Condorcet Details

1. The Ballot

The proposed form of the ballot is as shown here (as discussed in the How it Works example):

Most
1
← Preferred →
2

3
Least
4
Alpha]
[party 1]
[]
[]
[]
[]
Beta
[party 3]
[]
[]
[]
[]
Gamma
Independent
[]
[]
[]
[]
Delta
[party 2]
[]
[]
[]
[]

  • For all ballots the candidates will be ordered identically but in an order randomly determined by the constituency’s Returning Officer.
Most
1
← Preferred →
2
3
Least
4
Preferred
Party
Alpha
[party 1]
[]
[]
[]
[]
[party 1]
[]
Beta
[party 3]
[]
[]
[]
[]
[party 2]
[]
Gamma
Independent
[]
[]
[]
[]
[party 3]
[]
Delta
[party 2]
[]
[]
[]
[]

  • For all ballots the candidates will be ordered identically but in an order randomly determined by the constituency’s Returning Officer.
  • For the Preferred Party section, there being no advantage to randomizing the party names, and given that they should be ordered the same on all ballots for all constituencies, an ascending alphabetic order is suggested.

Such a ballot will be compatible with an optical-reader approach, and can be adjusted as needed to accommodate particular optical-reader technology as might be employed, yet can be manually counted as well as circumstances warrant.

Instead of marking only one candidate (with an X or a tick-mark, as for FPTP) any number of candidates can be marked by filling-in the appropriate square in the column representing the voter’s preference (no more than one square per candidate).

  • These will be interpreted as the candidates being marked with the ordinal preference number associated with the given preference column.
  • All candidates need not be marked. Unmarked candidates will be regarded as if marked with the ballot’s lowest-preference option.
  • A candidate with a lower-numbered preference than another candidate means that the lower-numbered candidate is the voter’s more-preferred choice between the two candidates.
  • Candidates can have the same preference number as other candidates, indicating that a voter has no preference between or among them.

It has been observed in Australian elections (where voting is compulsory and all candidates on the ballot must be marked) that some voters just number candidates from first to last as they appear on the ballot without necessarily expressing their true preferences. This is called “Donkey Voting.”

This could be due to them not being familiar with all the candidates, though under Australia`s rules they have to rank them all anyway; it could perhaps also be due in part to being compelled to vote, and perhaps such voters have no opinion about all the candidates.

Listing the candidates randomly, instead of alphabetically, minimizes systemic benefits to candidates who sort earlier alphabetically, as would be the case with an alphabetic sort. (It also potentially affects pair ranking in limited circumstances, as will be seen later.)

(This would argue as well for randomizing the ballot order per poll, or even per ballot. For a manual process this would very likely be unmanageable, but if desperately desired it could be supported for an optical-reader count.)

In our case where compulsory voting is NOT proposed, and where, as proposed, not all candidates need to be marked, it seems unlikely that anyone who “bothers” to vote would then not make an effort to vote his or her true preferences, such that donkey voting should be unlikely.

2. The Count

Once the polls close on election night the ballot boxes are opened and the count ensues.

Tally sheets should be pre-printed showing all the candidate pairs, and distributed to each poll. Such a form might look like this:

Candidate​Pairs
B more-
preferred
than A
No-
Preference
A more-
preferred
than B
Total
A:Alpha
vs
B:Beta
A: Alpha
vs B: Gamma
A: Alpha
vs B: Delta
A: Beta
vs B: Gamma
A: Beta
vs B: Delta
A: Gamma
vs B: Delta

The tally-form would be used to tally batches of ballots for each poll, to accrue totals for the poll, and for the returning-officer for the constituency, recording the results for the constituency itself.

When counting the voting officer examines each ballot in turn, and for each candidate pair on the tally sheet tallies up the preferences:

  1. If both candidate A and candidate B are unmarked, or where both are marked the same, enters a tally-mark in the “No Preference” column; otherwise
  2. For each candidate A, who is marked with a lower number than a given candidate B, or where B is unmarked, enters a tally mark in the “A more-preferred than B” column; otherwise
  3. For each candidate A, who is marked with a higher number than a given candidate B, or where A is unmarked enters a tally mark in the “B more-preferred than A” column;

Each pair gets exactly one tally mark for each ballot.

The total number of tally marks in the preference/no-preference columns for each row should equal the number of accepted ballots tallied; this gives a cross-reference to help identify discrepancies in the count.

To facilitate the count the pairs are listed based on ballot order so that in comparing A vs. B only those candidates following the given candidate A on the ballot need be examined.

  1. More than one tally sheet can be used, at the discretion of the voting officer, who might select batches of say, twenty, or fifty, or whatever number of ballots, and group them, as well as their respective tally sheets, and then do another similar batch, until done.
  2. Batch tally sheets mean that errors occurring in the count can conceivably be identified in terms of a given batch, and then only that particular set of ballots need be recounted, rather than the entire poll.
  3. Given that with many candidates there will be a great many pairs and a greater likelihood of error, batches would be a recommended best practice for a manual process.
  4. In the event that multiple tally sheets are used in this way the voting officer and assistant will add the counts from each batch tally sheet into a combined result for the poll.
  5. In the event that a computer and optical reader are used, as each ballot is passed through the reader it is tallied and recorded according to the algorithm noted above for the manual process. The output of this would be a report similar to the tally form noted above.
  6. Counting by optical reader will be fast, and though a reader per poll could be used it is probably fast enough that one installation at each polling station could be shared among all the polls at the given station, with each counting their respective polls in a single batch per poll once the polls have closed.
  1. Regardless of whether counted as a manual process, or using a computer and optical reader, the count for the given poll is then communicated to the constituency Returning Officer, where the action then flows.
  2. Instead of a single value per candidate, as for FPTP, the report will be three values (“B more preferred than A”, “A more preferred than B,” and “No-Preference”) – for each pair.
  3. The constituency Returning Officer will then combine the given poll’s counts with those from all the other polls to accrue the counts for the electoral district as whole.
  4. Note that only one counting step is required for each poll; once the data are communicated to the constituency returning officer and the poll secured, they’re done.

3. The Outcome

Once the count for the constituency is determined we sort the pairs according to the counts, and then determine the overall results (these will be preliminary results on election night as we cannot obtain a final result until the absentee ballots arrive):

The pairs are ranked, or sorted, as follows:

  1. If the majority vote for a given pair is greater than the majority vote for another pair the given pair shall precede the other (majority vote, descending);
  2. If the majority vote for a given pair is equal to the majority vote for another pair:

    1. The pair for which the minority vote is less than the other pair shall precede the other pair (minority vote, ascending); and otherwise
    2. If the minority votes of the given pairs are equal then they retain the same relative order as they have on the tally sheet (which derives from the random order on the ballot).

This sorts the pairs by descending strength of preference, or, more specifically: firstly by decreasing strength of win (majority vote, descending), secondly by increasing weakness of loss (minority vote, ascending), and thirdly by tally-sheet order.

The noted third-order sort on tally-sheet order is a key reason for randomizing the order on the ballot, and is somewhat arbitrary.

It should be observed, however, that if there is no Condorcet winner there is already an inherent degree of arbitrariness, however well justified, in choosing any particular approach to breaking preference-cycles; this is not materially worse than that already inherent.

In any event, it can be of consequence only if there is no Condorcet winner, and then only if there is equality between the majority-vote values and between the minority-vote values, and even then does not necessarily affect the outcome. (It can affect the outcome only if the relative order between these particular two pairs affects whether or not either of them is “affirmed” when evaluating the final result.)

It is also extremely unlikely in practical elections (except possibly when considering lowest-preference candidates, who are not likely to be a factor in any case): an electoral district in BC, for example, averages on the order of 34,000 registered voters; if we get a 60% turnout we would have roughly 20,000 ballots. For this particular scenario to occur we need that at least two different pairings of candidates get, out of our assumed 20,000 ballots, exactly the same number of majority votes, and exactly the same number of minority votes.

  1. Each pair establishes a relation between its two candidates in which one candidate is more preferred than the other, or they are preferred the same (tied).
  2. For a given pair consisting of candidate X and candidate Y:

    1. If candidate X is the majority candidate (and is not tied) then we consider that X is more preferred than Y, or: X → Y; this “more preferred” relation is transitive, meaning that if X → Y, and Y → Z then X → Z;
    2. If the pair are tied (“preferred the same as”) then X ↔ Y. This relation is also transitive, meaning that if X ↔ Y, and Y ↔ Z then X ↔ Z.

Once the pairs are ranked (sorted) we evaluate each pair in turn, from strongest preference to weakest preference, in order to rank the individual candidates themselves with respect to the other candidates:

  1. As we evaluate each pair we consider whether the relation it represents conflicts with or else augments the information we’ve got from previous (stronger-preference) affirmed pairs.
  2. If it conflicts we omit it from further consideration, and otherwise we “affirm” it. Note that “more preferred than” conflicts with “preferred the same as,” and vise versa: X → Y, and X ↔ Y cannot both be affirmed. We will only get a “conflicting” pair where there is a preference cycle in the candidate ranking; omitting conflicting pairs breaks any such cycle.
  3. The Ranked-Pairs rationale (for breaking preference cycles) is that: a stronger preference should prevail over a weaker preference in any case where we can’t keep them both.

In the end we have a complete ranking of the individual candidates with no preference cycles remaining:

  1. This will either uniquely identify one candidate for whom no other candidate is more preferred, who is then the Ranked-Pairs winner (and the Condorcet winner as well if in the original data there was no preference cycle involving this candidate); and otherwise
  2. If there is more than one the election is tied.

4. Recounts

Current practice (for BC Elections) is that if the result is “close,” which means the successful candidate beats the next runner-up by less than 1/500 of the accepted ballots, we automatically trigger a recount.

This is because in a close race a counting error of a few votes one way or the other can be significant and produce an entirely different outcome. The count should be exact, of course, but errors do happen, or disputes can arise regarding the evaluation or rejection of particular ballots, so if we’re closer than some presumed margin of error, we must apply extra diligence in verifying the result.

With Ranked Pairs such errors can factor into every candidate pairing. But, on the other hand, if we have two or more candidates who have little support among the voters, such candidates are likely to be vying for last-place, which means that the votes between pairs of such candidates should all be expected to be close – and we don’t want to force un-needed recounts.

The proposal, then, is that an automatic recount would be required if:

  1. There is more than one candidate in first-place (a tie); otherwise
  2. For any pair involving the indicated winner, the majority-vote vs minority-vote is less than, say, 1/500th of the total number of accepted ballots.

Next: How It Works

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