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*Julian D. A. Wiseman*

**Abstract:** This paper contains a worked example of PR-Squared, which is a new electoral system. It typically elects a majority government; it elects one local MP from each constituency each of whom is dependent on the local vote; yet still ensures that if two parties receive equal votes then they receive equal seats. Hence it has the advantages of first-past-the-post, and yet still has the 'fairness' of proportional representation.

**Contents: **
Publication History,
Summary of PR-Squared,
Candidates and Votes,
Party-level results,
Candidate-level results,
The winners, and a comparison with First-Past-The-Post,
Final observations.

**Publication history:** Only here. Usual disclaimer and copyright terms apply.

**[ This paper has been superseded by PR-Squared: A New Description, effective September 2001. ]**

Before perusing this worked example, readers are advised to read either the original description of PR-Squared, or the description of PR-Squared in the context of New Zealand.

**Summary of the operation of PR-Squared**

PR-Squared is a new electoral system. It typically elects a majority government; it elects one local MP from each constituency each of whom is dependent on the local vote; yet still ensures that if two parties receive equal votes then they receive equal seats. It works as follows:

As now, the country is divided into a large number of single-member constituencies;

As now, each party fields at most one candidate in all or some constituencies;

As now, each voter casts 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. The rounding is upwards for those parties with the largest fractional seat allocation, down for those with the smallest (the 'largest remainder' rule);

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 in non-marginal seats, with marginal seats being "rearranged" to ensure that parties receive the required number of MPs.

**A specific example**

We now proceed to a fully-worked example of PR-Squared. For simplicity, this example is 'small', with only three parties, seven constituencies, and sixty-two voters.

The three parties, colour-coded Red, Blue and Yellow, put forward candidates as follows:

Constituency: | Palatine | Capitoline | Aventine | Cælian | Esquiline | Viminal | Quirinal | |

Red | Mr A | Mr B | Ms C | Dr D | Miss E | Dr F | Mr G | |

Blue | Mrs H | Dr I | Mr J | Mr K | Miss L | Dr M | Mr N | |

Yellow | Mr O | Mr P | Mr Q | Dr R | Miss S | Dr T | Mrs U |

(The explicit naming of candidates is to emphasise that PR-Squared is **not** a list system: MPs are chosen by voters not by parties.)

These candidates receive votes as follows:

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

Red | 6 | 5 | 5 | 3 | 2 | 4 | 3 | 28 |

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

Yellow | 0 | 0 | 2 | 1 | 1 | 6 | 4 | 14 |

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

**The number of seats has each party 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. The Reds are to receive at least 3 seats (remainder of 0.98); the Blues at least 2 (remainder 0.03); and the Yellows at least zero (0.99). Two seats remain; these go to the parties with the largest remainders (Yellows' 0.99, and Reds' 0.98). So the Reds end with 4 seats (giving a narrow majority), Blues 2 seats and the Yellows 1 seat.

Observe that this result is partly proportional. As with PR, equal votes give equal seats. But double votes do not give double seats — they give quadruple seats. Thus the Reds, with 45% of the vote, won 57% of the seats. This explicit non-proportionality gives parties an incentive to form coalitions before an election rather than after: it prevents power shifting from the ballot box to the negotiating table.

So now we know how many seats each party has won, **which party has won which seat?**

Let's guess. If the first four seats were allocated to Red (Palatine, Capitoline, Aventine and Cælian), the next two to Blue (Esquiline and Viminal), and the last to yellow (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 this number (a process called 'happiness maximisation').

We guessed at a seat assignment of 'RRRRBBY'. Is there better? There are 105 ways to assign seats in a four, a pair and a single (= 7!/4!2!1!). These have happinesses as follows:

RRRRBBY 28; RRRRBYB 31; RRRRYBB 21; RRRBRBY 27; RRRBRYB 30; RRRBBRY 34; RRRBBYR 35; RRRBYRB 27; RRRBYBR 25; RRRYRBB 20; RRRYBRB 27; RRRYBBR 25; RRBRRBY 22; RRBRRYB 25; RRBRBRY 29; RRBRBYR 30; RRBRYRB 22; RRBRYBR 20; RRBBRRY 28; RRBBRYR 29; RRBBYRR 26; RRBYRRB 21; RRBYRBR 19; RRBYBRR 26; RRYRRBB 19; RRYRBRB 26; RRYRBBR 24; RRYBRRB 25; RRYBRBR 23; RRYBBRR 30; RBRRRBY 23; RBRRRYB 26; RBRRBRY 30; RBRRBYR 31; RBRRYRB 23; RBRRYBR 21; RBRBRRY 29; RBRBRYR 30; RBRBYRR 27; RBRYRRB 22; RBRYRBR 20; RBRYBRR 27; RBBRRRY 24; RBBRRYR 25; RBBRYRR 22; RBBYRRR 21; RBYRRRB 21; RBYRRBR 19; RBYRBRR 26; RBYBRRR 25; RYRRRBB 17; RYRRBRB 24; RYRRBBR 22; RYRBRRB 23; RYRBRBR 21; RYRBBRR 28; RYBRRRB 18; RYBRRBR 16; RYBRBRR 23; RYBBRRR 22; BRRRRBY 23; BRRRRYB 26; BRRRBRY 30; BRRRBYR 31; BRRRYRB 23; BRRRYBR 21; BRRBRRY 29; BRRBRYR 30; BRRBYRR 27; BRRYRRB 22; BRRYRBR 20; BRRYBRR 27; BRBRRRY 24; BRBRRYR 25; BRBRYRR 22; BRBYRRR 21; BRYRRRB 21; BRYRRBR 19; BRYRBRR 26; BRYBRRR 25; BBRRRRY 25; BBRRRYR 26; BBRRYRR 23; BBRYRRR 22; BBYRRRR 21; BYRRRRB 19; BYRRRBR 17; BYRRBRR 24; BYRBRRR 23; BYBRRRR 18; YRRRRBB 16; YRRRBRB 23; YRRRBBR 21; YRRBRRB 22; YRRBRBR 20; YRRBBRR 27; YRBRRRB 17; YRBRRBR 15; YRBRBRR 22; YRBBRRR 21; YBRRRRB 18; YBRRRBR 16; YBRRBRR 23; YBRBRRR 22; YBBRRRR 17.

So the maximum happiness is 35, when the seats are assigned as "RRRBBYR". In this assignment, thirty-five voters will have voted for their local MP.

Let us bring the results into a table, and for comparison also show which party would have won each seat under First-Past-The-Post.

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

Red | 6 | 5 | 5 | 3 | 2 | 4 | 3 | |

Blue | 4 | 3 | 2 | 5 | 5 | 0 | 1 | |

Yellow | 0 | 0 | 2 | 1 | 1 | 6 | 4 | |

PR-Squared winner | Mr A | Mr B | Ms C | Mr K | Miss L | Dr T | Mr G | |

Red | Red | Red | Blue | Blue | Yellow | Red | ||

FPTP winner | Red | Red | Red | Blue | Blue | Yellow | Yellow |

Several observations follow.

In six out of the seven seats, the PR-Squared winner was the same as the FPTP winner. In practice, PR-Squared will assign the non-marginal constituencies to the FPTP winner; but will re-assign 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**: governments are chosen by nation-wide vote totals, and all votes are equal, be they cast in a deep-red constituency, a deep-blue constituency, or a marginal constituency.With a small example, it is possible to list every combination of parties. With hundreds of constituencies the assignment would have to be performed by a computerised algorithm. This author, by no means a professional programmer, wrote code that performed the assignment for the UK's 1997 general election in about a minute. The software then took almost hour to confirm that it had done the correct assignment (running on a 1995-vintage laptop).

MPs are very dependent on the local vote. If a single voter in Quirinal had switched from Red (Mr G) to Yellow (Mrs U), then Mr G would have lost to Mrs U. Such a switch would have made Quirinal a 'less-marginal' constituency — 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 without a local vote cannot win. The voters have the power to dismiss an individual MP.

In such a small example, there is a fairly high probability of a tie. But equally, in First-Past-The-Post with 8 to 10 voters per constituency, there is a fairly high probability of a tie. With several tens of thousands of voters per constituency, this is much less of a risk for both electoral systems.

There is no need for the constituencies to be of equal size. Because governments are chosen by nation-wide vote totals, the fairness of PR-Squared is not reduced by having unequal constituency sizes.

Thus PR-Squared has the majoritarian effectiveness of first-past-the-post, the equal-seats-mean-equal-votes independence of geography of proportional representation, yet still every MP is a local MP dependent on the local vote.

*Julian D. A. Wiseman, March 2000*

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