Arend Lijphart, Electoral Systems and Party Systems: A Study of Twenty-Seven Democracies, 1945-1990 (Oxford University Press, 1994) Appendix A: Proportional Representation Formulas PR formulas can best be explained by classifying and subclassifying them: the first classification distinguishes between list PR, in which voters cast their votes for party lists of candidates, and the single transferable vote (STV), in which voters vote for individual candidates. List PR can then be classified into highest averages (divisor) and largest remainders (quota) systems.' And these can be classified further according to the particular divisor or quota that they employ. I shall use a few simple examples to illustrate their operation. Table A.1 shows the only two highest averages methods that are in actual use for the allocation of seats to parties: d'Hondt and modified Sainte-Lagu6. Seats are awarded sequentially to parties having the highest I average' numbers of votes per seat until all seats are allocated; each time a party receives a seat, its 'average' goes down. These 'averages' are not averages as normally defined but depend on the given set of divisors that a particular divisor system prescribes. The dhondt formula uses the integers 1, 2, 3, 4, and so on. As Table A. 1 shows, the first seat (indicated by the number in parenthesis) goes to the largest party, party A, whose votes are then divided by 2. The second seat is given to party B, because its 'average' (29,000 votes, its original vote total) is higher than C's and D's and also higher than A's votes divided by 2. The third seat goes to A because its vote divided by 2 is higher than B's vote divided by 2 and higher than C's and D's votes; and so on.' The final seat allocation to parties A, B, C, and D is 3, 2, 1, and 0 seats. TABLE A.l. Illustrative examples of the operation of two highest averages formulas in a six member district with four parties   TABLE A.2. Illustrative examples of the operation of three largest remainders formulas in an eight member district with four parties Seats allocated using D'Hondt divisors of 1, 2, 3, 4, etc. Party Votes Total seats (v) v/l v/2 v/3 A 41000 41,000(l) 20,500 (3) 13,667 (6 3 B 29000 29,000 (2) 14,500 (5) 9667 2 C 17000 17,000(4) 8500 1 D 13000 13000 0 TOTAL 100,00 Seats allocated using modified Sainte-Lague divisors of 1.4, 3, 5, 7, etc. Party Votes Total seats (v) v/1.4 v/3 v/5 A 41000 29,286 (1) 13,667 (3) 8200 2 B 29000 20,714 (2) 9,667 (5) 5800 2 C 17000 12,143 (4) 5667 1 D 13000 9,286 (6) 1 TOTAL 100000 Note: The order in which seats are awarded sequentially to parties is indicated by the numbers in parentheses. The Sainte-Lague formula, in the original form proposed by its inventor, uses the odd-integer divisor series 1, 3, 5, 7, and so forth. In practice, it is used only in a modified form which uses 1.4 instead of 1 as the first divisor. Its sequential procedure for allocating seats to parties is identical to that of the d'Hcpdt method. In the example of Table A.1, the first five seats are awarded to the parties in exactly the same order as in the d'Hondt method, but the sixth seat is won by party D instead of A; the final distribution of seats therefore becomes 2, 2, 1, 1. The three most common largest remainders (LR) formulas, using the Hare, Droop, and Imperiali quotas, are shown in Table A.2. In all quota systems, the first step is to calculate a quota of votes that entitles parties to a seat; a party gets as many seats as it has quotas of votes; any unallocated seats are then given to those parties having the largest numbers of unused votes (remainders). The Hare quota is the oldest and simplest of the quotas: it is simply the total number of valid votes divided by the number of seats at stake in a district. The Droop quota divides the total number of votes by the number of seats plus 1, and the Imperiali quota by the number of seats plus 2. The quickest method for calculating the results is to divide each party's votes by the quota, which yields the number of quotas each party has won.' Parties then receive one seat for each full quota; any seats that cannot be allocated in this way are given to the parties with the largest fraction of a quota. In the first example of Table A.2, based on the us of the Hare quota, parties A, B, C, and D have 3, 2, 1, and 1 full Hare quotas respectively and are therefore given 3, 2, 1, and 1 seats-a total of 7 seats-in the initial allocation. Since 8 seats are available, one more seat has to be allocated: it goes to the party with the largest remainder of votes, that is, the largest remaining fraction of a quota. This is party C with a remaining 0.36 of a Hare quota; and the final distribution of seats becomes 3, 2, 2, 1. Apart from the use of different (lower) quotas, the procedure for allocating seats to parties with the Droop and Imperiali methods is exactly the same. The LR-Imperiali formula has been used exclusively in Italy, and Italy has also used (in the 1948 and 1953 elections) an even lower quota, also-rather confusingly-referred to as the Imperiali quota: the number of votes divided by the number of seats plus 3. 1 shall refer to the latter as the reinforced LR-Imperiali quota (and to the corresponding formula as reinforced LR Imperiali). Both of the Imperiali quotas run the risk of allocating more seats than are actually available; this would happen, for instance, with the reinforced Imperiali quota in the example of Table A.2. The Italian rule for such cases is that the results be recalculated with the use of the next higher quota. TABLE A.2. Illustrative examples of the operation of three largest remainders formulas in an eight member district with four parties Hare quota = 100,000 [votes]/8 [seats] = 12,500 Party Votes Hare Full quota Remaining quotas seats seats Total seats A 41000 3.28 3 0 3 B 29000 2.32 2 0 2 D 13000 1.04 1 0 1 TOTAL 100000 8 7 1 8 Droop quota = 100,000 / (8 + 1) = 11,111 Party Votes Droop Full quota Remaining quotas seats seats Total seats A 41000 3.69 3 1 4 B 29000 2.61 2 0 2 C 17000 1.53 1 0 1 D 13000 1.17 1 0 1 TOTAL 100000 9 7 1 8 Imperiali quota = 100,000/(8 + 2) = 10,000 Party Votes Imperiali Full quota Remaining quotas seats seats Total seats A 41000 4.1 4 0 4 B 29000 2.9 2 0 2 C 17000 1.7 1 0 1 D 13000 1.3 1 0 1 TOTAL 100000 10 8 0 8 The examples of Tables A.1 and A.2 were selected not just to illustrate the different procedures but also to show that the choice of PR formula can affect the allocation of seats. Such differences do not always appear; in a seven-member district, for instance, the four parties would be awarded exactly the same 3, 2, 1, and 1 seats, respectively, by all five methods. Where differences do appear, however, they are not random but systematically affect the degree of proportionality and the electoral opportunities for small parties. These differences occur within the two groups of quota and divisor systems rather than between them. Among the quota systems, proportionality decreases as the quota decreases; this is illustrated in Table A.2 where the use of the Droop quota instead of the Hare quota causes the small party C to lose a seat and the largest party A to win an extra seat. This is somewhat counter-intuitive because one would expect smaller parties to benefit from a lower quota. How can this result be explained? The explanation is that lowering the quota will increase the remainders for the larger parties at a faster pace than for the smaller parties (illustrated in Table A.2 for the shift from the Hare to the Droop quota) and/ or that the number of remaining seats is reduced, which means that fewer of the remaining votes qualify for a seat in the final allocation (illustrated for the shift from the Droop to the Imperiali quota in Table A.2). Disregarding these remaining votes harms small parties because they are a large portion of the small parties' votes-and, of course, the entire vote total of a party that does not win any seats-but only a relatively small portion of the larger parties' votes. As a result, the seat shares of the larger parties will tend to be systematically higher than their vote shares, and the smaller parties will tend to receive seat shares that are systematically below their vote shares. The maximum disadvantage for small parties occurs when there are no remaining seats to be allocated at all. Although it is a divisor formula, the d'Hondt method can also be interpreted as a particular kind of quota formula--and thus be compared with the other quota formulas. Its purpose can be said to be to find a quota that will allow the allocation of all available seats in the first allocation and to disregard all remainders. This quota equals the last ,average' to which a seat is awarded: 13,667 votes in Table A.l.' When the parties' votes are divided by this quota, their quotas are 3.00, 2.12, 1.24, and 0.95 and the seat distribution 3, 2, 1, and 0 -without the need to honour any of the remaining votes; note especially party D's large unused fraction of 0.95 of a quota.' The pure Sainte-Lague formula can be interpreted in a similar way. Its quota is twice the last of the 'averages' to which a seat is awarded. For each quota of votes that a party has won, it is awarded one seat, and all remaining votes of half a quota or more are also honoured. If all remainders were so honoured, a strong bias in favour of the small parties would result-just as the d'Hondt rule of ignoring all remainders entails a bias against the small parties. By setting a boundary of half a quota above which remainders do, and below which they do not, qualify for a seat, Sainte-Lague treats all parties in an even-handed manner. However, the modified Sainte-Lague deviates from this high degree of proportionality by raising the first divisor from 1 to 1.4 and thereby making it more difficult for small parties to win their first seats. The formula operates almost like d'Hondt as far as a party winning its first seat is concerned, because the distance from 1.4 to 3 is proportionaby nearly the same as the distance from 1 to 2; if the first divisor were 1.5, the first-seat procedure would be exactly like d'Hondt. But for winning seats thereafter, modified Sainte-Lague works like pure Sainte-Lague. STV is more difficult to compare with the other PR formulas because voters cast their votes for individual candidates, in order of the voters' preferences, instead of party lists. Table A.3 presents a hypothetical example that, while it is very simple, illustrates all of STV's basic rules. In a three member district, there are 100 voters and seven candidates (P to V). In the top half of the table, the voters' preferences are summarized: there are 15 ballots with candidate P as the first preference, Q as the second preference, and R in third place, with no further preferences indicated; several other ballots also contain three preferences, but the rest indicate only one or two preferences. Like LR systems, STV requires the choice of a quota, which in practice is always the Droop quota. However, it is defined in a slightly different way from the LR Droop quota: the quotient arrived at by dividing the total vote by the number of seats plus 1 is rounded up or, if the quotient is an integer, 1 is added. In the example of Table A.3, the LR Droop quota would be 25, but the STV Droop quota is 26.6 TABLE A.3. Illustrative example of the operation of the single transferable vote in a three-member district with seven candidates Droop quota = [100/(3 + 1)] + I = 26 15 ballots P-Q-R 20 ballots S-T 15 ballots P-R-Q 9 ballots T-S 8 ballots Q-R-P 17 ballots U 3 ballots R-P-Q 13 ballots V Candidate First Second Third Fourth Fifth Sixth count count count count count count P 30 -4=26 26 26 26 26 Q 8 +2= 10 +5 =15 15 15 15 R 3 +2=5 - 5 = 0 0 0 0 S 20 20 20 +9=29 -3=26 26 T 9 9 9 -9 = 0 0 0 U 17 17 17 17 17 17 V 13 13 13 13 13 -13=0 Non-transferable ----- ----- ----- ----- + 3 = 3 + 13 = 16 Candidates elected: P, S, and U. In the first count, the ballots are arranged according to first preferences. If a candidate has a Droop quota or more of these first preferences, he or she is elected: candidate P with 30 votes in the example of Table A.3. In the second count, P's 4 surplus votes are transferred to the next lower preferences, half to Q and half to R, because the original 30 ballots with P as first preference were also split equally between Q and R as second preferences. Since the second count does not yield another candidate with the Droop quota necessary for election, the weakest candidate (R) is eliminated and his or her 5 votes transferred to the next preference on the ballots (Q) in the third count. This procedure has to be repeated in the fourth count with the elimination of candidate T and the transfer of his or her votes to candidate S-who now exceeds the Droop quota and is elected. In the fifth count, S's 3 surplus votes should be transferred to the next preference, but because no further preferences are indicated on the ballots, these votes become non-transferable. In the sixth count, the weakest candidate is again eliminated (candidate V); only four candidates are left, and candidate Q is next in line for elimination; this means that no further calculations are necessary and that candidate U is the third candidate to be elected. Because STV voters vote for individual candidates, they can vote for candidates of different parties. In order to compare STV with the other PR formulas, we therefore have to make the simplifying assumption that party votes lost by transfers to candidates of other parties are gained back by transfers from other parties, or-what is effectively the same assumption, but one that is easier to work with-that voters cast their votes entirely within party lines. (In Malta, most voters actually follow this party-line voting behaviour.) Table A.3 serves to exemplify this situation, too, if we assume that candidates P, Q, and R belong to party A, candidates S and T to party B, and candidates U and V to parties C and D respectively. The result is obviously very similar to that of LR-Droop: parties A and B have one full Droop quota and win one seat each, and the third seat goes to party C which has the largest remainder of votes.' The most significant exception from party-line voting or offsetting transfers is when parties conclude alliances with each other and encourage their voters to vote for their own candidates first but then turn to the candidates of the allied party or parties. This opportunity, which is an inherent feature of STV, is similar to the possibility of connected lists or apparentements that some list PR systems offer. It is examined as a separate variable in Chapter 5. On the basis of the above arguments, the PR formulas can be classified in three groups: LR-Hare and pure Sainte-Lague (although the latter is only a theoretical possibility) are the most proportional formulas; the d'Hondt and the two LR-Imperiali formulas are the least proportional; and LR-Droop, STV (which invariably uses the Droop quota, too), and modified Sainte-Lague are in an intermediate category.'