[EM] (4) APR: Steve's 4th dialogue with Kristofer

steve bosworth stevebosworth at hotmail.com
Fri Jul 17 12:54:28 PDT 2015

Re:  (4) APR: Steve's 4th dialogue with

> From:
election-methods-request at lists.electorama.com
> Subject:
Election-Methods Digest, Vol 133, Issue 2
> To:
election-methods at lists.electorama.com
> Date: Thu, 2 Jul 2015
12:01:24 -0700

> 1.
Thresholded weighted multiwinner elections
> (Kristofer

Date: Wed, 01 Jul 2015
22:14:11 +0200
> From: Kristofer Munsterhjelm
<km_elmet at t-online.de>
> To: Election Methods Mailing
List <election-methods at electorama.com>
> Subject: [EM]
Thresholded weighted multiwinner elections
> Message-ID:
<55944A13.7060800 at t-online.de>
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questions will follow each element of what Kristofer wrote:

>K:  I think I
see why the cloning attack is possible in two-stage weighted 
voting. If I'm right, then it is possible to make voting methods that

> produce results that fit weighted voting better -- at least
when the 
> voters are honest. However, I'm not sure if it is
possible at all if 
> enough voters are strategic.

>>S: Am I
mistaken in believing that, in practice, APR's 'weighted multiwinner
elections' would not be vulnerable to the threats either of effective
'cloning' or effective 'strategic voting'?  This practical
invulnerability would seem to arise from the facts that APR's
election of reps to a large national assembly would allow all citizen
to rank as few or as many of all the thousands of candidates in the
country.  Accordingly, for example, the portion of all the perceived
clones would be elected who were discover to be, for example, among
the 435 most popular candidates in the USA.  Each elected candidate
would simply have a weighted vote in the assembly equal to the number
of votes that each had received directly or indirectly from citizens.
 At the same time, how could any citizen or group of citizens be able
to have enough reliable knowledge about how enough other citizens
will rank candidates in order to be confident enough of having a
'strategy' that would have the effect of producing anything other
than an honest result?  In any case, what rational motive would any
citizen or group of citizens have in ranking other than their favored
candidates when their honest voting would instead guarantee that each
of their votes will only strengthen the elected reps they favor more
than the other reps? 

>K:  It might turn
out that the only way of 
> making weighted voting work is
through either varying the number of 
> winners (like in party
list) [...]

>>S:  Is not this
what APR does, or have I misunderstood you here?.

>K: [...] or by an
unconventional (nondeterministic) voting system or the Asset version
of this.

>>S:  Is not this
what APR does, or have I misunderstood you here?> 

>K:  First, why the
cloning attack is possible: when we use (call them semi-
majoritarian[1]) methods like IRV or Plurality,
if candidates represent  parties who can clone as many as they
want, then I think the strategic equilibrium gives each party a
number of seats equal to their number of Droop quotas. If we instead
use an unweighted multiwinner method (STV, 
> Schulze STV,
etc), then the Droop proportionality criterion gives each 
party at least their Droop quotas' worth in seats without strategy.

> Simply, when the number of winners is fixed, the
> methods have an implicit threshold of a
Droop quota, and the unweighted 
> multiwinner methods have an
explicit threshold of the same.

 I understand that, above and below, you are exploring the logic of
these theoretical possibilities but why would we actually want to use
such needlessly less proportional and much more complicated methods
than those offered by APR for electing the legislative assembly of a
large nation?  

me if I am mistaken, but perhaps your next post (also copied below)
provides part of your answer to this question.  It seems to worry
that APR might elect a so-called 'unpalatable set' of reps for the
assembly.  If this is part of your answer, do you still accept that
APR would always have the best chance of electing an assembly that
would be entirely palatable to its voting citizens.  This is because
each citizen could  rationally see that she has a rep in the assembly
who most likely will both  qualitatively and mathematically
(proportionately) represent her views?

your definition of 'unpalatable' something other than simply 'what
you happen subjectively to dislike?

more simplicity and clarity for our next dialogue, I think it would
be best for me to delay commenting on the relevant details of the
following parts of  your 2 posts only after you have had a chance to
answer these questions.

look forward to your replies.


>K:  In that light, the cloning strategy in three-seat
26: A1 > A2 > B > C > D
> 26: A2 > A1 > C >
B > D
> 25: B > C > A1 > A2 > D
> 10: C >
B > A2 > A1 > D
> 5: D > A1 > A2 > B > C
works because the A-group has more than two Droop quotas (a Droop
> being 23 here).
> Perhaps we'd want to
have no threshold at all, something like minimax 
> Approval
where we want to find the outcome that satisfies the voter who 
likes it the least the most. But it'd seem unreasonable to elect {A1,
> C} in this extreme variant of the above:
> 1000: A1
> A2 > B > C > D
> 1000: A2 > A1 > C > B >
> 500: B > C > A1 > A2 > D
> 2: C > B >
A2 > A1 > D
> 1: D > A1 > A2 > B > C

> and hence, that implies there should be some kind of
threshold, but that 
> it should be lower than a Droop
quota[2]. A lower threshold would mean 
> that the assembly
would be more broad than deep: it would choose a 
> compromise
among popular candidates to make room for more specialized 
> For instance, in a two-seat election
with an augmented LCR example like
> 50: L > C > R >
> 40: R > C > L > S
> 10: C > R > L >
> 20: S
> the method could elect {LR} with a
high threshold, but instead elect 
> {CS} if the threshold were
lower. In return, the candidate C would have 
> a much greater
weight in the latter case than either L or R would have 
> in
the former.
> The next thing to do would be to find a
proof of concept method that 
> would have a tunable threshold
just to show that it'x possible (at least 
> under honesty).
Here's one that reduces to Bucklin (if one picks a 50% 
> ==
> Let X be a set of
winners and t a threshold (in number of voters). Then 
evaluate(X, t) is a function that works on the ballots and returns a

> list of numbers in sorted order from least to most.
evaluate(Y, t) is 
> "better" than evaluate(X, t) if
the first number where the output from 
> evaluate(Y, t)
differs from evaluate(X, t) is one where evaluate(Y, t)'s 
number is greater than evaluate(X, t)'s. The best set is the one that
> better than every other, and that set wins.
Evaluate itself just counts the ranks of the ballots (of the
> in the set X), where first place is n, second
place is n-1, all the way 
> to nth place is 1 (and unranked is
0). It then sorts these and removes 
> the t worst.
> Here's the two-seat LCR example above with a
Droop quota (40) and with a 
> threshold of 10. {LR} wins with
a Droop quota and {CS} with the 
> threshold of 10. To skip,
just search for #. I have omitted other sets 
> like CL, CR,
LS, etc.
> evaluate({LR}, 40):
> rank number: 4
3 2 1
> 50: *L C R S
> 40: *R C L S
> 10: C *R L
> 20: S
> So we have 50 4s, 40 4s, 10 3s and
20 0s, or
> 20 0s, 10 3s, 90 4s
Threshold of 40 removes the 40 least, so the output is
> 80
> evaluate({CS}, 40):
> rank number: 4 3 2
> 50: L *C R S
> 40: R *C L S
> 10: *C R L S
20: *S
> So we have 50 3s, 40 3s, 10 4s, 20 4s, or

> 90 3s, 30 4s
> Threshold of 40 removes the 40
least, giving 50 3s and
> 30 4s.
> So here {LR}
with 80 4s wins over {CS} with 50 3s and 30 4s.
> With
a threshold of 10:
> evaluate({LR}, 10):
> As
above, before truncating, we have 20 0s, 10 3s, 90 4s
But now we can only remove 10 worst. So the final
> output is
10 0s, 10 3s, 90 4s
> evaluate({CS}, 10):
> 90
3s, 30 4s.
> Removing 10 worst gives
> 80 3s, 30
> So now {CS} with 80 3s and 30 4s wins over {LR}
with 10 0s
> first, because 3 is clearly greater than 0.

> #
> Hence it's possible to make something
that has a tunable threshold under 
> honesty.
But here's a problem. In the single-seat case, this behaves like
> it may violate the majority criterion to elect a
candidate that has more 
> second-place votes. That is, in
something like
> 55: A>B>C
> 35: B>C>A
10: C>B>A
> Borda will elect B, and with a threshold of
say, 10, so will the method 
> above:
> evaluate({B}, 10)
gives (truncated) 55 2s, 35 3s
> evaluate({A}, 10) gives
(truncated) 35 1s, 55 3s
> evaluate({C}, 10) gives (truncated)
45 1s, 35 2s, 10 3s
> All well and good. But for Borda, a
strategic majority can force a 
> winner by acting like an
unreasonable group or an aggregate of different 
> smaller
groups so that the method considers the majority winner to be 
the broad-support candidate. The problem with this is that if that's
> general property, then it might turn out that a Droop
quota can force 
> the election of a particular candidate in a
weighted voting method with 
> a lower threshold just by
pretending to be very unreasonable or to be 
> many different
smaller groups.
> It certainly is possible for the
proof of concept method. For instance, 
> the L-first voters
can truncate after L, after which the outcome changes 
> from
{CS} to {LS}. Part of this, however, is due to that the method 
above passes LNHelp but not LNHarm (because it's Bucklinesque and 
because voters are assigned completely to their first preferences).
> the strategy is particular to certain weighted
voting/tunable threshold 
> methods, then there's no problem.
But if it's general (and it might 
> intuitively be), that's
another matter.
> If it is general, I can think of
three ways to keep weighted voting:
> 1. Let the number of
seats/winners be adjustable.
> 2. Accept the Droop quota.
3. Use a (randomized) consensus method.
> Some
combination of 1 and 3 might also be possible.
> The
first option is to let the number of winners be adjustable. The 
general idea would be to have the same threshold and if a party
> itself, it does get another winner - but the number
of winners also 
> increases so that there's no benefit. Or,
conversely (if one accepts the 
> Droop quota), a candidate who
gets more than a Droop quota's worth 
> removes as many seats
as he has Droop quotas. E.g. in the clone example 
> above,
without cloning, the outcome would then be {A, B}, and with 
cloning, it would be {A1, A2, B}. Again, there's no benefit. Both of

> these make weighted voting more like party list PR.

> The second option is to just accept the Droop quota and use
> multiwinner method as basis. The cloning is still
possible, but at least 
> one gets more varied winners without
cloning and there's no need to 
> engage in strategic
nomination. With IRV, a party could split itself too 
> thin by
overestimating its support, but that won't happen with say, STV. 
A party just has to field enough candidates (say enough to fill the

> council) and doesn't have to worry about fielding too
> The third option is to use consensus. If it's
an Asset variant, we could 
> just "stick the candidates
in a room and have them negotiate until they 
> agree"
with a supermajority, but there's an incentive for the status quo 
to block the consensus. That can be fixed with randomized consensus -

> some kind of variant of the methods Jobst and Forest Simmons
> proposed[3]. Randomized consensus can be done either
before the election 
> (as a general method) or after (as a
variant of Asset), but the former 
> is much less likely to
> Basically, these methods consist of every
voter (candidates in case of 
> Asset) voting for a consensus
outcome as well as for a favorite outcome. 
> If fewer voters
than the threshold disagree about the consensus outcome, 
then it is picked, otherwise a random favorite outcome is picked. As

> random ballot is suboptimal but gives nobody an advantage,
so everybody 
> would want to find a consensus outcome to the
extent they can work 
> together to do so. The equivalent of a
cloning attack would be for a 
> majority candidate to stick to
his guns and only propose a council 
> consisting of his own
party. But if he tries to block the consensus, 
> then it falls
through to random ballot which favors nobody and only 
degrades the quality of the solution.
> Generalizing
this to multiwinner might be trickier because we want 
something that with high probability returns a low-threshold
> and that is strategically unbiased. The consensus
ballot can simply be a 
> set of winners. If fewer voters than
the threshold disagree about the 
> consensus, then it is
picked. But the fallback is hard. If we just do 
> "pick a
winner from voters' first preferences at random, eliminate, 
repeat", then that favors Droop quota cloning because only one
of the 
> clones get eliminated at once. "Pick a proposed
winner set at random" 
> favors extreme outcomes. It would
be fair, but have very high variance: 
> one could end up with
a council controlled completely by a single party, 
> it's just
which party would control it that would be random in a 
low-threshold unbiased manner.
> ---
[1] These are methods that never elect candidates with zero first 
preference votes and where giving someone more first preferences
> helps. I think the observation above holds for all of
these; it does at 
> least seem to hold for Plurality and
> [2] We could also argue that there should be a
threshold less than a 
> Droop quota by saying that if the
threshold is a Droop quota, then 
> (under strategy) the
outcome for weighted voting and unweighted voting 
> is the
same, so why bother with weights? Thus the point of having a 
weight should be to make weighted voting be more representative with

> fewer seats than ordinary unweighted voting can be. If, on
the other 
> hand, a Droop quota is good enough (and we just
want weighting to handle 
> excess beyond the Droop quota),
then we shouldn't use semi-majoritarian 
> methods but instead
unweighted multiwinner methods to decide upon the 
> winners in
the first stage.
> [3] 



> Date: Mon, 13 Jul
2015 20:33:50 +0200
> From: km_elmet at t-online.de
> To:
ElectionMethods at VoteFair.org; election-methods at electorama.com
CC: stevebosworth at hotmail.com
> Subject: Re: [EM] 19) APR:
Steve's 19th dialogue with Richard Fobes
> On
07/12/2015 06:05 AM, Richard Fobes wrote:
> > On 6/30/2015
7:39 PM, steve bosworth wrote:
> > > ...
> > >
>>>S: I accept that your method might mathematically at most
> > > 'nearly full proportionality'. However, APR
offers the advantage of
> > > 'full proportionality'. Do
you dispute this?
> >
> > Yes I dispute this. As I
have said before (and I think someone else
> > made a
similar point), your APR method does not achieve full
> >
proportionality. Specifically, with APR, not every voter is
> > by hisher first choice.
> >
> No method can achieve 100% percent proportionality. If you want
to say
> > that APR gets as close as is easily possible,
then I'll agree with that
> > on the condition that you also
acknowledge that VoteFair ranking also
> > can (if desired)
achieve that same high level of proportionality.
There's one aspect of this I was going to address in a reply of my
> to Steve, but I have been busy. Still, I can at least
mention it here, 
> since it's relevant to what you're saying,
and then I'll try to get that 
> reply done at some point.

> Suppose all voters rank every
candidate, and suppose you pick an 
> unpalatable set of
candidates as winners. Then you can assign voters to 
> each of
those winners: each voter is assigned to the winner that he 
ranks first (of those in the winner set), like APR would do. (Call
> procedure "weighted assignment".) The
candidates will have weights 
> proportional to their support
among the winning set. But this hardly 
> seems like a good
outcome, since the winning set only consists of 
> unpalatable
> Concretely: If we have
40: A > B > C
> 60: D > E > F
> then
{AD} with 40% to A and 60% to D is better than {BE} with 40% to B 
and 60% to E, even though in both cases, every voter's vote "counts"
> the sense of influencing a winner's weight.
Thus simply assigning every voter to the winner he prefers the most
> not in itself provide a good result if the method that
picks the winners 
> to begin with is lacking. And IRV is not
exactly the best of methods :)
> > 
> One possible
Condorcet approach could be:
> Define the number of
voters that are penalized when moving from one set 
> of
winners (say {ABC}) to another (say {DEF}) as the number of voters 
who prefer someone in the first set to someone in the second set
> ranks one of the former above one of the latter).

> Say {ABC} beats {DEF} if fewer voters are penalized by going
from {DEF} 
> to {ABC} than by going from {ABC} to {DEF}.

> Let a penalty CW be the set that beats every other.

> Multiwinner IRV most likely does not pick penalty CWs.
Inasfar as 
> penalty CWs are good things, this is a mark
against using multiwinner 
> IRV for picking the winner set --
even though every vote contributes to 
> adjusting weights no
matter what winning set was picked, as long as 
> every voter
ranks every candidate. And if we pick a penalty CW using a 
Condorcet method, there's nothing stopping us from calculating
> as above using that penalty CW set.
For that matter, you could use VoteFair proportional ranking to find
> winning set, and then use weighted assignment as a second
stage if you 
> want weighted voting. Since weighted assignment
works for any set that 
> doesn't contain candidates nobody
ranks first among those in the set, 
> you can use the output
from IRV, VoteFair PR, STV, Schulze STV, or 
> whatnot
for the second stage. Some of these can be better than IRV-at 
large: for instance, if I'm right about thresholds and that
> IRV gives each party a number of seats equal to
its Droop quota support 
> in cloning equilibrium, then using
STV would directly give that kind of 
> proportionality whereas
multiwinner IRV would only do so when the 
> parties are all
strategizing. Of course, if Droop proportionality is 
undesirable, then using STV would be bad, but so would using IRV be.

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