[EM] A few more Bucklin variants, because why not?

[Etjon Basha]

Dear gentlemen,


A while ago I did write here about the Iterated Bucklin
<https://electowiki.org/wiki/Iterated_Bucklin> method on which I’ve
recently had a chance to think and generalize about a bit more. Maybe some
of the below could be novel or otherwise of interest.


First, and for our purposes today, let's define the *Serious Candidates Set*
in the context of a ranked ballot, to include those candidates who would
win an approval count if they served as the approval cutoff across all
ballots.


In the [2:A>B, 3:C>A, 4:A>B] election as an example, the Set would include
A and B only, as applying the cutoff at C would still elect B.


I’ve been checking some random simulations from Kevin Venzke’s
votingmethods.net, and here are some properties of this Set that I
*suspect*:

1.        If there is a Condorcet Winner, this Set should always include
them.

2.        Otherwise, this Set should always partially overlap with the
Smith Set.

Now, quite a few methods emerge once the Serious Candidate Set is isolated
(by actually checking the approval winner once every candidate is used as a
cutoff). The five below allow truncation and equal ranking, and have been
checked (again courtesy of votingmethods.net) to ensure that they are
different from one-another and the 40-odd other methods Kevin has
aggregated over there.


So, which member of the Serious Candidate Set should be elected?

1.        Electing the Serious Candidate that wins their cutoff count by
the most approvals. Rather obvious but not too much of an improvement over
Approval (if any at all). Terrible Later No Harm failures, though this is
in the context where truncation is allowed. Fails Condorcet.

2.        Electing the Serious Candidate that wins their cutoff count by
the *least* approvals. A bit counterintuitive, but winning by the least
means that the winner had to “dip” the least into each approver’s rankings.
If this is not compliant with Later No Harm, it should at least fail
rarely. It would fail Later No Help spectacularly though, indeed having a
huge incentive to always rank your least favorite candidate that is still
likely to win last, instead of leaving them unranked. Unfortunately, I’ve
seen it elect the Condorcet Loser at least once.

3.        Electing the Serious Candidate that wins their cutoff count by
the most approvals *compared to the runner up*. May fail Condorcet the
least. Likely the most sensible of the bunch.

4.        Iterated Bucklin (now fitting into this generalised family) will
always elect a member of the Set, but it seems to be neither of the three
above with consistency. I cannot seem to find the pattern the method lands
on.

5.        Electing the Serious Candidate that wins the election if the
cutoff is set at the FPP winner. If the FPP winner is in the Set to begin
with, they will be elected. Otherwise, again a method that elects a winner
from the set through no obvious pattern. Of particular interest to me since
it’s the only method in here that can be hand-counted with relative ease
(it’s just an FPP count and an approval count after that).

For reference, standard Bucklin may not always elect members of the Set so
cannot be retconned into this tree. I've tested quite a few other methods,
and there are some for which I'm still to find a failure to elect from the
Serious Candidates Set, including Borda and, unsurprisingly, many approval
variations and Approval-Condorcet hybrids.

Just some preliminary thoughts above, hopefully of some interest.


Best regards,

Etjon Basha
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