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PR-Squared: UK, May 2010

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 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.

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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:

PR-Squared: small example


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.

The UK’s May 2010 election, if the electoral system were PR-Squared

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.

Using data taken from at 18:20 BST on Friday 7 May 2010 (after 649 seats declared) gives the following nationwide totals.

Votes for
one more
Liberal Democrat6,827,93846,621bn127.73128+33,26157
UK Independence Party917,832842bn2.312+181,8460
British National Party563,743318bn0.871+263,2190
Scottish National Party491,386241bn0.661+287,9746
Democratic Unionist Party168,21628bn0.080+459,4068
Plaid Cymru165,39427bn0.070+461,4763
Sinn Fein171,94230bn0.080+456,6905
Social Democratic & Labour Party110,97012bn0.030+503,7673
Ulster Conservatives and Unionists - New Force102,36110bn0.030+510,8780
English Democrats64,8264.2bn0.010+543,2670
Alliance Party42,7621.8bn0.010+563,3731
Respect-Unity Coalition33,2511.1bn0.000+572,2870
Traditional Unionist Voice26,3000.7bn0.000+578,8950
Sylvia Hermon (Independent)21,1810.4bn0.000+583,8131
OthersSmallSmallSmall0≈ +604,6230

Comparison: FPTP versus PR-Squared

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

— Julian D. A. Wiseman
Paris, 7th May 2010

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