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How votes become seats

In proportional representation systems there are five key mathematical formulae through which votes are translated into seats. In list PR systems, see List PR, there are 'highest average' methods (D'Hondt and Sainte-Lagu ) and 'largest remainder' methods (Hare and Droop). The single transferable vote form of PR, see Single Transferable Vote, almost always uses the Droop quota.

Examples of each method are given below:

Table 1 Seat Allocation by Two Highest-Average Formulas in a Six-Member District with Four Parties

Party

Votes
(v)

D'Hondt Allocation

v/1

v/2

v/3

Total
Seats

A

42 000

42 000 (1)

21 000 (3)

14 000 (6)

3

B

31 000

31 000 (2)

15 500 (5)

10 333

2

C

15 000

15 000 (4)

7 500

 

1

D

12 000

12 000

 

 

0

 

 

 

 

 

 

TOTAL

100 000

 

 

 

 

Party

Votes
(v)

Modified Sainte-Lague Allocation

v/1.4

v/3

v/5

Total
Seats

A

42 000

30 000 (1)

14 000 (3)

8 400

2

B

31 000

22 143 (2)

10 333 (5)

6 200

2

C

15 000

10 714(4)

5 000

 

1

D

12 000

8 571 (6)

 

 

1

 

 

 

 

 

 

TOTAL

100 000

 

 

 

 

Table 2 Seat Allocation by Two Largest-Remainder Formulas in a Six-Member District with Four Parties
Party	Votes	Hare Quotas (a)	Full Quota Seats	Remaining Seats	Total Seats
A	42 000	2.52	2	0	2
B	31 000	1.86	1	1	2
C	15 000	0.90	0	1	1
D	12 000	0.72	0	1	1

TOTAL	100 000	6.00	3	3	6

Party	Votes	Droop Quotas (b)	Full Quota Seats	Remaining Seats	Total Seats
A	42 000	2.94	2	1	3
B	31 000	12.17	2	0	2
C	15 000	1.05	1	0	1
D	12 000	0.84	0	0	0

TOTAL	100 000	7.00	5	1	6

Table 3 Seat Allocation by Single Transferable Vote in a Three-Member District with Five Candidates

Droop Quota = [ 100 / ( 3 + 1 ) ] + 1 = 26
Preferences
23 ballots P, Q, T
23 ballots P, R, S
16 ballots Q, R
5 ballots R, S
20 ballots S, T
8 ballots T, Q, R
5 ballots T

Candidate

First Count

Second Count

Third Count

P

46

- 20 = 26

26

Q

16

+ 10 = 26

26

R

5

+ 10 = 15

+ 8 = 23

S

20

20

20

T

13

13

- 13 = 0

Nontransferable

-

0

+ 5 = 5

Candidates elected: P, Q, and R

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