[EM] Vote transfer problems - plurality rule

[Abel Stan]

Hello everyone,

here I am with another musing on an uncommon type of mixed system, which is
basically the mixed (two tier) equivalent of STV with group voting ticket
only: Voters vote for individual candidates in their local districts, and
their votes get transferred to their party if it is not "used up". There
are two main variants, one is with only the losing candidates votes getting
used as list votes (imagine STV with no surplus vote transfer, only
transfer from elimination), the other one is the plurality surplus version:
instead of using a quota, the few countries which used or use this system
define surplus votes as (votes of plurality candidate) - ((votes of second
candidate) + 1 ). Since there are usually not enough list seats to make
this kind of system even close to proportional, it's not hard to see that
the second version is probably quite bad. But it is also intuitive for
anyone familiar with STV that some sort of surplus vote system should be
used in general. I have grappled with this topic and have before
suggested a quota system, but even then proportionality cannot be achieved
unless the right number of seats is used and districts/turnout are equal
enough, and there are some problems arising from plurality. I might write
about this but it is generally very impractical with (seat linkage) MMP
being a thing. And as the response of Markus Schulze on my last email
shows, the proper and efficient solution for the problems of MMP (if the
number of seats is not fixed) is already found, but not widely known and
sadly, not implemented (which lead to failures of two vote MMP in multiple
countries).

Nevertheless, while I would say these vote transfer systems are better left
as a curiosity, I think the research of Daniel Bochsler (2015 paper on
"bending the rules")  and Johannes Raabe on this subject does raise a few
points in its favour. Even single vote MMP is not perfect, it can be
manipulated, but the same strategies don't work in the standard versions of
these vote transfer systems. No party wins by running winning candidates
and losing candidates on different lists, unless there is vote subtraction
involved (this was the case in Italy). There are two other tactics, which
are less likely to lead to actual abuse of the system, so maybe under some
specific conditions, vote transfer systems can be more robust against
manipulation than MMP, while being almost as proportional (or more, if it
depends on the manipulation) One is when there are two many list seats, but
this is usually very unlikely and in practice there are always too few. But
if there were too many, the value of transfer votes would be worth more
than votes for plurality winners. Imagine an STV system where the first
candidate got elected with a quota of 33% but the next candidates may all
start from a quota of 25%. The second tactic is when the surplus votes are
not taken into account at all. In this case the parties will kind of try to
get those surplus votes by running a clone list candidate in safe seats
(stronghold-split tactic). This tactic has been theorized, but not really
observed, but this problem is not there at all if the winning candidates
receive a surplus counted from the second place candidate anyway. So you
could say that the same way, the worst a two-vote MMP can do (and maybe
under extreme presumptions, a one vote MMP) is become parallel voting
(MMM), the worst a vote transfer system without full plurality-second
surplus votes can become is the system with this surplus formula. While
this will effect any quota-based system too, it actually seems not too bad,
since it's still just surplus winners votes, so it's better than completely
manipulated MMP, even if very inefficient (lots of list seats, few district
seats).

But I have thought of a concerning hypothetical regarding this system, that
might indicate that it can be WORSE than parallel voting, even with
perfectly "fair" single-winner districts. Let me show two examples. Both
rely on defining how many district and list seats are is not important,
since adding list seats will not make the result converge to (and never
reach) proportionality (like parallel voting) or cross the point of
proportionality (like vote transfer systems are usually treated/excepted to
do in theory).

Example 1: The district result is coincidentally the same as the
proportional result, so parallel voting (in any ratio) is proportional, but
the "losers + plurality surplus" model of vote transfer (which again is a
theorized equilibrium of the stronghold-split strategy) would actually make
it more disproportional, no matter how many list seats there are.

Parallel voting Losers + plurality surplus
District(s) Party A Party B Party A Party B Party A Party B
1 60% 40% 60% 40% 20% 40%
2 30% 70% 30% 70% 30% 40%
3 10% 90% 10% 90% 10% 80%
wins 1 2 sum 100% 200% 60% 160%
% 33% *67%* % 33% *67%* 27% *73%*
Example 2: The districts give a disproportional result, and even though
adding the transfer votes with list seats would slightly mitigate it, it
would not mitigate it as much as parallel voting would. So again, parallel
voting would with more seats converge to proportional, but the vote
transfer compensation will not even converge, let alone fully compensate.
The interesting thing is that this is with only two parties! With more
parties, maybe exact proportionality would never be reached, but for two
parties usually the transfer votes would give some percentage on the other
side of the parallel voting ones, meaning that with the right weights of
the two, you could get proportional. It seems that is not always the case.

Parallel voting Losers + plurality surplus
District(s) Party C Party D Party C Party D Party C Party D
1 60% 40% 60% 40% 20% 40%
2 25% 75% 25% 75% 25% 50%
3 25% 75% 25% 75% 25% 50%
4 10% 90% 10% 90% 10% 80%
wins 1 3 sum 120% 280% 80% 220%
% 25% *75%* % 30% *70%* 27% *73%*

Example 3: I wanted to have one example for a two-party setup, where the
majority party gets less seats on both sides of the system. I suspect this
is impossible. The plurality vote transfer rule seems inherently to work in
the opposite direction as gerrymandering would in this case. I think
because it's mathematically linear, you cannot "gerrymander both ways": You
can gerrymander the districts for the minority party, but then the transfer
votes will always serve the majority party. Because you have to pack the
voters of the majority into fewer, therefore on average higher margin
districts, they will always receive enough surplus votes to have the
majority of transfer votes. I don't think there is a strategy where you
have parts of the minority party in small margin seats, and some of it in
very high margin seats because of the linearity. In the example, the ratio
of transfer votes will not change if you remove the second district, which
is the average of the first two.

Parallel voting Losers + plurality surplus
District(s) Party E Party F Party E Party F Party E Party F
1 52% 48% 52% 48% 4% 48%
*2* *71%* *29%* *71%* *29%* *42%* *29%*
3 90% 10% 90% 10% 80% 10%
wins 3 0 sum 213% 87% 126% 87%
% 100,0% 0,0% % 71,0% 29,0% 59,2% 40,8%
Example 4: Just to illustrate the already intuitive effect that more than 2
parties can have even if all the districts are won by a clear 51% majority,
here the strongest performing party (list) may receive less seats from both
the districts and the compensatory tier than they would be entitled to in
PR, so again, there is no combination ratio that would make it proportional
for this party.

Parallel voting Losers + plurality surplus
District(s) Party G Party H Party I Party J Party G Party H Party I
Party J Party
G Party H Party I Party J
1 51% 0% 0% 49% 51% 0% 0% 49% 2% 0% 0% 49%
2 51% 0% 0% 49% 51% 0% 0% 49% 2% 0% 0% 49%
3 26% 51% 12% 11% 26% 51% 12% 11% 26% 25% 12% 11%
4 26% 51% 12% 11% 26% 51% 12% 11% 26% 25% 12% 11%
5 26% 51% 12% 11% 26% 51% 12% 11% 26% 25% 12% 11%
6 26% 0% 51% 23% 26% 0% 51% 23% 26% 0% 25% 23%
7 26% 0% 51% 23% 26% 0% 51% 23% 26% 0% 25% 23%
wins 2 3 2 0 sum 232% 153% 138% 177% 134% 75% 86% 177%
% *28,6%* 42,9% 28,6% 0,0% % *33,1%* 21,9% 19,7% 25,3% *28,4%* 15,9% 18,2%
37,5%

To me these were not intuitive maybe exactly because I've been thinking of
such systems for too long, although maybe for some of you it is obvious,
still wanted to share. My (admittedly very impractical) approach at the
moment for the ideal vote transfer mixed system would be to not use any
district-based (%) quota, but to take the district winner with the lowest
amount of votes and say they have 0 surplus votes, and every other winners
surplus votes would be above this benchmark. This would of course need a
flexible amount of seats, and probably a lot since this system would almost
be perfectly proportional (the error would come from any list-PR
apportionment being applied to a modified vote total, where the rankings of
parties may be different than originally) therefore there will be a lot of
low-vote plurality winners. The "perfect" solution in my mind which would
totally counteract even the stronghold split strategy would be to take the
district second with the lowest amount of votes, but in my opinion, this is
unnecessary if everyone knows there will be as many extra seats as it takes
for almost perfect proportionality under the lowest winner-model. Unless
there are any objections, I think I would post this model in more detail
here, but if someone sees a critical flaw, please let me know.

Best,
Abel
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