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PR-Squared in New Zealand

Julian D. A. Wiseman

Abstract: PR-Squared is a new electoral system. It typically elects a majority government; it elects one local MP from each constituency each of whom is dependent on the local vote; yet still ensures that if two parties receive equal votes then they receive equal seats. Hence it has the advantages of first-past-the-post, and yet still has the 'fairness' of proportional representation.

Contents: PDF version, Publication History, Introduction, The 1996 election under PR-Squared, Positive Qualities of PR-Squared, Summary

Publication history: A version of this paper describing PR-Squared in the context of the UK was sent to the The Independent Commission on the Voting System on 14th February 1998. Both that paper, and this one describing PR-Squared in the context of New Zealand have been privately circulated since. The acrobat version is the original, the HTML versions contains minor alterations for simplicity of formatting. Usual disclaimer and copyright terms apply.

[ This paper has been partially superseded by PR-Squared: A New Description, effective September 2001. ]

PR-Squared is a new electoral system. It typically elects a majority government; it elects one local MP from each constituency each of whom is dependent on the local vote; yet still ensures that if two parties receive equal votes then they receive equal seats. It works as follows:

The country is divided into a number of single-member constituencies;

Each party fields at most one candidate in all or some constituencies;

Each voter casts a single vote in favour of a single candidate;

The votes for each party are totalled nation-wide;

The key rule: each party is allocated seats in proportion to the square of its nation-wide vote;

As only a whole number of seats can be won, the seat allocations must be rounded. The rounding is upwards for those parties with the largest fractional seat allocation, down for those with the smallest (the 'largest remainder' rule);

It is now known how many seats each party has won, but not which constituencies. Constituencies are allocated to the parties in the manner that maximises the nation-wide total of the number of voters who voted for their local MP. Equivalently, define a "happy voter" to be a voter who voted for his or her MP, and then assign seat winners so as to maximise the nation's total "happiness". In practice this will be First-Past-The-Post in non-marginal seats, with marginal seats being "rearranged" to ensure that parties receive the required number of MPs.

As a first example, imagine that three parties split the vote in proportions 24:18:10. The squares of these are 576:324:100, which total 1000, so seats would be split 57.6% : 32.4% : 10%, a majority for the largest party of 15.2% of the seats.

The following table shows the 1996 New Zealand General Election in detail: [table showing 1999 election result now also available]

PartyVotesVotes SquaredUnrounded seatsActual seats
New Zealand First276,60376.51bn9.3810
Act New Zealand126,44215.99bn1.962
Christian Coalition89,7168.05bn0.991
Aotearoa Legalise Cannabis34,3981.18bn0.150
United New Zealand Party18,2450.33bn0.040
(Uses party votes: assumes that, under PR-Squared, voters would have voted for the candidate from party for which they voted in 1996; and also assumes a total of 120 seats.)

The above table assumes a house of 120 MPs. In a house of 99 MPs, the unrounded totals would have been 49.74 / 34.51 / 7.74 / 4.43 / 1.62 / 0.81 / 0.12 / 0.03 (in the same order as the table), giving an actual seat allocation of National 50 / Labour 34 / NZFirst 8 / Alliance 4 / ActNZ 2 / CC 1, a minimal National majority.

We lack constituency-by-constituency results for a house of either 99 or 120, but do have constituency-by-constituency results for a 65-part division of the country. So, for the purposes of this example, let us counter-factually assume that New Zealand was divided into the same 65 single-member constituencies. (Further, we continue to assume that voters would have voted for the candidate from the voter's preferred party.)

In a house of 65 MPs, the unrounded totals would have been 32.66 / 22.66 / 5.08 / 2.911.06 / 0.53 / 0.08 / 0.02, giving an actual seat allocation of National 33 / Labour 23 / NZFirst 5 / Alliance 3 / ActNZ 1.

And who would have won which seat? As a randomly-chosen example, in Rakai the National Party candidate would have received 14180 votes, against 7956 for the (second-placed) Labour Party candidate. This would have been sufficient to ensure that the 'happiness-maximisation' allocated this seat to the National candidate in this constituency; and if in this constituency fewer than 4445 of those who voted National had stayed a-bed that day, then this seat would still have been held by the same candidate.

From this, we can see that PR-Squared has many positive qualities:

Fairness: PR-Squared explicitly ensures that equal votes give equal seats, and that nearly-equal numbers of votes give nearly-equal numbers of seats.

Stable government: the squaring of votes penalises small parties and penalises party splintering. Hence coalitions have an incentive to form before the election, rather than negotiating for power after the election. Minority factions within a coalition would have a strong incentive not to undermine that coalition, because if there was a split then the faction would lose most or all of its seats in the following election. Indeed, this is similar to the "exaggerative effect" of FPTP, an effect that penalises small parties and hence prevents them holding the balance of power. The absence of such an exaggerative effect leads to coalition politics in which power moves from the ballot box to the post-election negotiating table.

Voter choice: Under FPTP there are many seats in which the result is such a foregone conclusion that there is little purpose to voting. Under PR-Squared, all votes have the same effect on each party's total representation, irrespective of where they are cast. But, because of the squaring, there is little purpose in voting for tiny parties that are highly unlikely to win any seats anywhere.

Constituency MPs: Although parties are elected by their nation-wide results, every MP is local and dependent on local votes for re-election. Unlike MMP or AMS, there is one type of MP, and that one type of MP is a constituency MP.

Simple voting mechanism: voters just place a mark by the desired candidate. There is no requirement to sort a large number of candidates into a preferred order.

Parties' representation is independent of the geographical distribution of the votes. Thus boundary commissions become far less important. Moving a seat's boundary might still cause a rearrangement of seats, but not a change in any party's total number of seats. From the viewpoint of a party seeking a national mandate, boundary changes are irrelevant.

Party headquarters are denied the power and patronage that would result from a 'list system';

All votes carry equal weight, even if constituencies vary in size;

There are no "wasted votes", because parties gain or lose power on the basis of the nation-wide vote totals.

In summary, PR-Squared is an electoral system in which voting is simple, in which voters choose a local MP, which gives stable government, but which treats equally parties with dispersed and parties with concentrated support. It may not be perfect, but it is materially less imperfect than any other electoral system. In other words, PR-Squared is as good as it gets.

Julian D. A. Wiseman, 13th December 1998

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