# [EM] pairwise, fairness, and information content

Craig Carey research at ijs.co.nz
Fri Aug 16 03:52:05 PDT 2002

```
At 2002\08\15 23:44 -0700 Thursday, Richard Moore wrote:
>Craig Carey wrote:
>> The definition of monotonicity was wrong. In general (i.e. for some
>> number of winners and candidates), all 3 rules ought be rejected for
>> failing a perfect method.
>
>Hmm, I wonder what was wrong about the definition of monotonicity. It
>certainly called for a never-negative response to any single ballot
>substitution. Perhaps there is a miswording somewhere; I didn't spot
>one on my final proofreading though. Of course, monotonicity in the
>pure mathematical sense would also allow for methods that have a
>never-positive response to a ballot substitution; but by convention
>that's not what we mean when we talk about election methods being
>monotonic. In fact, all the second case does is reverse the
>categorization of the ballots, so that what we wished to count as an
>"A>B" ballot is effectively a "B>A" ballot and vice versa.

Hopefully the public definition of Monotonicity allows changes to any
number of 'ballot papers' (preference lists) but it requires that no
paper change except for having the preference of a given candidate
either [also given beforehand]:

* move to the left and towards the 1st preference (this include making
the preference appear on a paper), xor

* move away from the first preference (and this includes the cases
where the preference drops off the end).

Your definition did not keep the other papers unchanging.

Also your rule is against the style of STV in that gets the treatment
of coalitions handled wrongly. For exampl, (and this is a rough
example with the other papers missing):

Suppose
* X is of the Red party, and
* Y is of the Orange party.

Then this is to be allowed:
[1]   (Red_A  Red_B  X  Y  Orange_C Orange_D),     X loses, Y wins

<----alters to------>

[1]   (Orange_C Orange_D  Y  X  Red_A  Red_B),     Y loses, X wins

But the 'swapping' "monotonicity" rule did not allow it.
A coalition is indicated by the marking of that papers in a way that is
not well understood.

X can lose the 1st, and Y can lose the 2nd, because their similar
candidates are soaking up their votes. I presume that 3 candidates is
enough to show the rule undesirable.

>
>I wonder what "perfect method" would fail to honor the unanimous
>choice of the voters? What "perfect method" would give negative weight
>to a preference on any ballot?
>

"honour the unanimous" is attempting to introduce a rule and so it
would seem to be able to remove some other rule.

I have not defined the weight of preferences. Maybe it could be done,
and I guess that such a rule should pass my IFPP preferential voting
method.

>The third criterion I gave might be tighter than necessary from a

I can't remember what the 3rd criterion was, I do recall remembering
that it is fit to be forgotten. With your rules being ones you
devised yourself, it would help em if they were repeated (hopefully
in an increasingly clear way that you yourself had to say that they
were no good).

...
>criteria, and has no correlation to pairwise. At any rate, removing
>the third criterion would allow for arbitrary treatment of ballots.

This was the 3rd rule (please quote those for me):
----------------------------
>3. Permuting the ballots shall not change the result. This eliminates
>methods that arbitrarily weight ballots differently. With this
>restriction, it is not necessary to represent the ballots as an
>ordered set; it is sufficient to know the total number of ballots for
>each classification.
----------------------------

Rather than edge towards a conclusion that Approval is best (and IRV
is rejected because the attention is on the permuting of the earlier
preferences), I can say the rule is rejected for being vague. Or

It apparently says this:  If candidate A wins the 1st then candidate A
wins the 2nd:

(1st)   (A B C D E F G)     <-- an added paper to unspecified system
<--->
(2nd)   (B C D E F G A)

>Perhaps you would like to specify a suitable replacement for this
>criterion, but I am not willing to simply drop it.
>

No, it vanished by itself. It is also an 'undesirable' rule in 2 candidate
elections. A hard to explain rule that voters would not see as protecting
their fairness interests since it is incompatible with
truncation resistance.

...
>probabilities. I was using "as often as" to indicate a one-to-one
>correspondence between objects in two categories, but if you prefer to
>think probabilistically, then the equivalent would be a uniform
>distribution.
>
>My loose use of the phrase "as often as" could be applied for instance
>in a statement like, "Integers are even as often as they are odd."
>

Comparing infinities?.

I regard the topic as being something where the entities are like
gemstones of any colour and the rules ought ought be setting limits
on the slopes of the faces. The EM List has lots of people that
both talk about probabilities and that are incomprehensible and that
do not also get results just about ever. The could correspond to
vibrating gemstones like Ruby and Topaz and Diamond (avoiding any
reference to the Green or Turquoise colours), and so on, with vibrating
humming faces that are blurry and fuzzy.

....

>threshold. That means that there is a correlation between this method
>and pairwise.
>

Why not solve the problem rather than pursue something that it is not
defined and presumably not desired: "correlation".

...
>1/2. As I said, my proof did not require M to be unbiased: Lo, this
>method also correlates with pairwise. It will disagree with pairwise

You started out (if I recall right) implying that pairwise comparing
is important in 3 candidate elections.

>>  Richard still has not admitted that there is no
>> need to use pairwise comparing. It is not in the text above so Richard
>> either is wrong or will be expecting that the text above is wrong,
>> unless that dictator idea somehow contradicts.
>
>We really do have a communications problem here! I haven't said there
>*is* a need to use pairwise comparing. The closest I came to saying

We ought reject it.

...

>What I *have* been saying is that you are wrong when you say pairwise
>comparison contains no information about who should win an election,

I am uninterested in correlations. If they are probabilities then they
correspond to hypervolumes which people are not interested in. Theorists
write about that here, but get hindered when I ask for the formulae.

Considerations of lengths (linear dimensions) are a replacement and
such measures can compare 2 failing rules under the light of the rule
that they are failing. E.g. the comparison of 2 winner STV variants
under a test of monotonicity.

I have some Ada 95 code that gives wrong STV winners under Matlab and
correct winners outside of Matlab and I need to debug it. It seems to
be identical code getting the same input, and the face of it, it could
be an uninitialised variable. Windows and FreeBSD do not initialise
memory and Linux does, so Linux is a bad system to debug uninitialised
variables under.

>unless you want to completely disregard one or more of the criteria I
>listed.

They are all rejected. As the rules get more complex, then an ideal
method may be needed, and thus at some point progressing in the
design of rules requires that some partial progress in designing
axioms, have been done.

...
>And I'm not sure why you still think "that dictator idea" has any
>bearing on the matter.
>No sense beating a red herring after it's dead.

It is not clear that pairwise comparing is down and out amongst those
that liked it.

>Once more for the record: A counterexample would have to fail the
>correlation test and pass requirements 1 through 3. Dictatorship

"Tests" of "correlation" ?. If there is an axiom there then write it
up as an axiom. Previously it was used to advance the idea that
no matter how useless pairwise comparing us, it accidentally
correlates with my theories and so it contains information and thus
pairwise comparing is not rejected. I didn't realize it was a rule too.

>passes the correlation test (weakly), but fails requirement #3 since
>we need to identify the dictator's ballot. If we alternately define
>dictatorship so that the dictator's choice isn't anywhere in the set
>of ballots, then it fails requirement #2.
>

The word "dictator" is in a singular form, thus there is an attempt to
constrain the number of winners. Is some idea of dictators really
eliminated given that rule eliminating it (in your words) is also on
the way out.

Why does Richard not make a checklist of my rules:
1. P3: Right number of winners
2. P2: no change of the set of winners when smudging the preference of
a paper out over the next
3. P1: improved monotoniciy. P1 is the same as truncation resistance
and monotonicity (any number of winners), except that P1 does not
require the existence hidden temporarily existing ballot papers,
since it can impose its requirement in a single check.
4: P4: not needed while 3 or less candidates.
5: Embedding of SNTV/FPTP
5. PP: an aim of proportionality to 'mop up' the remaining degrees of
freedom. PP is a "sum (to subtotals if the paper names the candidate),
sort (the subtotals), and select (the required number corresponding to
the right number of most positive subtotals)" idea.

http://www.ijs.co.nz/ifpp.htm and http://www.ijs.co/nz/quota-13.htm

Also Richard can patch us into the latest reasoning of the philosophical
school of "Social Decision Theory" and what is mathematics of vibrating
"probabilistic" [hyper-D] gemstones is.

You might want to avoid P1 and monotonicity since it can involve quite
real quantifier logic. Simplifying logic equations can be time consuming
even though the meaning of the equations maybe will plot in a triangle.

Why not try to reject STV/IRV via an attempt to reject the P2-implied
idea that such methods can be factorised. Once you reject P2 then
factorisation has gone and then you are no longer faced with a reality
of 3 candidate Richie's "IRV" being a method with extremely simple
mathematics.

---------------------------------

At 02\08\15 17:00 -0700 Thursday, Alex Small wrote:
>Craig Carey said:
>>> See, no use of the number of winners. Yet at this list the persons that
>> write tend to restrict the number of winners to be 1, when it seems far
>> from obvious how the mathematics of fairness for voters and candidates,
>
>
>It tends to be inconvenient to have more than 1 mayor, governor, sherriff,
>or other executive officer for a given constituency.  That simple fact,
>rather than any axiom of mathematics, is the motivation for studying
>single-winner methods.  Of course, concerns of fairness (founded in either
>math, ethics, or politics) lead to the study of multi-winner methods for
>legislatures and similar bodies, but on this list the greatest
>intellectual interest seems to be in single-winner methods.
>

The document of Alex Small was sighted flowing along inside of a council
sewer while the men were flow of the water in the pipe.

At 02\08\15 16:56 -0700 Thursday, Alex Small wrote:
>Craig-
>
...
>Would you be willing to describe in simple, layman's terms, the election
>procedure that you would recommend for selecting a single winner from a
>set of two candidates?
>

A "procedure"?.

I am not, and I have not been, making such recommendations.

>I know that you prefer to discuss these issues in a more technical,
>rigorous manner, but if the study of election methods is to have any
>relevance to the rest of the world (a desirable but perhaps not essential
>thing) then the methods should be amenable to a simple statement for the
>layman.  The justification for selecting that method over others may, of

We disagree over what a layman is: you Americans seem to regard it as
something that may or may not get to hear of the fantastic mistakes.

>course, be highly technical.
>

"very very very very ... very very very very very highly technical".

Check the writing of Mr Moore: he is not rejecting my axioms but just
like yourself (Alex) can seem to get the browser operating properly
and take it out to the ijs.co.nz website. I was certainly saying that
80 years of social decision theory have brought nothing (save vibrating
crystals and boring proofs, etc.).

I presume Alex precisely means that rules like my P2 are rejected. So
Alex now has no rule requiring that in a 3 candidate election with
candidates, A,B,C, adding the same quantity of FPTP (A),(B), and (C)
papers, leads to no alteration in the set of winners.

If you can't understand the basics of the topic Alex, then perhaps you
could unsubscribe. Those laymen may want more from you there. Whatever
they are, they could object to having a very very vague categorization
of their personality problems [dislike the "technical"] justify ...
justify what, Alex?. Justify something relevant to them?.

It is American from start to finish: the recipients are stupid thus
the experts are no better. Britain harbours a different ideal: the
message of the messenger is rejected. E.g. the supercomputer demands
of Approval or the 'unfair at the beginning and unfair at the end'
pairwise comparing thing.

>Also, I realize that according to many people there is no "perfect"
>election method but most of us have our own preferred method.
...

So what if you can use adjective. One what shape is that commenting
and if not commenting on a shape then is it pinning a shape down or
then also not about satisfying the fairness expectations of
indivuduals. So, indeed what is the talk of "laymen" for when the
background to it can be that it is a conscious intention to have them
unfairly disposed of under some Alex Small doctrine. Are you studying
physics?.

Craig Carey.

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