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*Julian D. A. Wiseman*

**Abstract:** Had PR-Squared been used in the UK’s May 2010 election, there would have been a fairer assignment of seats, but still no single party with a majority of seats.

**Publication history:** only at www.jdawiseman.com/papers/electsys/pr2_uk_201005.html.
Usual disclaimer and copyright terms apply.

**Contents:**
Summary;
PR-Squared: small example;
The UK’s May 2010 election, if the electoral system were PR-Squared;
Comparison: FPTP versus PR-Squared.

PR-Squared is a new electoral system. Though it was originally designed for the UK’s House of Commons, it would also work well in other countries, particularly those seeking to increase the majoritarian-ness of a vanilla electoral PR system. PR-Squared 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:

As now, the country is divided into single-member constituencies;

In each constituency each party may field at most one candidate;

As now, 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 (also known as the method of odd numbers, Webster’s method, and the method of Saint-Lagüe).

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’.

Palatine | Capitoline | Aventine | Cælian | Esquiline | Viminal | Quirinal | Totals | |
---|---|---|---|---|---|---|---|---|

PartyA | 6 | 5 | 5 | 3 | 2 | 4 | 3 | 28 |

PartyB | 4 | 3 | 2 | 5 | 5 | 0 | 1 | 20 |

PartyC | 0 | 0 | 2 | 1 | 1 | 6 | 4 | 14 |

Totals | 10 | 8 | 9 | 9 | 8 | 10 | 8 | 62 |

Winner | PartyA | PartyA | PartyA | PartyB | PartyB | PartyC | PartyA |

We start with a small example with only three parties and seven constituencies, in which votes are as in the table on the right.

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 PartyA (Palatine, Capitoline, Aventine and Cælian), the next two to PartyB (Esquiline and Viminal), and the last to PartyC (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: PartyA wins Palatine, Capitoline, Aventine and Quirinal, PartyB takes Cælian and Esquiline and PartyC Viminal.

What would have happened if, for the May 2010 election, the UK’s electoral system had been PR-Squared? To compute this it is neccessary to map the votes under FPTP to PR-Squared.

Happily, the method of voting under PR-Squared is the same as under the current first-past-the-post: one tick in one box for one candidate standing for one party.

However, in Thirsk and Malton the death of a candidate has delayed the election until three weeks after other constituencies have voted, as reported by the BBC. For these purposes Thirsk and Malton has been ignored, and the House of Commons deemed to have one fewer MP: 649 instead of 650.

Using data taken from news.bbc.co.uk/1/shared/election2010/results/ at 18:20 BST on Friday 7 May 2010 (after 649 seats declared) gives the following nationwide totals.

Voting | PR-Squared | FPTP | ||||
---|---|---|---|---|---|---|

Party | Votes | Votes² | Unrounded seats | Rounded seats | Votes for one more unrounded seat | Seats |

Conservative | 10,706,647 | 114,632bn | 314.06 | 314 | +33,076 | 306 |

Labour | 8,604,358 | 74,035bn | 202.83 | 203 | +30,867 | 258 |

Liberal Democrat | 6,827,938 | 46,621bn | 127.73 | 128 | +33,261 | 57 |

UK Independence Party | 917,832 | 842bn | 2.31 | 2 | +181,846 | 0 |

British National Party | 563,743 | 318bn | 0.87 | 1 | +263,219 | 0 |

Scottish National Party | 491,386 | 241bn | 0.66 | 1 | +287,974 | 6 |

Green | 285,616 | 82bn | 0.22 | 0 | +383,167 | 1 |

Democratic Unionist Party | 168,216 | 28bn | 0.08 | 0 | +459,406 | 8 |

Plaid Cymru | 165,394 | 27bn | 0.07 | 0 | +461,476 | 3 |

Sinn Fein | 171,942 | 30bn | 0.08 | 0 | +456,690 | 5 |

Social Democratic & Labour Party | 110,970 | 12bn | 0.03 | 0 | +503,767 | 3 |

Ulster Conservatives and Unionists - New Force | 102,361 | 10bn | 0.03 | 0 | +510,878 | 0 |

English Democrats | 64,826 | 4.2bn | 0.01 | 0 | +543,267 | 0 |

Alliance Party | 42,762 | 1.8bn | 0.01 | 0 | +563,373 | 1 |

Respect-Unity Coalition | 33,251 | 1.1bn | 0.00 | 0 | +572,287 | 0 |

Traditional Unionist Voice | 26,300 | 0.7bn | 0.00 | 0 | +578,895 | 0 |

Sylvia Hermon (Independent) | 21,181 | 0.4bn | 0.00 | 0 | +583,813 | 1 |

Others | Small | Small | Small | 0 | ≈ +604,623 | 0 |

Some observations and comments on the results and on the hypothetical.

Assuming unchanged votes, the Conservatives would still be the largest party, and still slightly short of a majority. A coalition of any two of the big three parties would have a majority.

Labour won 26% more votes than the Liberal Democrats, but under the current system won 353% more seats. Some might consider this a repeat of the unfairness of the 1983 election. But under PR-Squared Labour would have won 59% more seats than the Liberal Democrats: the squaring causing the disparity in seats to be little more than twice the disparity in votes.

The Liberal Democrats won 7.4× as many votes as the UK Independence Party, but under FPTP seats were 57-to-0. PR-Squared would have awarded two seats to UKIP: deliberately disproportionately small, but fairer than none.

The British National Party won 14.7% more votes than the Scottish National Party, but FPTP gave 0 seats to the larger of these two parties and 6 seats to the smaller. Even though the BNP is distasteful, this seems unjust. PR-Squared awards one seat to each.

Under PR-Squared the sectarian parties of Northern Ireland would get no seats: instead the large nationwide parties would compete for the votes of the Northern Irish. There being no sectarian prizes to win might reduce the sectarian-ness of N.I. politics.

Under PR-Squared an extra vote to any of the three largest parties is worth about a thirty-two-thousandth of an unrounded seat. So a voter favouring one of the three largest parties should vote for it. But a vote for the fourth largest party would be less than a fifth as powerful: for it to be worth voting UKIP, a voter would have to like it at least 5½ times as much. For the BNP, the required factor is 8 times as much. PR-Squared asks voters to choose from amongst the plausible governments: small parties will wither away.

— Julian D. A. WisemanParis, 7 ^{th} May 2010www.jdawiseman.com |

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