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This page has been largely superseded by PR-Squared and the 2015 General Election.

PR-Squared: A New Description

Julian D. A. Wiseman

Abstract: This paper contains a new description of PR-Squared, superseding and incorporating the previous worked examples and technical notes, and is also available as a printable PDF. PR-Squared is a new electoral system that typically elects a majority government; it elects one local MP from each constituency each of whom is dependent on the local vote; yet it still ensures that equal votes mean equal seats. Hence it has the advantages of first-past-the-post, and yet still has the ‘fairness’ of proportional representation.

Publication history: Only here. Usual disclaimer and copyright terms apply.

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PR-Squared is a new electoral system designed for the UK’s House of Commons. It typically elects a majority government; it elects one local MP from each constituency each of whom is dependent on the local vote; yet it still ensures that equal votes mean equal seats.

PR-Squared works as follows:

We start with a simple example with only three parties and seven constituencies, in which votes are as follows:


The number of seats each party has won is calculated from the parties’ nation-wide vote totals: 28, 20 and 14. The seven seats are allocated proportional to the squares of these, giving an unrounded allocation of 3.98, 2.03 and 0.99, and hence a rounded allocation of 4, 2 and 1.

But which party has won which seat? Let’s guess. If the first four seats were allocated to Red (Palatine, Capitoline, Aventine and Cælian), the next two to Blue (Esquiline and Viminal), and the last to Yellow (Quirinal), then 28 voters across the nation would have voted for their MP. We say that, under this seat assignment, 28 voters are ‘happy’. PR-Squared allocates seats by maximising happiness. A computerised algorithm quickly shows that the maximum happiness is 35: Red wins Palatine, Capitoline, Aventine and Quirinal, Blue takes Cælian and Esquiline and Yellow Viminal.

We bring this into the table, along with the First-Past-The-Post winner:

FPTP winnerRedRedRedBlueBlueYellowYellow3:2:2
PR² winnerRedRedRedBlueBlueYellowRed4:2:1

Observe that:

Let’s consider an actual election result. In the UK 1983 general election the vote for the three large parties split the proportion 44.5% to 28.9% to 26.6%. Seats would have been allocated in proportion to the squares of these numbers: 1980.25, 835.21 and 707.56. Scaling the ratio of the squares so that they total 650 seats gives 365.4, 154.1 and 130.5, which round to an actual seat allocation of 365, 154 and 131: a majority of 80 for the largest party†2. Note that equal votes give equal seats, and nearly equal votes give nearly equal seats (unlike the first-past-the-post, in which the second-largest party received 1.09 times as many votes as the third largest, but 9 times as many seats).

As a further example, the following table shows the 1997 UK election in detail:

Liberal Democrat4,724,62622,322bn51.8752
Referendum Party811,679659bn1.532
Scottish National Party617,260381bn0.891
Labour Co-operative599,423359bn0.831
Ulster Unionist Party258,34967bn0.160
S. D. & L. P.192,06037bn0.090
Plaid Cymru161,03026bn0.060
Sinn Fein126,92116bn0.040
D. U. P.107,34812bn0.030
UK Independence Party106,00111bn0.030
(counting Miss Betty Boothroyd, the Speaker, for Labour)

And who would have won which seat? As a randomly chosen example, in Sedgefield the Labour Party candidate received 33526 votes, against 8383 for the (second-placed) Conservative Party candidate. This would have been sufficient to ensure that the happiness maximisation allocated this seat to the Labour candidate. Indeed, this seat would have still been won by Labour if fewer than 23710 of those who had voted Labour had stayed in bed that day: the winning MP’s majority could be said to be 23710.

By-elections are slightly more awkward under PR-squared than under FPTP. If a seat should become empty, there are two possibilities:

This second possibility requires care. PR-Squared gives parties an incentive to negotiate coalitions before an election, and to court votes without regard to geography; it gives candidates an incentive to encourage both the local and the national votes; and gives voters an incentive to vote for a party with a realistic chance of forming a government. Care must be taken to ensure that the rules about by-elections do not create any unwanted new incentives.

In the UK, an armed terrorist and racketeering organisation has a political front. Such support as it has is geographically narrow, and under FPTP it wins only a small number of constituencies (usually two, in 2001 four). Under PR-Squared, it is highly unlikely that any seats would be won by the terrorists’ front. But in certain constituencies it might win a local by-election, which gives the terrorists an incentive to ‘cause’ a by-election. This is not an idle concern, as both bombings and assassinations have continued during their ‘cease-fire’. So death by murder should not cause a by-election. But if resignations cause a by-election, then the terrorists would have an incentive to kill the sitting MP’s children one by one, until the MP is persuaded to resign. These are not incentives that an electoral system should be creating. So, if there are to be FPTP-style by-elections, they should be triggered only by the death from natural causes of a sitting MP†3.

Hence PR-Squared has many advantages.

Julian D. A. Wiseman, London, September 2001


†1 Also known as the method of odd numbers, Webster’s method, and the method of Saint-Lagüe. The original definition of PR-Squared rounded by the Largest Remainder rule: the difference between the two will only very rarely be more than one seat for any party. Eg., consider dividing 17 seats with the squares of the votes in the ratio 30:3:3:1:1; or 21 seats with the same ratio; or 20 seats with 67:6:6:6:2:2:2:1:1.

†2 We counter-factually assume that the vote totals would be the same under PR-Squared as under FPTP, as such an assumption is near enough to be true for these purposes. However, there is one particular difficulty with mapping FPTP results onto PR-Squared: under FPTP, not all parties stand in all seats. This is because the parties have no incentive to stand where there is no chance of winning. But for completeness PR-Squared needs a rule to cover the case in which, in a particular constituency, no seat-winning party put forward a candidate (i.e., all the candidates are from small local parties). However, whilst there ought to be a rule, in practice, the parties’ incentive is such that this rule would never be needed. Parties want votes, and any party with a chance of winning seats anywhere will want every vote it can get in any constituency. But there should be a rule nonetheless, and it would work as follows. In each constituency in which a party did not put forward a candidate, it is assumed that the party did put forward a candidate (a ‘deemed null candidate’), but that this candidate received zero votes; and any seat won by a deemed null candidate would remain empty.

†3 This also raises the issue of the death of a candidate just before an election. What should happen if a candidate dies after being nominated but before the polling date? To withdraw the candidate from the poll denies voters the opportunity to vote for that party, which seems unfair. The natural course is to allow the dead candidate to remain a candidate: for electoral purposes the candidate will be deemed to have died immediately after the election. If the dead candidate is elected, the by-election rules then come into play.

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