[EM] SNTVlike reduction for weighted majoritarian voting
Kristofer Munsterhjelm
km_elmet at tonline.de
Wed Jun 24 06:52:01 PDT 2015
While writing my reply to Steve Bosworth, I think I found a problem with
some types of weighted voting.
Let's consider a very general setup for weighted voting, consisting of
two stages. In the first stage, one uses some kind of election method to
reduce the field of C candidates into one of W winners, W <= C. In the
second stage, one assigns voters to the winners to give the winners
different weight (proportional to the number of voters assigned to them).
When we split the weighted voting process in two like this, it's
possible to use a whole bunch of different election methods for the
first stage. The APR proposal uses IRV, for instance. Similarly, it's
possible to use a bunch of different assignment methods for the second
stage: the simplest and most obvious one is to assign each voter to the
winner he voted first (or gave highest rating, if it's a rated method).
But if the first stage is to use a majoritarian method (i.e. a method
ordinarily used to find a single winner or social ordering), and that
method is cloneproof, then we might get a strategy that (unless I'm
wrong) severely compromises the weighted aspect.
If there are W winners and W is fixed, then any majority party has an
incentive to field a certain number of clones to crowd out the others.
For instance, say the initial election is something like:
52: A > B > C
25: B > C > A

10: C > B > A
with two winners. A and B win, so A gets a weight of 52/87 and B gets a
weight of 35/87.
Then A has an incentive to clone itself into A1 and A2 and tell its
supporters to split, voting for these evenly:
26: A1 > A2 > B > C
26: A2 > A1 > B > C

25: B > C > A2 > A1
10: C > B > A2 > A1
Now B has been pushed off, and A has gone from controlling 60% to
controlling 100%. In a council with a majority vote, that makes no
difference since 60% is enough, but hopefully it should be clear how the
strategy can be generalized.
Each party faces pressure in two directions. If there was a threeseat
election like
52: A > B > C > D
25: B > C > D > A
10: C > B > A > D
5: D > A > B > C
Then B would face pressure from C and D which it could alleviate by
cloning; but doing so would permit A to clone even more widely in turn
and perhaps push many of B's clones off. If I am correct, the
equilibrium turns out to be an SNTVlike strategy, which in essence
reduces the system to party list proportional representation.
An intuitive way of considering this is: if a party can split itself
into k parts without any part having less support than the winner with
the least support, then it is to that party's benefit to do so; doing so
will push someone who's not of that party out of the winning group. A
party's weight can be used no matter whether it's distributed or
concentrated as long as the party leadership controls all the
candidates. Furthermore, this could be used to avoid upper thresholds
like the Asset limit in APR: a party simply fields enough candidates
that no single candidate exceeds the upper threshold.
So, again unless I'm wrong, majoritarian systems do not seem to be
suitable as the first stage of a weighted voting process, because they
would give unfair benefit to organized groups of candidates who could
coordinate and clone as shown above.

This problem doesn't occur in party list PR itself because the number of
parties is not fixed beforehand. So if you start with (Plurality for the
sake of simplicity)
52: A
25: B
10: C
with 10 seats using Webster/SainteLague, the share is
A: 6, B: 3, C: 1
If A tries to clone:
26: A1
26: A2
25: B
10: C
A1: 3, A2: 3, B: 3, C: 1
to no benefit.
However, trying to run party list PR with majoritarian systems like
Condorcet doesn't work very well, at least not with the allocation
mechanism given above[1]. Since the majoritarian systems can put
candidates with no FPP votes first in their social ranking, you end up
with the weird result that the candidate that the election method
unambiguously considers the best corresponds to a party that, according
to the allocation method, has no voters associated with it and should
thus have no seats.
I think I have found some Condorcetlike methods that handle weighted
voting and party list PR, and I may write about them later. The
exhaustive versions of the methods have exponential runtime in the
number of parties, but (thankfully) not in the number of seats.

[1] I suspect that the reason Plurality works for party list is closely
related to the reason that it's not a good singlewinner method. A
singlewinner method gives a social ordering that places the candidate
that would be best when you *have* to pick only one (according to its
logic), then the one that would be second best, and so on. That's why
cloning the winner (if the method is cloneproof) has all the clones of
the original winner appearing after the winner: if the original winner
would not be able to stand, the clone is the next best. But for party
list PR, you'd want a group representing many opinions, not just copies
of the best compromise. If that reasoning is right, then a proportional
ranking method would fit much better, and that's sort of what one of the
Condorcetlike methods become.
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