +0200 Wednesday, Jobst Heitzig wrote:
 >I would like to defend my opinion that voters should have the option
 >to express "undecidedness" about certain pairs of alternatives, or, in
 >other words, to "abstain" from some but not all pairwise decisions.
 >
 >First of all, I think one should distinguish at least to situations:
 >1) Few alternatives (less than 10, say), no possibility for write-ins
 >2) Many alternatives and possibly write-ins
 >
 >
 >1) In the first situation, voters know (the names of) the alternatives. A
 >voter might be well-informed to make up a complete ranking. She also might
 >not care at all. Or she might have strong feelings concerning some
 >pairings but at the same time feel unsufficiently informed about some
 >other alternatives. In this last case, she either might want to assure
 >that these unknown alternatives only get a chance when they have a strong
 >support, or she might want to delegate the evaluation of these
 >alternatives to the other voters - this being perhaps mostly a question of
 >how much she trusts the other voters to make a good or correct decision.
 >
 >Let me give an example:
 >Two extreme candidates A,B, and an "intermediate" one C.
 >14 voters have A>C>B
 >35 voters have A>B and don't know what to make of C
 >14 voters have B>C>A
 >35 voters have B>A and don't know what to make of C
 > 2 voters have C>A=B.
 >This might occur when, before the election, the media focused on A and B,
 >so that many voters are uninformed about the moderate candidate C.
 >


The list (not just Jobst Heitzig) is writing about using an '=' in preference
lists. And that is in a rather full failure to comment on whether or not
P2 is imposed.

P2 is that rule which is is passed if and only if all these P2 subrules like
these are passed:

    P2(0): {t:(A), t:(B), t:(C), t:(D),...};
    P2(1): {-(N-1)t:(A), t:(AB), t:(AC), t:(AD),...} [a different 't'];
    P2(2): {-(N-2)t:(AB), t:(ABC), t:(ABD),...};
    P2(3): {-(N-3)t:(ABC), t:(ABCD),...}; and so on.

Jobhst may have been considering having an interpretation of "=" that is
consistent with P2. I argue for P2 by saying that it seems to be essential
if methods are derived. However I refer to the deriving of simple 3 candidate
methods and not 2 candidate methods, so it has not been commented on at the
Election Methods List. That is because each poster that has spotted the vital
essential P2 rule will be at a loss of what to say about it in the next k
weeks, where every subscribers is encouraged to estimate a truest value for
k.

I comment on When there is an equals sign in the list of symbols for
candidates. Although the Election Methods List has been creating a lot of
messages about "=" signs as if a treat to the use of plain preferential
ranked papers, it is too infinitesimal in nature to even need to be
considered if P2 is imposed. The Alternative Vote method passed the P2
test.

All this talk about "=" symbols in preference lists is talk about something
infinitesimal if P2 is made an axiom.

Above this was written:

 > 2 voters have C>A=B.


With P2 giving the natural meaning to "=", that is able rewritten as this:

       x   (CAB)
       y   (CBA)     :   x+y=2

Both x and y are Real-valued so the "=" is denoting something infinitesimal
and something that the list does not seem to want to understand. That
interpretation holds no matter how many candidates or winners there are
(subject to there being at least 3 candidates).

Can people write less messages on the "=" notation.

Not only do authors here write using numbers all the time but they can't
identify when something is as wide as a point in space or not. The authors
want the topic to be infinitesimal (via considering just examples) but even
there competence is low.

Perhaps the previous author would accept or reject P2.





At 01.09.26 11:19 +0200 Wednesday, Jobst Heitzig wrote:
 >I would like to defend my opinion that voters should have the option
 >to express "undecidedness" about certain pairs of alternatives, or, in
 >other words, to "abstain" from some but not all pairwise decisions.
 >
 >First of all, I think one should distinguish at least to situations:
 >1) Few alternatives (less than 10, say), no possibility for write-ins
 >2) Many alternatives and possibly write-ins
 >
 >
 >1) In the first situation, voters know (the names of) the alternatives. A
 >voter might be well-informed to make up a complete ranking. She also might
 >not care at all. Or she might have strong feelings concerning some
 >pairings but at the same time feel unsufficiently informed about some
 >other alternatives. In this last case, she either might want to assure
 >that these unknown alternatives only get a chance when they have a strong
 >support, or she might want to delegate the evaluation of these
 >alternatives to the other voters - this being perhaps mostly a question of
 >how much she trusts the other voters to make a good or correct decision.
 >
 >Let me give an example:
 >Two extreme candidates A,B, and an "intermediate" one C.
 >14 voters have A>C>B
 >35 voters have A>B and don't know what to make of C
 >14 voters have B>C>A
 >35 voters have B>A and don't know what to make of C
 > 2 voters have C>A=B.
 >This might occur when, before the election, the media focused on A and B,
 >so that many voters are uninformed about the moderate candidate C.
 >

Zero winners elections can receive the same papers. The plain STV-like
ballot papers define a list of candidates that should be dealt with with
ruthlessness if that is to the advantage of candidate of a preference
nearer the 1st preference. Jobst, you don't seem to be using a wording
that is independent of the number of winners. Do we want us to understand
that the meaning of a paper sloshes around as the number of winners is
varied? (e.g. a re-election is required if a council decided that 10
candidates would be needed even they initially wanted 8).


...
 >Let us analyse this in the following way. The matrix of strict preferences
 >is this:
 >

Why should anybody be interested in what that matrix is given that perfect
and desirable ideal methods that do not get scraped into the bin by courts,
are not based on any pairwise comparing ideas. "Matrix" means 2x2 and tensors
would be needed to generalise the theory.

The veil of having all ideas be one winner might be more than just tarpaulin
over a ship, but covering up for a half rusted missing theory.

----

If my quota-13.htm page (URL below) is looked at, it will be seen that the
P2 rule as the reduction of the dimensionality of the 15 paper problem to
the problem of solving a method for only the papers (AB),(B),(C). So every
author here that believes that P2 is too tough to think about or clearly
reject, is spelling out a position to the list of either never advancing
beyond deriving a 2 candidate method. One wonders if the list can even do
that. Even getting the 2 candidate problem solved using axioms is quite as
simple as it might seem.

What was the purpose behind writing about "=" whilst not commenting on the
rule I named P2 ?.


Craig Carey
http://www.ijs.co.nz/quota-13.htm