Optional Preferential Voting
Craig Turner, 18 January 2025
--

I have been reviewing the algorithm that the South Australian Electoral
Commision use for Legislative Council elections. In this paper I will be
recommending a tweak to that algorithm.

My intended audience for this note is the SA EC. The reader will need at least
a basic understanding of multi-member proportional representation elections.
In particular, you would need to understand that when a candidate is elected,
all of the ballots that contributed to their election receive a discount
(derived from the Quota), and are then redistributed to other candidates.

Here, I will discuss two algorithms,

    alg/1, which is what I infer to be used at the moment by the SA EC.

    alg/2, a similar algorithm that I learnt at university.

Some notes about me,

    Whilst at university in the 90s, I acted as a scrutineer for Flinders
    student elections over several years. At that time, those elections ran on
    a multi-member optional preferential system. When allocating preferences
    that system carried voting values to several decimal places - five places
    from memory.

    During my studies, I took a lecture presented by the late Professor Dean
    Jaensch, who was a guru in the mechanics of electoral systems. This
    lecture presented several algorithms (e.g. FPTP, single member
    preferential, PR multi-member compulsory, PR multi-member optional). At
    the time, I noted the algorithm he presented for PR multi-member optional
    elections to be the same as we had been using in student elections.

    I have acted as a scrutineer at state and federal elections.

Until recently, the South Australian Legislative Council (upper house) was
elected through a multi-member proportional representation system that
insisted that electors nominated preferences against all candidates. I know
this as a /Compulsory Preferential/ system. Here, voters were presented with a
ballot paper that was split by a horizontal line. They were asked to fill out
the section above the line or below the line only.

    To vote /below the line/, the voter would need to put numbers against all
    of the candidates who were listed in the election. This gave the voter
    fine-grained control of the direction of their vote, but some people found
    it to be laborious to number all of the boxes, or confusing.

    For convenience, the system allowed voters to instead vote for a single
    party /above the line/. This would automatically populate the voter's
    ballot from a templates submitted to the Electoral Commission by their
    chosen political party before election day. It was as though the voter had
    voted below the line, in the manner recommended by their favourite party.

Around 2018, the model was changed,

    The above-the-line/below-the-line split was gone. Ballot papers now listed
    all candidates, similar to the old below-the-line model. (update: this is
    incorrect, see [2])

    Voters could choose to fill in preferences against all boxes (as before)
    or only some of the boxes (this is new).

This new model fits into a category of Proportional Representation that I know
as the /Optional Preferential/ model.

I learnt of the change from compulsory-preferences to optional-preferences
recently, and it rekindled my interest in the topic. I looked at a document on
the SA Electoral Commission website that shows how preferences were
redistributed for the 2022 Legislative Council election. [1]

There were details in that spreadsheet that caught my attention,

    1. The candidates on the last pane are elected with quotas quite a lot
    lower than the quota required of the other candidates.

    2. Candidates elected half way through the order have reached the same
    quota as early candidates.

These observations tell me that the count algorithm used by the SA EC is
different to the algorithm I knew from my time at Flinders.

Having studied the spreadsheet further, I have reached this conclusion: that
the SA Electoral Commission adapted their previous Compulsory Preferential
algorithm to the new system with minimal changes.

I am not familiar with how election rules are set but as far as I know it is
the commission's prerogative to select an algorithm that is both good and
practical.

Here, I will ask that the SA EC consider a reform that would bring their
system closer to a technically-pure Single Transferable Vote algorithm.

These are the algorithms I wish to contrast,

    Alg/1: This is the Adapted Compulsory Preferential algorithm to calculate
    the results of an Optional Preferential vote, which is what the SA EC
    appear to use currently. This algorithm calculates the Quota at the start
    of the count only. Each candidate is required to reach that original
    quota. The election/elimination process continues until only two
    candidates remain. Then, those two candidates are also elected.

    Alg/2: This is the Optional Preferential algorithm that I learnt at
    Flinders. Here, we calculate the Quota at the start of the vote, and then
    again before each Elimination. As a result, some candidates will be
    elected with a lower quota than other candidates.

It may seem strange that Alg/2 allows some candidates to be elected with fewer
votes than earlier candidates. But this is a necessary consequence of the
Optional Preferential model. In that model, when a voter decides not to fill
out more boxes, they are signaling this intent: that any remaining part of
their vote should be removed from consideration. Effectively, they have asked
to convert their vote from a formal vote to an informal vote for whatever
remains of the count. Hence, it follows that the Quota should drop, because
the quota is derived from the count of formal votes.

Here, I am making the argument that the Electoral Commision should reform
their processes to replace Alg/1 with Alg/2 for future elections. That would
serve these goals,

    1) To respect the wishes of those voters who have communicated that they
    wish any remaining voting power of their ballot to be removed from
    consideration.

    2) To ensure that preferences given against successful candidates are
    passed through at a value that is proportionate to the number of votes
    that are active in the system at the time of the distribution.

To summarise, these are the disadvantages of use of Alg/1 for these elections
compared with Alg/2.

    1) It dilutes the ballots of people who have contributed to the successful
    election of early candidates more than those ballots would be diluted if
    the Quota was recalculated.

    2) It benefits candidates who do not receive strong preference flow from
    successful candidates in a way that is disproportionate to the intent of
    the voters at large, particularly if those candidates can survive to be
    one of the final two.

Alg/2 requires slightly more work than Alg/1: there is an extra step to
recalculate the quota before each Elimination. This is not time-intensive.
Making that calculation would take about a minute each time. It might add as
much as 30m of extra overhead in total to an election count. This is a small
amount of time compared to the many days that the team spend managing piles of
ballot papers.

I hope I have demonstrated the following,

    That Alg/2 honours the /Single Transferable Vote/ goal more effectively
    than Alg/1.

    That Alg/2 is about the same amount of work as Alg/1.

    That a move from Alg/1 to Alg/2 is a simple reform.

I will be presenting this note to the SA Electoral Commision, and I hope they
give it due consideration. @SAEC - please adopt Alg/2 for legislative council
elections.

--
:1
    At
    https://www.ecsa.sa.gov.au/?view=article&id=636:2022-state-election-results-and-statistics-downloads&catid=12&highlight=WyJsZWdpc2xhdGl2ZSIsImNvdW5jaWwiLDIwMjJd
    click on "Legislative Council distribution of preferences". Study the
    excel file within that archive, which focuses on distribution of
    preferences.

:2
    (4 February 2025) This is not correct. I have learnt that there is still
    an above-the-line / below-the-line dynamic to ballot papers. In the new
    model, electors can fill in some of the above-the-line boxes to get a
    template vote,, or some of the below the line. This earlier error is not
    important to the key points in this paper about the counting algorithm.