[EM] An Approval/Condorcet combination method

[Gervase Lam]

Demorep has suggested ACC(?) (Approval Completed Condorcet) as a way of
combining Condorcet and Approval.  He has also mentioned the "50% majority
rule" many times, which I think is a part of ACC.

The thing that bothered me about the method was the usage of the 50%
majority.  This makes things black and white when in fact there is a
region of grey in between.

A few months ago this got me thinking, specifically about Fuzzy Logic.
This was the in thing around about the time I started university.

In Computing, a value is either 0 or 1.  0 could represent a candidate
losing while 1 could represent a candidate winning.  There is no in
between.  The value is a Boolean value.

Fuzzy Logic allowed any number between 0 and 1.  In other words, the
number is a probability value.

In a Condorcet pairwise contest, a full vote goes to one candidate or the
other, not a partial vote.  In other words, a Boolean value is involved.
It would be nice to change it to be more like Fuzzy Logic.

In order to try to do this, why not scale the number of votes in each
pairwise contest by the Approval percentage?  That is multiply the
Approval percentage by the number of votes in each pairwise contest and
then use a Condorcet Method on the result?  The Approval percentage could
be seen as the sum total of the Utility values that all of the voters have
given to a candidate divided by the number of voters.  That is, the
"average Utility" of a candidate.

For example, if candidates A and B had 32% and 28% Approval respectively
and the pairwise votes for A and B are 45 and 55, the result after
multiplication would be 1440 and 1540.  Therefore, B would be the winner
of the pairwise comparison.

In fact, there isn't a need to have an Approval percentage.  Just use the
number of Approval votes.  Using the example above, if the total number of
voters were 50, A would get 16 * 45 = 720 and B would get 14 * 55 = 770.

One (dubious) way of looking at this is to think of one ballot.  On the
ballot are a set of candidates who were approved and a set of candidates
who were not.  The Boolean values would be 1 for approved and 0 for not
approved.

The person who voted on the ballot is effectively multiplying the pairwise
votes for the disapproved candidates by 0 and the approved candidates by
1.  This results in 0 votes for the disapproved candidates and 1 "vote"
for the other candidates.  In order to break the tie between the Approved
candidates, the pairwise contest is used (i.e. the Condorcet method is
used).

This multiplication by the 0 or 1 is a bit like the AND operator in
Boolean mathematics.  The candidates must be Approved AND do well in the
pairwise contests to win.  An ideal combination of Approval and
"Condorcet".

It could be said that having to do well in both Approval and "Condorcet"
is harsh.  One way around this would be to replace the "Boolean AND
operator" (i.e. multiplication) with the "Boolean OR operator" (i.e. the
plus sign).  In other words, instead of multiplying the number of votes in
each pairwise contest by the number of Approval votes, the two numbers
would be added together.

I personally think that using multiplication is better as candidates
should really "pass" both Approval and "Condorcet" tests rather than one
or the other.  Demorep has mentioned that the problem with Approval is
expressiveness, which Condorcet has much more of, while Condorcet has a
mandate problem.  In other words, just because a candidate is voted highly
on two Ranked/Condorcet ballots, does it mean that the candidate is
Approved in both the ballots?

I don't know if there are any big holes in the methods, which I suppose
could be called Approval Scaled Condorcet and Approval Added Condorcet
respectively.  Or, more specifically, you could say Approval Scaled Plain
Condorcet (ASPC), Approval Added Plain Condorcet (AAPC), Approval Scaled
Ranked Pairs (ASRP) etc...

Thanks,
Gervase.

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