How It Works

Condorcet/Ranked-Pairs is simple and easy for voters (though somewhat more work for election officials).

It consists of a single round of voting in which each voter casts a single ballot expressing his or her order of preference among the candidates.

These ballots are tallied in a single counting round, then analysed in a single evaluation round to determine the final ranking amongst them.

Candidates are NOT “eliminated” in this process, and there is no “weighting” of ballots or preferences: all preferences for all candidates are holistically considered.

1. One Voting Round

Each voter casts a single ballot expressing his or her order of preference among the candidates. The voter marks candidates (no more than one mark per candidate) as being preferred more, less, or the same as other candidates, or can leave any of them unmarked; unmarked candidates are interpreted as if marked with the ballot’s lowest-preference option.

Such a ballot might look like this:

Most
1
← Preferred →
2

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

In a (closed list) Condorcet MMPR general election the voter would also mark his or her party-preference for the party-list election like this:

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]
[]
[]
[]

We can see that irrespective of this voter’s candidate preferences he or she prefers [Party 2] for the party-list election. This choice has no bearing on the outcome of constituency elections, but contributes to the party’s standings when allocating list-seats to achieve proportionality in the legislature.

  1. For MMPR a party-list preference is only relevant for a general election, and would not be included for by-elections.
  2. In a MMPR system general election, party-list preferences from all valid ballots in all constituencies are accrued to determine desired party-proportionality for the legislature as a whole.

    Once the constituency elections are determined and the party seat-count is compared against this desired proportionality, additional representatives (from party-supplied lists) may then be elected to achieve desired proportionality for the legislature as a whole.

We can see from this ballot that:

  • Alpha is this voter’s most-preferred candidate.
  • This voter is indifferent between Gamma and Delta, but prefers them each more than Beta and less than Alpha.
  • Beta is this voter’s least-preferred candidate (need not be marked at all).

2. One Counting Round

Once the polls close, the ballots are then tallied in terms of a “round robin” competition — matching each candidate, one-to-one, against each other candidate.

The above example ballot would be tallied like this:

Candidate
Pairs
B more-
preferred
than A
No-
Preference
A more-
preferred
than B
Total
A: Alpha
vs
B: Beta
1
1
A: Alpha
vs B: Gamma
1
1
A: Alpha
vs B: Delta
1
1
A: Beta
vs B: Gamma
1
1
A: Beta
vs B: Delta
1
1
A: Gamma
vs B: Delta
1
1

Here we see the pairwise, one on one, match-ups between each candidate and each other candidate — all determined from this single ballot. Again, as for the originating ballot itself, we see that:

  • Alpha is this voter’s most-preferred choice.
  • This voter is indifferent between Gamma and Delta, but prefers them each more than Beta and less than Alpha.
  • Beta is this voter’s least-preferred choice (need not be marked at all).

We proceed as for the sample ballot to tally each ballot cast, to accrue an overall result for the constituency as a whole:

Candidate
Pairs
B more-
preferred
than A
No-
Preference
A more-
preferred
than B
Total
A: Alpha
vs
B: Beta
27
6
67
100
A: Alpha
vs B: Gamma
48
0
52
100
A: Alpha
vs B: Delta
22
8
70
100
A: Beta
vs B: Gamma
60
1
39
100
A: Beta
vs B: Delta
80
2
18
100
A: Gamma
vs B: Delta
22
0
78
100

By inspection we can see immediately that in this example:

  • Alpha wins every pairing in which he or she occurs; Alpha beats every other candidate and is the overall winner, the Condorcet winner, in fact, by definition.
  • We can also see that Beta loses to every other candidate, and is the Condorcet loser, in fact, by similar definition.
  1. From each ballot we get exactly one tally mark for each candidate-pair, indicating for each given pair which candidate the particular voter prefers more than the other, or else that he or she has no preference between them.
  2. Tallying ballots in this way is more work for election officials than for first-past-the-post, where they would just put each ballot in a pile for one candidate or another and then count the ballots in each pile. In an election with many candidates or a great many ballots this would likely require a computerized tallying solution, but in any case this extra effort doesn’t impact the voters themselves.
  3. The winner of a given match-up is the candidate whose preferred-more-than value is greater than the preferred-more-than value of the other candidate in the match-up. If these values are equal, the match-up is tied. The No-preference value for a given match-up is not relevant for determining who wins the given match-up, but becomes relevant later when we “rank” the pairs.
  4. It is worth noting that with FPTP (and IRV), Beta, even though losing every one-to-one contest against every other candidate, might very well win, and Alpha, though winning every one-to-one contest with each other candidate, might very well lose.

3. Ranking the Pairs

As noted, if there are no preference cycles in the results, we can see by inspection who wins the election. In the more general case we would apply the Ranked-Pairs (Condorcet Completion) method, which will identify a winner regardless of any preference cycle, as well as establishing a complete ranking among all the candidates.

The first step is to sort the pairs in descending strength of preference, i.e. from most-strongly preferred, to least-strongly preferred (as described on the “Notes” tab):

Majority
Candidate
Minority
Candidate
Majority
Vote
Minority
Vote
Delta
Beta
80+2 = 82
18+2 = 20
Gamma
Delta
78+0 = 78
22+0 = 22
Alpha
Delta
70+8 = 78
22+8 = 30
Alpha
Beta
67+6 = 73
27+6 = 33
Gamma
Beta
60+1 = 61
39+1 = 40
Alpha
Gamma
52+0 = 52
48+0 = 48

→ is preferred-more-than : ↔ is preferred-the-same-as

  1. The candidate who wins the given match-up is the “majority” candidate for that pair. If the match-up is tied, it makes no difference, and we arbitrarily pick a majority candidate. The “other” candidate is then the “minority” candidate.
  2. The “majority vote” is the number of votes in which the given majority candidate is more-preferred-than the given minority candidate plus the no-preference value. The “minority vote” is the number of votes in which the minority-candidate is more-preferred-than the given majority candidate plus the no-preference value.
  3. We express the candidates in each pair showing the majority-candidate first, in the form:

    1. [majority-candidate][minority-candidate] — if there is a winner of the pair;
    2. [majority-candidate][minority-candidate] — if the pair is tied (the distinction between majority vs minority candidate is irrelevant in this case).
  4. The pairs are then sorted (or “ranked”, hence “ranked-pairs”) in order of decreasing voter preference, as follows:

    1. First, by descending majority-vote (strongest win),
    2. Secondly, where the majority-vote values are the same, by ascending minority-vote (weakest loss), and
    3. Finally, if both the majority- and minority-vote values are the same the pairs retain the same relative order as on the tally-sheet.

4. Ranking the Candidates

Once the pairs are ranked we evaluate each pair in turn to rank the individual candidates themselves for the final outcome:

Starting at the first pair, as sorted (“ranked”), in order of decreasing voter preference, we then accrue the relationships defined by each pair, skipping any pair that conflicts with previous (stronger-preference) “affirmed” pairs, and otherwise affirming them — a stronger preference should prevail over a weaker preference in any case where we can’t keep them both:

Majority
Candidate
Minority
Candidate
Affirm
Accrued Result
Delta
82
Beta
20
DeltaBeta
Gamma
78
Delta
22
GammaDelta
Beta
Alpha
78
Delta
30
GammaDelta
Beta
AND
AlphaDelta
Beta
Alpha
73
Beta
33
GammaDelta
Beta
AND
AlphaDelta
Beta
Gamma
61
Beta
40
GammaDelta
Beta
AND
AlphaDelta
Beta
Alpha
52
Gamma
48
AlphaGamma
DeltaBeta

→ is preferred-more-than : ↔ is preferred-the-same-as

There were no preference cycles in this example so all of the pairs are affirmed. (A preference cycle cannot occur until at least the third pair so the first two pairs are always affirmed.)

We can also do this analysis graphically:

Ranking Graph
  1. Each candidate is represented as a node in a directed graph;
  2. Each pairwise relationship is depicted by an arrow pointing from its majority candidate to its minority candidate; (If it is a tie the arrow goes both ways.)
  3. We iterate in-order through the ranked-pairs (i.e.: from strongest to weakest):
    1. adding arrows to recognize the relationship indicated by each pair; except that
    2. if the arrow would create a cycle (i.e.: a “contradiction”) we omit it — a stronger preference should prevail over a weaker preference in any case where we can’t keep them both.
    3. An arrow between two nodes can, for clarity, be removed if one node can be reached from the other along a longer arrow-path.
  4. Finally, any node that has no arrows directed to it represents a most-preferred candidate; if there is only one such node it uniquely identifies the most-preferred (winning) candidate, and otherwise we have a tie.
  1. Each pair establishes a relationship 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 transitive, as well, meaning that if X ↔ Y, and Y ↔ Z then X ↔ Z.
  3. As we evaluate each pair we consider whether the relation it represents conflicts with or else augments the information we’ve gotten from previous (stronger-preference) affirmed pairs. 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.
  4. We will only get a “conflicting” pair where there is a preference cycle in the candidate ranking; omitting conflicting pairs breaks any such cycle. The Ranked-Pairs rationale is that: a stronger preference should prevail over a weaker preference in any case where we can’t keep them both.
  5. Note that it is still possible to end-up with a tie in any position. This is not a preference conflict to be removed, but an indecisive electorate. If we’re electing the “n” most-preferred candidates this would only present a problem when the tie affects the determination of these n candidates. As for other systems this would either entail another election, or, depending on the enacting legislation, perhaps a tie-breaking mechanism such as a coin-flip would be used — this is outside of the scope of ranked-pairs itself, however.)

By either technique we obtain a final candidate ranking as follows:

  1. Alpha (most-preferred candidate) — the Winner!
  2. Gamma
  3. Delta
  4. Beta (least-preferred candidate)

In the end, the candidate who beats every other candidate, one on one, is the winner — and will be, hands down, the candidate preferred by the majority of voters.

Next: Preference Cycles

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