Re: Dec 15 voting amendment draft
Raul Miller wrote:
On Fri, Feb 14, 2003 at 05:33:37PM +1100, Clinton Mead wrote:
Monotonicity Criteria.
If
(i) There is an election X in which option A wins,
(ii) There is a vote V that ranks option B above option A.
(iii) There is an election Y which is identical to election X except
that vote V has option A and option B swaped on their preference list.
then
Option A must win election Y.
I have a few quibbles about quantifiers (for this to be correct,
(ii) should begin "For any vote V" and (iii) should begin "For
all elections Y"), but essentially this is correct.
I'm not sure why "For any vote V" is required.
I'll illistrate with a simple example, additive identity.
Additive Identity Criteria.
If
(i) There is an element A.
then
A + 0 = A
So if there exists 5, then 5 + 0 = 5
I could also say.
For any A,
A + 0 = A
However applying your suggestion of replacing "There is" with "For any"
to this example.
If
(i) For any element A.
then
A + 0 = A
Doesn't seem to make gramatical sence to me. As you point out, the
meaning is the same, I just find my definition easier to use in proofs.
Smith Criteria.
That definition was also too relaxed -- mind you, in this relaxed form
the proposal complies with this criteria where if it was expressed in
exact form the proposal wouldn't. But I guess it doesn't really matter,
since that's not what Clinton is writing about.
I'm not sure how relaxed you believe it is.
It is relaxed, as you point out, with regards to supermajorities, but it
isn't relaxed with regards to non-supermajority elections.
Lets take this election.
2 AB
Smith criteria says the winner shall come from the set {A, B}.
Smith criteria also says the winner shall come from the set {A}.
If it fails once, it fails totally, so the only way for smith criteria
to be satisfied is for A to be the winner.
The criteria implies the innermost unbeaten set, without having to
explictly state it, which simplifies proofs, IMHO.
You may have relised this, however your previous response about your
interuptation gave me the contrary view, so I thought I should clarify it.
Also, another criteria.
Participation Monotonicity.
If
(i) There is an election X in which option A wins.
(ii) There is a vote V ranks option A over option B.
(iii) There is an election Y identical to election X except that it has
an additional vote V.
then
Option B must not win election Y.
I did a web search on "participation monotonicity", the only hit I found
was http://www.wiwi.uni-bielefeld.de/~imw/papers/332/convexfuzzygames.pdf
Where did you find this criteria? [Did you make it up?]
I have seen it before, and I have noticed it has been talked about on
this boards with particularly in regard to quorums (that is, it should
not be advantagous for people not to vote). Admitedly, I didn't check
how it was refered to, so I just made a name up on the spot and gave it
a definition I felt was reasonable.
One way of modifying the current proposal to satisfy this criteria
would be to make quorum and supermajority failures be transitive. But,
if you recall, my last attempt at that violated monotonicity (a much
worse failing, in my opinion). Also, Anthony Towns has made a fairly
convincing case that supermajority failures should not be transitive.
I agree generally with this paragraph. I think what is causing the
problems is the implimentations of the default option, in particular how
it is treated very speically. I am still of the opinion that in this
election.
100 ABD
99 BAD
Where A has a 50:1 supermajority requirement, and D is default, B should
win, as opposed to A. (Election 1)
I don't think options with large supermajorities should easily win just
because no-one wants to "further discuss" any more.
Also, in this election.
4 ABD
3 BDA
2 DAB
Where D is default.
I believe D should win, that is any cycles should favour the default
option. (Election 2)
These opinions somewhat influences how I design my election methods.
In my opinion, "participation monotonicity" is something we should
minimize, but I'm not convinced we should entirely eliminate it -- I
don't think it's more valuable than monotonicity, quorum, supermajority,
nor than preference ranking of options (nor than condorcent voting
for elections where all votes satisfy quorum and supermajority), and I
think we'd have to effectively give up some or all of these to guarantee
"participation monotonicity".
That said, if you can show that the Feb 7 proposal does not minimize
failure along this dimension within the above constraints, or if you
can show that there's a better set of constraints, I think we'd all be
interested in hearing about that.
Thanks,
I agree that monotonicity is more important, but I still think
participation monotonicity is still very important, as encouraging
people to not vote is never a good thing. Really, we should be looking
for a method if possible that satisfies both, along with quorums and
supermajorities. I'm not convinced thats impossible yet.
I'm going to present two methods. I would appresiate if anyone can point
out if these fail the above three criteria, or if anything else is wrong
with them.
For simplicies sake, let an (X:Y) supermajority be X/Y below (that is
(3:2) = 1.5).
To give the default option preference as in Election 2, let the
supermajority requirement of the default option < 1.
Firstly, the method I prefer, which gives my prefered result in Election 1.
---
(1) A defeats B if more votes prefer A over B than prefer B over A.
(2) A defeats B by (X,Y), if A defeats B, where X is equal to the number
of votes that prefer A over B, and Y is the number of votes that prefer
B over A,
(3a) Defeat (X1,Y1) is greater than Defeat (X2,Y2) if X1 is greater than X2.
(3b) Defeat (X1,Y1) is less than Defeat (X2,Y2) if X1 is less than X2.
(3c) Defeat (X1,Y1) is greater than Defeat (X2,Y2) if X1 is equal to X2
and Y1 is less than Y2.
(3d) Defeat (X1,Y1) is less than Defeat (X2,Y2) if X1 is equal to X2 and
Y1 is greater than Y2.
(3e) Defeat (X1,Y1) is equal to Defeat (X2,Y2) if X1 is equal to X2 and
Y1 is equal to Y2.
(4a) A superdefeats B, if A has a greater supermajority requirement than
B, and number of votes that prefer A over B multiplied by X is greater
than the number of votes that prefer B over A, where A has a
supermajority requirement of X,
(4b) A superdefeats B, if A has an equal supermajority requirement as B,
and A defeats B.
(4c) A superdefeats B, if A has a smaller supermajority requirement than
B and B does not superdefeat A.
(5a) S is an undefeated set, if every option in set S superdefeats every
option not in set S, and S is non-empty.
(5b) S is the smith set, if there is no subset of S not equal to S which
is an undefeated set.
(5c) A is considered, if A is in the smith set, and there is no B such
that B is in the smith set and B has a smaller supermajority requirement
than A.
(6a) A has a beatpath to B of strength X, if A defeats B by X,
(6b) A has a beatpath to B of strength X, if 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.
(6c) A has a beatpath to B of strength 0, if there is no X such that A
has a beatpath to B of strength X
(7) 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.
(8) A is a finalist if A is considered and B is considered, and there is
no B such that B has a beatpath win to A.
(9) A is a winner if A is a finalist and there is no B such that B is a
finalist and the casting vote prefers B over A.
---
Now, a method I don't like as much, which leans towards the current
implimentation, replace rule 4 as below.
(4a) A superdefeats B, if A is not the default option, and B is not the
default option, and A defeats B.
(4b) A superdefeats B, if A is not the default option, and B is the
default option, and number of votes that prefer A over B multiplied by X
is greater than the number of votes that prefer B over A, where A has a
supermajority requirement of X,
(4c) A superdefeats B, if A is the default option, and B is not the
default option, and B does not superdefeat A.
---
Thanks.
Clinton
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