[EM] "Conditional Approval" variant (needs a name)

[Kevin Venzke]

CONDITIONAL APPROVAL VARIANT

This is a three-rank Approval variant.  Its aim is to reduce the potential for regret, in voting
or not voting for certain candidates.  Voters rank all candidates either Favorite, Compromise, or
Unacceptable.  It differs from MCA (Majority Choice Approval) in that Compromise rankings are not
necessarily regarded in the absence of a majority favorite.  This makes potential for regret lower
than in MCA.

I don't think this method is summable, as for one count of the ballots to suffice, a (2^C) by C
matrix must be maintained ("C" being the number of candidates).  For the following "technical"
description, though, I'll use this approach.

METHOD

The matrix has 2^C columns.  They represent every possible set of the candidates, including an
empty and full set.  There are C rows, each corresponding to a candidate.  The cells contain the
number of voters who approve the row candidate given the column set of past "leaders."

For each ballot, add 1 to the cells in every row corresponding to a candidate ranked "Favorite." 
Then add 1 to every cell (row X, column Y) such that X corresponds to a candidate ranked
"Compromise," and Y corresponds to a set of candidates containing at least one candidate ranked
"Unacceptable" on the ballot.  That's it.

Resolution: We maintain a set of "leaders," which begins empty.  Follow these steps: 1. Find the
cell with the most votes in the column corresponding to the leader set.  2. If the candidate (to
which the row of that cell corresponds) is in the leader set, that candidate is elected and we
stop.  3. Add the candidate to the leader set and go to step 1.

WHAT IT MEANS

Every candidate ranked "Favorite" will always be "approved" by the voter.  But those ranked
"Compromise" will only be approved if it looks like a candidate ranked "Unacceptable" will win.  I
say "looks like," because by subsequently approving the compromises, someone else may take the
lead.

One limitation to keep in mind is that "Compromise" approval can't be withdrawn, even if a
"Favorite" takes the lead.  The voter will have to make predictions about what will happen, and
decide based on that whether to list "Compromises."

EXAMPLES

I have generated these randomly with a program.

8: D > C > AB
8: A > CD > B
6: BC > D > A
4: B >  > ACD

Here, Conditional Approval disagrees with the Condorcet winner, which is D.  In "CA": B is the
initial leader with 10 votes.  The first two blocs hate B, though, so they offer support for C and
D.  This causes C to lead with 22 votes.  The 4-bloc hates C, but they have no compromises and C
wins.

Note that the first 8-bloc could defend the CW by not listing C as a compromise.  (The 6-bloc's
compromise votes for D are never cast.)


5: C > E > AD
2: C > DE > A
3: A > C > DE
9: D > E > AC

C is the CW, but E wins.  D leads initially with 9.  The 5- and 3-blocs hate D and offer support
for E and C.  C takes the lead with 10 votes.  The 9-bloc, however, hates C and approves E, giving
E 14 votes.  The 2-bloc never uses its compromises.  The 5-bloc could've defended the CW by
omitting E as compromise.


Here, three candidates lead, and the original leader wins:
9: B > D > CE
8: E > B > CD
6: D > CE > B
5: CE >  > BD

No CW.  E leads with 13.  The 9-bloc responds by approving D, who then has 15 votes.  The 8-bloc
then approves B, resulting in 17 votes.  Then the 6-bloc makes its concessions, restoring E to the
lead with 19 votes.


Interestingly, this method doesn't seem to satisfy the Weak FBC.  In the (very?) unusual situation
that you know that if your favorite leads, a compromise that you detest will take the lead and
win, it may be to your advantage to not support your favorite at all.  Example:

4: C > D > AB
6: A > BD > C
6: BC > D > A
7: D > AB > C

C leads with 10 votes.  In response, B takes the lead with 19 votes.  The 4-bloc is excluded, as
they detest the winner.  It might've been a safer bet for them to not vote for C at all, knowing
that C is so controversial.  Voting for D (the CW) might've ensured a minimally acceptable result.


Any thoughts?


Kevin Venzke
stepjak at yahoo.fr


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