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Re: April 17th Draft of the Voting GR

>> On Fri, 18 Apr 2003 17:54:46 -0400,
>> Raul Miller <moth@magenta.com> said: 

 >> > I've also made explicit that only undropped defeats are used in
 >> > determining the schwartz set.

 > On Fri, Apr 18, 2003 at 04:05:06PM -0500, Manoj Srivastava wrote:
 >> Whoa there. I think A transitively defeats option C if a defeats B
 >> and B defeats C, whether or not the defeat is dropped.

 > Why do you think that?

	Because I think that defeats are a lower level construct than
 the Schwartz set. A defeats B is well defined (whetrher or not we are
 going to use the Schwartz set later on to determine a
 winner). Similarly, A can transitively defeat B, using the common
 definition of transitive, irrespective of the later actions. Note
 that in case of circular ties, a may transitively defeat C, and C may
 transitively defeat A. 

	This does not help us determine the winner, of course, so we
 seek to break the tie, such that one of the above statements
 is no longer true. In other words, A transitively beats C, but after
 adjusting the schwartz set, C no longer transitively beats A
 (considering only the undropped defeats). 

	In order to define this last condition, I am introducing the
 concept of unconditional defeats, like shown below (I think we had
 been straining the definition of a transitive defeat -- we were
 certainly defining it so that it caused me some confusion). 

	Of course, now I see that transitive defeats are become
 irrelevant, and this devolves down to what you had proposed, but I
 still find this version less confusing.


Replace A.6 with:

   A.6 Vote Counting

     1. Each voter's ballot ranks the options being voted on.  Not all
        options need be ranked.  Ranked options are considered
        preferred to all unranked options.  Voters may rank options
        equally.  Unranked options are considered to be ranked equally
        with one another.  Details of how ballots may be filled out
        will be included in the Call For Votes.
     2. If the ballot has a quorum requirement R any options other
        than the default option which do not receive at least R votes
        ranking that option above the default option are dropped from
     3. Any (non-default) option which does not defeat the default option
        by its required majority ratio is dropped from consideration.
        a. Given two options A and B, V(A,B) is the number of voters
           who prefer option A over option B.
        b. An option A defeats the default option D by a majority
           ratio N, if V(A,D) is strictly greater than N * V(D,A).
        c. If a supermajority of S:1 is required for A, its majority ratio
           is S; otherwise, its majority ratio is 1.
     4. From the list of undropped options, we generate a list of
        pairwise defeats:
        a. An option A defeats an option B, if V(A,B) is strictly greater
           than V(B,A). 
        b. (A,B) is a defeat of option B if option A defeats option B.
        c. An option A transitively defeats an option C if A
           defeats C or if there is some other option B where A
           defeats B AND B transitively defeats C.*2
     5. We construct the Schwartz set from (a subset of) the list of
        pairwise defeats.*1

         -  If (A,C) is an undropped defeat then option A
            unconditionally defeats option C.
         -  If (A,B) is an undropped defeat, and option B
            unconditionally defeats option C, then option A
            unconditionally  defeats option C.
         -  An option A is in the Schwartz set if for all options B,
            considering only the undropped defeats, either A
            unconditionally defeats B, or B does not unconditionally
            defeat A.

        If there are defeats between options in the Schwartz set, we
        drop the weakest such defeats from the list of pairwise
        defeats, and return to step 5.

           a. A defeat (A,X) is weaker than a defeat (B,Y) if V(A,X)
              is less than V(B,Y).  Also, (A,X) is weaker than (B,Y)
              if V(A,X) is equal to V(B,Y) and V(X,A) is greater than
           b. A weakest defeat is a defeat that has no other defeat
              weaker than it.  There may be more than one such defeat.
     6. If there are no defeats within the Schwartz set, then the winner
        is chosen from the options in the Schwartz set.  If there is
        only one such option, it is the winner. If there are multiple
        options, the elector with the casting vote chooses which of those
        options wins.  
*1 Please note that in case of circular ties, unconditional defeats
   cannot be determined, and thus the tie must be broken in order to
   determine the outcome.
*2 Please note that in the face of circular ties, A may transitively
   defeat C, and, simultaneously, C may transitively defeat A. 

Check me if I'm wrong, Sandy, but if I kill all the golfers... they're
gonna lock me up and throw away the key!

Manoj Srivastava   <srivasta@debian.org>  <http://www.debian.org/%7Esrivasta/>
1024R/C7261095 print CB D9 F4 12 68 07 E4 05  CC 2D 27 12 1D F5 E8 6E
1024D/BF24424C print 4966 F272 D093 B493 410B  924B 21BA DABB BF24 424C

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