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PR-Squared: Bettering Israeli Politics

Julian D. A. Wiseman

Abstract: Israel has recently elected the Eighteenth Knesset, and learnt, once again, that power comes not from the ballot box, but from the post-election negotiations. This paper describes a better electoral system, PR-Squared, that would typically elect a majority government, that would elect one local MK from each constituency each of whom is dependent on the local vote, whilst still ensuring that equal votes mean equal seats.

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

Contents: Abstract; Publication history; Summary; Small example; Knesset 2009; Qualities.


PR-Squared is a new electoral system. It was originally designed for the UK’s House of Commons, but would 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 MK 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 Party1 (Palatine, Capitoline, Aventine and Cælian), the next two to Party2 (Esquiline and Viminal), and the last to Party3 (Quirinal), then 28 voters across the nation would have voted for their MK. 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: Party1 wins Palatine, Capitoline, Aventine and Quirinal, Party2 takes Cælian and Esquiline and Party3 Viminal.

Knesset 2009

Let’s consider the result of the election held on 10th February 2009 for the Eighteenth Knesset. The following table shows for non-small parties:

÷ 106
Votes for
one more
Yisrael Beiteinu394,577155,69112.101215+17,895
Labor Party334,900112,1588.72913+20,286
United Torah Judaism147,95421,8901.7025+39,269
United Arab List-Ta'al113,95412,9861.0114+47,509
National Union112,57012,6720.9814+47,911
New Movement-Meretz99,6119,9220.7713+51,983
The Jewish Home96,7659,3630.7313+52,958
The Green Movement-Meimad 27,737 769 0.06 0 0 +89,522
OthersSmallSmallSmall00Large, but
≤ 113,903


This has good qualities.

Julian D. A. Wiseman, New York, February 2009

Various technicalities, such as the rules for bye-elections, are discussed in PR-Squared: A New Description.

For party i, given that the Knesset is of total size 120 seats, ‘Votes for one more unrounded seat’ = Sqrt[ Votesi² + (∑Votes²)/(120×(1–Votesi²/(∑Votes²)) – 1) ] – Votesi. This has a small-party maximum value of Sqrt[ (∑Votes²)/(120–1) ], though in theory a larger value could occur for a party that has an unrounded allocation of almost 120–1.

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