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.] ______________________________________________________________________ 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. ______________________________________________________________________ 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. ______________________________________________________________________ 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
