# [EM] Minimally improving Approval

Kevin Venzke stepjak at yahoo.fr
Wed Feb 11 20:22:02 PST 2004

```Dear anyone,

The most important issue to me is how to minimally improve Approval.

By "minimal" I really mean that I want to avoid a pairwise matrix, or a similar
number of values to be maintained, and I want to avoid a large or unpredictable
number of counts of the ballots.  (It would be nice if I could insist on a
single count, but I think it isn't possible.)

By "improve" I mean to reduce the likelihood that a voter will regret
the way he voted.  This tends to mean improving the Condorcet efficiency.

I use three-slot MCA ballots.  This permits finding a first-preference winner
and an approval winner (most 1st+2nd ratings), and declare them the two finalists
after just one count of the ballots.  I would much prefer a method that didn't
privilege someone other than the approval winner, but privileging the FP winner
is an easy way to ensure the method meets Majority Favorite.

MCA is the simplest method using this model, and only needs one count.  But
I think it is strategically equivalent (or nearly so) to Approval.  I believe
this because the middle slot ("approved") is only useful if you think your
preferred candidates can win if they have an immediate majority, but not
otherwise.  I can't imagine a scenario where that makes any sense.

"FPW vs AW":

Another obvious and simple method is to do a second count, and elect the winner
of the pairwise contest between the FPW and AW.  But this concerns me in a
scenario such as:

7 A>B
5 B
5 C
3 D>C

A is the FPW and B is the AW.  A wins the pairwise contest.  However, A suffers
a loss to C which is stronger than the A>B victory.  That might simply be an
unintuitive result that we could accept.  But I would prefer to unambiguously
improve Approval, and electing A over B doesn't appear to be an unambiguous
improvement.

Much worse is the possibility that the A voters don't really have a second preference
for B, and only vote for him in the hopes that he will be an easily defeated AW.
(This would also work if they voted A>D.)

"Withdrawable Approval":

The rest of my ideas concern how to avoid this problem.  Here is one good method,
but it might be vulnerable to human error in a hand count: On the second count,
count FPW>AW (pairwise) votes as bullet-voting for the FPW.  (They are trying
to make the FPW win.)  AW>FPW votes are counted towards every approved candidate
except for the FPW (if he was indeed approved), because they are trying to get
anyone but the FPW to win the second count.  AW=FPW votes approve every approved
candidate.  If the FPW wins, he is elected; if anyone else wins, the AW is elected.

So in the scenario above, the result of the second count is that C has 8 votes,
A has 7, B 5, and D 3.  The stronger C>A victory succeeds in preventing A>B.

"No Worse Losses":

A more straight-forward method, but one which is more likely to make the voter
regret: Elect the FPW if he beats the AW, and this victory is stronger than any
defeats suffered by the FPW to anyone else.  This can be done with two counts,
where the second count involves finding the FPW's pairwise contests, which means
(2*N)-1 values to be maintained.  ("N" is the number of candidates.  We may be
able to ignore some candidates in the second count, if mathematically they could
not possibly beat the FPW.)

This creates some regret for voters voting C>FPW>AW or AW>FPW>C, where C is some
non-finalist candidate.  Exceptions can be made in the count to prevent voters
from having an unwanted effect on the pairwise contests.  But making these
exceptions complicates the count, and also probably makes the method identical
to the previous method (which doesn't require the 2*N tallies).

"FPW & AW//MinMax":

In the "No Worse Losses" method it's possible for the FPW to lose even when he
beats the AW pairwise, and the AW's worst loss is worse than the FPW's.  I
could accept this since I think it is safer to privilege the AW.  But we could
instead elect whichever of the FPW and AW has a lower MinMax score.  Finding
all of those pairwise contests, however, requires about 4*N tallies in the second
count.  I also wonder if this method increases the likelihood of regret, and
the complexity of the situations where it occurs.

"Majority FPW vs AW":

I think this method is as simple as we can get without paying attention to
non-finalist candidates.  Elect the FPW if he has a majority-strength victory
over the AW, else the AW wins.  This needs only 2 values to be maintained in
the second count.

(Optionally, I think FPW=AW ballots which approve both should be counted to both
candidates, to help the FPW reach a majority.  The important thing, I think, is
that ballots approving neither the FPW nor AW should be assumed to prefer the AW
in a method as crude as this, since it makes offensive strategy harder.  They
don't need to actually be counted for the AW; it's enough to abstain.)

It is possible, I imagine, that the FPW could still suffer a loss stronger than
his victory over the AW.  But if the FPW>AW win is majority-strength, then it
isn't possible that a larger majority (or any majority) will complain about
the FPW beating the AW.  (Actually, this is the whole reason that "majority" is
a magic number.)

Does anyone have any ideas on this topic?

Kevin Venzke
stepjak at yahoo.fr

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