PR-Squared and the 2015 General Election

Julian D. A. Wiseman

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

Publication history: Only at www.jdawiseman.com/papers/electsys/pr2_2015.html. Usual disclaimer and copyright terms apply.

There is a PR² group on Facebook. This was not created by, nor is it controlled nor endorsed by, the author, who is not a member of Facebook. Nonetheless, its existence might be of interest to some readers.

PR-Squared: how it works

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 and that fewer votes cannot mean more seats.

PR-Squared works as follows.

• The country is divided into a large number of single-member constituencies.

• In each constituency each party may field at most one candidate.

• Voters cast 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, which is done using the method of major fractions^†1.

• 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 (FPTP) in most non-marginal seats, with marginal seats being rearranged to ensure that parties receive the required number of MPs.

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 Red were allocated the first four seats (Palatine, Capitoline, Aventine and Cælian), Blue the next two (Esquiline and Viminal), and Yellow the last (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 wins Viminal.

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

Observe that:

• In six out of the seven seats, the PR-Squared winner is the same as the FPTP winner. In practice, PR-Squared assigns most non-marginal constituencies to the FPTP winner; but reassigns some of the marginal constituencies to make the national totals work. This does not diminish the importance of a vote cast in a marginal constituency: the government is chosen by nation-wide vote totals, and hence all votes are equal, whether cast in a deep-red, a deep-blue, or a marginal constituency.

• MPs are dependent on the local vote. If a single voter in Quirinal had switched from Red to Yellow, then Red’s Quirinal candidate would have lost to Yellow’s Quirinal candidate. Such a switch would have made Quirinal a less-marginal constituency, so Yellow would hold it and would instead lose Viminal (one voter switching isn’t enough to change the seat totals; if three switched from Red to Yellow, then Reds would win one seat fewer, and Yellows one more). Hence MPs are truly local; a candidate requires a local vote, and the voters have the power to dismiss an individual MP.

• Constituencies being of unequal size is much less unfair than under FPTP. Because the government is chosen by nation-wide vote totals, the fairness of PR-Squared is not reduced by having unequal constituency sizes. Relatedly, there is no purpose to gerrymandering.

1983 and 2015 UK general elections

Let’s consider an actual election result. In the UK 1983 general election the vote for the three large parties split 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× as many votes as the third largest, but 9× as many seats).

As a further example, the following table uses data from the UK general election in May 2015.

Relative to the current system of FPTP, the effect on the two largest parties would be modest. But compare UKIP and the SNP. UKIP had 2.67× as many votes as the SNP, but the current electoral system gave the SNP 56× as many seats. Doubtless the SNP could construct arguments about why this is legitimate, but not arguments likely to persuade neutral observers. Compare to PR-Squared, under which a party cannot have both fewer votes and more seats. Yes, under PR-Squared the votes per seat are not identical (antepenultimate column), but that is deliberate, to penalise slightly the small parties and thereby encourage coalitions to be formed before the election.

The last column of the table illustrates the voters’ incentives. It shows how many more votes a party would need for its unrounded PR-Squared seat count to be +1 higher. To gain one seat, smaller parties need more votes than do larger parties (sometimes excepting the largest party). Consider the position of a voter who is willing to vote for either the Conservative party or for UKIP. One more vote for the Conservative party is worth +1⁄35105 of a seat. One more vote for UKIP is worth +1⁄50574 of a seat. A Conservative vote is 1.44× as powerful as a UKIP vote. So if 1.44 Conservative seats would be preferred to 1.0 UKIP seats, vote Conservative. But if the relative fondness for UKIP is sufficient that 1.0 UKIP seats would be preferred to 1.44 Conservative seats, vote UKIP. The equivalent ratio between Labour and the Liberal Democrats is 2.49× and between Labour and the SNP is 3.96×. Under PR-Squared life would be harder for small parties than for large, but not impossibly hard.

Hypthetical coalitions in the 2015 UK general election

The previous table shows the result of the 2015 UK general election under PR-Squared, assuming that everything else had stayed the same. Which, of course, it wouldn’t. Indeed, the purpose of PR-Squared is that it doesn’t: the purpose is to encourage pre-election coalition-making rather than post-election. Let’s further demonstrate this incentive by considering two possible pre-election coalitions.

We start with a ‘Devolution Party’, presumably favouring the de-centralisation of the UK state. Assume that the Liberal Democrats, the Scottish National Party, Plaid Cymru and the SD&LP formed such a coalition. Omitting some irrelevant rows, that might have looked like the following.

Forming that coalition before the election, making it visible to the electorate, has taken its PR-Squared seats from 26 to 71. PR-Squared rewards an earlier coalition agreement.

And a Lib-Lab coalition would have had more votes than the Conservatives, and hence more seats, even if not quite a majority.

However, a coalition cannot be an entire sham. Two criteria are relevant. First, in any constituency a party may put forward at most one candidate. So the parties trying to form a coalition need to agree on a single candidate in each constituency. Without this much agreement, a putative coalition is no such thing.

Second, a coalition must convince voters. Some might; some won’t. As an example of a coalition that failed to convince the UK electorate, go back to the 1997 general election. There was a coalition between a pro-business pro-efficiency grouping, and a euro-sceptic grouping. This coalition was called ‘The Conservative Party’, and the electorate didn’t like it. But in 2015 a coalition of left to centre-left pro-devolution parties might have the internal coherence to convince the electorate. PR-Squared rewards the honesty of showing that coalition to the electorate, agreements and warts and all.

This honesty would have other advantages. After the 2010 election the Liberal Democrats went into coalition with the Conservatives. The Conservatives were much the larger of the two parties, so did most of the governing. LibDem voters, many of whom saw themselves as at least partly anti-Conservative, afterwards seemed appalled to have ‘voted for’ a mostly-Conservative government. If this coalition had been formed before the 2010 election, then the 2015 election might have been much less of a disaster for the LibDems. Much less.

By-elections

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

• The first possibility is that by-elections don’t happen. Empty seats would remain empty until the next general election.

• The second possibility is that a FPTP-style by-election is held.

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, there has been an armed terrorist and racketeering organisation with a political front. Such support as that front has always been geographically narrow, and under FPTP it wins only a small number of constituencies, usually four. Under PR-Squared, it is unlikely that it would win any seats. But in certain constituencies it might win a local by-election. Its terrorist branch has been inactive since September 2001—may it stay that way. But the rules should not create an incentive for such organisations to ‘cause’ a by-election. So death by murder must 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.

Advantages of PR-Squared

Hence PR-Squared has many advantages.

• PR-Squared elects a government, rather than a collection of parties who then negotiate for power. If parties are sufficiently close in policies to be likely to form a coalition, then they have a strong incentive to do so before rather than after the election, and thus to do so in a manner visible to the electorate. For example, if five parties split the vote equally, then the seats would split equally. But if two of these parties could form a coalition before the election, and still hold 40% of the vote, then that coalition would win four-sevenths of the seats, a functional majority. Thus PR-Squared keeps power at the ballot box rather than at a negotiating table.

• Likewise, voters have an incentive to vote for a large party that has a plausible chance of forming a government. A vote for a small party with no chance of forming a national government, whether a small extremist party or a small local party, would be a wasted vote.

• But every MP is still a local constituency MP, dependent on the local vote. An unpopular or corrupt MP could lose, even if that MP came from a large party.

• PR-Squared’s voting mechanism is very simple: put a single cross by a single candidate. There would not be any need for parties to distribute ‘voting sheets’ that show how to order complicated lists of preferences.

• PR-Squared extends voter choice, at least relative to the UK’s current electoral system, but only to parties with a chance of forming a government. In Liverpool there is little purpose in voting under FPTP; the Labour vote is so overwhelming that it could be weighed rather than counted. But under PR-Squared, voting is important everywhere. An additional Labour vote won’t make any difference in Liverpool, but it might make the difference somewhere else, as might a vote for the Conservatives or the LibDems or the SNP. However, whilst there is an extension of voter choice, it is a limited extension: a vote for the communists or the fascists would still be a wasted vote.

Further reading

• PR-Squared: a three-party system (PR-Squared has ‘stable’ outcomes with three parties, probably not with four, and five or more large parties would soon merge into fewer).

• PR-Squared: UK, May 2010 (a fairer assignment of seats, but still no single party with a majority of seats).

• Non-Intuitive Features of Electoral Systems (or the ways in which your favourite system can misbehave).

Footnotes

†1 Also known as the method of odd numbers, Webster’s method, and the method of Saint-Lagüe. The pre-2001 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.

†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.