From electowiki

Quota proportional representation squared (QPR2) is an electoral system designed to produce semi-proportional election results across a region by electing two representatives in each of the region’s districts. The 1st seat in every district is awarded to the party or candidate which receives the most votes, similar to first-past-the-post voting (FPTP). The 2nd seat is awarded to one of the remaining district parties or candidates so that squared proportionality is achieved across the region, using a calculation that aims to award parties their seats in the districts where they had their strongest performances, relative to the Droop quota.

The system is very similar to "PR squared", devised by Dr. Julian Wiseman.


The first suggestion for squaring votes before assigning seats in proportion to these squares appears to have been made by Julian Wiseman in 1998, proposing a version of the system to elect the UK Westminster parliament.[1] In May 2010, Rod Stringer[2] offered improvements that would make the system more practical and politically acceptable,[3]and this version of the system was later simplified.

Objectives and rationale

Stringer believes that a combination of political vested interest and inability to agree on one of the various true proportional systems means that it is unlikely that the UK will ever abandon FPTP for Westminster. A resolute attachment to single-member constituencies and the presumed desirability of majority governments are also factors, especially for the UK's most successful party, the Conservatives.

PR squared is a system that is a relatively small change from the perspective of voters and the major parties, has some benefits for almost all parties, and accordingly may stand a higher chance of achieving consensus for its adoption. While not closely proportional, PR squared can be shown to be approaching about half the disproportionality of FPTP, according to certain common measures, in particular treating third and fourth parties which compete UK-wide more fairly.

The prospect of majority government on a minority of votes remains quite possible, and in large part PR squared is advocated for pragmatic reasons. It is essentially a "half-way house" between FPTP and PR, retaining much of the former's structure and familiarity, while going a considerable distance towards proportionality.

Similar systems

Aside from the novel idea of allocating seats in proportion to the square of the votes, the system has some resemblance to the Zweitmandat in Baden-Württemberg, the Sekihairitsu used in Japan, and Dual member proportional proposed in Canada, known as "best loser" systems.

PR squared may not fit that definition, however, since it is hard to view as a loser a candidate who gains a strong second-place in a two-member seat, and these are among the candidates the system elects.

QPR2 described

Constituency structure

The current single-member seats would be amalgamated into ideally units of two, although accommodation could be made to retain three island seats as single-member and, exceptionally, a few three-member seats could also exist, mostly in the cities or where a county currently has an odd number of seats. This constituency structure would mimic the constituency structure that in fact existed continuously in England from 1265 to 1885 and which did not finally disappear in some places in the UK until 1950.

Ballot structure

Major parties would most likely run two candidates in each constituency, although it is not a requirement. If two, candidates for a party could run together as primary and secondary in an effectively closed list, similarly as in the Dual member proportional system, or they could run separately. Parties might choose the latter option in an open seat or in a hopeless seat, to effectively combine a primary with the general election.

It is recommended that voters employ a preferential 1,2,3, etc vote, for reasons discussed below, although an "X" would also be deemed a valid vote.

The ballot size would not be materially increased over that of the current FPTP ballot.

National vote count

Counting would proceed in a constituency similarly to FPTP, except where candidates from the same party were running separately, in which case they would first have their combined vote total aggregated and assigned to their party. The party vote totals and those for any individual independent candidates in the constituency would then be forwarded to the national counting centre, one for each of the four UK constituent nations.

Upon receipt, these totals would be aggregated party-wise and the squares of the totals computed, ultimately producing,in each nation, party percentages relative to the sum of these squares. The normal vote percentages would naturally also be calculated and published, just as now.

National seat computation

In each nation, the squared percentages calculated above would be applied to the national seat total to estimate the number of seats won by each party. Largest remainder would be used to assign fractional seats. Note that the ultimate result may vary slightly from these figures, for reasons outlined later.

For example, the first "target" computation for a simple one nation house of 650 could be:

Example election
Party %Votes %Seats Unrounded Seats Rounded Seats
A 40% 56.1% 364.9 365
B 30% 31.6% 205.3 205
C 15% 7.9% 51.3 51
D 10% 3.5% 22.8 23
E 5% 0.9% 5.7 6

An alternative example election result:

Example election
Party %Votes %Seats Unrounded Seats Rounded Seats
A 35% 46.4% 301.8 302
B 30% 34.1% 221.8 222
C 20% 15.2% 98.6 98
D 8% 2.4% 15.8 16
E 7% 1.9% 12.1 12

First members elected

Again, similarly to FPTP, the candidate who tops the poll will be declared elected by the local returning officer. In the case of two running on a closed list the primary candidate will be elected. Additionally, any other candidate (or list) that exceeds the Droop quota will also be declared elected. In a few cases, mostly inner-city Labour seats, a party may obtain two Droop quotas - more than 66.67% - in which case both candidates of that party are elected, and the seat closed. Similarly, another group of constituencies will be closed by virtue of the fact that two different parties each obtain a quota. A little over half the MPs should be elected in this manner, and election night coverage by TV networks would initially not look too different to what it does under FPTP.

In the case where two party candidates are running separately, and their combined party vote either wins a plurality over all other party votes, or it exceeds the quota, a further STV-style count must be employed later, once the full complexion of the constituency is discerned, to distinguish the winning candidate(s). Such instances should be comparatively rare since, in viable seats,, most parties would usually adopt the closed "tandem" list variant of presenting candidates.


  • Party A (Smith [primary], Jones [secondary])

as opposed to what could be termed the "unicycles" approach

  • Party A (Jones)
  • Party A (Smith)

Subsequent members elected

The novel part of the process really begins here. At the national count centre, the ordinary party vote percentages in each constituency for the first and second ranked parties are then normalised relative to the Droop quota, disregarding any integer part.

For example, if Party A obtained 45% and Party B obtained 25%, these would translate to approximately 0.35 and 0.75 "remainder" quotas respectively, with Party A of course already having won one seat. Normalised remainder quotas, instead of raw remainder votes, are employed to control for constituency electorate size and/or turnout effects.

The national count centre and TV networks would rank these remainders separately for each party from highest to lowest, including the constituency name, as results become available. As election night develops, these rankings become indicative of those parties which will win their available second seats, and where they will likely win them. As fresh results arrive, they will be "slotted" into the correct order, so the table rankings will continuously change, attracting interest from viewers and commentators.


Party D - Seats available: 8
Constituency Remainder Quota
Blankshire West 0.91
Oldtown North 0.83
Bracketshire South 0.71
Newtown East 0.60
... ...
... ...

In simple terms, the number of seats already won by each party on the first count is deducted from their national seat allocation and the n best remainder quotas for each party are assigned to the relevant constituencies as the second seats. Some parties may end up winning both seats as a result, while strong second places will also be rewarded.

One issue is the order in which these final allocations are made. Depending on the order, the actual candidates elected may vary slightly. It is suggested that the parties are prioritised in ascending order of size, i.e. the smallest party has all its allocations filled first, then the next smallest, etc. See also the Anomalies section below.


It may be that a local independent or micro-party, with no prospect of national seat allocation, makes a breakthrough to quota or plurality in a particular constituency, and hence wins a seat. This is an admirable feature, not a bug, of the system, although its incidence may be only marginally more frequent than under FPTP.

In such case, the major parties' national seat allocation would require to be re-computed, after deducting a seat from the national total initially available. Such adjustments would be automatically calculated by national counting centre and TV network computers.

National thresholds?

The system, by virtue of the mathematics of squaring, will generate nominal "natural thresholds" for each of the four nations. The actual figures will vary slightly, depending on the precise distribution of votes for the parties, but they are approximately:

  • England 2.5%
  • Scotland 7%
  • Wales 8%
  • Northern Ireland 10.5%

Especially for the smaller countries, these may seem high, until it is recognised the thresholds are not absolutely "hard", nor indeed guaranteed. The additional features of the system afford an alternative route to winning a seat or seats, by virtue of topping a local poll or obtaining a quota. In this respect, the system makes it marginally easier for an independent compared to FPTP. Approximately 33.34% compared to perhaps 35-40% under FPTP.

Conversely, a national party in England on 5% could still win no seats if its vote was so uniformly distributed that it obtained no constituency first or second places. But that, of course, is little different to the current system in the UK.

If an explicit threshold or barrage was thought appropriate - say 5% in England - that could be implemented, leaving the smallest parties with their only route into the legislature being via very strong individual constituency performances.

Why ranked ballots?

While not absolutely essential, they would have utility as follows:

  • in a minority of cases where a winning party's candidates ran separately in a constituency, it may be more appropriate to employ a kind of STV to identify the winning candidate instead of the alternative, a kind of SNTV.
  • the ranked ballots could be subsequently recounted to compute a nationwide two-party preferred vote, as occurs in Australia. In a UK context that may be of value in the event of a hung parliament.


Every vote counts equally

Parties would be incentivized to seek votes from Land's End to John’O'Groats, not just in the "marginals". Good Tory candidates with effective campaigns in Liverpool or Glasgow, even if not elected there, would be as crucial to the national outcome as good Labour candidates in Surrey. Parties would thereby reconnect with the voters, and moribund party organisations in previous electoral deserts would be revived.

Ludicrously, in both 2005 and 2010, a majority of all voters found themselves represented by parties they did not vote for in their constituencies. Under PR-Squared in 2010, around three-quarters of voters would have been represented by an MP from the party of their choice.

Electoral bias is removed

Since the national vote totals are the overwhelming determinant of the election outcome, the boundaries, and size, turnout and efficiency effects become irrelevant. All parties would be presented with a level playing-field. The sisyphean task of tinkering with the boundaries to partially redress electoral bias is made redundant, and boundaries could more properly reflect actual communities.

Safe seats are reduced

While parties may still have their core constituencies, every candidate there and elsewhere might have to compete equally against others from the same party for the privilege of being elected, especially if the parties were to adopt the open list format of running candidates as opposed to the "tandem" closed list format.

Party-wise, the tendency for FPTP to produce clean sweeps in sub-regions would be reduced, meaning hitherto reasonably safe seats would become vulnerable, especially for secondary candidates on "tandem" lists.

More evenly-spread representation

The tendency under FPTP for sub-regions to produce wipe-outs for a party would be somewhat reduced, and parties would enjoy a fairer geographical distribution of their seats. Pockets of minority support would still gain representation.

Overall majority possible

Under the system, simulations of the elections of 1979, 1983, 1987, 1992, 1997, 2001 and 2019 would have still produced overall majorities, while landslides would be moderated.

No practical possibility of "wrong-winner" election

No other constituency-based system offers this, including FPTP. Under simulations, PR squared gets the very close election of February 1974 "right", whereas under FPTP, the national vote plurality winners (the Conservatives) were reduced to second place in seats. Similar FPTP "inversions" occurred in 1951 and 1929.

Regularity of swing

How does PR squared measure up against FPTP's fabled ability to 'throw the rascals out'? Unlike pure PR where only about six seats would switch between Labour and Conservative for every 1% swing - inviting deadlock, or preferential systems such as AV and STV where unpopular parties may be crushed owing to those systems' invitation to 'gang-up' against them, PR-Squared would operate with almost the regular, quiet precision of Big Ben.

Under FPTP, on average since 1974, a net 13.1 seats have changed hands between the two main parties for every unit of swing, although it has varied widely, and been in long-term decline in the UK. In 2001 each unit of swing "swung" just 3 net seats (fewer than PR would deliver!), whereas in 1992 and 2010 it swung around 20.

PR-Squared would deliver an almost identical average of 13.4 seats, but unlike FPTP, would do this more consistently and indefinitely, ranging from 11 to 15, according to simulations.

Treats third (and smaller) parties more fairly

At almost every election in the past the Liberal Democrats (and its predecessor parties) would have won significantly more seats than under FPTP; extremists, splinters and micro-parties would win few, if any.

Smaller parties or independents may occasionally break through in a particular constituency, just as they can under FPTP. Closed list forms of PR make this impossible, and very difficult even under semi-open lists.


Since the "wasted vote" disincentive under FPTP largely falls away under PR squared, it might be expected that turnout would rise if voters are informed that their vote really does count, even in hitherto electoral deserts for their preferred party.

Reduced tactical voting

Similarly to the turnout benefit, one might hypothesise that tactical voting would be reduced. If voters understand their votes have a direct effect on the national outcome, they may feel encouraged to vote more honestly for what they want, instead of (under FPTP) guessing which party has the best chance of locally defeating what they don't want.

Retains small constituencies

Unlike mixed-member proportional systems, there would neither be two classes of MP, nor “zombies” (those "killed-off" in a constituency yet "resurrected" via a list). Every MP would have to face the voters directly and would represent a small tangible constituency.

Coalition preferences revealed

In the case of a hung parliament, coalition building is placed in the hands of the voters, if the ranked ballot add-on is adopted, to calculate the two-party preferred vote.


Single-member seats abolished

It is recognised that, in a multi-party democracy, a more proportional and representative electoral system cannot be created from exclusively single-member constituencies, except with huge loss of credibility. (as in Wiseman's original proposal, re-assigning single-member constituency losers as winners)

A minimal increase in district magnitude to two (essentially reverting to the previous long-standing UK configuration) would, under PR-Squared, make huge strides towards representativeness and proportionality, with few minor anomalies undermining its credibility.

More hung parliaments

According to simulations (see below) the elections of October 1974, 2005 and 2015 would have produced hung parliaments, in contrast to overall majorities under FPTP. Given the closeness of votes in those three elections, these outcomes do not seem unreasonable.

Of the thirteen elections since February 1974, FPTP produced 10 overall majorities and 3 hung parliaments. Simulations indicate that the outcomes under PR squared would have been 7 and 6 respectively. It's worth noting that, due to long-term changes in the operation of FPTP in the UK, since 2010 the ratio of majority to hung parliaments has been even, and the current (as of 2023) opposition Labour party faces an unprecedented challenge to secure a majority at the next UK election. It seem the UK is having more hung parliaments anyway.

Not PR

PR-Squared does not explicitly seek close proportionality, although simulations indicate that it goes a lot further towards that outcome than FPTP.

Moreover, PR-Squared can deliver more proportional outcomes than some supposedly PR systems! For example, the Welsh Additional Member System has produced Gallagher Indexes of disproportionality in excess of 10% in four out of six elections since 1999. Even higher values can be found in elections to the European Parliament conducted in Ireland under STV. [4]

Simulations suggest that PR-Squared would have delivered Gallagher Indexes of less than 10% in nine out of thirteen elections for the UK parliament since 1974. (and compared to only one under FPTP)


While simulations indicate that around 60% of the seats would be declared individually, throughout election night, just as under FPTP, the final allocation of seats would not be made until all votes are counted, when it would occur practically instantaneously. However a clear indication, especially for smaller parties, would become apparent long before the conclusion of counting, and computer forecasts would no doubt be continuously updated, similarly for most electoral systems including FPTP.

Possible anomalies

In 1994-5, Professor Anton Buhagiar suggested improvements to Malta's STV system to increase proportionality and eliminate "wrong-winner" elections. After considering various methods of first assigning party seats to multi-member constituencies so that strict national proportionality was maintained, he leaned towards the use of a "Priority Queue". This is a rather complex and convoluted method, particularly in the UK with 650 seats to allocate (or rather about 40% of them), and Buhagiar conceded that anomalies will arise whichever method is chosen.[5] Buhagiar's paper was in fact the seed for Stringer's suggested improvements to Wiseman's original version of PR-Squared.

Buhagiar describes the anomaly (if seats are assigned partywise, before he considered the priority queue alternative) as follows:

"The proposed order for the party scans in the partywise distribution was that determined by the size of the nationwide first count vote, as mentioned in iv) above. The largest party has all its seats assigned first to the districts, then the next largest, and so on, until finally one assigns the seats of the smallest party. By the time one reaches the scan for the smallest party, most of the district seats will have been already filled, with the result, say, that such a party will be awarded its seat in a district where it does not the have highest number or the highest district percentage of votes.

The problem is that in the partywise distribution, where seats are distributed by party, all the seats of a given party will have a higher priority over the choice of district than any seat of a smaller party. It could therefore happen that a seat which was marginal for the larger party could be assigned to a district, which should have been assigned to a less marginal seat of a smaller party.

Conceivably one can alter the order, specified in iv) and vi) above, in which the parties are scanned for the partywise distribution of seats. If for example one were to start with the smallest party first, and end up with the largest, the partywise seat distribution in the districts might turn out to be unfair on some candidates of the larger parties. The problem is that whatever the order of the party scans, priority in the partywise distribution is determined by party size only, without any other consideration whatsoever."

PR Squared examples

Suppose the two-member constituency vote result in Newtown East was as follows:

Newtown East
Party %Votes Remainder Quota
Party A 55.0% 0.65
Party D 20.0% 0.60
Party C 15.0% 0.45
Party B 10.0% 0.30

We see that Party A is first and has exceeded the quota, entitling it to the first seat, with 0.65 remainder quotas. Party D is second on 20%, with 0.60 remainder quotas, and has not yet been allocated a seat.

But now consider Party D's table of remainder quotas (and assume that Newtown East is indeed Party D's fourth best local result, with a total of 8 further seats to be allocated to Party D)

Party D - Seats available: 8
Constituency Remainder Quota
Blankshire West 0.91
Oldtown North 0.83
Bracketshire South 0.71
Newtown East 0.60
... ...
... ...

If Party D is the smallest national party awarded seats, and the partywise smallest first allocation is adopted, and Party A's 0.65 remainder quota is also somewhere amongst its best remainders entitled to seats, then Party D will be awarded the seat despite having a slightly smaller remainder quota than Party A in Newtown East. The next best result of Party A's list of remainders would move up that list, replacing the position of the unsuccessful candidate in Newtown East.

Simulations indicate that whatever method is employed, including Buhagiar's preferred Priority Queue, such anomalies cannot be avoided entirely, and are just subjectively more or less "unfair" to the particular candidates affected. Simulations also suggest that only a handful of allocations would meet such conflicts (usually fewer than 20 in a house of 650, or about 3% of the seats). Of these 3%, many, if not most, of the largest remainders would belong to the smaller parties in any case. The number of actual remainder quota "inversions" might be counted on the fingers of one hand.

An alternative resolution of these approximately 20 seat conflicts would be to follow that method recommended for the Dual Member Proportional System. In this case, simply award the seat to the party with the largest remainder quota, and the party denied the seat would utilise its next best reminder quota for its next viable allocation. Under this method the partywise order of allocation is not relevant.

Whichever method is adopted, simulations show that around 97% of the declarations will be straightforward, employing either FPTP or full quota, or best remainder quotas.

Such rare blemishes could be argued to be far outweighed by the list of overall major improvements offered by PR^2 compared to FPTP.


Note that Buhagiar was considering seat allocations to achieve the closest overall proportional result in Malta. PR squared does not have that constraint and indeed, by the nature of the system, the largest and smallest parties are significantly over and under represented respectively. It seems fairer that most of such anomalies should fall on the largest parties, which already benefit hugely from the system by virtue of their seat bonuses. Therefore the simplified smallest party first allocation is suggested.

It could be justified that the only guarantee of a seat is to win a plurality and/or a full quota in a constituency, and candidates affected by the anomaly would be "secondary" candidates by definition, if they were running in a closed list tandem.

Such anomalies are hardly any more unfair than those which currently pass without comment under FPTP.


  • the Liberal Democrats winning most votes across the city of Oxford in 2010, but winning neither of its two parliamentary seats.
  • the common scenario where candidate X loses a seat with, say, 47%, while candidate Y wins another with only 30%.

Possible additional features

Addressing lack of candidates

For all serious parties, maximising their national vote share is imperative under this system. But on occasion, for a variety of reasons, a party may not be able to field a full slate in every constituency, thereby seriously handicapping itself under PR-Squared.

In such cases, a party-only box could be permitted on the ballot in a candidate-less constituency. While not benefiting the party locally, it would still garner votes that would contribute to its national seat-total, helping to elect candidates elsewhere. Of course, the usual rules on lost deposits would apply to such "empty" boxes, to deter national no-hopers from running an "empty" box in every constituency.

See also the case of the Speaker below.

The Speaker

The case of the Speaker presents a possible difficulty. Representing a two-member seat, if the Speaker, under his former party colours, was one of a pair from the same party, that party could hardly stand down in his favour (as happens under FPTP), losing both an MP and crucial national votes, although standing would risk the Speaker's defeat. A similar self-interested view would undoubtedly be taken by the other parties. Under single-member FPTP in the UK, Opposition parties frequently stand against the Speaker, invariably to no effect. There are several possible solutions.

  • adopt the Irish solution, where the Ceann Comhairle is automatically deemed re-elected as a member of the House, reducing the number of electorally-contested seats in his/her multi-member constituency by one. Under PR squared, this would result in a single-member contest between the parties in the Speaker's former constituency.
  • simply deem the Speaker an ex officio or co-opted Member of the House, without any need to continue to represent a constituency, thereby restoring the former constituency to normal electoral competition. The current Westminster convention seems a bit farcical, considering the last Speaker to be defeated in his constituency was Richard Onslow in 1710, and it's difficult to see how a Speaker can vigorously pursue or defend the interests of their constituents while remaining, as s/he must, completely neutral towards the government of the day.
  • adopt an "empty" box mechanism on the ballot paper (see previous section) to permit one or more parties to still win crucial national votes, without risking the defeat of the Speaker.

Death of a candidate

In the UK this has occurred in Thirsk and Malton in 2010 and South Staffordshire in 2005 and on six other occasions since 1918.

With single-member FPTP this results in a deferred election, held some weeks later, after the affected party has had time to re-nominate a replacement candidate. This would not be possible under PR squared, due to the direct interaction between votes in every constituency and the overall national result. The whole election outcome could not be delayed for weeks waiting for one final result.

Parties could nominate substitute candidates in advance, in the small print somewhere on the ballot, as occurs in several European countries. That would be the simplest and obvious answer.

Three or four member constituencies?

While the basic proposal envisages reorganising into uniformly two-member constituencies, that may be impractical in some respects. Several natural units in the UK, such as cities, councils or counties currently have an odd number of single-member constituencies. To insist on units of two would result in either breaking these natural units or increasing malapportionment. (Such malapportionment, however, would have no impact on the national outcome, unlike with single-member FPTP.)

There is no reason, in principle, why PR^2 could not accommodate a few 3-member seats. The trade-off would be that the quota in such seats would be somewhat lower at 25%, and some third-placed candidates could secure election in a three-seater. Some may find this inequitable; others may shrug it off as just a facet of a flexible system, of scant interest.

A four-member seat would have a quota of 20%, which might be thought too low.

Alternatively, the "quota" could be fixed at 33.34% irrespective of the number of members to be elected in a constituency.

Quasi single-member?

The fanatical attachment to the sacred single-member district by FPTP supporters is strange on many levels.

  • The single-member district is a relative new arrival on the British political scene, only uniformly used since 1950. For over 600 years, until 1885, the double-member system was the norm, and its junior equivalent still exists in many places at local government level. A return to the previous constituency structure could hardly be described as radical or revolutionary.
  • The supposedly sacred constituency is dismembered about every 10 years, at the behest of the Boundary Commissioners. Constituencies disappear, are created, renamed, or may have their political complexion transformed beyond recognition, without a single vote having changed. The irony of course is this havoc is wreaked because of the very existence of FPTP constituencies in the first place, and the unavoidable problem they create - creeping malapportionment.
  • The voters themselves are clearly indifferent. Survey evidence reveals that only about 20% of voters can name their MP, or their constituency.[6]
  • The contortions undertaken to "preserve" the constituency-link in hideous hybrid systems such as the German system, which eventually spun out of control as a credible system. "Preserve" them by halving their number and making them twice as large? Sounds logical.

It might seem that these exclusive imaginary turfs exist more in the minds of the MPs themselves, appealing to their vanity, and the satisfying discovery that the long-term trend has been for these principalities to become safer and safer for incumbents.

But suppose a change was actually agreed upon and PR-squared or a similar small district magnitude (>1) system was adopted. There would naturally be excitement and understandably some apprehension as the first election under the new system approached. A period of transition to adjust would naturally be required.

So why not permit the first-elected in the new two-member constituency to choose what could be re-imagined as his or her "home constituency", for example the northern part. S/he almost certainly knows where the most votes are. The runner-up gets the other part.

Would the voters care, or even notice?



A simulation of previous recent UK elections gives:

PR^2 Simulation
Election Conservative Labour LibDem,etc Nationalist UKIP/Brexit/Ref Green Outcome
2019 364 200 26 40 1 1 Con Majority 78
2017 312 282 9 28 1 Hung
2015 318 212 14 46 39 3 Hung
2010 299 200 120 10 2 1 Hung
2005 232 280 106 9 1 Hung
2001 208 355 68 12 Lab Majority 47
1997 189 384 55 12 1 Lab Majority 109
1992 329 235 58 12 Con Majority 7
1987 337 197 95 4 Con Majority 24
1983 346 158 126 3 Con Majority 42
1979 336 248 32 7 Con Majority 37
1974 Oct 243 294 62 24 Hung
1974 Feb 272 268 71 12 Hung

(for simplicity, results in Northern Ireland are not shown, as they are outside of the mainland party system. Also for simplicity, the Scots and Welsh nationalists are combined in the above table.)

The usual caveats apply to such simulations, such as the voters may well vote differently under a different system (in particular, the "wasted vote" argument falls away), smaller parties did not always put up a full slate of candidates, and smaller parties may have done slightly better owing to breakthroughs in particular constituencies.


PR squared should also increase the number of voters who have an MP of their first choice party, compared to FPTP. This is due to a combination of the slight increase in the district magnitude, and the quota algorithm assigning seats more efficiently than FPTP in terms of the geography of support.

We see this quite starkly in an example simulation from Wales.

Firstly, the seats under each system.

Wales 2010
Labour Conservative LibDem Plaid Cymru
Votes 36.2% 26.1% 20.1% 11.3%
FPTP seats 26 8 3 3
PR^2 seats 21 11 6 2

Next, how many of a party's voters have an MP from their choice of party.

Wales 2010
Total Voters Represented Labour Conservative LibDem Plaid Cymru
42.6% 75.7% 35.2% 17.6% 21.6%
72.3% 98.8% 83.0% 54.4% 34.9%

Note that despite Labour obtaining five fewer seats under the PR^2 simulation (21 compared to 26 out of 40), almost all of its voters in Wales (ex Powys) would have had an Labour MP. Other parties also see dramatic improvements in the representation of their voters.

In this respect, PR squared seems almost comparable with STV, despite a far smaller district magnitude.

In terms of the overall disproportionality of the example result in Wales, the Gallagher Index is reduced from 22.9% to 13.1%.

Comparison with FPTP for Westminster

Under PR squared, it can be shown that a 1% swing between the major parties will result, on average, in about 2% of the seats changing hands, or around 13 seats.

This is very interesting as it is almost identical to the average shift currently seen under FPTP.

By comparison, pure PR systems would only shift about 1% of the seats, or 6 seats.

Marginal seats

Under FPTP the election is for practical purposes only fought in the long-term decreasing number of marginal seats. As the Jenkins report indicted:

"The problem with FPTP is that 70% of the voters are considered by the parties to be either so 'sufficiently saved' or so 'irredeemably damned' as to be safely ignored..."

Under FPTP many classic bellwether marginals, such as the two Bury seats (and previously the two Prestons) often flip from 2:0 to 0:2, whereas under PR-Squared these would likely become reasonably safe and stable 1:1 for the parties.

Conversely, the new marginals might be found particularly in regions where previously parties enjoyed clean sweeps on relatively low shares of the vote, e.g. 100% of the seats for 50% of the votes.

Of course, under PR-Squared, which seats are marginal, where they are, and how many there are, becomes largely irrelevant to the overall result, and the long-term operation of the electoral system.


By-elections could be held identically as under FPTP, or if preferential voting was adopted as the norm, by the Alternative Vote.


There is polling evidence that a significant proportion of voters imagine that single-member FPTP actually operates in a similar fashion to PR squared, which of course is not the case.

Under FPTP a vast number - sometimes even a majority - of all votes elect no-one and have no impact on the result, whereas with PR squared almost all votes will directly affect the national outcome. But many voters believe that their vote under FPTP, even if objectively ineffective, in some mysterious way somehow still helps elect "the Prime Minister", their preferred "party", or increases its seat total.[7]

It would be fairer to voters to adopt a system that actually gives them what they think they're already getting!


Related Systems

Quadratic voting


  1. Wiseman, Julian (September 1, 2001). "PR-Squared: A New Description".
  2. using the internet handle RodCrosby
  3. Stringer, Rod (May 10, 2010). "AS PR BECOMES CENTRE STAGE – WHAT ABOUT THIS".
  4. Gallagher, Michael (January 18, 2023). "Election Indices" (PDF).
  6. "Audit of Political Engagement 10, The 2013 Report" (PDF).
  7. Report by NOP Research Group, Jenkins Report (1998), Vol. 2