[EM] Two approval ballot methods using approval opposition

[Kevin Venzke]

Hello,

The first method I ever came up with was two-slot MMPO, which I'll 
call "MMAO" for "Minmax(approval opposition)." In MMAO, the candidate
with the least greatest approval opposition from another candidate is
elected.

I designed this method because I wanted B to win on these ballots:

25 A
25 AB
25 BC
25 C

But in Approval, this is a three-way tie. In MMAO, B's score is 25 while
A and C both have 50.

The major downside of MMAO is that it can elect a candidate who could
not possibly beat some other candidate pairwise. In the above scenario,
if you add a million "A" and a million "C" votes, B still wins.


I was thinking recently about the tendency to want to do something
special when more than one candidate receives majority approval. I guess
a large reason why this desire exists is that this scenario is where
Approval could fail to pick a majority's favorite. But another reason
might be that there is necessarily overlap in support between the
majority-approved candidates. There is no way to guess which one would
defeat the other pairwise. Clearly both candidates are decent, and it
might be a bit unsatisfying to just elect the most-approved.

So how about this?: If more than one candidate is majority-approved,
elect the one of these with the least greatest approval opposition.
Call this method Majority Approval,Minimum Approval Opposition or "MAMAO."

This preserves Approval's satisfaction of FBC and clone independence (that
is, if we use Woodall's view of Approval), and also reduces the negative
point of Approval, that approving an additional candidate never helps
but may hurt a preferred candidate. Using the AO measure, it's possible
that approving another candidate will hurt the AO score of a candidate
you don't approve, and move the win to a candidate that you did approve.

Here, AO doesn't create a problem of electing candidates who could 
not possibly beat another candidate, since majority-approved candidates
could always pairwise beat each other. We know that because the overlap
in support would always be enough to make one candidate or the other
the only majority-approved candidate, if the overlapping voters decided
to pick one candidate or the other.

This principle can be extended to all pairwise contests, using a two-slot
Condorcet principle with the "tied at the top" rule.

Specifically: Disqualify every candidate X for whom there is some candidate
Y such that X's approval is less than Y's approval opposition to X (that
is, the number of voters approving Y but not X). It isn't possible that
all candidates are disqualified. Then elect the candidate with the best
AO score. (Use approval to break ties.) Call this method "tCMAO."

I ran some simulations to see how often these rules differ from Approval
given one particular setup: The case that all ballots are A, AB, B,
BC, or C (i.e., no AC ballots), with a random number of each type. This
is meant to simulate a spectrum of voters, with A and C at opposite
ends of the spectrum.

The results are interesting. MMAO, MAMAO, and tCMAO *never* disagree with
Approval when the Approval winner is B. They only disagree with Approval
in order to move the win *to* B.

Percentage of elections in which each method disagreed with Approval
(in order to elect B):
MMAO: 11.3%
MAMAO: 2.7%
tCMAO: 9.6%

It's surprising that tCMAO comes so close to matching MMAO. In some
trials the MMAO winner had under 10% of the approval of the Approval
winner. In the other two methods, it's impossible for the winner to have
under half of the approval of some other candidate.

A philosophical question would be whether there is any justification
for electing someone other than the Approval winner when we're using
approval ballots. Everything being equal, a more-approved candidate is
more likely to be the CW than a less-approved one. But on the other
hand, AO figures may substitute for a guess regarding what voters' lower
preferences might have been if they were using rank ballots.

Now for an example or two:

28% A
5% AB
12% B
34% BC
21% C

B and C have majority approval (and therefore one doesn't defeat the
other according to the "tied at the top" rule); the approval winner 
is C. But MAMAO and tCMAO both elect B.

25% A
22% AB
2% B
25% BC
26% C

Only C has majority approval. But tCMAO elects B. It finds that C doesn't
clearly beat B, and then notes that C has 47% opposition from A while
B's worst contest is 26% opposition from C.

Here's an example where tCMAO elects a candidate with barely half the
approval of the Approval winner:

34% A
30% AB
5% BC
31% C

A has 64% approval, but B wins with 35%. A doesn't clearly defeat B
according to the "tied at the top" rule, and A faces 36% opposition
from C while B faces only 34% from A.

I think that's all I'll say for now. Any thoughts?

Kevin Venzke



	

	
		
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