# [EM] Proportional representation through Bucklin-STV/Asset

Abd ul-Rahman Lomax abd at lomaxdesign.com
Mon Mar 29 12:21:35 PDT 2010

```At 06:13 AM 3/29/2010, Raph Frank wrote:
>On Mon, Mar 29, 2010 at 4:02 AM, Abd ul-Rahman Lomax
><abd at lomaxdesign.com> wrote:
> > I'd like to get specific comment on this method. I use the Hare quota,
> > generally, with Asset methods, because it provides natural consequences for
> > inability to compromise. The loss of a seat is not particularly harmful if
> > this is a large-member election.
>
>In practice, if there is a reasonably large number of voters, this
>effectively means that at least 1 seat will be unfilled and the Hare
>quota with one seat guaranteed to be unfilled is pretty much
>equivalent to the Droop quota, but with 1 fewer seat.

Yes. So, if you want 30 seats, you might set the seat count at 31.
What could happen is that *if every remaining Asset holder agreed,*
then you'd get an extra seat. Big deal. 31 members instead of 30. No harm.

>In practice, the optimal quota, in terms of maximising representation
>would be somewhere between the Hare and the Droop quota.

The problem is that "representation" isn't being well-defined. I'm
claiming that representation is defective if the one represented did
not choose the representative, directly or indirectly.

>With 9999 voters and 5 seats, the Hare quota is 1999 votes (assuming
>rounding down) and the Droop quota is .

No rounding with proportional Asset, except maybe to .001 vote,
something that could not reasonably affect results. It's possible to
have no rounding at all, the active vote for a ballot is always a
rational number between 0 and 1.

Droop quota is 1666.5.

>   This means that all 5 seats
>can be taken by 9995.  Thus, if even 5 voters refuse to compromise,
>then only 4 seats can be filled and so, with high probability, only
>7996 will be represented.  The Droop quota is 1667 and if that is
>used, then 5 seats would mean that 8335 voters are represented.

If one voter (elector) does not compromise, then a seat is not
elected. In this example, the number of seats being elected is low. I
assume that this is a district election, and that this is only rough
proportional representation for large parties. Given that it is
possible to have practidally complete proportional representation,
where every voter is ultimately represented, isn't that a tad limiting?

It can easily be argued that if we want elected representation (with
"losers" and "winners," i.e., some people are represented and a
minority are not), then the Droop quota should be used, and that is
exactly why it is used, we still think that representatives should be
elected and have practically no concept of chosen representation,
though it's a no-brainer with property rights, a huge disconnect that
I noticed many years ago.

>This means that the Droop quota is likely to result in higher representation.

It depends on what happens with the remaining votes! I mentioned this
above, but I know it could easily be missed. Suppose the Hare quota
is used, and election is per district. Perhaps four seats would be
elected (could be less, even, though less would probably be unusual).
Suppose there are ten districts, the desired Assembly size is 50
seats. Okay, we've elected 40 of them. Then the candidates in first
position on the remaining ballots can, across the entire
jurisdiction, amalgamate votes to create additional seats. They might
nail down all but a few.

The question for me is what should be done with the "dregs." Those
who can't or won't compromise. I'd say that these candidates, while
they certainly represent some voters, may not be good choices for the
Assembly precisely because of that. Asset is unique in that it could,
in theory, elect people from outside the original candidate set. More
seats might be created by such compromises. We can assume with a
large election, that a seat would still be vacant. What if the
remaining electors hold an election, using a decent methd, for the
remaining seat, which is then given some restricted rights in the
Assembly, the right to participate in deliberation, but the voting
power is impaired to a fractional vote, being the fraction that
approved this candidate, in the end.

If direct voting by electors is allowed, the problem goes away. The
extra seat really doesn't matter, it is more a matter of convenience
than anything else. The election for the last seat could still be
held, but it would not convey any unjust voting power.

What ordinary STV does is to elect seats with, at least for the last
election or last few, unjust voting power. Because there have been
losers, there must be winners, who gain representation out of
proportion (though only by a little) to their actual support in the electorate.

>However, an alternative would be to use something like
>
>0.75*Hare + 0.25*Droop = 1917
>
>Filling all 5 seats would require 9585 votes.  This could be
>accomplished even if there are 414 holdouts.

It is possible to elect a last seat or seats with a lower quota, but
not so low as to create a serious inequity. But I prefer to keep it
straight. If you can get enough voters or their representatives
(electors) to agree, you gain a seat. If you don't, you don't. There
is a justice to the small loss of representation. (It's not small if
it is a completed election with a five-member target. But it's small
if its one seat out of 50, and the whole issue of voting power
largely vanishes if direct voting is allowed, which can only be done
if seat voting power (the number of voters that the seat represents)
is equal for all seats -- unless it's devalued as described. The
tradition of a peer assembly is very strong, people are uncomfortable
with unequal voting power, which is why I'm interested in Asset
instead of pure Delegable Proxy.

>Obviously, there is a trade-off.  Maybe the rule would be
>
>a*Hare + (1-a)*Droop
>
>with a being decided by the previous election.  If all 5 seats were
>filled, then a can be made slightly larger.  Otherwise, it is reduced
>slightly.
>
> > If it is an election of a limited number of
> > members from each district (which means loss of proportional
> representatin),
> > then I presume a Droop quota would be used, because gaining a full slate
> > would be important. On the other hand, one could use the Hare quota for
> > district elections, then allow the Asset electors with votes remaining from
> > a district election to amalgamate across the entire legislative
> > jurisdiction, thus providing small-minority representation state-wide. I
> > like that, isn't it interesting? District elections, but no loss
> of minority
> > representation!
>
>Sounds reasonable.  However, using Hare nationally would still mean
>that there is 1 unfilled seat.

Strictly, "nationally" might apply to some nations, but not to the
United States, as far as any reasonable futur is concerned. State
Assemblies could be possible, and it would only be a State
Constitution involved. Town Councils could be done in many places.

As I've pointed out, it is possible to fill that remaining seat,
assigning it a fractional vote. An isolated, special-function seat
with lowered voting power would not be a problem, compared to larger
disparities. But, more to the point, it's just one seat out of a
large Assembly, presumably, and vacant seats pop up all the time, for
lots of reasons. With Asset, a vacanat seat could normally be filled
quickly, it's one of the advantages.

>It also depends on how you calculate the Hare quota in each district.
>It could be calculated nationally after all the votes are counted, or
>calculated locally based just on the votes in that district.

Interesting problem. I would favor setting the Hare quota
Assembly-wide, which would even out disparities in district
populations. In fact, it coudl allow large variation in the district
sizes, while remaining fair. A high-population district would
naturally elect more representatives.

Hint: This is actually the same as a statewide election, it just
amalgamates first by district. It's really the same as pure Asset, as
long as there is a single quota.

>Calculating it nationally would mean that each seat is equal to the
>same number of voters.  However, calculating is locally would make the
>system easier to manage.

Not. It's just a number the total number of voters, divided by the
maximum number of seats.

>Also, if the quota is different between districts, then it creates an
>incentive to game the system, by electors choosing which quota to use.

Of course, bad idea. It would also create differences in the number
of voters represented by each seat.

> > 1. Q = V / N. (This can be done with the Droop quota to make it more
> > deterministic. I oppose it for reasons I won't detail here.)
> >
> > <snip>
> >
> > 5. Because there have been no eliminations, all elections so far
> can be seen
> > as rigorously correct and fair.(skipped original sections)
>
>So this is just standard PR-STV, up to elimination?

Yes. Standard PR-STV is rigorously fair up until eliminations begin.
All seats elected with the Hare quota are obviously qualified. Using
the Droop quota, it's a little bit shaky, if you want to use the
Droop quota, you should allow for the possibility that the remaining
electors agree! And then you get an extra seat.

> > 10. The ballots are now treated as Bucklin ballots. The second rank is
> > counted. ("Second rank" means "second active rank.") Seats are assigned
> > whenever a candidate, in a round of counting, gains a quota of votes, and
> > those ballots are devalued accordingly. In this case, an elected candidate
> > might be in a lower position on the ballot. The candidate is marked as
> > elected, but the higher position candidate remains active, and may attain
> > election through votes from other ballots, to the extent that any voting
> > strength is left. As a winner is found, any ballots counted for that
> > candidate are devalued as before.
>
>Since this method would be pretty difficult to hand count, it would
>probably be worth using Meek's method.

I prefer vote-for-one asset, with FAAV as a tweak to deal fairly with
overvotes. (Fractional Approval Asset Voting). Very, very simple to count.

If the first elections are done by district, the number of candidates
elected is small at first, so the math should not get too hairy.
That's my off-the-cuff impression. The problem is in the
devaluations, and the fact that there can be many candidates in
asset. The devaluations are unique to a set of ballots with elected
candidates, and that number can be moderate. It would get hairier and
hairier with more elected candidates per district. With three as the
max, we'd normally see only two elected (or less). So we'd need to
know, for all the new candidate totals, how many were

"virgin," unused for election.
used for election of the first candidate
used for election of the second candidate
used for both.

The first class of ballots would have a value of 1 full vote.
The second would have a value of the extra over the quota, divided by
the total for the first candidate.
The third would have the same for the second elected candidate.
The last would have a value of the first fraction times the second.

It's not that bad at all. Assembly-wide, it would be pretty bad! So
the Assembly-wide part of the election, in which electors holding
unassigned votes participate, would be pure Asset, no more ballot amalgamation.

The scheme I describe could accomodate huge numbers of candidates.
The district results are simply a list of elected candidates plus
standing votes remaining by candidate. The math has been done
locally. (Any not-fully used ballot is assigned, at the fractional
value, to the candidate in first position, regardless of whether that
candidate has been elected or not.)

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