proportional list
Dear all,
I would like to draw your attention to a pre-print I have uploaded to arxiv.
https://arxiv.org/abs/2506.12318
A House Monotone, Coherent, and Droop Proportional Ranked Candidate
Voting Method
Ross Hyman
Subjects: Theoretical Economics (econ.TH)
A Ranked candidate voting method based on Phragmen's procedure is
described that can be used to produce a top-down proportional
candidate list. The method complies with the Droop proportionality
criterion satisfied by Single Transferable Vote. It also complies with
house monotonicity and coherence, which are the ranked-candidate
analogs of the divisor methods properties of always avoiding the
Alabama and New State paradoxes. The highest ranked candidate in the
list is the Instant Runoff winner, which is in at least one Droop
proportional set of N winners for all N.
If I were to rewrite the paper for an audience interested primarily in
single-winner elections. I would emphasise the following things:
A Droop proportional list has the property that the top N candidates
in the list, for any N, satisfy the Droop proportionality criterion
for N winners.
A Droop proportional list is a candidate list in which independence of
irrelevant alternatives is not a desirable election criterion. For
proportionally to be complied with, the weight of a ballot's input in
deciding the relative ordering of candidates A and B should depend on
the placement of other candidates on the ballot, since these other
candidates can be elected to a high position on the list and reduce
the weight of the ballot for deciding lower positions.
In general, the Condorcet winner cannot always be at the top of a
proportional list. It is easy to devise ballot sets where two
candidates each have more than a third of the vote, so they must be in
the top two positions, and neither is the Condorcet winner.
The Instant Runoff winner can always be at the top of a proportional
list. There will always be a Droop proportional compliant set of N
winners, for any N, that includes the IRV winner. I prove this in the
paper. I also suggest in the paper that this property of the IRV
winner is a way to give quantitative meaning to the term "core
support."
If one is only interested in a single-winner election, and one wants
to choose from the top of a Condorcet list or an IRV-topped Droop
compliant list, in which circumstances is one preferable to the other?
My answer to this is that it depends. If the voters are not polarized,
and can be represented by a fixed single peaked distribution with a
median that well characterizes the distribution, the Condorcet
candidate is probably better, to prevent center squeeze.
But if the electorate is divided into many factions, as I believe is
the case in the U.S., with those factions dynamically responding to
the presence and absence of candidates and the possibility of
candidates, in such a way that a fixed voter distribution in the
absence of the candidates is a theoretical abstraction with no
practical meaning, I think the IRV winner can be better, as there is
always support for it, no matter how many factions there may be.
Best,
Ross
On 2025-06-23 19:05, Ross Hyman via Election-Methods wrote:
Dear all,
I would like to draw your attention to a pre-print I have uploaded to arxiv.
https://arxiv.org/abs/2506.12318
A House Monotone, Coherent, and Droop Proportional Ranked Candidate
Voting Method
Ross Hyman
Subjects: Theoretical Economics (econ.TH)
A Ranked candidate voting method based on Phragmen's procedure is
described that can be used to produce a top-down proportional
candidate list. The method complies with the Droop proportionality
criterion satisfied by Single Transferable Vote. It also complies with
house monotonicity and coherence, which are the ranked-candidate
analogs of the divisor methods properties of always avoiding the
Alabama and New State paradoxes. The highest ranked candidate in the
list is the Instant Runoff winner, which is in at least one Droop
proportional set of N winners for all N.
As I understood it from my first glance, the main differences between
your method and Aziz's methods are
- vs bottom-up: yours is top-down and thus elects the same top
candidate as the base method (QPQ/IRV).
- vs top-down: yours doesn't need to go through every solid coalition.
Is that correct, or did I miss some other properties?
If I were to rewrite the paper for an audience interested primarily in
single-winner elections. I would emphasise the following things:
A Droop proportional list has the property that the top N candidates
in the list, for any N, satisfy the Droop proportionality criterion
for N winners.
A Droop proportional list is a candidate list in which independence of
irrelevant alternatives is not a desirable election criterion. For
proportionally to be complied with, the weight of a ballot's input in
deciding the relative ordering of candidates A and B should depend on
the placement of other candidates on the ballot, since these other
candidates can be elected to a high position on the list and reduce
the weight of the ballot for deciding lower positions.
In general, the Condorcet winner cannot always be at the top of a
proportional list. It is easy to devise ballot sets where two
candidates each have more than a third of the vote, so they must be in
the top two positions, and neither is the Condorcet winner.
The Instant Runoff winner can always be at the top of a proportional
list. There will always be a Droop proportional compliant set of N
winners, for any N, that includes the IRV winner. I prove this in the
paper. I also suggest in the paper that this property of the IRV
winner is a way to give quantitative meaning to the term "core
support."
This is why I think house monotonicity comes with a cost. In a
left-center-right situation like the one you described, you can either
have center squeeze (if your proportional ordering is, say, Left > Right
Center), or you can have disproportionality (Center > Right > Left).
So IMHO, for a general multiwinner method, unless there's a particular
reason, house monotonicity is not desirable. Party lists require house
monotonicity so you have no choice there, but with a setting like STV,
there shouldn't be any reason to require house monotonicity.
(It would be interesting to find out how parties solve this problem in
countries with party list. No matter how the party draws up the list,
those who do so are faced with the same trade-off since the party list
format forces house monotonicity.)
-km
Hi Kristofer,
You are correct. The Aziz top-down method, the Phragmen bottom up
method, and the Phragmen top down method are based on
Jefferson/D'Hondt and share its properties of Droop proportionality
and coherence. There are differences in which candidates they elect,
based on the fact that the Phragmen methods can elect from "imperfect"
solid coalitions.
You are also correct that the main reason house monotonic lists are
required to be produced are for proportional representation elections.
I can think of three situations where they would be useful: 1) a
party's production of a closed list that voters vote on in the
election, 2) a party's production of a list that voters can choose to
select instead of ranking candidates in and STV election, 3) an
organization's production of a list that it promotes to voters in an
STV election.
I looked and could not find any definitive examples of a party or
organization creating a proportional list through a ranked-candidate
election of its members.
The examples I quoted in the paper, from Voting Matters, were the
closest I could find. It is unclear from those papers if the proposed
elections described in those papers were ever held.
I would be interested to learn examples of how parties and
organizations create their proportional lists.
Best,
Ross
On Tue, Jun 24, 2025 at 1:28 PM Kristofer Munsterhjelm
km-elmet@munsterhjelm.no wrote:
On 2025-06-23 19:05, Ross Hyman via Election-Methods wrote:
Dear all,
I would like to draw your attention to a pre-print I have uploaded to arxiv.
https://arxiv.org/abs/2506.12318
A House Monotone, Coherent, and Droop Proportional Ranked Candidate
Voting Method
Ross Hyman
Subjects: Theoretical Economics (econ.TH)
A Ranked candidate voting method based on Phragmen's procedure is
described that can be used to produce a top-down proportional
candidate list. The method complies with the Droop proportionality
criterion satisfied by Single Transferable Vote. It also complies with
house monotonicity and coherence, which are the ranked-candidate
analogs of the divisor methods properties of always avoiding the
Alabama and New State paradoxes. The highest ranked candidate in the
list is the Instant Runoff winner, which is in at least one Droop
proportional set of N winners for all N.
As I understood it from my first glance, the main differences between
your method and Aziz's methods are
- vs bottom-up: yours is top-down and thus elects the same top
candidate as the base method (QPQ/IRV).
- vs top-down: yours doesn't need to go through every solid coalition.
Is that correct, or did I miss some other properties?
If I were to rewrite the paper for an audience interested primarily in
single-winner elections. I would emphasise the following things:
A Droop proportional list has the property that the top N candidates
in the list, for any N, satisfy the Droop proportionality criterion
for N winners.
A Droop proportional list is a candidate list in which independence of
irrelevant alternatives is not a desirable election criterion. For
proportionally to be complied with, the weight of a ballot's input in
deciding the relative ordering of candidates A and B should depend on
the placement of other candidates on the ballot, since these other
candidates can be elected to a high position on the list and reduce
the weight of the ballot for deciding lower positions.
In general, the Condorcet winner cannot always be at the top of a
proportional list. It is easy to devise ballot sets where two
candidates each have more than a third of the vote, so they must be in
the top two positions, and neither is the Condorcet winner.
The Instant Runoff winner can always be at the top of a proportional
list. There will always be a Droop proportional compliant set of N
winners, for any N, that includes the IRV winner. I prove this in the
paper. I also suggest in the paper that this property of the IRV
winner is a way to give quantitative meaning to the term "core
support."
This is why I think house monotonicity comes with a cost. In a
left-center-right situation like the one you described, you can either
have center squeeze (if your proportional ordering is, say, Left > Right
Center), or you can have disproportionality (Center > Right > Left).
So IMHO, for a general multiwinner method, unless there's a particular
reason, house monotonicity is not desirable. Party lists require house
monotonicity so you have no choice there, but with a setting like STV,
there shouldn't be any reason to require house monotonicity.
(It would be interesting to find out how parties solve this problem in
countries with party list. No matter how the party draws up the list,
those who do so are faced with the same trade-off since the party list
format forces house monotonicity.)
-km
On 2025-06-24 23:37, Ross Hyman wrote:
I looked and could not find any definitive examples of a party or
organization creating a proportional list through a ranked-candidate
election of its members.
The examples I quoted in the paper, from Voting Matters, were the
closest I could find. It is unclear from those papers if the proposed
elections described in those papers were ever held.
I would be interested to learn examples of how parties and
organizations create their proportional lists.
Yeah, I wasn't trying to imply that they did use proportional orderings,
more just commenting that even if the party leadership decides the order
(or some internal party procedure, consensus, or similar does), the
decision-makers still have to deal with the same problem.
-km