Re: current A.6 draft [examples]
Firstly, I mistakenly defined the strength of defeats in my last
definition, so I've changed rule (3) to match my implimentation.
Strength is now measured by how many votes prefer A over B. In the
previous definition I defined the strength of defeats as the difference
between how many votes prefer A over B and how many votes prefer B over
A. The fixed definition of strength of defeats matches my implimentation
seems to match the current draft (except where default options and
supermajorities are involved).
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Definition of CCSSD
(1) A defeats B if more votes prefer A over B than B prefer over A.
(2) A challenges B if more than or an equal number of votes prefer A
over B than prefer B over A.
(3) A defeats B by X, where X is equal to the number of votes that
prefer A over B, if A defeats B.
(4) A superchalleges B, where A has a supermajority requirement of (X:Y)
if the number of votes that prefer A over B multiplied by Y is greater
than or equal to the number of votes that prefer B over A multiplied by X.
(5) A is considered if A superchallenges all options with supermajority
requirements less than A.
(6) A is considered if A challenges B, where B has supermajority
requirements greater than or equal to A.
(7) A has a beatpath to B of strength X, if A and B are considered and A
defeats B by X, or if A, B and C are considered and A defeats C by Y and
C has a beatpath to B of strength Z, where X is equal to the minimum of
Y and Z.
(8) A has a beatpath to B of strength 0 if there is no non-zero X such
that A has a beatpath to B of strength X
(9) A has a beatpath win to B if the largest X such that A has a
beatpath to B of X is greater than the largest Y such that B has a
beatpath to A of Y.
(10) A is a finalist if A is considered and there is no B such that B
has a beatpath win to A.
(11) A is a winner if A is a finalist and there is no B such that the
casting vote prefers B over A.
---
Raul Miller wrote:
On Fri, Dec 06, 2002 at 03:13:59PM +1100, Clinton Mead wrote:
I basically have a lot of questions, starting with: What do the other
letters stand for (CSSD?). I ask, because the system you propose is
further from pure Clone-proof Schwartz Sequential Dropping than mine is.
If you remove rules (2), (4), (5) and (6) about challenges, superdefeats
and considering, and remove the considered conditions from (7) and (10),
I believe it is functionally equivilent to CSSD. Correct me if I'm
wrong. The definition above is the way I've implimented it (ignoring any
further mistakes on my own behalf), hence its the way I've described it.
http://www.barnsdle.demon.co.uk/vote/sing.html mentions the equivilance
of the beatpath and CSSD methods.
The B voters as a whole prefer A over B, and being sincere voters,
propose A and vote sincirely for it. If the B voters were insincere,
by voting B over A, they would of recieved a more preferable result.
Why do you say that "B wins" is more preferable to "tie between B and D"?
If the person with the casting vote chooses "B", the outcome is the same.
If the person with the casting vote chooses "D", we continue with further
discussion on how to pick a better option [perhaps "A" will win in the
next election, perhaps "B" will win, or perhaps some better option "C"
will win].
Consider this slightly different vote, under the A.6 draft method, that
doesn't produce a tie.
A requires 3:1 supermajority, D default.
4 ABD
1 ADB
1 BDA
1 DAB
D defeats A 6:5 (due to supermajority)
A defeats B 6:1
B defeats D 5:2
Drop B defeats D
D > A > B
D wins.
D has won, despite being ranked last on a majority of votes.
So instead of the vague opinion I've given in previous posts, heres an
attempt of a proof that selecting a method selecting D as the winner is
undemocratic.
---
Axioms
Democratic Axiom - "If a method does not pass one of the following
axioms, then the method is undemocratic"
Majority Wins Axiom - "In a non-supermajority election, if is an option
A solely ranked first on a majority of votes, then if option A must win."
Minority Loses Axiom - "In a non-supermajority election, if there are
options A and B, and option B is solely ranked last on a majority of
votes, then if option B must not win."
Supermajority Election Axiom - "In a supermajority election, the
non-supermajority majority wins and minority loses axioms must hold on
the set of non-supermajority options".
Applying 'Supermajority Election Axiom' on 'Majority Wins Axiom' and
'Minority Loses Axiom', we get this derived criteria.
Supermajority Majority Wins Derived Criteria - "In a supermajority
election, if there is a non-supermajority option A solely ranked first
on a majority of votes, then option A must win."
Supermajority Minority Loses Derived Criteria - "In a supermajority
election, if there are non-supermajority options A and B, and option B
is solely ranked last on a majority of votes, then option B must not win."
---
Proof that A.6 draft is undemocratic.
---
In this election, using the A.6 draft method.
"""
A requires 3:1 supermajority, D default.
4 ABD
1 ADB
1 BDA
1 DAB
"""
Non-supermajority option D is solely ranked last on a majority of votes.
Since there are non-supermajority options B and D, and option D solely
ranked last on a majority of votes, non-supermajority option D must not
win. (Supermajority Minority Loses Derived Criteria).
The draft method is not democratic if it does not follow 'Supermajority
Minority Loses Derived Axiom' (Democratic Axiom)
Therefore, draft method is undemocratic if option D wins.
Since D wins according to draft method, draft method is undemocratic.
---
Proof that CCSSD is not undemocratic
---
Let option A be a non-supermajority option who is ranked solely first on
a majority of votes.
As there are no options with supermajority requirements below A, A is
considered by definition of CCSSD rule (5).
Since A is ranked solely first on a majority of votes, A defeats all
other options.
Hence A has a beatpath to all other options, and there is no beatpath
from any other option to A.
Since A is considered, and no option has a beatpath to A, and all other
options are beaten by a beatpath from A, therefore option A is the sole
winner.
Therefore, CCSSD satisfies 'Supermajority Majority Wins Derived
Criteria' and 'Majority Wins Axiom'.
-
Let option B be a non-supermajority option who is ranked solely last on
a majority of votes, and let option A be another non-supermajority
option in the election.
As there are no options with supermajority requirements below B, B is
considered by definition of CCSSD rule (5).
Since B is ranked solely last on a majority of votes, all other options
defeat A.
Hence all other options have a beatpath to B, and B has no beatpath to
any other option.
Since B has no beatpath to any other option, all other options have a
beatpath win to B.
Since all other options have a beatpath win to B, option A must have a
beatpath win to option B, therefore, option B is not a finalist, due to
CCSSD rule (10).
Since option B is not a finalists, then option B is not a winner.
Therefore, CCSSD satisfies 'Supermajority Minority Loses Derived
Criteria' and 'Minority Loses Axiom'.
-
Since CCSSD satisfies 'Supermajority Majority Wins Derived Criteria' and
'Supermajority Minority Loses Derived Criteria', CCSSD satisfies
'Supermajority Election Axiom'.
Since CCSSD satisfies 'Majority Wins Axiom', 'Minority Loses Axiom' and
'Supermajority Election Axiom', CCSSD satisfies 'Democratic Axiom'.
Therefore, CCSSD is not undemocratic.
---
Since A.6 draft method is undemocratic, and CCSSD is not undemocratic,
CCSSD is less undemocratic than the A.6 draft method.
---
Unless one of my four axioms are unreasonable (point out if they are), a
step in my proof is flawed, or if I've mis-calculated the result of this
election, the current draft method is undemocratic.
CCSSD would select B as the winner in the above election, as only B and
D would be considered, since A does not beat all non supermajority
options and can not be considered by rule (5), and since B defeats D, B
is the winner.
---
Let's start with a justification of why this approach to supermajority
handling is better.
Thanks,
Hope the above helps more than my previous ramblings.
Thanks
Clinton.
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