[EM] Droop fails the Markus Schulze Rule

[Craig Carey]

It seems to me that numbers created by computers ought
 decide the significance of the difference between Droop
 and Hare. Coding up such a program would immediately run
 into the problem that a "wasted vote" is not defined.

In this message I give a definition of vote wastage after
 taking note of the idea that STV has transfer values that
 are not always equal to 1.



At 21:16 18.10.99 , Markus Schulze wrote:
>Dear participants,
>
>I don't agree with Donald that the task of an election method
>is to allocate voters to candidates. To my opinion, the task
>of an election method is to allocate to a given combination of
>opinions of the voters a set of n winners (where n is the number
>of seats).
>
>Thus the number of wasted votes cannot be defined as the number
>of voters who cannot be allocated to a winning candidate.
>The number of wasted votes must rather be defined as the number
>of voters in the largest set A with the following property:
>Independently on how the voters of this set A vote, the election
>result will always be the same.
>
That might not be satisfactory.

Mr Schulze's definition is not about methods, but about
 methods and particular ratio of voting paper counts.
The definition seems to be the shortest distance between
 a point that represents the ballots/counts, and the nearest
 boundary to some 2 candidates' win-lose regions, inside the
 simplex.

 'Distance' doesn't mean Euclidean distance. For example, suppose
 only 4 types of papers are allowed. Then to alter the election
 paper ratios so that the point representing the election, moves
 from one vertex to another (e.g. from (1,0,0,0) to (0,1,0,0)),
 a 100% change of papers will be required.
 Moving the point representing the election from a vertex to the
 centre of that vertices's opposing face is a shorter Euclidean
 distance but it still a 100% change of the votes
 (for example: from (1,0,0,0) to (0,1/3,1/3,1/3)).

Suppose Mr Davison considered all this and then coloured the
 regions containing points where the FPTP boundary was a closer
 boundary than the boundary of the method under consideration
 (STV). It doesn't matter where the 2 faces are: they will have
 points on both sides. Replacing FPTP with a competing STV
 doesn't alter the fact that large volumes contain points that
 favour the nearer win-lose surface, even when the two interior
 surfaces are permitted to almost merge.

The idea of Mr Shulze needs to take account of hypervolumes,
 but it seems to be understood too well.


>[I want to add that (in so far as most election methods don't
>guarantee that a voter cannot be punished for going to the polls
>and voting sincerely) the concept of wasted votes cannot really
>be used as a criterion. It is only a heuristic like Blake
>Cretney's aim to find the "best guess for the best candidate."
>The reason: The concept of wasted votes implicitly presumes that
>every voter wants to be counted; but a voter who worsens the
>election result (by going to the polls and voting sincerely)
>rather wants to be ignored than counted. This is also the reason
>why Michael Dummett rejects the concept of wasted votes.]
>
>Markus Schulze

When a voter casts a vote, that vote shifts the point
 representing the election outcome by some amount towards the
 point representing the voter's vote. Given that the point of
 the election outcome exists if the voter does vote, then if the
 voter doesn't turn up and vote, the voter is effectively casting
 a negative vote and the election's point goes back to where it
 was. So not voting is little different from voting, in that the
 point representing the election outcome is moved by some distance.

Mr Schulze may be wrong when saying voters want to be ignored:
 That seems to presume that the voter has not got a full knowledge
 of the method and the paper counts. Some of the persons advocating
 their positions had a knowledge of an instance where number made
 their stance correct.

In any case, if a voter wants candidate A firstly then B second, and
 the voting method would make A lose and C win if A was voted
 for, then instead of not voting, the voter might wish to vote for
 candidate D just to cause the method to make B win, if that was
 a way to get the desired result.
This is all part of the rule, (Q1), below.

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

It should be possible to define wasted votes and find out if
 Mr Donald Davison is wrong or in the right, and by how much,
 in the Hare-STV vs. Droop-STV discussion.

Note that since FPTP is a one of the worst methods at wasting
 votes, and since it satisfies monotonicity and (P1), then
 wasting of votes has nothing much to do with those two
 definitions or rules.

STV wastes votes because of its use of "transfer values".

That is well known about STV. The complex CPO-STV method is a
 variant of STV that is said by its designer(s) to be designed
 to minimise loss of the effect of votes.

That "transfer value" idea can be generalised so it applies to
 all methods. That is done below.

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

Examples of what seems to be wasting of the power of votes.


[1] There would be a "wasted vote" if a change led to this:

 (1)
    10  (A B C D E F)   : {B}   = {A,B,C,D,E,F}.(set of Winners)

 (2)
     7  (A B)
     3  (A F)           : {B,F} = {A,B,C,D,E,F}.(set of Winners)

System (2) shows that 30% of the vote can elect candidate F.
In system (1), that 30% got lost when being "transferred" over
 preferences (B C D E).


[2] There would be a wasted vote if a change led to this:

 (1)
    10  (A B C D E F)   : {C,D,E,F} = {A,B,C,D,E,F}.Winners

 (2)
     3  (C D E F)
     7  (B)             : {B}       = {A,B,C,D,E,F}.Winners


It seems to be too strong to require that methods not have any
 instance of votes being wasted due to preceding losing
 preferences. For example, votes get wasted over the preference
 for B, in the (B C) paper, in this election:

(1)
    5  A .
    4  B C       <-- Votes are not transferred to C
    2  C A             : STV, IFPP, FPTP: {A} = Winner

(2)
    5  A .
    4  C
    2  C A             : STV, IFPP, FPTP: {C} = Winner




A definition of a rule:
--------------------------------------------------------------
                     RULE (Q1)                  19-Oct-1999

 Regard a voting paper as a binary number, with each preference
 replaced by 1 if the candidate won and by 0 if the candidate
 lost. Then this rule is not satisfied if the voter could
 increase the value of the binary number by altering the paper
 in any way (including dividing it). Alterations are however
 constrained by a requirement that the weight of the paper be
 not altered by any factor less than 0 or greater than 1.

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

Violations of (Q1) can be integrated. The (Q1) rule is too
 strong to be used to reject preferential voting methods
 for any instance of noncompliance.


If the rule applies to a single paper, then the weight of the
 paper is the distance of the point representing the election
 to the face opposing the paper's vertex. The rule defines a
 region around the election point that extends outwards by that
 distance. (Q1) says that that region shall not intersect a
 win-lose boundary.

The relative hypervolume for a particular method could be
 calculated. A computer could do that. The value found could
 be compared against the value found for FPTP. Perhaps all
 methods with a (Q1) failure figure that exceeds FPTP's for
 any number of winners and candidates, could be ruled out.


Having an aim of minimising the wastage of votes, measured
 by points where (Q1) is not satisfied, isn't able to coexist
 particularly well with another aim, say proportionality.

Possibly "proportionality" can be defined to be (Q1).

In my last message I gave a seemingly different way to find
 a proportional method. I am fair from clear that the two
 formulations would give the same result.

Has anybody got the ideas 'vote wastage' and 'proportionality'
 defined well enough, to be able to explain the essentials
 of the difference(s) between the two?.


G. A. Craig Carey, Avondale, Auckland, 19 Oct 1999