Unusual behaviour regarding default options and supermajority requirements.
Assume a set of votes as below.
CBA
CBA
CBA
ACB
BAC
Where A is the default option, and both B and C have a supermajority
requirement of 2:1.
B v A = 4 v 1
C v B = 4 v 1
C v A = 3 v 2
Noting that C is the condorcet winner.
Going through the steps as in the Nov 16 draft.
1. Trivial
2. All options beat the default option.
3. No quorums.
4. C is eliminated as it does not beat the supermajority requirement over A.
5. B defeats A, so B wins.
Not only has the condorcet winner lost, but this system has a distrubing
property if the election is recounted, with the same set of votes, there
is a new winner.
Since B won, we'll make B the default option and give it supermajority
protection of 2:1 vs all other options.
Then, re-run the election, assuming the same votes.
1. Trivial
2. A doesn't beat the default option B, so A is eliminated.
3. No quorums.
4. C satisfies the supermajority requirement of 2:1 over B, so stays in
the running.
5. C beats B, so C wins. (as it should of in the first vote, in my
humble opinion).
So the current vote counting system, as far as I can tell, has the
property that re-running an election with identical votes can produce a
different result. Assuming that supermajority protection is generally
designed to promote stability, it instead promotes instability, forcing
people to re-raise identical amendments to finally produce a sencical
result.
It might of been that I've just misunderstood the wording of the Nov
16th draft, and if so, either I've made a big mistake in the way I've
read the draft or a little clarification is nessacary.
---
One way to impliment supermajority and quorums is to just multiply or
add to the votes between the default and non default options.
For example, in the above election.
B v A = 4 v 2 (multiply A's vote by supermajority requirement)
C v B = 4 v 1
C v A = 3 v 4 (multiply A's vote by supermajority requirement)
Remove weakest proposition, A > C
C > B > A
C wins.
---
However, even the above method has problems.
Assume the a supermajority of 3:1 now.
B v A = 4 v 3
C v B = 4 v 1
C v A = 3 v 6
Remove weakest proposition, B > A
A > C > B
A wins.
However, if the C voters realised this outcome, with a little strategy
they could defeat the supermajority.
The C voters would propose a vote between A and B.
B v A = 4 v 3 (with A's supermajority protection)
B wins.
Now, B gets supermajority protection, but the C voters propose a vote of
B v C.
C v B = 4 v 3 (with B's new supermajority protection)
C wins.
Here, the C voters have made their option the winning option by
proposing two election, one of which they were intentionally not involved.
Of course A voters could counter this by proposing C as an option in the
vote, but this is the sort of strategy we are trying to avoid.
---
So heres another proposal, that seems to work well in this case.
(i) If the default option defeats all other options, considering its
supermajority and quorum benefits, than it is declared the winner.
(ii) Otherwise, conduct the ballot normally, ignoring supermajorities
and quorums.
This proposal has the property that the default option is not treated
specially unless it has quorum and supermajority benefits.
It also stops a 'default option loop', when for example, we have a
situation where
A > B
B > C
C > A
and A is the default option.
Here, B is eliminated, because it is does not defeat the default option
(step 2) and C is the winner (as it beats A).
Then, the election is re-run, with same votes, however C is now the
default option.
Here, A is eliminated, as it does not defeat the default option, and
since B beats C, B is the winner.
And the loop continues.
In effect, the default option is at a disadvantage, as its strong
defeats are removed early, and weaker defeats come to eliminate it. I
assume this is in contrast to the intention of the default option.
Using my suggested procedure, where the default option comes into play
only if the default option defeats all other options (with the benefit
of any supermajority and quorums), stops the loop caused by early
elimination of non-default options, while still requiring one option to
beat the supermajority.
I don't see any problem with ignoring the supermajority or quorum is one
option defeats it. This is because, like illustrated in the main
example, even though option C does not beat the super majority, under
the current system, option C votes could just use two elections, the
first with A v B, to get B elected (as B beat the supermajority), and
then B v C to get C elected. Eliminating options that don't beat the
supermajority doesn't stop them from being elected, it just makes it a
hassle and requires strategy for them to be elected. So, therefore,
ignoring supermajorities and quorums after one option defeats the
default option with the protection of quorums only prevents voters
resorting to strategy.
---
In my opinion, having repeated elections with identical votes produce
different results and requiring repeated elections to get sencical
results is worse than, dare I say it, non-monotonicity! *shock, horror*
Anyway, there's my slightly incoherent two cents, hope it helps.
Clinton
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