Quota Borda system

The Quota Borda System (QBS) is a PR electoral system for use in multi-member constituencies. It is based around determining solid coalitions and electing candidates from them using the Borda count. It was devised by Michael Dummett and published in his 1984 book Voting Procedures.

Procedure

According to Schulze, the way the Quota Borda system is described is somewhat convoluted, but can be boiled down to these points:

• The election procedure proceeds in rounds, called "stages", and terminates when all s seats have been elected.
• First, determine the Borda scores of every candidate in the election.
• Then, in the kth stage, starting from k=1, and ending at k = number of candidates:
  • For each solid coalition of k candidates, with support exceeding at least one Droop quota:
    • Let q be the maximum number of Droop quotas its support exceeds, and let x be the number of candidates in that coalition who have been elected at a prior stage.
    • If q > x, elect the candidate in that solid coalition with the highest Borda count that hasn't been elected at a prior stage.
• If the process ends without every seat having been filled, fill the remaining seats with the unelected candidates with the highest Borda scores.

It's only necessary to elect one candidate at a time from each coalition because either there is only one unelected candidate left in the coalition, or some of them must have been elected in earlier stages. For example, if the coalition {A, B} exceeds two Droop quotas in the second stage, then either A or B exceeds one Droop quota in the first stage. Thus either A or B must have been elected in the first stage, so it's only necessary to elect the other one in the second stage.

The procedure can be generalized to base methods other than Borda by using that base method's order of finish instead of Borda's.

Analysis

Choosing candidates to be elected from solid coalition supported by Droop quotas ensures that the method passes proportionality for solid coalitions. Using a single-winner Borda count to decide which candidate in each coalition is to be elected will, in the absence of strategic voting, tend to elect the candidate closest to the median voter.

Thus all of QBS's proportionality comes from its PSC compliance; subject to this proportionality constraint, it will elect the most centrist candidates possible.

For instance, if voters and candidates are distributed on a standard normal distribution, every sufficiently large solid coalition supported by voters to the left of the center will elect the rightmost candidate within the coalition's "slice" of the normal distribution. In the same way, every solid coalition supported by voters to the right of center will elect the leftmost candidate in that coalition.

The purpose of this behavior is to reduce polarization and sectarianism while still remaining broadly proportional.

In addition, the method can be made more consensus-based at the expense of proportionality by stopping the count at an earlier stage than the number of candidates. Doing so will fill the remaining seats with the Borda winners who haven't yet been elected.

Criticism

Nicolaus Tideman argued that QBS can deny representation to minority groups that support irrelevant alternatives. Three-seat example:

1: c1, c2, c3, e1, e2, e3, e4, d1

1: c2, c3, c1, e1, e2, e3, e4, d1

1: c3, c1, d1, c2, e1, e2, e3, e4

6: e1, e2, e3, e4, c1, c2, c3, d1

4: e1, e2, e3, e4, c1, c2, c3, d1

5: e1, e2, e3, e4, c1, c2, c3, d1

6: e1, e2, e3, e4, c1, c2, c3, d1

7: e1, e2, e3, e4, c1, c2, c3, d1

8: e1, e2, e3, e4, c1, c2, c3, d1

9: e1, e2, e3, e4, c1, c2, c3, d1

Had d1 not run, there would've been a Droop solid coalition for c1-3, guaranteeing one of them a seat. But instead, e1-3 all win. This is an example where the Expanding Approvals Rule and STV, two other common PSC-compliant methods, would elect one of c1-3.

Comments

Imagine a six-seater constituency in a plural society of three dominant groups, where the three groups are roughly 30:30:30.  (There were many such constituencies in pre-war Bosnia.)  Success in a QBS depends on a good number of top preferences and/or a good Modified Borda Count MBC score; see below.

Lest their members/supporters split the vote, the matrix vote – like RCV (PR-STV) – prompts all parties to nominate only as many candidates as they think might get elected.  At the same time, the MBC element of a QBS encourages the voters to submit a full ballot.  Accordingly, in a 6-seater 30:30:30 constituency (in Bosnia), each faction could expect to win 2 seats; at the same time, those parties which do not fall into one of the country’s three ethno-religious categories (like Bosnia’s Social Democrats) might also hope for some success.

Now in many countries, not least those democracies which make decisions in binary votes, societies tend to divide into two: left- or right-wing, socialist or capitalist, and so on.  Likewise, in many societies already divided, each ethno-religious grouping tends itself to divide into two, to have a more radical and a more moderate party; (this was true both in Northern Ireland and in Bosnia).  Accordingly, in a 30:30:30 constituency, each of the two main parties in each ethno-religious grouping, might like to nominate 2 candidates; but no grouping would want to nominate more than 4.   Meanwhile, others like the Social Democrats might also have a good chance.  So that’s 14 candidates already, but not too many more.

Come the vote, every voter would be encouraged, by the MBC element, to cast a full ballot of 6 preferences.  In this way, QBS entices voters to cross the gender gap, the religious divide and even the sectarian chasm; the methodology is ideally suited to plural societies, and especially conflict zones.

In a six-seater constituency, the analysis proceeds as follows, counting:

(a)        all the candidates’ 1^st preferences;

(b)       all the candidate pairs’^[1] 1^st and 2^nd preferences; and

(c)        all their MBC scores.

At each stage, if there are still candidates to be elected, the count proceeds to the next stage.  In the analysis:

Part I

         stage

(i)        all candidates with a quota of 1^st preferences are deemed elected;

          (ii)        all pairs of candidates with two quotas of 1^st/2^nd preferences are elected;

then, in

Part II, in which any candidates elected in Part I, in stages (i) or (ii), are no longer counted,

          (iii)       candidates with the highest MBC scores are elected.

QBS has only one count, albeit of three different types of totals: (a), (b) and (c); next, in the analysis, three different stages.

Further reading

• Emerson, Peter (2016). From Majority Rule to Inclusive Politics. Cham: Springer International Publishing. doi:10.1007/978-3-319-23500-4. ISBN 978-3-319-23499-1.