[EM] STV+AV

[Raph Frank]

One possible way of combining AV + STV is to allow equal ranks.  This
method becomes a method that is very similar to approval in the single
winner case.

When determining if a candidate is elected, all candidates at the rank
share the remaining vote strength, but when determining if a candidate
should be eliminated, the candidates at the rank get the full strength
of the ballot, like in approval.

A feature of STV is that the proportionality for solid coalitions
criterion isn't affected by the elimination ordering, so you can use
any rule that you want.  This assumes that candidates who exceed the
quota are elected (and elected candidates are immune from
elimination).

Assuming there was a ballot:

A: 1
B: 1
C: 2
D: 2
E: 3
F: 3

In round 1, the ballot would count as

For electing:
A: 0.5
B: 0.5
Others: 0

For eliminating
A: 1
B: 1
Others: 0

So, the vote is shared when determining if someone has reached the
quota, but is full strength when determining who to eliminate.

If a candidate has more than a quota of "electing" votes, he is deemed
elected.   Otherwise, the candidate with the lowest "eliminating"
votes is eliminated.

If B was eliminated and A was elected with 133.3% of the quota (so 75%
of the voting strength consumed), then the ballot would count as:

For electing:
A: 0.75
B: 0 (eliminated)
C: 0.125
D: 0.125

To eliminate
A: 0.75 (irrelevant since already elected)
B: 0.75 (irrelevant since already eliminated)
C: 0.25
D: 0.25
Others: 0

This means that if all party supporters vote for party members as
ranks 1 and 2, then the party is guaranteed to gets its share of the
seats.  This follows from the solid coalition criterion.

However, when determining the intra-party ordering, rank 1 candidates
are considered approved and rank 2 candidates are disapproved.  The
most approved party candidates (up to the limited number of seats the
party gets) are the last to be eliminated, so they will be the ones
elected.

Standard PR-STV becomes IRV in the single seat case.  However, this
system becomes a method that is very similar to approval in the single
seat case.

This means that for parties that get 1 seat, the intra-party decision
is made by IRV, but with this method, the intra-party decision is made
by approval (or at least an approval-like method).

Assuming Meek's method for the transfers, then the rules are

For both vote types (for election and for elimination)

Vote strength passed to the next rank

- the amount that would be passed to the next rank if all candidates
at the current rank were ranked in order
- This is the product of (1 - keep factor) for all candidates at the
current rank times the vote passed into this rank
- Nothing is passed to the next rank unless all candidates at the
current rank are elected or eliminated

Sharing (for election)

- vote strength not passed to the next rank is shared between
candidates at the current rank in proportion to their keep factors

Sharing (for elimination)

- vote strength not passed to the next rank is given at full strength
to all candidates at the current rank