[EM] Minimally improving Approval

[Forest Simmons]

How about this:

A preliminary ranking of the candidates by approval scores is modified by
correcting the single greatest "discrepancy" (if there is any) unless
correcting this discrepancy would create another discrepancy of the same
or greater magnitude.

A "discrepancy" is a pair of candidates who (according to a majority of
ballots) is ranked in the wrong order.  Correction of a discrepancy is
accomplished by swapping the positions of the two candidates involved.

The highest ranked candidate of the final ranking is the winner.

Alternately, the corrections of discrepancies (by the same rule) could be
continued until no further correction could be made without introducing
another discrepancy of the same or greater magnitude.

The pairwise matrix with approval scores along the diagonal would be
sufficient for this method.

I'm not sure that it would be simple enough to be called minimal
improvement.  For that matter, I'm not even sure that it is any
improvement.

If the swaps were limited to adjacent pairs, then it could be compared to
the standard method that drill sergeants and PE teaches use to get folks
lined up in descending order of height, with the "greatest discrepancy
first" rule removing the inherent ambiguities of that process familiar to
all who have participated in it.

Also if the swaps were limited to adjacent pairs, the proviso about
introduction of new discrepancies could be dropped.

However, I believe that limiting swaps to adjacent pairs might do more
violence to the FBC than the other versions do, since one might worry
about one's favorite blocking the progress of one's compromise up through
the ranks without having the steam to go as far up as compromise would if
it could only get past favorite.

Finally, I know this method is something like Steve Eppley's MAM, so
forgive me if one of my variants duplicates one of his. In any case I
don't think he ever proposed it for three slot ballots.

Forest

On Thu, 12 Feb 2004, [iso-8859-1] Kevin Venzke wrote:

> 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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