Re: current A.6 draft [examples]
Raul Miller wrote:
It's not fair to base an argument on an axiom which is known to be false.
On Sun, Dec 08, 2002 at 04:45:07PM +1000, Anthony Towns wrote:
It doesn't matter whether the axiom is false as written: it's trivial
to salvage its intended meaning (by either dropping quorum requirements,
or qualifying the axiom to apply only when the quorum requirements have
already been met for all the options listed).
I can accept, with that clause added, the axiom that the option ranked
first must win the election.
Quorum requirements was not intended to be in the scope of my argument.
I didn't mention quorum requirements, but I should of mentioned
explicitally that they were out of its scope. The argument deals with a
set of votes, a set of options with optional supermajority requirements,
one of which may optionally be the default option. It doesn't deal with
quorum requirements, supression of free speech, violence, vote rigging
or anything else that may effect the democraticness of a method.
Admitidly, it was reasonable to assume quorum requirements to be part of
the arguments scope, but I did not intend that.
I am uncomfortable this for the axiom that the option ranked last must
lose. It's just too arbitrary. For example, consider also a ballot with
only one option (not that our current system allows this). The resulting
statement is rather akward to accept as being true without proof.
Myself wrote:
> 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."
Note the minimum inclusion of two options A and B for 'majority loses'
axiom to have an effect. These are intended to be two different options.
Personally, I'm more unconfortable the idea of an option not losing when
a majority of votes rank it last and there are other non-supermajority
options. But which is more unconfortable is for debate.
--------------------------------
In any case, later on I'll define another criteria in my opinion an
election system should follow, and will attempt to prove that CCSSD (and
newly defined DPCCSSD) does follow and the Dec 7 draft does not. This
criteria 'Consistancy', is basically that if an option wins when it is
not the default option, it should win when it is the default option.
---
Firstly, I'm going to define "Default Protection CCSSD", which I will
call DPCCSSD. Its the same as my old CCSSD, except for rule 9.
Plain CCSSD is defined at
http://lists.debian.org/debian-vote/2002/debian-vote-200212/msg00020.html.
Its all leading up to another proof, I'll get there...
---
Definition of "Default Protection Considered Clone-Proof Schwartz
Sequential Dropping" (DPCCSSD).
(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
(9a) A has a beatpath win to B if A and B are both not default options
and 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.
(9b) A has a beatpath win to B if A is the default option, and there is
a non-zero X such that A has a beatpath to B of strength X.
(9c) A has a beatpath win to B if B is the default option, and there is
non-zero X such that A has a beatpath to B of X, and there is no
non-zero 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.
Consistancy Criteria - "If election X and election Y have identical
votes and supermajority requirements, and election X has a default
option of A, and election Y has a default option of B, and B is the
winner of election X, then B must be the winner of election Y."
---
Consider this election, with no supermajority requirements.
4 CBA
3 BAC
2 ACB
B defeats A 7:2
C defeats B 6:3
A defeats C 5:4
---
Default option A.
-
Dec 7 Draft: Drop weakest non default defeat (C > B)
B > A > C
Dec 7 Draft: B wins.
-
CCSSD:
All considered.
Strongest beatpath A to B = 5:4
Strongest beatpath B to A = 7:2
Beatpath win from B to A.
Strongest beatpath A to C = 5:4
Strongest beatpath C to A = 6:3
Beatpath win from C to A.
Strongest beatpath B to C = 5:4
Strongest beatpath C to B = 6:3
Beatpath win from C to B.
C has no beatpath win against it.
C wins.
CCSSD: C wins.
(note this result is the same as plain CSSD).
-
DPCCSSD:
All considered.
A has a Beatpath Win to B (since A is default and has a beatpath to B).
A has a Beatpath Win to C (since A is default and has a beatpath to B).
DPCCSSD: A wins.
-
Summary
A Default
Draft: B Wins.
CCSSD: C Wins.
DPCCSSD: A Wins.
--
B Default.
-
Draft:
Drop weakest non default defeat (A > C)
C > B > A
Draft: C Wins.
-
CCSSD: C Wins.
-
DPCCSSD: B Wins.
---
All results, summerised.
Method | Default A | Default B |
---------|-----------|-----------|
Draft: | B Wins | C Wins |
CCSSD: | C Wins | C Wins |
DPCCSSD: | A Wins | B Wins |
----------------------------------
---
Consistancy Criteria means that immediately re-running an election with
the default option of the previous winner never produces a different
result, and also that being the default option never negatively affects
and options chances of winning an election. The current draft proposal,
in some cases, (such as above) disadvantages the default option, which I
will assume not the aim of protecting supermajority defeats.
--
Now, the proof.
Consistancy Axiom: "If a method fails Consistancy Criteria, it is not
consistant, otherwise it is consistant."
Consistancy Criteria: "If election X and election Y have identical votes
and supermajority requirements, and election X has a default option of
A, and election Y has a default option of B, and B is the winner of
election X, then B must be the winner of election Y."
Ties are out of this proofs scope.
-
Draft method.
Consider the election with votes below and no supermajority requirements.
4 CBA
3 BAC
2 ACB
Let X be the election above with default option A, and Y be the election
above with default option B.
Since,
Election X and election Y have identical votes and supermajority
requirements,
Election X has the default option of A.
Election Y has the default option of B.
B is the winner of election X.
Therefore, for draft method to be consistant, B must be the winner of
election Y.
However, C is the winner of election Y, according to the draft method.
Therefore, draft method is not consistant.
-
CCSSD:
The result of an election is independent of the default option. (As the
default option is not mentioned in the CCSSD definition).
Let election X and election Y be elections that only differ by their
default option.
Let election X have a default option of A, and election Y have a default
option of B.
Since results are independent of the default option, election X and
election Y produce the same winner.
Let the winner of election X and election Y be either A, B or C.
If the winner is B,
Election X has a default option A,
Election Y has a default option B,
B is the winner of election X.
Therefore B must be the winner of election Y.
Which is true.
Therefore, method is consistant for winner B.
Similarly consistant for winner A (swap A with B, X with Y when subbing
into consistancy criteria definition).
If C is the winner, there are no consistancy criteria to pass, as
consistancy criteria only deals with default option winners, and C is
not a default option of either election.
Therefore, consistant for C as winner.
Therefore, consistant for all winners.
Hence, CCSSD is consistant.
---
DPCCSSD:
Let election X and election Y be elections that only differ by their
default option.
Let election X have a default option of A, and election Y have a default
option of B.
Without loss of generality, assume election X produces the winner B or C.
Without loss of generality, assume that if election X produces the
winner C, then election Y produces the winner C or D.
(Both the 'Without loss of generality' I'm sure are true, basically
taking any other conditions is just swaping letters around. It will take
a while to prove it though, try to find a counter example, and you
should find all counter examples just involve swaping letters and doing
different subsitutions.)
Since C and D are not default options, if election X produces the winner
C, DPCCSSD is consistant. (Non default winners are not in the
consistancy definitions scope).
Since election X produces the winner B, in election X, B must have a
beatpath win to all other options, and therefore, B must have a non-zero
strength beatpath to all other options.
Beatpaths are uneffected by default options.
Therefore, in election Y, B must have a non-zero strength beatpath to
all other options.
By DPCCSSD rule (9b), B must have a beatpath win to all options in
election Y.
Therefore no options have a beatpath win to B.
Therefore, B is the winner of election Y (rule 10 and 11).
Since
Election X has default option A,
Election Y has default option B,
B winner of election X
Therefore
For DPCCSSD to be consistant, B must be the winner of election Y.
Which it true.
Therefore DPCCSSD is consistant with B the winner of the election X.
Hence, due to without loss of generality assumptions, DPCCSSD is consistant.
---
Summary.
Draft is not consistant.
CCSSD is consistant.
DPCCSSD is consistant.
---
Back into opinion.
The problem with the current draft method is that it resolves defeats in
favor of the default option. In many cases, this is determental to the
default option. At least CCSSD never penalises the default option, and
DPCCSSD resolves all cycles in favor of the default option. This
provides real protection for the default option, as it can not be
defeated in a cycle due to a particular arrangement of strengths of
defeats. I've been getting the vibe from most people that a preferred
method would favour the default option, however in some cases draft
seems to discourage a default option win.
I've proven that the draft causes options which recieve a majority of
last preferences can be selected ahead of non-supermajority options, and
above I've attempted to prove the draft method is inconsistant. Assume
that proof is reasonably sound, what property or results does the draft
method have over CCSSD and DPCCSSD? The only thing I've heard bad about
CCSSD is this result.
Raul Miller wrote:
The ballot has options A, B and F, A has a supermajority requirement
of 3:1, B has a 1:1 majority requirement and F is the default option.
60 voters vote ABF
40 voters vote BAF
10 voters vote F
B defeats F by 100 (100:10)
A defeats F by 100 (100:10)
A defeats B by 60 (60:40)
CCSSD produces B as the winner, Draft method produces A as the winner.
If you insist on A as the winner, which I think is not ideal in my
opinion, its easy to change the definition slightly.
Create "I Don't Like This Version Of Default Protection Considered
Clone-Proof Schwartz Sequential Dropping" (IDLTVODPCCSSD)
By changing rule (5) to this.
(5) A is considered if it has no supermajority requirement, or
superchallenges the default option.
The result of the above election would be A.
On a unrelated note, you can change rule (6) from
(6) A is considered if A challenges B, where B has supermajority
requirements greater than or equal to A.
to
(6) A is considered if A superchallenges B, where B has supermajority
requirements greater than or equal to A.
If that tickles your fancy. However, deleting rule (6) will violate
consistancy (as an option not considered due to the absense of 6 could
be considered by (5) in a recount with a changed default option.)
Changing rule (6) will not effect any of my two proofs, changing rule
(5) will violate consistancy (as different options would be considered
dependent on the default option). This is why I don't like the above
version of IDLTVODPCCSSD, but at least this method only violates
consistancy when different considered options result (that is, when
there are supermajorities), unlike the current proposal which violates
consistancy when there are not supermajority options. It also does not
grant wins to options which recieve a majority of last preference votes
when there are other non-supermajority options. Personally, I like
DPCCSSD better that IDLTVODPCCSSD, but even IDLTVODPCCSSD seems to be
better than the draft, except for its name.
------------------
Now I ask the question, what properties does the draft method which are
better than CCSSD, DPCCSSD, and IDLTVODPCCSSD, except for ease of
pronunciation. I'm not saying my method is perfect, its just that, until
someone points of a flaw in it, its hard to fix. And I'm not a self
critical sort of person, unfortuantly, so I need others to help. And
please, could someone think of a better name?! How does 'Badcore CSSD'
sound? 'CSSD w00t!' maybe? 'Smooth CSSD'? Hopefully not 'Inherently
flawed and incredible dodgy CSSD', but I'll let you decide.
Thanks
Clinton
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