[Election-Methods] Another method on approval ballots (approval-limited minimum opposition)

[Chris Benham]

Kevin Venzke wrote:

>So this compromise occurred to me:
>
>"Elect the candidate to whom the greatest opposition is the least 
>(breaking ties in favor of greatest approval), whose approval is at least
>as high as the greatest opposition to him."
>  
>

This didn't interest me much when I first saw it because I can't see 
that getting voters to do nothing but fill in
approval ballots and then not elect the Approval winner can be seriously 
justified.

But I think for methods that also collect ranking information it could 
be the basis of  a potentially useful set:
"the set of candidates whose approval score is higher than  their 
maximum approval opposition score".

If we interpret approval as ranking above bottom or equal-bottom, then 
we have a concept that is immune
to Irrelevant Ballots and implies Plurality and Minimal Defense. The set 
always includes the approval winner
and I think also the CW if there is one. (And of course we have a 
maybe-useful criterion: "the winner must
come from the set.")

So one idea for a Condorcet method: "elect the member of the set that is 
highest-ordered by MinMax (Margins)".
[I'm not sure if we need to do it that way or if we can just first 
eliminate (and drop from the ballots) the non-members
of the set.]
This should lack WV's random-fill incentive, and I think meet mono-add-top.

Another alternative, for a method that hopefully meets FBC, would be to 
use MinMax (PO) instead of  MM(M).

In the classic  49A, 24B, 27C>B scenario the set is {B}.

What  do you think?

Chris Benham



>Hello,
>
>I was thinking recently about how one might design a method aimed to
>minimize potential for regret at least for supporters of the median
>candidate. By "regret" I mean especially the situation that supporters
>of the median candidate give the election away by voting also for a 
>second preference.
>
>I had an idea worth sharing, I think...
>
>On cast approval ballots I like to guess that the median candidate is
>the one to whom the greatest opposition is the least. (The greatest
>opposition to a candidate X is defined as the size of the largest group
>of voters who approve a common candidate and disapprove X. In other
>words, if you remove all ballots approving X, what is then the highest
>approval score of any candidate?)
>
>I like this measure because typically supporters of the median candidate
>(when there is also a "left" and a "right" candidate) can't hurt this
>candidate by also approving the "left" or "right" candidate that is their
>second choice. It only hurts the median candidate sometimes when the
>greatest opposition to the second choice is the median candidate (so
>that this second choice is turned into the median candidate). But
>usually we'd expect that the greatest opposition to the second choice
>is coming from the opposite side of the spectrum, not the median.
>
>The trouble with always electing this "median" candidate is that he might
>have very little approval:
>
>49 A
>1 AB
>1 BC
>49 C
>
>B would be the "median" candidate with just 2 approval.
>
>Assume that approved candidates would have been ranked if a rank ballot
>had been used. Assume also that disapproved candidates would not have
>been given any ranking. Given these assumptions, a candidate in an
>approval election might have been the Condorcet winner if and only if
>his approval score is higher than the greatest opposition to him.
>
>So this compromise occurred to me:
>
>"Elect the candidate to whom the greatest opposition is the least 
>(breaking ties in favor of greatest approval), whose approval is at least
>as high as the greatest opposition to him."
>
>I don't have proofs, but simulations of mine couldn't find any monotonicity
>or FBC failures with this. (Actually I first tried "elect the candidate
>with the least max opposition IF his approval is at least as high as etc.,
>otherwise elect the approval winner," but this had both problems.)
>
>Compared to Approval this makes a difference in a scenario like this:
>
>30 A
>25 AB
>15 CB
>30 C
>
>Approval scores are A 55, B 40, C 45.
>Max opposition scores are A 45, B 30, C 55.
>
>Approval elects A. This method ("ALMO") identifies B as closest to 
>"median" and sees that B has enough approval to possibly be the Condorcet
>winner on rank ballots, and so elects B.
>
>(Naturally you can argue that this isn't an improvement, or that
>"opposition" isn't a useful concept.)
>
>Any thoughts?
>
>Kevin Venzke
>
>
>
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>
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