[Date Prev][Date Next] [Thread Prev][Thread Next] [Date Index] [Thread Index]

Re: Request for comments [voting amendment]

Raul Miller wrote:
This is not a full draft.  In this post, I'm only including
text for replacing A.6 of the constitution.  I wanted to
also rewrite the changes to A.3, but I've got to run some
errands tonight and I'm not going to have time to write up
a full draft.

Please let me know of any flaws in the following partial draft:

I'd like some clarifications...

  A.6 Vote Counting

    1. Each ballot orders the options being voted on in the order
       specified by the voter.  Any options unranked by the voter are
       treated as being equal to any other unranked options and below
       all options ranked by the voter.

Definition: A "ballot" consists of a ranking A>B>C>D>... of options submitted by a voter. It defines a total ordering of options for a particular voter (i.e., for any pair of options A and B, we can claim that a particular voter feels either A>B, A<B, or A=B, and iff A>B, then B<A).

Let |A>B| be the number of voters who voted A>B.
Similarly, for |A<B| and |A=B|.

(Obviously, |A>B| + |A=B| + |A<B| = total number of voters, because of the definition of a ballot.)

    2. Options which do not defeat the default option are eliminated.

       Definition: Option A defeats option B if more voters prefer optio
       A over option B than prefer option B over option A.

Let A>>B ("A defeats B")        if |A>B| > |B>A|
Let A==B ("A ties B")           if |A>B| = |B>A|
Let A<<B ("A is defeated by B") if |A>B| < |B>A|
(Note: A==A, for all options A)

Eliminate all options A if Default>>A.

Clarification:  What if Default==A?

    3. If an option has a quorum requirement, that option must defeat
       the default option by the number of votes specified in the quorum
       requirement or the option is eliminated.

Does is mean:

Eliminate A if |A>Default| < Quorum


Eliminate A if |A>Default| - |Default>A| < Quorum

Again, how do we deal with that |A=Default| case?

    4. If an option has a supermajority requirement, that option must
       defeat the default option by the ratio of votes specified in the
       quorum requirement or the option is eliminated.

(?) Eliminate A if |A>Default| / |A<Default| < Supermajority Ratio

Again, what about the |A=Default| votes?

    5. If one remaining option defeats any other remaining options,
       that option wins.


"Condorcet Winner"

If there is a remaining option A, such that for all remaining options B, either A=B, or A>>B.

    6. If more than one option remains after the above steps, we use
       Cloneproof Schultz Sequential Dropping to eliminate any cyclic
       ambiguities and then pick the winner.  These represent a procedure
       and must be carried out in the specified order:

       i. All options not in the Schultz set are eliminated.

          Definition: An option C is in the Schultz set if there is no
          other option D such that C is in the beat path of D AND D is
          not in the beat path of C.

Let A>>>B mean there is a possibly empty sequence C, D, ..., E, F of remaining options such that A>>C, C>>D, ..., E>>F, F>>B

Then B is on the beat path of A.

The Schultz Set = { A | A>>>A }

Note: Because A==A, it isn't the case that A>>A, so if the Schultz set includes A, then there must be a B!=A such that A>>B>>...>>A. Since A>>B, we then have B>>...>>A>>B, so B is also in the Schultz set. Therefore, the Schultz Set can't be a Singleton Set.

          Definition: An option F is in the beat path of option G if
          option G defeats option F or if there is some other option
          H where option H is in the beat path of G AND option F is in
          the beat path of H.

       ii. Unless this would eliminate all options in the Schultz set,
	   the options which have the weakest defeat are eliminated.

	   Definition: The strength of a defeat is represented by two
	   numbers: the number of votes for the defeated option and the
	   number of votes for the defeating option.

	   The more votes in favor of a defeated option, the weaker
	   the defeat.	Where two pairs of options have the same number
	   of votes in favor of the defeated option, the fewer votes in
	   favor of the defeating option, the weaker the defeat.

So if we have two defeats A>>B and C>>D, then we lexigraphically compare (|B>A|, -|A>B|) and (|D>C|, -|C>D|)

What do you mean by "options with the weakest defeat"?

My understanding was that we were removing defeats from consideration. If, for example, there were four options A, B, C, D in the Schultz Set, we'd initially be looking at the following set of defeats, in strongest to weakest order:


After eliminating the weakest defeat (in this case, B>>D, we no longer consider it when determining the Schultz Set, as if we had declared B==D, so that neither B>>D or D>>B held.

So what do we really want to eliminate here?


What if we had |D>C| = |B>D|, |D=C| = |B=D|, |D<C| = |B<D|, so that neither D>>C nor B>>D was weaker than the other? Do we eliminate both defeats?

       iii. If eliminating the weakest defeat would eliminate all options
	    in the Schultz set, a tie exists and the person with the
	    casting vote picks from among these options.

Hmmm, if we had:

Defeats = {A>>B, B>>C, C>>A}

then the Schultz set is {A, B, C}. Eliminating the weakest defeat (C>>A) would break all the cycles, so !(A>>>A), !(B>>>B), !(C>>>C), so the Schultz set is now empty. Do we want to the casting-vote-caster to decide in this case, or do we want to say that A wins?

I think it would be better to declare a tie iff all the DEFEATS would be eliminated, not all the options. Since all the defeats would be eliminated only if they were all of equal strength, there shouln't be a problem.

       iv. Otherwise, a new schultz set is found, with those weakest
           defeats eliminated,

       v. If this new schultz set contains only one option, that option

As defined, the Schultz set can't be singleton. Maybe I'm misinterpreting teh Schultz set.

       vi. Otherwise, these steps (i-vi) are repeated with this new
           schultz set.


Reply to: