[EM] The "Single Contest" method

[Kevin Venzke]

Hi all,

After playing a bit more I am unsure what further improvements I could
make to "SC," so I will finally define it now.

1. Rank ballots. Equal ranking and truncation are allowed. At the same
time, the voter explicitly places an approval cutoff.
2. If there is a strict majority favorite (strict! no tying candidates)
then this candidate wins.
3. Otherwise, elect the winner of the pairwise contest between the pair
of candidates who minimize the number of voters that approved neither 
one.
4. If pairs are tied on this measure, break the tie for the pair whose
winner has the most voters on his side in that contest (i.e. WV). (You
can also use winner's approval instead of winner's votes-for, but this
doesn't seem to be as good in most cases.)

So, this method determines an instant runoff to perform by letting the
voters' approval cutoffs suggest what the electorate thinks the most
important contest actually is. This is not monotone, but the limits of
approval make it difficult to do anything insincere as a result of it. 
I have also "improved" it (both aesthetically and performance-wise) in
comparison to an approval distance measure by allowing approval for both
of the two candidates to count as a vote for that pair. (This is 
basically what you are doing: Using approval to vote for the runoffs 
you'd prefer. In effect you are voting for every pairwise contest to
be the "single contest," except for the contests between the candidates
you didn't approve.)

Because the rankings are used to resolve a single contest, there is
almost no strategy there. There is some compromise incentive due to
the majority favorite rule, but it appears so far to be a necessary evil.

I tried many things as a replacement for majority favorite, but most
made the method much worse. I tried allowing a non-strict majority
favorite to win, as well as adding on the front a Condorcet// or CGTT// 
or a DSC-ish limit (i.e.: disregard pairs of candidates including some
candidate with a majority-strength solid coalition against them).
Most of these bring compression, truncation, or burial incentive. I
also tried having no rule #2 of any type at all, which was disastrous.

Here is the best loosening of the majority favorite rule that I came
up with, which seems like an improvement in some scenarios I tested:

"1. (edit so that you can't vote equal rankings at all. I didn't define
this rule for that setting.)
2. If there is a strict majority favorite then this candidate wins.
*Otherwise*, see whether any candidates could become a strict majority
favorite if they counted their strict first preferences (X>...), plus
the second preferences (Z>X>...) on ballots where the first preference
(Z) received less than half the approval score that this candidate (X)
received in first preferences. If there is such a candidate, elect the
one who would have the highest total."

The idea is that if Z is unmistakably weak (to the point that no one is
even concerned that they may accidentally give him too much approval),
the voter can dump him and let X have the vote as a FP. But, in many
scenarios this rule makes very little difference.

I really would like to find something that is a little more generous
than majority favorite, but which can't be the focus of manipulation
by other voters.

It is possible to come up with similar methods when you see that the
trick is to do as much as you possibly can using approval and not 
rankings. MAIRO (i.e. if there are multiple majority approved then use
the rankings to have an instant runoff between the top two approved,
otherwise don't use the rankings at all) tests well a lot of the time
but I hardly think it could be a proposal, as the logic for selecting
the two finalists is manipulable by nominating multiple candidates.

There is also a method I call MAOIR (Max Approval Opposition Instant
Runoff) where we check for a majority favorite, and failing that, take
the contest between the approval winner and the candidate expressing
the most approval opposition to the approval winner. However, this 
method tests noticeably worse than SC and MAIRO.

So, it's perhaps not enough to pick any pair of finalists that you can
rationally think up. Using approval to pick a single pairwise contest
isn't the whole trick. You have to pick the contest "well," and maybe
there is room for discussion there. Maybe I got lucky and picked the
best criterion on my first try, though.

Thanks to Forest (for getting me thinking about nonmonotone metrics)
and Markus (for the original idea that just because you have information
doesn't mean you ought to use it) and Juho (for his criticism of the 
Smith set, which is quite related).

Let me know if you have any thoughts.

Kevin Venzke