[EM] STV and having quotas to eliminate losers

[Bart Ingles]

I should add that the losing quota does seem rule out some of the
worst-case AV results, so I would consider it an improvement over both
AV and FPP.

If voter utilities are on a scale of 0 to 1.0, n is the number of
candidates, then the worst-case (where all voters only have one
meaningful choice) winner's social utility under various systems would
be (I think):

FPP:                1/n
Two-round runoff:   1/(2n)
AV:                 2^(1-n)
Condorcet:          0
AV w / loser quota: 1/n
Approval:           1/n

-Bart Ingles



Craig Carey wrote:
> 
> 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