Common Seat Allocation Methods in Proportional Representation Systems and a Novel “Priority Allocation by Rank” Method

There are many examples of elections in Canada in which the successful parties who formed government earned a majority of seats that were disproportionate to their proportion of the vote. Some argue these outcomes signal the need to transform to a system of proportional representation. Several types of proportional representation systems exist. In some of those systems, determining how seats should be distributed among parties requires applying some form of seat allocation method. This article discusses two such methods, the largest remainder and highest average methods, and presents a novel method developed by the author, referred to as “priority allocation by rank.” Voting results from British Columbia’s 2020 election were used to create a hypothetical election scenario for the purposes of comparing seat allocations produced by these three methods. The results suggest that the priority allocation by rank method has potential to be a viable alternative to the other two methods.

Tim Sheaff

Tim Sheaff is a lawyer with the BC Ministry of Attorney General and was formerly legislative counsel. The opinions expressed in this article are his own and do not represent the Ministry of Attorney General or the Government of British Columbia.

Introduction

There are many examples in Canada of elections that resulted in the party forming government earning a majority of seats that were disproportionate to their proportion of the vote. For instance, in Ontario’s 2022 election, the Progressive Conservative Party won 67 per cent of the Legislature’s seats while being the preferred party of only 40 per cent of voters.1 Likewise, in Quebec’s 2022 election, the Coalition Avenir Québec garnered only 41 per cent of the vote but earned 72 per cent of the National Assembly’s seats.2 Some argue these seemingly unfair outcomes signal the need for electoral reform, including transforming from the current first-past-the-post (FPTP) electoral system, more formally known as single member plurality, to one of proportional representation (PR).3

FPTP should be readily familiar to Canadians, as it forms the basis for all federal and provincial elections; the country or a province is divided into electoral districts in which one member of Parliament or a legislature, respectively, is elected. It’s a simple design that is easily understood – the candidate who gets the most votes wins the spoils of the competition (an invitation to join the relevant jurisdiction’s legislative body). By contrast, PR systems use different models of varying complexity in their attempts to achieve proportional results.

PR has long been a topic of interest and debate for Canadians. Attempts have been made in Canada to institute PR, but, to date, none have been successful. For example, Ontario, Prince Edward Island and British Columbia have held referenda on the subject, with voters rejecting PR in each case.4 In 2015, the Governor General’s Speech from the Throne expressed the Liberal government’s commitment to “undertake consultations on electoral reform and … take action to ensure that 2015 will be the last federal election conducted under the first-past-the-post voting system.”5 The Liberal government’s goal, however, was ultimately abandoned.6 Recently, the Liberal Party of Canada, at its 2023 party convention, expressed its desire to try instituting PR again by passing a motion to add to its party platform the establishment of a National Citizens’ Assembly on Electoral Reform.7

Numerous variations of PR systems exist. As between BC’s three referenda in 2005, 2009 and 2018, the province put forward four different types: single transferable vote, dual-member, rural-urban and mixed-member.8 Amongst Prince Edward Island’s referenda in 2005, 2016 and 2019, the province proposed dual-member, mixed-member, preferential voting and FPTP plus leaders.9 Ontario, in its 2007 referendum, sought the public’s opinion on the mixed- member system only.10 Another PR system used in many international jurisdictions, though not submitted to Canadian referendum voters, is the party list system (List PR).11

Under List PR, a jurisdiction is divided into regions, referred to as multi-member districts (MMDs), from which several members are elected.12 For each MMD, parties put forward a list of candidates, and on election day voters select their preferred party. After voting concludes, seats in the MMD are proportionally allocated to parties using some predetermined seat allocation method. Presently, there are two general categories of methods used – largest remainder (LR) methods and highest average methods. Each party’s allocated MMD seats are then distributed amongst its candidates according to their ranks on the party’s list. In a List PR system that uses a closed list, parties determine their candidates’ rankings before the election. When an open list is used, voters also identify their preferred candidate when selecting their preferred party, and candidates’ rankings are determined by voter preferences.

This article sets out to do several things. First, it describes the LR and highest average methods. More specifically, this article discusses two variants of the LR method (differentiated by their use of the Hare and Droop quotas, which are explained in greater detail below) and the D’Hondt method, a variant of the highest average method. Next, this article introduces a novel method developed by the author as an alternative to the LR and highest average methods, referred to as the “priority allocation by rank” (PAR) method.13 For the purposes of explaining these three methods, each of them is presented in the context of their use in the List PR system. This article then discusses an experiment conducted by the author (primarily as a proof of concept of the PAR method) that used BC’s 2020 election results to create a hypothetical election scenario that applied List PR to compare seat allocations produced by employing the three methods.

Description of LR method

The LR method allocates seats in an MMD over several steps.14 First, a quota is determined for the MMD. A quota is a bloc of votes that, for each time it is exceeded by a party’s vote total, will guarantee the party one seat. Two commonly used quotas are the Hare quota and Droop quota. The Hare quota is calculated by dividing the MMD’s total votes by the number of MMD seats.15 Slightly more complicated, the Droop quota equals the MMD’s total votes divided by the sum of the number of MMD seats plus one, after which the resulting quotient is increased by one.16 For the purposes of this article, the phrases “LR (Hare)” and “LR (Droop)” are used to distinguish between LR methods that apply these quotas, respectively.

Next, each party is awarded one seat for each bloc of votes within its vote total that is equal to the quota. Those blocs of votes are then subtracted from the party’s vote total, and the party’s remaining votes are its remainder. If there are any remaining unallocated seats after the parties have been given their quota allocations, the remaining seats are distributed to parties one at a time, in order starting with the party having the largest remainder, until all remaining seats have been allocated. For illustrative purposes, an example of seat allocations using the LR (Hare) and LR (Droop) methods are provided in Tables 1 and 2.

Description of D’Hondt method

The D’Hondt method is a highest average method that employs an iterative process by which seats are allocated to parties in an MMD one at a time.17 For simplicity, each time a seat is allocated in the iterative process is described in this article as a “round.” In each round, the party with the most votes attributed to it is awarded the round’s seat. The votes attributed to a party for a round is determined by dividing the party’s total MMD votes by the number of seats awarded to the party in all previous rounds (if any) plus one.18 If a party has not been awarded any seats, then the votes attributed to it would simply be its MMD vote total.

As an example of how this process would unfold for a party, suppose that a party that earned 5,000 votes in an MMD is awarded its first seat. For the next and subsequent rounds, that party’s MMD votes will be divided by two (two being the sum of the party’s one awarded seat plus one), resulting in 2,500 votes being attributed to the party. If that party is later awarded a second seat, the party’s attributed votes will be 1,667 (its MMD votes divided by three) for each proceeding round until it earns a third seat. The process of dividing the party’s MMD votes continues for each additional seat the party earns. A complete example of how seats are allocated among many parties using the D’Hondt method is shown in Table 3.

Table 1 – Example seat allocation using the LR (Hare) method for an MMD with 10 seats and 10,000 votes

Quota = 1,000	P1	P2	P3	P4	P5
Votes	4,650	2,550	1,675	775	350
Full quotas [total votes in quota blocs]	4 [4,000]	2 [2,000]	1 [1,000]	–	–
Remainder votes	650	550	675	775	350
Remaining seats allocated by largest remainder	1	–	1	1	–
Final allocation	5	2	2	1	–

Note: “P” means party. Quota is calculated as 10,000 / 10 = 1000.

Table 2 – Example seat allocation using the LR (Droop) method for an MMD with 10 seats and 10,000 votes

Quota = 910	P1	P2	P3	P4	P5
Votes	4,650	2,550	1,675	775	350
Full quotas [total votes in quota blocs]	5 [4,550]	2 [1820]	1 [910]	–	–
Remainder votes	100	730	765	775	350
Remaining seats allocated by largest remainder	–	–	1	1	–
Final allocation	5	2	2	1	–

Note: “P” means party. Quota is calculated as 10,000 / (10 + 1) + 1 = 910 (910.09 rounded to the nearest whole vote).

Table 3 – Example seat allocation using the D’Hondt method for an MMD with 10 seats and 10,000 votes

	P14650 votes		P22550 votes		P31675 votes		P4775 votes		P5350 votes
Round	Div	R o u n d Vote	Div	R o u n d Vote	Div	R o u n d Vote	Div	R o un d Vote	Div	R o u n d Vote
1	1	4650	1	2550	1	1675	1	775	1	350
2	2	2325	1	2550	1	1675	1	775	1	350
3	2	2325	2	1275	1	1675	1	775	1	350
4	3	1550	2	1275	1	1675	1	775	1	350
5	3	1550	2	1275	2	838	1	775	1	350
6	4	1163	2	1275	2	838	1	775	1	350
7	4	1163	3	850	2	838	1	775	1	350
8	5	930	3	850	2	838	1	775	1	350
9	6	775	3	850	2	838	1	775	1	350
10	6	775	4	638	2	838	1	775	1	350
Final allocation	5		3		2		–		–

Note: “P” means party, “Div” means divisor (the number of seats that have been allocated to a party plus one), and “Round Vote” means the votes attributable to a party in a round. Bolding in the table indicates which party has been awarded a round’s seat.

Description of PAR method

As noted earlier, the PAR method is a novel method created by the author as an alternative to the LR and highest average methods. In developing the PAR method, the author’s overall guiding principle was to limit the use of mathematical formulae while still achieving a proportional distribution of seats that is consistent with the other methods.

Under the PAR method, each party is assigned a rank based on its MMD vote total. The party with the most votes is assigned the highest rank. Seats are then allocated to parties in priority order according to rank. When it’s a party’s turn to receive seats, the party is allocated a percentage of the MMD’s seats that is closest to its percentage of the MMD vote (subject to the caveat below); for example, a party that gets 25 per cent of the MMD vote should receive a share of the MMD’s seats that is closest to 25 per cent. This continues until all MMD seats are allocated. As a caveat, if, when it’s a party’s turn to receive its seats, there are not enough unallocated seats remaining to fulfill all of the party’s percentage share, the party will only receive the remaining number of unallocated seats, or, similarly, the party will receive no seats if none remain. An example allocation using the PAR method is presented in Table 4.

Sometimes, there could be unallocated MMD seats that remain after all parties have been given their percentage shares.19 In such cases, as a supplemental step, one of the remaining MMD seats will be allocated to each party in priority order by rank (again stopping when there are no more MMD seats left to allocate). An example of such scenario is provided in Table 5.

Converting a party’s percentage of the MMD vote to seats is the only aspect of the process that requires a mathematical calculation. To make this conversion, the party’s percentage share of the MMD vote (its MMD votes divided by the total MMD vote) is multiplied by the number of MMD seats. The product of that multiplication in almost all cases is likely to be a mixed number (such as 2.54 or 3.12). That mixed number is then rounded up or down, as appropriate, to the nearest whole number. That whole number represents the party’s percentage share of the MMD’s seats.20

As an example of this calculation, consider an MMD with five seats and 6,000 total votes, in which a party garnered 3,300 of those total votes. That party’s share of the MMD vote is 3,300 divided by 6,000, or 55 per cent; consequently, the party should be awarded a share of the MMD’s seats that is closest in proportion to 55 per cent. The direct proportional equivalent of 55 per cent of the five MMD seats is 2.75 seats; however, because there is no such thing as a partial seat, 2.75 is rounded up to three (being the nearest whole seat). The party’s share of the MMD’s seats is, therefore, three seats.

Comparing the PAR, LR and D’Hondt methods

Using BC’s 2020 provincial election results,21 the author conducted a test to examine how the PAR, LR (Hare), LR (Droop) and D’Hondt methods compare in allocating seats in a List PR system. The author combined the province’s 87 electoral districts into 19 test MMDs (11 MMDs consisting of five seats and eight MMDs consisting of four seats). In each case, the electoral districts forming an MMD were adjacent. Test voting results were derived for each test MMD by adding the votes for each electoral district that formed  the MMD. Votes were divided into five categories, as delineated by the province’s reported election results: four categories representing votes attributable to the New Democratic Party, Liberal Party, Green Party and Conservative Party, respectively, and one category named “Other” consisting of the votes of all other candidates. For the purposes of this test, the Other category was treated as though it were its own party.

Applying each method, seats were allocated to parties in each test MMD based on the test voting results, after which the seats in all test MMDs were added to derive the parties’ theoretical provincial seat allocations. The results of this experiment are summarized in Table 6, which shows (1) the parties’ total provincial votes and shares of the total provincial vote, (2) the parties’ actual provincial seat allocations and actual shares of the province’s seats, and (3) the parties’ theoretical seat allocations using the four methods and derived theoretical shares of the province’s seats.

As can be seen, the parties’ respective theoretical seat allocations produced by the four methods are more proportionally aligned with the parties’ provincial votes as compared to the parties’ actual seat allocations. Moreover, the PAR method’s seat allocations are not only consistent with those produced by the other three methods, but they are also closer in proportion to the parties’ provincial votes as compared to the seat allocations produced by the LR (Droop) and D’Hondt methods.

Table 4 – Example seat allocation using the PAR method for an MMD with 10 seats and 10,000 votes

	P1	P2	P3	P4	P5
Votes	4,650	2,550	1,675	775	350
Parties’ rankings	1	2	3	4	5
Share of MMD vote	46.5%	25.5%	1.675%	7.75%	3.5%
Share of MMD vote multiplied by number of MMD seats	4.65	2.55	1.675	0.775	0.35
Parties’ proportional shares of seats	5	3	2	1	–
Final allocation	5	3	2	–	–

Note: “P” means party.

Table 5 – Example seat allocation using the PAR method for an MMD with 10 seats and 10,000 votes when there are remaining unallocated seats after parties have been allocated their seat shares

	P1	P2	P3	P4	P5
Votes	5,400	3,400	450	400	350
Parties’ rankings	1	2	3	4	5
Share of MMD vote	54%	34%	4.5%	4%	3.5%
Share of MMD vote multiplied by number of MMD seats	5.4	3.4	0.45	0.4	0.35
Parties’ proportional shares of seats	5	3	–	–	–
Initial allocation of parties’ shares	5	3	–	–	–
Supplemental allocation of remaining unallocated seats	1	1	–	–	–
Final allocation	6	4	–	–	–

Note: “P” means party.

Conclusion

People are bound to have different views about which seat allocation method is best. For lawmakers, in choosing their favourite method, factors under consideration would likely include not only which one produces their most preferred proportional distribution but also which one is easiest to explain to, and most easily understood by, the voting public. There are examples of lawmakers expressing concern about the latter issue of understandability. For example, in the context of the Droop quota, one opposition member during second reading in the BC legislature on the Electoral Reform Referendum 2018 Amendment Act, 2018,22 (which formed part of the legal basis for the province’s 2018 referendum on PR) stated, “I have yet to hear anyone, after all the hours of debate that have gone on in this House, actually explain the Droop formula — how that works.”23 As stated by another, “I don’t think it’s fair that we ask voters to go back to college or high school and learn formulaic, mathematical equations to understand if they’re going to get a representative of their choosing to represent them in Victoria.”24

This author hypothesizes that the PAR method’s use of rankings and allocations of shares of seats nearest to parties’ respective proportions of the vote is at least as understandable as the LR method’s use of quotas and comparisons of largest remainders and the D’Hondt method’s process of iterative divisions and is perhaps even more easily understood. This hypothesis requires further testing.

Table 6 – Comparison of actual BC provincial voting results, actual provincial seat allocations and theoretical seat allocations derived using the PAR, LR (Hare), LR (Droop) and D’Hondt methods

	NDP	LIB	GRN	CON	OTHER
Total provincial votes	898,384	636,148	284,151	35,902	29,047
Proportional share of votes	47.7%	33.8%	15.1%	1.9%	1.5%
Actual provincial results
Actual provincial seat allocation	57	28	2	–	–
Proportional share of actual seat allocation	65.5%	32.2%	2.3%	–	–
PAR method
PAR theoretical seat allocation	45	30	11	1	–
Proportional share of theoretical seat allocation	51.7%	34.5%	12.6%	1.1%	–
LR (Hare) method
LR (Hare) theoretical seat allocation	39	30	16	1	1
Proportional share of theoretical seat allocation	44.8%	34.5%	18.4%	1.1%	1.1%
LR (Droop) method
LR (Droop) theoretical seat allocation	46	32	8	1	–
Proportional share of theoretical seat allocation	52.9%	36.8%	9.2%	1.1%	–
D’Hondt method
D’Hondt theoretical seat allocation	48	34	5	–	–
Proportional share of theoretical seat allocation	55.2%	39.1%	5.8%	–	–

Note: NDP, LIB, GRN and CON mean New Democratic Party, Liberal Party, Green Party, and Conservative Party, respectively.

Additionally, the results of the experiment presented in this article suggests both that List PR will produce more proportional results compared to FPTP and that the PAR method is capable of being a viable alternative to the LR and highest average methods.

Further testing is needed to confirm if this holds true across many election scenarios. Such additional testing does not necessarily need to use voting data from actual elections; instead, such tests could likely be completed using computer-generated random scenarios, since the methods are algorithmic functions that can be applied to any data set to extract results for comparison purposes. Of course, randomized scenarios that fit more closely with typical election results are bound to provide data that is more relevant.

Finally, while this article focused on seat allocations in the context of List PR, the LR and highest average methods may also form part of other PR systems.25 Consequently, there is potential for further examination of whether the PAR method could have application to such other PR systems as well.

Notes

1. Elections  Ontario,  General  Election  Summary  of Candidates Elected and Valid Ballots Cast, online: <https://results.elections.on.ca/en/publications>.
2. Élections Québec, Results of October 3, 2022 General Election, online: <https://www.electionsquebec. qc.ca/en/results-and-statistics/general-election- results/2022-10-03/>.
3. Editorial Board, “Ontario’s election produced a result that is unfair and unrepresentative. The voting system needs to be changed”, Toronto Star (9 June 2022), online: <https://www.thestar.com/opinion/ editorials/2022/06/09/ontarios-election-produced-a- result-that-is-unfair-and-unrepresentative-the-system- needs-to-be-change.html>; Alyssia Rubertucci, “Groups, parties demand electoral reform following Quebec’s ‘distorted’ election results”, CityNews Montreal (5 October 2022), online: <https://montreal.citynews. ca/2022/10/05/quebec-election-results-reform>.
4. Lawrence Leduc, “The Failure of Electoral Reform Proposals in Canada” (2009), 61:2 Political Science, 21-40; Elections BC, Report to the Chief Elector Office: 2018 Referendum on Electoral Reform, online: <https:// elections.bc.ca/docs/rpt/2018-CEO-2018-Referendum- Report.pdf>;  Referendum  Commissioner,  Report of Referendum Commissioner to the Speaker of the Legislative Assembly of Prince Edward Island pursuant to s. 7(1) of the Electoral System Referendum Act (August 2019), online: <https://wdf.princeedwardisland. ca/download/dms?objectId=b6e9b581-38b8-4622-9110-c2a4db23fb69&fileName=2019-referendum- commissioner-report.pdf>.
5. Canada, Making Real Change Happen, Speech from the Throne to Open the First Session of the Forty-second Parliament of Canada, online: <https://www.canada. ca/en/privy-council/campaigns/speech-throne/making- real-change-happen.html>.
6. Leah Schnurr, “Canada abandons electoral reform in reversal of Trudeau pledge”, Reuters (1 February 2017), online: <https://www.reuters.com/article/us-canada- politics-reform-idUSKBN15G5AD>.
7. Liberal Party of Canada, 2023 Official Party Policies, online: <https://2023.liberal.ca/wp-content/uploads/ sites/565/2023/05/Policy-Resolutions-2023-National- Convention_OFFICIAL_ENG.pdf>.
8. Supra note 4, Leduc and Elections BC.
9. Supra note 4, Leduc and Referendum Commissioner; Chief Electoral Officer (Prince Edward Island), 2016 Annual Report of the Chief Electoral Officer, online: <https://www.electionspei.ca/sites/www.electionspei.ca/ files/2016Plebiscite_CEO_Report.pdf>. 
10. Supra note 4, Leduc.
11. As a testament of its popularity, in 2017, twenty of the thirty OECD countries were using some form of the party-list system; see Lydia Miligan and Geoffrey Alchin, Proportional Representation in Practice: An International Comparison of Ballots and Voting Rules, (Fraser Institute, 2018), online: <https://www.fraserinstitute. org/sites/default/files/proportional-representation-in- practice.pdf>.
12. Ibid, Miligan and Alchin.
13. The author reached the conclusion that the PAR method is novel following extensive research. It remains possible that the PAR method has been articulated in literature that was not uncovered by the author.
14. Michael Gallagher, “Comparing Proportional Representation Electoral Systems: Quotas, Thresholds, Paradoxes and Majorities” (1992), 22:4 British Journal of Political Science; Martynas Patašius, “Suitability of the Single Transferable Vote as a Replacement for Largest Remainder Proportional Representation” (2022), 14:8 Symmetry; Parliamentary Research Branch, Electoral Systems, by Brian O’Neil, BP-334E (Canadian Library of Parliament, May 1993), online: <https://publications. gc.ca/site/fra/9.565029/publication.html>.
15. As an equation: Hare quota = total votes / total seats.
16. As an equation: Droop quota = total votes / (total seats + 1) + 1.
17. Electoral System Referendum Act, SPEI 2018, c 25; Dylan Difford, “What is the difference between D’hondt, Sainte-Laguë and Hare?”, Electoral Reform Society, online: < https://www.electoral-reform.org.uk/what-is- the-difference-between-dhondt-sainte-lague-and-hare>.
18. As an equation: n = party’s votes / (seats allocated during all previous rounds + 1), where n equals the number of MMD votes attributed to a party for a round.
19. This situation would occur if, when the parties’ proportional shares of the MMD’s seats are calculated, downward rounding results in the parties’ collective proportional shares of seats to be fewer than the total number of MMD seats.
20. As an equation: party’s share of seats = (party’s votes / total votes) x number of seats, rounded to the nearest whole seat.
21. Elections BC, October 24, 2020, Provincial General Election, Report to the Chief Electoral Officer, online: <https://www.elections.bc.ca/docs/rpt/2020-provincial- general-election-report.pdf>. 
22. S.B.C. 2018, c. 55.
23. “Bill 40: Electoral Reform Referendum 2018 Amendment Act, 2018”, 2nd reading, Legislative Assembly of British Columbia Official Report of Debates, 41-3, No 174 (October 31, 2018) at 1705 (Michelle Stilwell), online: <https://www.leg.bc.ca/documents-data/debate- transcripts/41st-parliament/3rd-session/20181031pm- Hansard-n174>.
24. “Bill 40: Electoral Reform Referendum 2018 Amendment Act, 2018”, 2nd reading, Legislative Assembly of British Columbia Official Report of Debates, 41-3, No 173 (October 30, 2018) at 1540 (Ellis Ross), online: <https:// www.leg.bc.ca/documents-data/debate-transcripts/41st- parliament/3rd-session/20181030pm-Hansard-n173>.
25. Supra note 11, Miligan and Alchin.