Main index | Electoral System index | About author |

*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 www.jdawiseman.com/papers/electsys/pr2_israel_2009.html.
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:

The country is divided into single-member constituencies, presumably numbering 120;

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 (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 MK.**Equivalently, define a ‘happy voter’ to be a voter who voted for his or her MK, and then assign seat winners so as to maximise the nation’s total ‘happiness’.

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

Party1 | 6 | 5 | 5 | 3 | 2 | 4 | 3 | 28 |

Party2 | 4 | 3 | 2 | 5 | 5 | 0 | 1 | 20 |

Party3 | 0 | 0 | 2 | 1 | 1 | 6 | 4 | 14 |

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

Winner | Party1 | Party1 | Party1 | Party2 | Party2 | Party3 | Party1 |

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.

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

The name of the party, and the number of votes received (source);

The square of each party’s votes, seats being proportional to this (shown in millions of votes²);

Each party’s unrounded number of seats, being 120 × Votes² ÷ ∑Votes²;

The rounded number of seats—the number of seats won by that party;

For comparison, the number of seats won under the current non-squared PR system;

The number of additional votes that would have caused the unrounded seats to be one larger.

Party | Votes | Votes² ÷ 10 ^{6} | Unrounded seats: PR² | Rounded seats: PR² | Seats: party-list PR | Votes for one more unrounded seat ^{†} |
---|---|---|---|---|---|---|

Kadima | 758,032 | 574,613 | 44.66 | 44 | 28 | +13,577 |

Likud | 729,054 | 531,520 | 41.31 | 41 | 27 | +13,504 |

Yisrael Beiteinu | 394,577 | 155,691 | 12.10 | 12 | 15 | +17,895 |

Labor Party | 334,900 | 112,158 | 8.72 | 9 | 13 | +20,286 |

Shas | 286,300 | 81,968 | 6.37 | 6 | 11 | +23,014 |

United Torah Judaism | 147,954 | 21,890 | 1.70 | 2 | 5 | +39,269 |

United Arab List-Ta'al | 113,954 | 12,986 | 1.01 | 1 | 4 | +47,509 |

National Union | 112,570 | 12,672 | 0.98 | 1 | 4 | +47,911 |

Hadash | 112,130 | 12,573 | 0.98 | 1 | 4 | +48,040 |

New Movement-Meretz | 99,611 | 9,922 | 0.77 | 1 | 3 | +51,983 |

The Jewish Home | 96,765 | 9,363 | 0.73 | 1 | 3 | +52,958 |

Balad | 83,739 | 7,012 | 0.55 | 1 | 3 | +57,844 |

The Green Movement-Meimad | 27,737 | 769 | 0.06 | 0 | 0 | +89,522 |

Others | Small | Small | Small | 0 | 0 | Large, but ≤ 113,903 |

This has good qualities.

Equal votes mean equals seats. Unlike the electoral system used for the UK’s House of Commons, or for either of the US legislative chambers, parties receiving approximately equal numbers of votes receive approximately equal numbers of seats.

Voters are incentivised to vote for large parties. Consider the position of a voter who is considering voting for either Likud or United Torah Judaism. An extra vote for Likud is worth about one thirteen thousandth of a seat. An extra vote for United Torah Judaism is worth about one thirty-nine thousandth of a seat. So the vote is three times as powerful if given to the larger party. And for that reason, the vote is more likely to go to the larger party. Thus small parties will wither away.

Parties are incentivised to negotiate coalitions before the election—coalitions that would hence be visible to voters. Imagine that five parties would split the vote equally, and hence each would win 20% of the seats. But if two parties could agree to merge (even if only for the purposes of the election), and could hold the 40% of the vote, that merged party would win four-sevenths of the Knesset, 69 seats. That would be a functional majority. Thus small parties will merge with or into larger ones.

There is no purpose to gerrymandering. Many counties with geographic constituencies, even countries that are otherwise honest, have gerrymandered constituencies. (See, for example, How to rig an election,

*The Economist*, 25^{th}April 2002, or Pyongyang on the Potomac?,*The Economist*, 16^{th}September 2004.) But under PR-Squared governments are chosen by nation-wide vote totals, so each party’s number of seats would be unaffected by having constituencies of unequal sizes or odd shapes.MKs are dependent on the local vote. If, in the first example, a single voter Quirinal had switched from Party1 to Party3, then Party1’s Quirinal candidate would have lost to Party3’s Quirinal candidate. Such a switch would have made Quirinal a less-marginal constituency, so Party3 would hold it and would instead lose Viminal (one voter switching isn’t enough to change the seat totals; if three switched from Party1 to Party3, then Party1 would win one seat fewer, and Party3 one more). Hence MKs are truly local; a candidate requires a local vote, and the voters have the power to dismiss an individual MK. That would help keep Israeli politics non-corrupt.

*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[ *Votes*_{i}² + (∑*Votes*²)/(120×(1–*Votes*_{i}²/(∑*Votes*²)) – 1) ] – *Votes*_{i}.
This has a small-party maximum value of Sqrt[ (∑*Votes*²)/(120–1) ], though in theory a larger value could occur for a party that has an unrounded allocation of almost 120–1.

Main index | Top | About author |