PR with weighted rep. power
In a post in September, Toby Pereira said the following about the
"Random Ballots" method I had implemented in quadelect:
This is very interesting. Thank you for doing this analysis. Random
ballots, as I understand your implementation, I think would not be
strategyproof. If there are c candidates to be elected, then every
voter ranks their top c. And then you sequentially pick c ballots at
random, electing the top-ranked unelected candidate on each ballot.
Is that correct? In that case, I think if a voter is fairly confident
that their favourite candidate will be elected anyway, it makes sense
for them to put other candidates above them that might not get
elected anyway. If my second favourite candidate is not very popular
among other voters, and my favourite is, I am likely to put my second
favourite top.
And I responded that that too is interesting, because that looks like
vote management (Hylland free riding), and I hadn't expected vote
management to be that robust a strategic feature.
But a few days ago I had a thought. Suppose we change the method to the
following:
- Pick c random ballots. For simplicity, disallow equal-rank.
- Let each candidate's voting power be the fraction of these ballots on
which he is the favorite. - Elect every candidate with positive voting power. (Note that there may
be fewer than c distinct winners.)
These candidates are the winners, and their votes have the given weight
in the assembly.
This turns the method into a sort of randomized party list. Is it
strategyproof?
The voting stage seems to be, at least, because in effect each candidate
has an infinite number of clones, and if you prefer A to B, you would
rather want to give an assembly seat to A (or a clone of A) than to B.
Note that the number of seats need to be variable, otherwise there would
be an incentive to vote for separate candidates to push opposition
candidates out of the assembly.
If there's an incentive to strategy for this method, it would be in how
voter weights add up to majorities (power indices being nonlinear). But
I'm not sure there's an incentive there either.
So if it is strategyproof, we have a weighted variable seat method that
is proportional without vote management/free riding; and we also have
results that say that Droop proportionality is incompatible with
complete immunity to free riding. The natural question then is: how
close to DPC can we go and have immunity to free riding? Is it possible
by just having a variable size assembly, or do we need some amount of
weighted representative power too?
I don't know what the answers would be, but they're interesting thoughts.
-km
I will ask a potentially very silly question and I own up to this: in which
way would this setup be superior to allocating singe-seat districts with
separate candidate pools and electing a random winner from each? Is it the
issue of variable voter participation?
Regards,
On Fri, 15 Nov 2024, 12:21 pm Kristofer Munsterhjelm, <
km-elmet@munsterhjelm.no> wrote:
In a post in September, Toby Pereira said the following about the
"Random Ballots" method I had implemented in quadelect:
This is very interesting. Thank you for doing this analysis. Random
ballots, as I understand your implementation, I think would not be
strategyproof. If there are c candidates to be elected, then every
voter ranks their top c. And then you sequentially pick c ballots at
random, electing the top-ranked unelected candidate on each ballot.
Is that correct? In that case, I think if a voter is fairly confident
that their favourite candidate will be elected anyway, it makes sense
for them to put other candidates above them that might not get
elected anyway. If my second favourite candidate is not very popular
among other voters, and my favourite is, I am likely to put my second
favourite top.
And I responded that that too is interesting, because that looks like
vote management (Hylland free riding), and I hadn't expected vote
management to be that robust a strategic feature.
But a few days ago I had a thought. Suppose we change the method to the
following:
- Pick c random ballots. For simplicity, disallow equal-rank.
- Let each candidate's voting power be the fraction of these ballots on
which he is the favorite. - Elect every candidate with positive voting power. (Note that there may
be fewer than c distinct winners.)
These candidates are the winners, and their votes have the given weight
in the assembly.
This turns the method into a sort of randomized party list. Is it
strategyproof?
The voting stage seems to be, at least, because in effect each candidate
has an infinite number of clones, and if you prefer A to B, you would
rather want to give an assembly seat to A (or a clone of A) than to B.
Note that the number of seats need to be variable, otherwise there would
be an incentive to vote for separate candidates to push opposition
candidates out of the assembly.
If there's an incentive to strategy for this method, it would be in how
voter weights add up to majorities (power indices being nonlinear). But
I'm not sure there's an incentive there either.
So if it is strategyproof, we have a weighted variable seat method that
is proportional without vote management/free riding; and we also have
results that say that Droop proportionality is incompatible with
complete immunity to free riding. The natural question then is: how
close to DPC can we go and have immunity to free riding? Is it possible
by just having a variable size assembly, or do we need some amount of
weighted representative power too?
I don't know what the answers would be, but they're interesting thoughts.
-km
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On 2024-11-15 03:24, Etjon Basha wrote:
I will ask a potentially very silly question and I own up to this: in
which way would this setup be superior to allocating singe-seat
districts with separate candidate pools and electing a random winner
from each? Is it the issue of variable voter participation?
Strategyproof methods usually aren't good in practice; in my case I'm
just using it to show you can have some kind of proportionality and not
be susceptible to free riding, even though Droop proportionality implies
some vulnerability to free riding.
Your method is probably superior in the sense that there's no need for
weighted representative power and it's still strategyproof. But it
relies on candidates not being able to run in multiple districts, which
means part of the problem has already been solved (compared to a
multiwinner PR method).
In a multiwinner PR method, a candidate A might have near-unanimous
support, and this could lead to free riding problems because some of A's
supporters would be better off ranking some other candidate first. But
in a single-member district system, A can only be elected in one of the
districts; in a sense, the free-riding problem is solved by that only
one district can vote for A.
If each district has a clone of A, then SMD random ballot and my system
produces pretty much the same result, except SMD random ballot embodies
A's voting weight in how many of his clones are elected.
Your idea could be used to make (also strategyproof) generalizations of
random pair, though. Say there are n seats and at least 2n candidates.
Establish a random order of these candidates. Pick 1/n of the ballots at
random, and elect the candidate of the first two that they prefer to the
other. Throw the ballots away and repeat for the next 1/n fraction of
ballots (and next two candidates in the established order). But, like
random pair, this fails Pareto.
-km