It’s great to see Borda seriously considered as in Erik Jochimsen’s PLERS!
Before commenting on improvements necessary for PLERS, readers may like to try out my new interactive spreadsheet at https://tinyurl.com/ElectionReformOz. It allows easy comparison to contrast and compare different vote counting sytems. It covers the main voting systems including Borda – of which PLERS is a variant, as is my DCAP system.
In a Preferential voting election where ALL preferences are filled in, Borda, PLERS and DCAP give exactly the same relative results. The main difference is that DCAP uses a metric which is more meaningful to voters because it rates each candidate with the percentage of voters who put that candidate in the top 50% of candidates.
However, with Optional Partial Preferential Voting, DCAP maintains exact linearity by “NORMALISING” each partial vote to fill all preferences left blank with a totally fair preference value that rigorously maintains the one-vote-one-value standard. To the best of my knowledge Borda and PLERS do not do that. PLERS makes an improvement by filling in blanks with the value of the worst preference number, but that diverges from strict linearity.
DCAP also has rigorous algorithms to linearly translate normalised CANDIDATE preferential Votes into normalised PARTY preferential votes to allow fair multi-member Proportional Representation. E.g., this will fix the shocking 12% error rate in the 2020 ACT election when 3 of the 25 MLAs elected were elected despite preferences proving that three of the eliminated candidates beat the three wrongly elected by the flaws inherent in traditional voting systems.
Now specific comments on PLERS: PLERS would be improved with modifications to give fair interpretation of votes which may otherwise be treated as informal, expiring, or exhausting after the first few preferences.
PLERS’ technique of replacing all blanks in Optional Preferential Votes with the preference value for the least preferred candidate changes PLERS from strictly linear to non linear because it unfairly biases against all the candidates who are not preferenced. Consideration of the PLERS example 2, 3 and 4 illustrates how ‘normalisation’ would improve PLERS.
PLERS Example 2 can be written: A:B:C:D:E:F = :2::1:3:4. PLERS says:
“Candidates A and C have not been given a preference, so are considered equally last. “
But what is equal last? Equal last is not a preference rating 6 out of 6, or 5 out of 6 for both A and C. A fair representation is to allocate (or “normalise”) that to 5.5 each, so that A:B:C:D:E:F becomes 5.5:2:5.5:1:3:4.
Note that if Candidate F was also not preferenced in the original vote (i.e. :2::1:3:-) then A to F would normalise to 5:2:5:1:3:5. The key to normalising is that the sum of the preference numbers in a normalised vote must add to the same as the sum of the numbers from 1 to the number of candidates. If votes aren’t correctly normalised, then the count is no longer precisely linear.
PLERS Example 3: A:B:C = 3:2:-
If there was a tie between Candidate A and B then this vote clearly favours B and should be permitted to count in a tie-break. Hence the fairest solution is to normalise this vote to A:B:C = 2:1:3, rather than declaring the vote informal. The objective of normalisation is to minimise wasted votes and to make a fair and just system.
PLERS Example 4: A:B:C:D:E:F = 1:Not You:4:3:3:2
If there was a tie between Candidate D or E and any other candidate it is just and fair to allow that vote to support C or D. Similarly, if there is a tie between C and any other candidate except B, this vote clearly rates C as not worse than 4th, certainly not 5th or 6th . Hence this vote can and should be allowed to influence the result. Hence the fairest solution is to ‘normalise’ this vote to A:B:C:D:E:F = 1:6:5:3.5:3.5:2.
Such normalisation is easily done by simple computer algorithms as a vote is lodged on-screen or as a ballot-paper is scanned.
Note that if PLERS were to use normalisation, then the PLERS candidate ranking would be the same as my DCAP ranking, though expressed in a different metric. The DCAP metric gives the percentage of voters who rank a candidate’s (Candidate Acceptance Percentage) in the top 50% of candidates. DCAP rankings of: 100%; 50%; and, 0%, corresponds to: Unanimous 1st Preferences; an average preference of (N+1)/2; and, Unanimous last preference: or in PLERS Ranking that corresponds to: N; N/2; and, 0.
So effectively DCAP and PLERS are directly equivalent apart from DCAP’s use of normalisation to put partial and full preferential votes on an equal footing.
Normalisation has the distinct advantage that it facilitates SPLIT Partial Preferential Voting where a voter faced with say 31 Parties fielding say 101 candidates (as in a Federal Senate Election in proportional representation of HSW’s 6 Senators), then voters can choose to vote, for example, using preference numbers 1, 2, 3, 4, 5, 6, 7 for their first few preferences then finish with 25, 26, 27, 28, 29, 30, 31 for the Parties they detest while omitting the preference numbers 8 to 24 for the Parties they know nothing about. The normalisation algorithms then fill in all the blank preference numbers on that vote with the “don’t know, don’t care” value of 16 (the average of the missing numbers).
This normalisation process is described in more detail in documents describing my DCAP vote counting system at https://tinyurl.com/ElectoralReformOz. Normalisation is totally fair when any blank preference numbers and resolvable anomalies (e.g., as in PLERS Examples 3 and 4 above) are normalised so that the sum of the normalised preference numbers (allocated to each candidate in the vote being normalised) equals the sum of the numbers from 1 to the number of candidates, (i.e. Sum=N*(N+1)/2 where N is the number of candidates).
In multi-member electorates, when voting for Candidates but attempting to achieve proportional representation of Parties, things get messy. It is oxymoronic (probably worse) to (allegedly) expect fair proportional representation from laws allowing optional preferential voting for Candidates rather than voting for Parties. PLERS appropriately calculates the proportional representation of each party by limiting each Party to the same number of Candidates being considered. However, this would be more accurate with normalisation which can then be applied (as in DCAP) to Parties with any number of Candidates giving more accurate, more linear results.
Concerning linearity, see my DCAP documents showing that Hare-Clark exaggerates the non-linearity of the eliminate-distribute (instant-run-off?) method of ‘counting’. For example, I’ve proved that in the ACT 2020 election, 3 of the 25 MLAs elected would have lost a one-on-one run-off election against a Candidate from a different Party who was eliminated in the (so-called) ‘count’ for their electirate. That could have far-reaching effects on whether a government actually earned the right to govern. Fortunately, the ACT 2020 the results were not even close. However, the non-linearity of current preferential vote ‘counting’ methods increases greatly as the candidates are more evenly matched, giving more chaotic results.
This applies in single-member electorates and even more so in multi-member electorates.
Election theory needs an overhaul. Arrow uses some poor assumptions and hence made wrong conclusions. E.g., Borda fails Arrows tests, but the fact is that Arrow failed to realise that Borda is absolutely correct in all circumstances where there is full preferential voting. And that’s why normalisation is necessary for accurate linear results using optional preferential voting.
Pundits say that Borda fails Arrow’s criteria, but I’ve proven Arrow wrong by showing that a narrow absolute majority may hide the fact that far more voters would be more accepting of a different candidate while the nominal absolute majority ‘winner’ was a far more convincing winner, on preferences, of the wooden spoon as the least preferred candidate. Why is it so? The problem is that current vote ‘counting’ systems are obsessed with first and near first preferences while totally ignoring last or near-last preferences which are, not always, but in some cases, more important if we want to achieve more stable consensus government.
A clear but extreme example is where there are 4 candidates where Candidate A gets 51% of first preferences but 49% of last preferences and another candidate gets zero% first preference but D gets 100% of second preferences. Then, no matter what other preferences Candidates B and C get, it is clear that D is actable to 100% of voters while A is unacceptable to 49% of voters and does not deserve election despite a narrow ‘absolute majority’. That’s why Borda is right and Arrow is wrong. My spreadsheet shows exactly that example – and allows readers to try other combinations also.