Re:
(5) APR: Steve's 5th dialogue with Kristofer

+0200

On 07/17/2015 09:54 PM, steve bosworth
wrote:

Hi Kristofer and all other readers,

Sorry,
I too have been delayed in replying sooner.

By the way, my mail client says your posts are in the Windows-1254

S:
Now, I’m using Windows 10 and Office
2013.

election-methods@electorama.com

S: Steve's questions will follow each element of what Kristofer wrote:

attack is possible in two-stage weighted

that

the

all if

of

S:
Please explain what you mean by ‘party-list equilibrium’ and how it
relates to APR.

example, one among the 435

a weighted vote in the assembly equal to the number of votes that each had
received directly or indirectly from citizens.

S:
Yes, I must accept the THEORETICAL possibility that strategic voting could
be used with APR.  However, am I correct
in understanding that all strategic voting requires the strategizer to have accurate
knowledge about how all other relevant people will be voting.  I do not see how any person or party could
acquire such knowledge in the above APR election of a ‘large national assembly’
which allows each ‘citizens to rank as few or as many of all the thousands of
candidates in the country’.  This is why
it seems to me that successful strategic voting would be impossible for
PRACTICAL purposes.  Your Taiwan example
does explain how parties might try to organize strategic voting but why should
we worry about it succeeding in the case of APR.

the two-stage voting method [….]

S: Yes, you did mention this but I do
not yet understand why you are discussing it.  APR’s counting of all the ranking in the
general election is a one-stage method with 4 counts:  These are explained in the following Endnotes
(4 and 9) to my article:

4. APR's use of Asset Voting
   (also see Endnotes 2 & 12) provides two ways in which a representative may
   also receive some votes from citizens indirectly:  Firstly, when none of the candidates ranked
   by a citizen have received enough votes to be elected, she can require her
   first choice but eliminated candidate to pass her 'default' vote on to the candidate
   he most trusts, e.g. the candidate highest on his pre-declared list.  He must sequentially do this until one of his
   favored candidates is elected.  All these
   available 'default' votes must be sequentially transferred, one by one,
   beginning with those held by the eliminated candidate who currently has the
   fewest number of votes.  If more than one
   eliminated candidate share this position of currently being the least popular,
   the order in which they will transfer the 'default' votes each holds will be determined
   by lot.

If and when any of these
default votes fail to help elect any representative after all the holders of
these default votes has made these provisional transfers, each must then be
given to the representative who has now been elected and is most favored by the
eliminated candidate who holds it.

Secondly, in response to the
possibility that a very popular representative may initially receive more than
10% of the country’s weighted vote, she must publish exactly how these 'extra'
votes will be non-returnably added to the weighted vote of her trusted fellow
representative(s).  This is to avoid any
question of a representative being in a position to ‘dictate’ to the
assembly.  The transferring of these
extra votes would proceed sequentially, starting with the representative who
had received the most votes above the 10% limit.

Consequently, the list of all
the elected candidates and their different weighted votes is finalized only
after two earlier ‘provisional’ counts have been completed.  The first produces a provisional list of the
pre-established number of elected candidates by counting all citizens' votes,
except those which had been given only to eliminated candidates.  The second provisional count would produce a
somewhat modified list by also counting the 'default' votes as described above.

The third count would include
all the transferred 'extra' votes from the above very popular representatives
who had received more than 10% of all the votes. Consequently, this third and
final list would contain all the pre-established number of representatives,
each with his or her finalized weighted vote in the assembly, none with more
than 10% of all the citizens’ votes.
Each citizen would know to which representative's weighted vote her vote
had been added.  Also see Endnote 9.

9. The FEC
   both ensures that each association’s APR general election ballot paper will be
   given to each of its registered voters at his or her local voting station on
   election day, and coordinates the countrywide
   counting of all citizens' rankings. This count determines both which candidates
   are elected and exactly how many weighted votes each representative will have
   in the House of Representatives.  Each
   will have a weighted vote exactly equal to the number of citizens whose votes
   helped to elect them.

All 435 elected candidates (congresspersons) would be discovered by
counting the rankings from all voting citizens in the country.  They would be found by sequentially eliminating the least popular candidate from the race,
one by one, until only the pre-established number of reps for each association
remain.  Again, each of these
representatives would have a weighted vote in the House of Representatives
exactly equal to the number of rankings (votes) each had received by the time
the last candidate had been eliminated, and all the 'default' and 'extra' votes
had been transferred (see Endnote 4).

S:
Does this make it any clearer that strategic voting would be practically
impossible using APR?

S:
What do you think?

S:
If you still think that the remaining parts of your reply to our 4th
APR dialogue are relevant, let me know and I will respond to them as best I
can.

Thank you,

Steve

On 10/01/2015 11:08 PM, steve bosworth wrote:

Yes. I mean the very opposite.

Firstly, no method is entirely invulnerable to strategic voting; that
was my point when mentioning Duggan-Schwarz. All you can do is find more
resilient methods, or balance resilience against other desirable
properties (like how good results you get under honesty).

Second, the cloning attack I mentioned specifically targets APR's
IRV-based election mechanism. What I tried to show is that APR is
vulnerable to cloning. Unlike STV's similar vulnerability, the IRV based
method used in ARV has a vulnerability that favors well-organized
participants, and so would give an advantage to parties that can organize.

A party list equilibrium is one where parties have an advantage over
non-parties to a point that the results from the methods mirror that
which would happen under party list. Parties are not explicitly made
part of the method (unlike party list) but because of the advantage of
belonging to a party, the method acts as if parties are formal parts of
the system.

Again, a good comparison is the former SNTV method in Taiwan. SNTV
itself has no mention of parties; there are just candidates and they win
according to their Plurality counts. Yet it did in effect work as a
party list system because everybody engaged in vote allocation.

Suppose I'm leader of a party X and I have absolutely no idea of how
voters not aligned with X will vote. Then I can still do the birthday
coordination I mentioned earlier to have my supporters evenly spread
their votes for the party's candidates.

This can never harm me (or my party), but it can help by displacing
other candidates. What candidates will be displaced? There are two
types: minor candidates who have less than a Droop quota, and members of
other parties that didn't coordinate and so appear to have less than a
Droop quota, yet would have had more if surpluses had been distributed
as in STV.

Here's a simple proof sketch for why it can never harm to spread the
votes evenly:

• Suppose that party X gets one candidate elected when it doesn't
  strategize.
• Now suppose that X tries to clone this candidate to win more. By
  cloning, every voter ranks all the clones next to each other (but not
  necessarily in the same order).
• Then the worst that can happen is that all the clones but one are
  eliminated, after which the election is just like if the party hadn't
  done any cloning at all.

So it works when the party gets one candidate without strategy. For more
candidates, the only danger is that say, k of n candidates won without
strategy, but spreading out the support means IRV eliminates more than
n-k in one go.

Let's say the party evenly distributes votes among the k candidates who
won (but not among the other n). It's clear that doing so won't push any
of the k candidates off the council because if the votes for the k
candidates were unequally distributed, one of them must have had less
votes before the even distribution than after. However, he still got
elected. So he'll still be elected when he gets more votes.

That leaves the other n-k. Suppose the votes are distributed evenly. But
at the point where n-k have been eliminated, all we're left with is a
group of k candidates. Since the votes are evenly distributed among the
n, they'll be evenly distributed among the k once n-k have been
eliminated, and by the argument above, these k will always be elected
anyway.

So the worst that can happen to party X is that they don't gain any new
seats by evenly distributing the votes. Hence it can never harm to clone
as long as the party ensures even distribution. W5.

More precisely: the party never loses by cloning if they can make the
distribution among their candidates more equal than it was without
cloning. The more information the party has about how others vote, the
better it can cancel out unevenness in their votes, but even if the
party knows absolutely nothing, cloning can't harm it.

Why does that matter? Since cloning can push off candidates that are not
as well-organized, the method favors groups that can pull off an
organized coordination campaign, i.e. parties.

(I also note that furthermore, the presence of organized cloning would
subvert the asset voting fix that you refer to. A party could spread the
support for its candidates so that nobody gets above the threshold.)

I discuss it because the "run IRV until k candidates remain and they're
the winners" method is equivalent to it, and splitting it into two
stages makes it more clear what the problem is.

Once only k candidates remain in IRV, they have a weight according to
the number of first place preferences they have on the ballots where
every other candidate (not in the set of the k) has been eliminated. But
that is precisely the weight that the second stage of a two-stage method
would assign them.

In an IRV-until-k method, the remaining k winners' weights matter,
that's true. But just as important is who those k are, because if
you're not one of the k, you don't get any weight at all. That's how the
cloning benefits party X: by pushing off people that aren't members in
favor of those who are.

Or as I said in the cloning example of my last post: "the first stage
weakens the second - the Z-voters' first preferences are no longer
counted, instead only their third preferences are".

This is IRV-until-k, right? I suppose that the "pre-declared list" is
that voter's ballot. Or do you mean something else? If it is something
else, how does the voter signal that he wants his vote to go down a
separate list instead of following the list given by his ballot?

An organized party can circumvent that restriction, as mentioned before.
It doesn't really matter because the party's intention will be very
clear when it engages in vote allocation, but the restriction can't
protect against a party who wishes to render it irrelevant.

It doesn't seem to invalidate the proof sketch.

For default votes: either they are the ballots themselves, in which case
there's no problem, or they're optional different lists, in which case
voters for X could just not use them.

For extra Asset reallocations: they don't come into play when cloning is
used because no single member amasses enough weight to go past the
threshold.

I think my cloning example in that mail is relevant. I showed an
instance where APR would first choose {X, Y, Z} as the winners to
distribute weight among. Then X clones and the outcome switches to {X1,
X2, Y}: Z is pushed off. Since the method moved from thinking Z should
be included into thinking Z should not, it was mistaken in at least one
of these cases. Which is it? Should candidates with less than a Droop
quota (Z in this case) always be retained, or should they be excluded
beforehand so that cloning has no effect?

More generally, I'd like to know how you define proportionality. Do you
have a method-independent criterion as to what it means for an outcome
to be proportional, and if so, what is it? Such a definition could help
answer the question above, and it would also explain how you would
conclude that my method sketches of an even earlier post were
necessarily less proportional than APR's IRV.

(6) APR: Steve's 6th dialogue with
Kristofer & Others

From: election-methods-request@lists.electorama.com

election-methods@electorama.com

elections

[….]

weighted

methods that

least when the

possible at all if

K: Yes. I mean the very opposite.

K:> Firstly, no method is entirely invulnerable to strategic voting; that

S:
Given these and your later words, it seems that either you are not understand
exactly how my APR is counted or I am not understanding the terms you are using.  Thus, in order for me fruitfully to address
your argument, please explicitly define, explain or comment on the following words,
phrases, and dialogues copied from all that is repeated after the +++++++++++
line:

the exact differences you have in mind
between IRV, STV, and APR;

vote allocation (is this simply
ranking?);

Droop quota (Strictly speaking, APR has
no use for the Droop quota.  APR may
elect some candidates who have received fewer votes than this quota.  This will occur as a result of some candidates
being elected with more votes than this quota and therefore receiving more
‘weighted votes’ in the assembly.);

spread the votes evenly (Does this mean
giving all the party voters’ 1st preferences to the most favored
candidate while they give the same number to 2nd or lower preference
clones?  In any case, I do not yet see
how this strategy would give this party more ‘weighted votes’ in the assembly,
or could ‘push off’ any opposing candidates.);

k of n candidates (k is the total
number the party’s most favored candidates while n is the total number of
candidates the party is running.  However,
I see that k seems to have a different meaning nearer the end of your post.);

n-k candidates (means the total number
of candidates less favored by the party.
If so, I do not understand how ‘IRV eliminates more than n-k in one go);

What ‘threshold’?  I do not see APR as having any ‘threshhold’.  APR simply continues to eliminate the
candidate that currently has the fewest votes until only the pre-established
number of candidates remain to be elected to each electoral association.

‘the first stage weakens the second’ (I
think you are referring more simply to APR’s 4-stage count explained in Endnote
4.)

Also,
please clarify the following questions.
These question have also been copied from also remains below the
++++++++++ line:

S:
I do not yet understand why you think an analysis of Single Non-Transferable
Voting (SNTV) is relevant to STV or APR.

[….]

clearer that strategic voting would be

S:  Again, I appreciate that your ‘proof sketch’
(especially if I fully understood it) may have shown how a strategizing party
will lose nothing in trying (except time, the money for some of the deposits
for some of its eliminated candidates, perhaps confusing its voters, etc.).  What I do not yet understand is how, in
practice, they will acquire the for knowledge needed to get any more benefits
than this.

K: > For default votes: either they are the ballots themselves, in which
case

S:  No. ‘Default votes’ are passed onto winning
candidates by the 1st choice candidate of the voter who has not
ranked any winning candidates.

K: > For extra Asset reallocations: they don't come into play when cloning
is

S:  By saying this, you seem to have forgotten
that APR does not use thresholds.  Instead,
APR requires very popular elected candidate to retain only up to 10% of all the
‘weighted votes’ in the assembly.  Any
votes received above this 10% limit must be non-returnably passed on by the
relevant MP to other MPs she trusts.

[….]

K:

S:  I do not yet understand these claims.  In APR, only the very popular elected
candidate (i.e. any with more than 10% of the weighted votes in the assembly)
would be allowed to ‘distribute weight’ to their less popular colleagues.  Also, it would help me if you could express
the same argument without relying on abstract mathematical symbols like X, X1,
X2.  In any case, do these 3 symbols
represent 3 different parties; 3 different candidates; or one most preferred
candidate, a 1st choice clone candidate, and a 2nd choice clone candidate?

K: > More generally, I'd like to know how you define proportionality. Do you

S:  For me, complete proportional representation
would be mathematically achieved when every citizen vote counts for one in the
assembly (no votes wasted), the percentage of each different MP’s vote in the
assembly is equal to the percentage of voting citizens who had voted for that
MP.  Qualitatively, representation would
be as complete as possible if the electoral system allows each citizen to rank
any of all the candidates which she sees as being able to represent a scale of
value similar to her own.  I see the
associational element of APR as structurally maximizing the chances that each
citizen will be able to see the largest number of such candidates that are
available in her society, i.e. APR seems to maximize the chances that each
citizen’s vote will be added to the weighted vote of the person in the society
who has become an MP and who most accurately and reliably will represent her
hopes and fears in the assembly.

I look forward to your clarifications so I can usefully address your whole
argument.

Steve
Bosworth

++++++++++++++++++++++++++++++++++++++++

K: > >> As for the cloning attack, I specifically found it while
analyzing APR's

See below.

mean by ?party-list equilibrium? and how it

the facts

citizens to rank

country.

clones

for example, one among the >>> > 435 most popular candidates in the
USA. Each APR elected candidate receives a weighted vote in the assembly equal
to the number of votes that each

voting

am I correct in understanding that all

election

country?  This

Your Taiwan example does explain how

worry

K: > Suppose I'm leader of a party X and I have absolutely no idea of how

S:  I do not understand this paragraph.

S:  Given my incomplete understanding, I still
can imagine that the above strategy might fail to elect any candidate other
than the ones it would have elected without strategizing.  However, I still do not understand how anyone
could acquire sufficient for knowledge to be confident enough to go to the
bother (time, expense, danger of confusing your voters, etc.) of try to
strategize.

K: > More precisely: the party never loses by cloning if they can make the

S:  I do not yet see how the relevant non-party
candidates might be ‘pushed-off’ in this way.

K:>
…. the method favors groups that can pull off an

S:  What ‘threshold’?  I do not see APR as having any
‘threshhold’.  It simply continues to
eliminate the candidate that currently has the fewest votes until only the
pre-established number of candidates remain to be elected to each electoral
association.

method [?.]

they're

S:
Here, k has a different meaning than given to it above.

K: > Once only k candidates remain in IRV, they have a weight according to

S:
I do not know if this is important for your explanation but some of the
candidates would also have some lower than ‘first place preferences’ before
they are eliminated.  Also, some winning
candidates would have been elected with the help of some lower than 1st
preference votes.

K:

that the second stage of a two-stage method

S:  Either you are not understanding APR’s method
of counting or at least I am not understanding you at this point.

These are explained in the following

two

citizens

of the candidates ranked by a citizen have

candidate

'default' votes must be sequentially

each

list" is

be

different

counts

citizens'

above.

the

all

the

all

countrywide

directly or indirectly

for

each of these representatives would have

votes

S:  Again, I appreciate that your ‘proof sketch’
may have shown how a strategizing party will at least lose nothing in trying
(except time, the money for some of the deposits for some of its eliminated
candidates, etc.).

K: > For default votes: either they are the ballots themselves, in which
case

S:  No, ‘default votes’ are passed onto winning
candidates by the 1st choice candidate of the voter who has not
ranked any winning candidates.

K: > For extra Asset reallocations: they don't come into play when cloning
is

S:  By saying this, you seem to have forgotten
that APR does not use thresholds.  It
requires very popular elected candidates to retain up to 10% of all the
‘weighted votes’ in the assembly.  Any
votes received above this 10% limit must be non-returnably passed on to other
trusted MPs.

4^th

K: > More generally, I'd like to know how you define proportionality. Do you

S:  For me, complete proportional representation
would be mathematically achieved when every citizen vote counts for one in the
assembly (no votes wasted), the percentage of each different MP’s vote in the
assembly is equal to the percentage of voting citizens who had voted for that
MP.  Qualitatively, representation would
be as complete as possible if the electoral system allows each citizen to rank
any of all the candidates which she sees as being able to represent a scale of
value similar to her own.  I see the
associational element of APR as structurally maximizing the chances that each
citizen will be able to see the largest number of such candidates that are
available in her society, i.e. APR seems to maximize the chances that each
citizen’s vote will be added to the weighted vote of the person in the society
who has become an MP and who most accurately and reliably will represent her
hopes and fears in the assembly.

This reply might be rough, and definitely is long in the coming, due to
that I had to deal with a lot of real world matters while working on the
post. I hope it is not too rough, though.

On 10/22/2015 01:01 AM, steve bosworth wrote:

To try to be simple:

IRV is a method where you repeatedly eliminate Plurality losers until
only one candidate remains. IRV-until-k repeatedly eliminates Plurality
losers until only k candidates remain.

STV is similar to IRV, except that it also distributes surplus votes. If
a candidate has more than a Droop quota's worth of first preferences, he
is automatically elected and the surplus (the number of voters he had
past a Droop quota) is distributed to the next preferences of the voters
who contributed to his election.

The part of APR I'm focusing on is the "eliminate until k remain and
then give them the weights according to the number of first place votes
they got" part. That is equivalent to IRV-until-k followed with giving
each candidate weight according to the number of first place votes they
get on a ballot set where everybody except those k have been eliminated.

Vote allocation is the SNTV strategy I have been mentioning. It may more
generally be called vote management: i.e. the coordination, by a party,
of how its supporters vote, for the purpose of gaining some benefit. See
https://en.wikipedia.org/wiki/Vote_allocation.

My whole point is that a party that has more than k Droop quotas' worth
of support can get k of its candidates elected by cloning. If every
party does this, then candidates with less than a Droop quota's worth of
support are pushed off because there isn't any room left.

A Droop quota in voter terms is v/(s+1) votes, where v is the number of
voters and s is the number of seats. In fractional terms, it's 1/(s+1),
e.g. 1/2 = 50% for a single-winner election, 1/3 = 33% for a two-winner
election and so on.

Droop quotas might be relevant even though the method does not
explicitly mention them. Consider single-winner Plurality phrased like
"count the votes, and whoever has the most votes wins". Then the
equivalent Droop quota (1/2 = 50%, a majority) is important in that any
candidate who gets more than this is automatically elected. Yet the
definition of Plurality does not explicitly mention majorities, let
alone Droop quotas.

For clarification, when I use single letters like X, Y, Z, I usually
mean candidates. When I use letters followed by numbers (e.g. X1, X2), I
usually mean clones of the same candidate (X). You can also consider
that to be clones provided by the same party, in which case X would be
the party.

Now, suppose the party is P and it fields four candidates: P1, P2, P3,
and P4.
Suppose it has the support of 100 voters. Then "evenly spreading" would
instruct the voters to vote in a way so that 25% of the voters vote for
any given clone in the first place, 25% of the voters vote for any given
clone in second, 25% of the voters vote for any given clone in third,
and 25% vote for any given clone in fourth. A simple way of doing this
is as follows:

25: P1 > P2 > P3 > P4
25: P2 > P3 > P4 > P1
25: P3 > P4 > P1 > P2
25: P4 > P1 > P2 > P3

I.e. they vote around a circle with each rotation in equal numbers.

By using dice or cards, it is possible to perform an even more
sophisticated allocation[1] so that after say, P1 is eliminated, 33% of
the ballots vote for the one of the three remaining clones in first, and
the same for second and third place. However, I don't want to confuse
the issue.

k usually is a number corresponding to a part of the whole, and n is a
number corresponding to the whole. For instance, the binomial choose
operator is usually written as "n choose k", which gives how many ways
one can pick k items from a group of n.

In the proof sketch, k is the number of candidates from a party who won,
and n is the number of candidates that party fielded.

In the term "IRV-until-k", k is the number of winners and n is the
number of candidates in total.

Correspondingly, n-k is the number of candidates who did not win, either
from the party (in the proof sketch) or generally.

I mean the Asset voting threshold. Say that any candidate who achieves
more than 10% of the vote will have to distribute his weight according
to a predeclared order. Then that 10% is the threshold. (It may be 20%,
15%, the argument still stands.)

No, I mean the two-stage reformulation of the weighted voting process.
The first stage determines who the winners are. The second stage
determines their weights. "The first stage weakens the second" is just
the observation that a candidate that has been pushed off by IRV can't
ever get a nonzero weight, so however fair the idea of giving weights to
winners is, it doesn't help if you get pushed off the winner list entirely.

That is because both SNTV and IRV-until-k is an adaptation of a
single-winner method to a multiwinner system; and because IRV is close
enough to Plurality that some of the weaknesses in the latter makes it
possible to do an attack in the former.

In some ways, performing vote management in IRV-until-k is less risky
than doing it in SNTV. If a party overestimates its support in SNTV, it
can risk losing all of its seats; however, in IRV-until-k, the excess
clones will simply be eliminated and then the count proceeds as if the
party didn't field those clones to begin with.

What I said above holds the key. The idea is that if a party evenly
spreads its votes, then it doesn't matter if they field too many clones.
Those clones will then simply be eliminated and then the count proceeds
as if they didn't field too many clones.

The voters who execute an even spreading strategy intend mainly to
distribute their own voters irrespective of what the other voters are
doing. That's why they don't need much information about how the other
voters are voting: all they intend to do is to push off other candidates
by making more than one candidate get elected at a lower weight where
only one used to be elected at a higher weight. (See below.)

The whole appeal of IRV, as I understand it, is that it reduces the
problem of electing a winner from a pool of 10 candidates to the problem
of electing a winner out of 9 candidates, and so on down to the problem
of electing a winner out of 2 candidates, at which point you can just
use majority rule. For IRV-until-k, it reduces the problem of electing
say, 3 winners out of 10 candidates to the problem of electing 3 winners
out of 9, then 3 out of 8, then ... down until 3 out of 3.

However, this very same elimination mechanic makes it risk-free to field
lots of clones, because the very worst thing that can happen is that
some clones get eliminated, and then you're reduced to a similar
situation as if you hadn't fielded those clones to begin with.

So we've established the risk-free nature of cloning (apart from time,
deposits, etc). How can a party benefit? Suppose that there are two
situations. Let's disregard what I've called the threshold for now: I
want to be as simple as possible so that the explanation won't be
confusing. That is, let's disregard the clause that says that candidates
who get too much weight must redistribute it.

So, there are two situations. In the first, the party fields one
candidate and he gets 20% of the weight, is elected, and represents that
party with 20% weight.

In the second, the party fields two candidates and (because of even
spreading), each candidate gets at least 10% of the weight and they're
both elected. If the attack fails, the remaining candidate gets 20% so
there's no risk. Assume it succeeds.

Now, cloning seems to have given the party no benefit. However, if there
are only 10 people in the assembly, then what happens is also that one
of the other candidates were pushed off to make room for the party's
second candidate.

We know that the sum of the two winners' weights will still be at least
20% because the party has 20% support. But if there were some voters who
first voted for the candidate who was pushed off, then for the party
that cloned, then those voters' votes will go to the party that cloned
since the candidate who was pushed off is no longer around to receive
any weight. So by cloning and pushing off, the party may receive more
than 20% support in total.

It doesn't need to know how others vote to do so. If it misjudged, then
one of its clones is eliminated and it gets another shot with the
remaining clones. So the whole point of even spreading is that the party
maximizes the chances of any given clone winning, to the extent it can
among its own supporters. Any further help provided by other voters is
just a bonus.

Why is even spreading the best choice? Well, suppose a party P has two
clones: P1 and P2. If the supporters of P vote only for P1 first, then
P2 will be eliminated right away. If the supporters of P vote only for
P2 first, then P1 will be eliminated right away. The more the voters
shift votes from P1 to P2, the better a chance P2 has to survive...
until its survival comes at the cost of P1. And that breakeven point
happens when P1 has the same number of first preferences, among P's
supporters, as P2 has.

In short, the party clones to maximize the number of different people
they can get into the assembly. They like getting lots of people with
low weight into the assembly more than getting a few people with high
weight in. Why? Because getting lots of their own reps in displaces reps
not in the party from the assembly, and the subsidiary votes of voters
who vote for the reps who were pushed off may then go to the party instead.

E.g. if the Republican party can give all of its candidates greater
support than the greatest support of independent libertarian candidates,
and all libertarians vote Republican before they vote Democratic, then
the Republican party can displace the libertarian candidates and add the
libertarian voters' weight to their own.

I see. That shouldn't be a problem, since if the party pushes off
representatives not aligned with them, the voter can only choose between
winners, i.e. can't make any of the reps who were pushed off get back on.

Here I use threshold in the general sense: a barrier above which the
rules change[2]. In this particular case, the change is that the
candidate must pass on the excess.

Alright, I'll try to rephrase it in the Republican-Libertarian example
above. It will be grossly simplified, of course: in reality there would
be more seats and more factions, but still.

52: Repub1 > Democrat1 > Libertarian > Moderate
25: Democrat1 > Moderate > Libertarian > Repub1
10: Libertarian > Moderate > Repub1 > Democrat1
5: Moderate > Democrat1 > Repub1 > Libertarian

These are all candidates. The Republican party has one candidate:
Repub1. Similarly, the Democratic party has one candidate: Democrat1.
The Libertarian and Moderate candidates are either the sole candidates
fielded by respectively Libertarian and some Moderate party, or
independents that hold respectively libertarian and moderate views.

In this pre-cloning election, the moderate is eliminated and one from
each group is elected: a Republican, a Democrat, and a Libertarian. The
Democratic winner also gets the weights from the moderate voters.

But now the Republicans clone. All the unaffiliated voters vote the
clones in the same order (perhaps Republican 1 is the more genuine one):

26: Repub1 > Repub2 > Democrat1 > Libertarian > Moderate
26: Repub2 > Repub1 > Democrat1 > Libertarian > Moderate
25: Democrat1 > Moderate > Libertarian > Repub1 > Repub2
10: Libertarian > Moderate > Repub1 > Repub2 > Democrat1
5: Moderate > Democrat1 > Repub1 > Repub2 > Libertarian

As before, the moderate is pushed off. But now the second Republican
displaces the Libertarian and the winners are

Republican #1, Republican #2, Democrat #1.

By cloning, the Republicans in effect acquired the libertarian voters'
weight. The libertarians find their weight going to their third choice
instead of their first simply because the cloning worked. Note again
that the Republican party did not have to know that everybody would
judge Republican 1 as more genuine than #2: even spreading worked just fine.

Does that help?

My question is, since the method changed its mind from electing a
Republican and Libertarian (and Democrat) to electing two Republicans
(and a Democrat) through no change of the voters' actual preferences,
when did it make the right call? Did it when it elected the Libertarian,
or when it elected two Republicans? It can't be both.

There is a third alternative: that the method should give less popular
candidates seats whenever there's room to do so, but that if there isn't
any room, then there isn't any, and that's no fault of the method
itself. But that alternative isn't unproblematic, either.

I don't know if the voting numbers are realistic, but again, I'm only
out to show why cloning gives a benefit when it does. You could probably
construct voter profiles, something like:

• The 52 Republican voters are mainstream Republicans and thus don't
  want anything to do with the libertarians,

• The 25 Democrats are cosmopolitan moderates who abhor the Religious
  Right and would rather have a libertarian if it came to that,

• The 10 libertarians don't like the mainstream parties much at all,

• And the 5 moderates find the Democratic candidate more appealing than
  the Republicans.

But that's not in itself all that relevant for the situation.

I've moved my reply to this to the vote-wasting thread since that seems
to be a more appropriate place for it.

[1] Obviously, such an approach would be a lot more suitable to mail
ballots or internet voting than physical ones, but I don't want to give
the impression that voters need to make use of clever randomization
procedures for the cloning procedure to work at all. Since cloning never
hurts, the closer the voters can get to a complete even spread, the
better, and simple rotation (as I have detailed it) will work a lot of
the time.

[2] E.g. CED definition 5: "a level or point at which something would
happen, would cease to happen, or would take effect, become true, etc".