[EM] STV and having quotas to eliminate losers

[Craig Carey]

At 11:53 13.10.99 , David Catchpole wrote:
>On Wed, 13 Oct 1999, Craig Carey wrote:
>
>> The Droop Quota...
>> 
>>       "[the] Droop Threshold is the number of ballots divided by
>>        (the number of seats plus 1), then adding 1"
>
>While this is the usual definition of Droop quota in most legislation, I
>prefer to think along the lines of Droop simply meaning "votes greater
>than the number of ballots divided by (the number of seats plus 1) mean a
>win for the candidate."



Notwithstanding that adding of 1 which might suggest to people who prefer
 the "1/N" of the Hare formula, the Droop Quota actually specifies use of
 1/N when used in the Alternative Vote method.

Some more details on how to fix STV and not merely make less probable, its
 unacceptable behaviour, but also get it nearer to passing the duality rule
 (which says that 'the method itself results from swapping winners with
 losers and negating votes').

STV could be improved by having quotas applied against losers.
(Doing that not merely reduces defects but makes the method more like
 First Past the Post (FPP, FPTP).)

Perhaps some of the proponents of electoral reform could identify for
 me, the grounds on which the method tends to be rejected.

Some may find the following arbitrary.


-------------------------------------------------------------------------

TWO QUOTAS

Define the Droop Quota Fraction For Winners to be such that, the
 (Droop Quota) = the (Droop Quota Fraction For Winners) times the 
 number of [valid] votes.

In STV, the Droop Quota Fraction For Winners is:

    1 / (number of winners that have still yet to be found)
 
This modified STV theory has applies a Quota to losers which can be
 called the Droop Quota for Losers.

The Droop Quota Fraction For Losers is:

    1 / (number of losers that yet to be eliminated)

 
Consider STV:
   * the initial number of candidates = NC;
   * the winners that need to be found = NW;

In each stage of vote transferring:

   * the number of candidates remaining = N
   * the number of winners have yet to be found = M
   * the number of losers have yet to be found = (N - M)

the two Quotas are:

   (Quota for Winners) = 1/(M+1)
   (Quota for Losers)  = 1/((N-M)+1)

(Neither STV nor the AV method has a quota for losers).

-------------------------------------------------------------------------

ELECTING ONE WINNER:

The proposed Quotas for a modified Alternative Vote method are
 found by substituting the values in:

   * the number of candidates remaining = N
   * the number of winners have yet to be found = M = 1
   * the number of losers have yet to be found = (N - 1)

the two Quotas are:

   (Quota for Winners) = 1/2,                       = 1/(M+1)
   (Quota for Losers)  = 1/N,       = 1/((N-1)+1)   = 1/((N-M)+1)

-------------------------------------------------------------------------

ELECTING TWO WINNERS:

* (Quota for Winners) = 1/(M+1), M = number of winners yet to be found
* (Quota for Losers)  = 1/((N-M)+1), N = number of candidates remaining

When two winners need to be elected, the Quotas would be:
 Case: M=2, i.e. 0 winners have found:
   (Quota for Winners) = 1/3,                       = 1/(M+1)
   (Quota for Losers)  = 1/(N-1),   = 1/((N-2)+1)   = 1/((N-M)+1)

 Case: M=1, i.e. 1 candidate has been picked a winner:
   (Quota for Winners) = 1/2,                       = 1/(M+1)
   (Quota for Losers)  = 1/N,       = 1/((N-1)+1)   = 1/((N-M)+1)


Information about what STV is, follows.
Each "if then else" in the method makes STV even more complex when
 the actual shape of the win-lose boundaries are being considered.

-------------------------------------------------------------------------

A DEFINITION OF STV 

The text is copied from pages here & around here:
  http://www.cix.co.uk/~rosenstiel/stvrules/index.htm
  http://www.cix.co.uk/~rosenstiel/stvrules/details.htm

(The details about 2 decimal places have been removed).

The basic idea is that at each stage, surpluses of winners are
 transferred and surpluses of losers are transferred. Rules determining
 when transferring stops don't seem to be clearly the rules that would
 be best.
Paragraph 5.2.5 makes it clear that losers are rejected by rejecting
 them one by one, and no quota exists that causes more than one loser
 to be rejected simultaneously. It is that feature of STV that is
 being said to be a bad feature.
 

---- Transferring of Surpluses and Finding Winners ----------------------
    1.2
       (iii) Election by quota, being the minimum number of votes which,
        if attained by as many candidates as there are places to be
        filled, leaves at most a quota of votes unused. This is the
        Droop Quota, being the total valid vote, divided by one more than
        the number of places to be filled. Thus, if seven representatives
        are elected together, and if each of seven candidates obtains a
        Quota of one-eighth of the votes, then at most one-eighth of the
        votes are unused;

    5.1.6
        Calculate the quota by dividing the total valid vote by one more
        than the number of places to be filled.

    5.2.2
        If one or more candidates have surpluses, the largest of these
        should now be transferred. However the transfer of a surplus or
        surpluses is deferred and reconsidered at the next stage, if the
        total of such surpluses does not exceed either:
          (a) The difference between the votes of the two candidates who
              have the fewest votes, or
          (b) The difference between the total of the votes of two or
              more candidates with the fewest votes who could be excluded
              under rule 5.2.5, and the vote of the candidate next above.

    5.3.7
        Calculate the total value of the transferable papers.
        If this exceeds the surplus, determine the transfer value of each
        paper by dividing the surplus by the number of transferable
        papers. If the total value does not exceed the surplus, the
        transfer value of each paper is its present value.


---- Eliminating Losers and Transferring their Voters ------------------

    5.2.5
        If, after all surpluses have been transferred or deferred, one or
        more places remain to be filled, the candidate or candidates with
        the fewest votes must be excluded. Exclude as many candidates
        together as possible, provided that:
          (a) Sufficient candidates remain to fill all the remaining
              places
          (b) The total votes of these candidates, together with the
              total of any deferred surpluses, does not exceed the vote
              of the candidate next above.

    5.2.7
        Exclusion of one or more candidates constitutes a stage in the
        count. If, after completing this, there are any surpluses to
        transfer, and not all the places have been filled, proceed as in
        paragraph 5.2.2. Otherwise proceed to exclude further candidates
        as in paragraph 5.2.5.


-------------------------------------------------------------------------

Has anybody else got some suggestions on whether or not paragraph 5.2.5
 (or the corresponding paragraph in other documents), should be
 reworded?. If it would be modified, then what would the new wording be?.
 Modifying STV is likely to always to end up with a method that is not
 optimal. So much so, I wouldn't consider it a significant problem if
 ties were resolved by a rule like this [hypothetical]:

    ...then all preferences in the parcel that was about to elect a
    Tory candidate (or one in list (1) of Schedule E) that would have
    had tied in outcome to within 2.3% of another candidate but lost,
    shall have all sub-parcels remarked in the way specified in
    Schedule E and the rules ...



Craig Carey