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

[Blake Cretney]

Craig Carey wrote:

> 
> In the this message, it shall be assumed that there is only one winner,
>  and also that the votes all sum to 1. (They have been rescaled).
> 
> Here's a definition of this maili list's 1 winner tweaked-AV method:
> 
>                       AV0 preferential voting method
> 
> For stage=1,2,3,... loop
>    If the number of candidates remaining is <= 1 then quit the loop.
>    Sort the remaining candidates by their 1st preference votes.
>    Identify those candidates which have less 1st preference votes than
>      the quota, and identify the candidate with the least 1st preference
>      votes.
>    Eliminate all the candidates that did not meet the quota and the
>      one with least votes.
> end loop.
> 
> The quotas are not defined in AVO
> 
> I shall call this method, where the quotas are undefined, the AV0 method.
> 
> The problem is, find the quotas that give the best method.
> It seems to be just the sort of thing the UK electoral reform group
>  (I forget the name of it), should have done and finshed many years or
>  even decades, ago.
> 
> ----------------------------------------------------------------------
>                       AV1 preferential voting method
> 
> AV1 is AV0 except that the quotas are defined.
> 
>  Quota =
>       1 / (number of remaining candidates)
> 
> Hence, if there are n candidates, then the quota at the 1st stage is
>  1/n (e.g. 1/10 if there are 10 candidates). The quota for the 2nd
>  stage =
>       1 / (n - 'number of candidates eliminated in 1st stage')
> 
> ----------------------------------------------------------------------

37 A C B
26 B A
24 C B A
13 D C

With 4 candidates, quota is 1/4, or in this case 25 votes.  C and D
are eliminated.  B wins.

But, if we change 2 votes from A C B to C B A, we get

35 A C B
26 B A
26 C B A
13 D C

D is eliminated.  Then, with a quota of 1/3=33 votes, B and C are
eliminated.  A wins.

Note that in regular AV, A wins both.

> I have no doubt that this AV1 method is major improvement over the
>  Alternative Vote (i.e. the 1 winner STV method).
> 
> It seems to me that there are two unsatisfactory areas with AV1:
> 
>  (i) Proving AV1 satisfies (P1) won't be simple most probably. I.e. it
>     it is not known to have good theoretical properties.

It's almost always easier to prove that a method fails a criterion
than that it passes one.  It makes sense to try to find a
counter-example before trying to find a proof.

> 
> The quota can't be above 1/3 because that could reject all
>  candidates, for example, when candidates have the 1st preference
>  counts of 1/3 after being perturbed by (3/v, -1/v, -2/v), and v
>  is large enough to take all three under the quota.

You are using the quota in a different way that I was.  In the method
I will call AVQ, all candidates must get more than 1/3 of the 1st
preference vote, or be eliminated right away.  If this would eliminate
all candidates, the plurality winner is chosen instead.  This
elimination is only done at the beginning, not at every step, and is
not based on the number of candidates.  Here is an example:

40 A
32 B
28 C B

Since B and C fall beneath the quota, they are both eliminated.  A
wins.  It is my contention, backed up by the evidence I gave
previously, that AVQ meets P1.

> >Here's a quote from Lord Alexander (speaking about AV)
> >
> >> I find this approach wholly illogical. Why should the second 
> >> preferences of those voters who favoured the two stronger
> >> candidates on the first vote be totally ignored and only those
> >> who support the lower placed and less popular candidates get a
> >> second bite of the cherry? Why, too, should the second
> >> preferences of these voters be given equal weight with the
> >> first preferences of supporters of the stronger candidates? 
> >
> The 2nd sentence is hard to understand and it does not appear to
>  really be obviously true. It might be about a problem that appears
>  in the 3 candidate election that reapears in all others.
> 
> Regarding his last sentence, he is getting close to saying votes
>  ought be wasted. Ideally the 100th preference should if possible,
>  have the same power and influence as a first preference, contrary
>  to the nature of STV.

That's why I favour Condorcet-type methods.  Let me point out that
both AV1 and AVQ suffer from vote-splitting, and therefore may be
considered to waste votes.

40 A
31 B C A
29 C B A

Here, B and C have split the vote.  

> >As well, I have never understood why you think that a method that
> >says a change from
> >C B A
> >to
> >C A B
> >can make the winner change from C to A, is not behaving reasonably. 
> 
> The alteration is: (C+B)-(CA+).
> 
> >If A gets more support, that can reasonably mean that A should win.
> 
> It doesn't matter what happens to A because the alteration is ruled
>  out by 'truncation resistance SPC' (and (P1)) because of what happens
>  to candidate C.
> 
> So the AV method doesn't always get the wrong winner through
>  emphasizing the wishes of supporters of a string of dead losers.

Are you aware of Arrow's theorem, which says that in any realistic
method, you are always going to have the possibility of spoiler
candidates, who do not win, but still alter the election result?

That doesn't mean, however, that the election has to be determined by
a "string of dead losers". AV tries to avoid this, but not necessarily
in the best possible way.

In particular, in the example

45 A B C
12 B A C
43 C B A

AV eliminates B, then elects A.  In this example, C is clearly acting
as a spoiler, since, if C was not running

45 A B
55 B A

B would win.

Since the C B A voters prefer B to A, we can say that these voters
are ill-served by the fact that their support is locked behind the
dead loser C, until it is too late to help the viable option B.  In
fact, SPC mandates this locking away.

As a criterion for excluding spoilers, I prefer the Local
Independence from Irrelevant Alternatives Criterion (LIIAC).  I have
no idea what they mean by "Local".  But in any case, this criterion
says that you should find the smallest non-empty set of candidates
such that no candidate outside the set is majority preferred to any
candidate in the set.  Then, the candidates outside the set should not
have any effect on the election outcome.

However we define "dead loser", we would not expect that a dead loser
would be preferred by a majority to a viable candidate.  LIIAC
guarantees that if you have a bunch of "dead" candidates, none of whom
are preferred to any of the viable candidates, none of the dead
candidates can affect the election outcome.

---

Blake Cretney