[EM] (P1) tweak up of AV meth. using random walk may fail

[Craig Carey]

I haven't commented closely on the Tideman methods or AVQ.


At 09:22 31.01.00 , Blake Cretney wrote:
....
>> It is my contention, backed up by the evidence I gave
>> >previously, that AVQ meets P1.
>> 
>> I am sure that would be wrong if AVQ was defined in a way that was
>>  an actual definition. 
>
>In what way is it not an "actual definition".  It is a procedure
>which you can carry out for any example you like.

Because I didn't figure out the quotas. The problem was that you
 referred to number "1/3" and described only a 3 candidate
 example. Just like here:

>1.  Check to see if any candidates get more than 1/3 of the 1st
>preferences.  If not, the winner is the candidate with the most 1st
>preferences (like FPP).  Otherwise, go to step 2.
>2.  Eliminate all candidates with 1/3 or under of the 1st
>preferences.  These candidates are eliminated from ballots in the same
>way candidates are eliminated in AV.
>3.  Hold an AV election between the remaining candidates.
> 
>Is there any part of that procedure that you do not view as
>sufficiently defined?


Suppose there are 20,000 candidates. It is highly unlikely that any
 would get more than 1/3 of the vote. Very unlikely. Rather than the
 method becoming incomprehensibly adequate at recovering support
 at deep preference levels, it just pick the FPP winner, i.e. it
 ignores all preferences after the 1st.

That method has got to be worse than AV and AV1, and STV can get
 support that is 5 to 20 preferences deep contrubuted to a candidate.


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

>
>> A method won't pass (P1) if all the surfaces do not meet properly and 
>> form a gentle piecewise-flat surface.  Any method with an "if then else" 
>> in it, comes under more suspicion of failing (P1). Your AVQ has this 
>> feature: "if all are rejected at any stage then one will be picked".
>
>Are these assertions based on anything other than intuition?  Isn't
>it possible that P1 does not relate to a smooth graph?  Isn't it
>possible that a smooth graph is totally meaningless?
>

Inside the simplex of possible elections (with the papers corresponding
 to vertices), an "if-then-else" positions a flat that quite possibly
 that has win regions that do not match up and align properly. So part
 of the flat becomes part of the surface of the win region. It gets less
 obvious to argue futher.

Anyway, AVQ is already shown to be almost identical to FPP.

...
>violates P1.  Otherwise, you will have to face the possibility that
>your intuition is incorrect, and that either AVQ gives a smooth graph
>or P1 does not imply a smooth graph.

What is that word "graph" mean: "a directed net", or "plot"?.

...
>> Ideally the 100th preference should if possible,
>> have the same power and influence as a first preference, contrary
>> to the nature of STV.
>
>Methods that consider candidates two-by-two (pairwise) have this
>property.  I'm going to describe such a method further down.

Condercet has that other problem of not finding enough winners.

>> Condorcet is a method that returns the wrong
>>  number of winners. As far as it goes (which isn't far) , it satisfies
>>  (P1).
>
>When I said Condorcet-type methods, I meant methods that meet the
>Condorcet criterion.  There are lots of suggested methods along these
>lines, however, I am going to describe one of the better ones,
>Tideman's method.
>

Now that AVQ was lost into history as a failed method ...

>Compare every candidate to every other to get one-on-one (pairwise)
>majority decisions.  That is, if there are candidates X and Y, you
>would find out how many people rank X over Y, and how many rank Y over
>X.  Then, the candidate with more votes is said to have a majority
>over the other, with a margin of whatever the difference is.
>
>For example, if 30 people vote X over Y, and 20 vote Y over X, I will
>say that X has a majority over Y with a margin of 10.  
>

There are 5 cases. You actually ignored two cases:
.......
...X...
...Y...
..X..Y..
..Y..X..

I do not know what is to be done with papers marked (...X...).

Also, until you add the word "Condorcet" (or whatever), there is no
 summation of counts on papers. The word "rank" can only apply to
 a single paper (since it is defined for [3 of the 5 cases for] a
 single paper). Your definition above seems to not contain the
 idea of adding counts of papers.

I will omit commenting on most of the rest.
Nethertheless is could be of real interest to whomever.


...
>> This is pairwise comparing idea, an idea of much reduced credibility
>>  (ref. Condorcet) due to the failure to remain plausible in any
>>  idiot's opinion when applied to extremely large problems.
>>
> 
>I think I have made clear with my example of Tideman's method that it
>is possible to have a pairwise method, meeting the Condorcet
>Criterion, which is not overly prone to ties.  Is that what you are
>referring to?

I suppose your statement saying a fixed Condorcet style method could
 be more likely truer than mine which said they all go extremely wrong
 for large numbers of candidates.

If real numbers are used then "ties" ought not be a problem.


...
>> I regard it as vital that the method get the right number of winners.
>>  That simplifies derivations from rules.
>
>Just because LIIAC says that certain candidates should not effect the
>result (and should not win), does not mean that all the remaining
>candidates must be declared co-winners.  In my example for Tideman's
>method, only D was excluded from having an influence by LIIAC, but a
>single winner was still chosen.  Tideman passes LIIAC, but without any
>indecision as a result.
> 

LIIAC uses pairwise comparing. Why should A win and B lose if A
 pairwise beats B ?. It shouldn't I presume. Maybe Condorcet never
 had an answer either. It not a tiny question when the reality of
 iterated shadow casting in the simplex is deleting some hoped for
 rules. Too much is lost.

Does this Tideman method ever elect the wrong number of winners?

...
>> STV's SPC should not be lost, since it is a tremendous simplifier of
>> the mathematics. Once SPC is imposed, STV ends up being the only
>>  known choice ?, for 20 or more candidates, and when there are less.
>> 
>
>Some methods that pass SPC are FPP, AV, and AVQ.  Are you really
>saying that you favour SPC just because it excludes lots of methods,
>and therefore simplifies your choice?  I feel that you must have some
>other justification for defending a criterion that you yourself admit
>results in significant problems.  All known methods that pass it
>either violate monotonicity or have vote-splitting.

The 1st sentence of what I wrote is most suspect [it said that SPC
 is important if without it, no solution can be found].

Here's another argument for STV and all similar methods (i.e. about
 STV) (any number of winners).

* * *
It seems to me that proportionality mops up degrees of freedom, and the
 only other important reality is what a voter can detect when in the
 voting booth. They don't understand that LIIAC is important. That is
 a sort of rule that a method designer might (falsely) allege important.

They can select a single candidate, and do these two things: (a) alter
 the preferences before that candidate's preference, and (b) alter the
 preferences after.

It is not possible to get the method invariant to permutations of the
 preceding preferences if they are all losers or(?) if they are all
 winners. That could be desirable; it is not obtainable.

However, one can get an invariance of the winner set on altering
 trailing preferences. So that (named "SPC" in this list) may as well
 be done. SPC is an easy consraint to meet, and a major method (STV)
 satisfies it.
* * *

Mr G. A. Craig Carey
Auckland