Arend Lijphart,
Electoral
Systems and Party Systems:
A Study of TwentySeven Democracies,
19451990
(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 SainteLagu6. 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 SainteLague
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
SainteLague formula, in the original form
proposed by its inventor, uses the oddinteger 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
seatsa total of 7 seatsin 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 LRImperiali
formula has been used exclusively in Italy, and Italy has
also used (in the 1948 and 1953 elections) an even lower
quota, alsorather confusinglyreferred 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 LRImperiali 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 sevenmember 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 counterintuitive
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' votesand, of
course, the entire vote total of a party that does not
win any seatsbut 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 formulaand
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
SainteLague 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 resultjust 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, SainteLague treats all parties in an
evenhanded manner. However, the modified SainteLague
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 firstseat procedure would be
exactly like d'Hondt. But for winning seats thereafter,
modified SainteLague works like pure
SainteLague.
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 threemember district with seven
candidates

Droop quota = [100/(3 + 1)] +
I = 26

15 ballots

PQR

20 ballots

ST




15 ballots

PRQ

9 ballots

TS




8 ballots

QRP

17 ballots

U




3 ballots

RPQ

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

Nontransferable









+ 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 Swho 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
nontransferable. 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, orwhat is effectively the
same assumption, but one that is easier to work withthat
voters cast their votes entirely within party lines. (In
Malta, most voters actually follow this partyline 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 LRDroop: 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 partyline 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: LRHare and pure
SainteLague (although the latter is only a theoretical
possibility) are the most proportional formulas;
the d'Hondt and the two LRImperiali formulas are the
least proportional; and LRDroop, STV (which
invariably uses the Droop quota, too), and modified
SainteLague are in an intermediate category.'