Approval/Plurality combinations

[Bart Ingles]

Approval's simplicity is tempting despite its drawbacks.  Even if you
hold a separate FPP vote to detect a majority winner, the two votes
together are simpler than full ranking and can be conducted on any
equipment, including punch-card ballots and paper "X" voting.  Of course
if you are going to hold both votes, it would make sense to come up with
a counting method that makes better use of the available information.

IMO the combination of an Approval and an FPP vote would yield most of
the useful information obtainable from ranked balloting.  While you only
have three ranking levels (first choice/middle choices/rejected), you
also gain a piece of information not available with ranking: the cutoff
between acceptable and unacceptable candidates (note DEMOREP's repeated
calls for a YES/NO vote to be used along with ranking).

***

I see two basic approaches to using this information: collated and
uncollated.  The collated approach would retain the information
contained in the relationship between a voter's first choice and his
Approval choices; in other words, increment an entry in a
two-dimensional array indexed by [first choice, approval choice] for
every approval vote.  This would be difficult to do with paper ballots,
and would require software modification to punch-card voting equipment. 
Election methods possible under this approach would include Condorcet
and a fairly accurate form of IRO.

The uncollated approach would simply tabulate the first-choice and
approval votes separately.  This should work on any existing equipment
and would be straightforward with paper ballots as well.  Election
methods possible under this approach include a simplified IRO and a
Plurality-Approval hybrid that should address some of the objections to
both.  The uncollated methods are described below.

QUESTION:  Have the following methods already been done?  If so, by whom
and what are they really called?

***

Simplified IRO:

The simplified IRO simply uses the plurality vote to choose the top two
finalists, and the approval vote to decide between them.  This is
equivalent to transferring losers' votes, since the finalists' plurality
totals are duplicated in the approval vote.  Caveats: if a loser gives
approval to both finalists, the two votes simply cancel out.  Likewise,
if one finalist gives an approval vote to the other, his vote is
canceled out.

***

Plurality-Approval Hybrid

The idea is to vary smoothly between FPP and Approval, using Approval
only as needed to obtain a majority winner.  This method works a little
like Borda, in that the second-choice totals are weighted and then added
to the first-choice totals.  A candidate's second-choice total is
defined as his approval total minus his first-choice total.  The
weighting used is the minimum required for one of the candidates to
reach a bare majority (50%), and is limited to a value between 0 and 1. 
The formula to calculate the candidate's total is
C = F + W(A - F)  where:

C = computed total  (always 50%, unless there is a first-choice majority
or no approval majority is possible)
F = first-choice total
A = Approval total  (note the "A - F" term is equivalent to
"second-choice total")
W  = Weighting factor  (see below)

If a candidate has a first-choice majority, the minimum weight factor is
0 (the Approval vote is unused).  If not, the easiest way to find the
winner is to work backwards from the totals for each candidate, to find
the weight factor required for that candidate to reach 50%.  The formula
to do this is:

W = (T/2 - F) / (A - F)

The candidate with the lowest weight factor is the winner, at 50%.  This
weight can then be used to compute the other totals using the first
formula.  If the lowest weight factor is greater than 1, then 1 is used
and the final vote is equivalent to Approval with no majority (using a
factor greater than one would give approval votes more weight than the
first-choice vote).

I played around with this method a little, using some simple examples
and obvious strategies.  The results seem to be similar to Approval (or
Condorcet or IRO for that matter, when corresponding strategies are
used).  It does seem to exclude some very weak Condorcet winners in
favor of one who is both the plurality and runoff winner, though.  More
later, if anybody's interested.