A Voting System for Optimal Proportional Multi-member Representation

Background

When it is suggested that an election be held to decide some matter, the default is usually a "show of hands", or possibly a secret ballot with the votes counted on the same lines: whoever gets the most votes wins. It seems obvious, impartial, and fair.

The problem arises when there are more than two answers for a question or more than two candidates for a position: if the electors are evenly split, the winner may actually have received support from only a minority; indeed, for V voters deciding between C candidates, the winner may need as few as (V+1)/C supporters. (See fault type A, below.)

Notation

In multi-round elections these values may vary from round to round.

The Ideal Voting Outcome

An ideal voting system would result in exactly equal representation for all participating voters, in the sense that they would have the same (small) portion of the attention of the representatives, who (we assume) only have "ears" for the voters whose votes actually helped elect them.

Clearly this is an asymptotic goal: something to strive for, but actually impossible to completely achieve:

How it can go wrong

Fixing the voting system

There have been many attempts to create voting systems that improve on "first past the post", which has the dual disadvantages of being "obviously correct" to the naïve layman, while actively disenfranchising everyone except the supporters of the successful candidate.

And that's just when the electorate has to choose a single candidate.

When the electorate has to choose multiple candidates, almost all systems are seriously flawed. :

Proposal

I propose the following system be adopted:

The core of this system is Preferential Voting, however votes are reweighted in each round to avoid bias against supporters of popular candidates

As with "single position" preferential voting, the lowest-polling candidate is removed on each round until the highest-polling candidate is electable, but:
  1. set the electability threshhold T as Vtotal/N (where Vtotal is total of votes for all candidates and N is number of positions still to be filled); and
  2. to discount their supporting voters' effect in subsequent rounds not by 100% (as currently) but by the ratio (Vx-T)/Vx (where Vx denotes the number of voters who current top preference is for candidate x and who still have lower preferences remaining, and T is Vtotal/N as above).
Under this system and taking last year's Internet NZ election of councillors as an example, at round 1 the numbers were:
    VJB=16     VJM=12    VCP=7
    VMW=16     VNJ=9     VSS=5
    VDC=15     VCS=9     VMP=1
    Vtotal=90
    N=6
    T=15

Under this system JB, MW and DC would have been immediately declared elected. (This differs from the Internet NZ system where it took 20 further rounds to elect the first 3 candidates.)

Then, anyone whose first preference had been for JB or MW would have 1/16th of a vote left to apply to subsequent rounds (the margin by which the number of voters for those candidates exceeded the electability threshhold T), and anyone whose first preference was for DC would have no vote left (having exactly matched the electability threshhold T).

Think of it this way: anyone who hasn't elected a candidate yet still has 1 vote (43 voters) and those who have elected a candidate have 2 votes between them; total 45. Which makes sense — exactly half the votes "used up" in electing half the positions.

Then for the second round, N=3.

Unfortunately we don't know the actual number of voters, so the rest of this example is hypothetical. But it's a fair bet to say that Vtotal would have been 45 or fairly close — remember the 43 disenfranchised voters from round 1 are still in full force, and the group that elected JB, MW & DC would almost certainly all have had other preferences.

So T would still be about 15, and nobody would be elected on the second round — JM was next highest, and even if all of those "2 votes" went to him he still wouldn't reach the threshhold.

N is decremented after rounds which elect a candidate; Vtotal and T are recomputed for each round, and take into account the fractional votes that some voters still have. (After multiple rounds some voters may have only a fraction of a fraction of a fraction of a fraction of a vote...)

Repeat until... we we only have one position (left) to fill (N=1), it becomes a matter of repeatedly dropping the lowest-polling candidate until there's only one left, and then declare that one elected. (This stage has exactly the same effect as the current system, although it's not described that way.)