an informal discussion of proof issues
This is an informal writeup on the logical properties of the voting
mechanics I proposed on 2002-12-08.
Assertion: For elections where quorum rules and supermajority rules are
not used, that proposal is equivalent to cloneproof ssd.
Definition: quorum rules are not used where every non-default option
gets the required number of votes preferring it to the default option.
Definition: supermajority rules are not used where none of the options
have supermajority requirements.
Proof -- by inspection. [Does anyone doubt this? I can go through the
details if anybody needs me to, but this would be rather verbose.]
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Analysis of quorum in the context of the election methods criteria
documented at http://www.electionmethods.org/evaluation.htm:
(1) Monotonicity criterion [Condorcet or Approval]
The way I look at it, not submitting a ballot is a neutral act,
favoring neither the default option nor any other option. Submitting
a ballot is an act which may rank one over the other. Submitting a
ballot ranking an option above the default option ranks that option
higher. Submitting a ballot ranking an option below the default option
ranks it lower.
Quorum rule satisfies monotonicity because quorum is only satisfied
by ballots which rank an option above the default option and quorum
will never be caused to not be satisfied by any such ballot once
it has been satisfied.
Criterion which are only satisfied by Condorcet:
(2) Condorcet Criterion
(3) Generalized Condorcet Criterion
Quorum rule does not satisfy these criterion.
For example: quorum is 45, A and B require 1:1 majority, D is the default
option, three votes are received: 2 ABD, 1 BAD. The election defaults.
(4) Strategy Free Criterion [Condorcet only]
(5) Generalized Strategy-Free Criterion [Condorcet only]
(6) Strong Defensive Strategy Criterion [Condorcet only]
(7) Weak Defensive Strategy Criterion [Condorcet or Approval]
If we classify "the election defaults" as "not a candidate", quorum
would satisfy these criterion.
If we classify "the election defaults" as "a candidate" then quorum
does not satisfy any of these criterion [the three vote
example, above, illustrates this failure for all criterion].
(8) Favorite Betrayal Criterion [Approval only]
The quorum portion of the rules satisfies this criterion, the proof is
essentially the same as for monotonicity. However, since the underlying
system does not satisfy this criterion the point is moot.
(9) Summability Criterion [Condorcet and Approval]
Quorum satisfies this criterion. Proof: we can add an additional row
and column to the tally table to represent quorum, and another set of
rows and columns to label which option is the default option, and have
each vote add 0 to the these, now all information needed to determine
the outcome of the vote is in this summable array.
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Supermajority
Supermajority rule comes into play when we have an option with a
supermajority requirement which receives a majority of votes preferring
it to the default option, but not a supermajority of such votes.
For this case, we introduce an artificial [non-democratic] defeat of the
supermajority option. Ultimately, we remove the option from the smith
set because of this artificial defeat.
(A) Monotonicity criterion
TODO: prove or disprove that this criterion holds.
Condorcet only:
(B) Condorcet Criterion
(C) Generalized Condorcet Criterion
Supermajority doesn't satisfy these criterion.
(D) Strategy-Free Criterion
(E) Generalized Strategy-Free Criterion
(F) Strong Defensive Strategy Criterion
(G) Weak Defensive Strategy Criterion [Condorcet or Approval]
Supermajority doesn't satisfy these criterion.
It would be interesting to fully define the cases where these criterion
are satisfied even with supermajority and the cases where these
criterion are not satisfied.
(H) Favorite Betrayal Criterion [Approval Only]
Supermajority probably does not satisfy this criterion because condorcet
does not satisfy this criterion, and some elections with supermajority
requirements will be resolved using condorcet.
(I) Summability Criterion
Supermajority satisfies this criterion, using a mechanism analogous to
that which was used for quorum: add additional rows and columns to the
array to record supermajority requirements, and add 0 to this part of
the array for each vote.
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Thus: I think that the issue to focus on (for the Dec 8 draft) is whether
or not monotonicity is satisfied in the context of a "superdefeat".
It's probably also worth figuring out whether there are any significant
issues involving (D)..(G).
Thanks,
--
Raul
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