Imperiali quota

The Imperiali quota is a formula used to calculate the minimum number, or quota, of votes required to capture a seat in some forms of single transferable vote or largest remainder method party-list proportional representation voting systems. It is distinct from the Imperiali method, a type of highest average method. It is named after Belgian senator Pierre Imperiali.

Ecuador is among the few countries that uses this allocation system, while Italy used it for its Chamber of Deputies from 1946 to 1993.

If a large number of party lists poll just over the Imperiali quota, it is possible for this method to distribute more seats than there are vacancies to fill (this is not possible with the Hare or Droop quotas). If this occurs, the result needs to be recalculated with a higher quota (usually the Droop quota). If it does not happen, Imperiali usually distributes seats in a similar fashion to the d'Hondt method—i.e., favouring larger parties.

Formula

The Imperiali quota may be given as:

${\frac {{\mbox{total}}\;{\mbox{votes}}}{{\mbox{total}}\;{\mbox{seats}}+2}}$

• Total votes = the total valid poll; that is, the number of valid (unspoilt) votes cast in an election.
• Total seats = the total number of seats to be filled in the election.

An example of use in STV

To see how the Imperiali quota works in an STV election imagine an election in which there are 2 seats to be filled and 3 candidates: Andrea, Carter and Brad. There are 100 voters as follows:

65 voters Andrea Carter

15 voters Carter

20 voters Brad

There are 100 voters and 2 seats. The Imperiali quota is therefore:

${\frac {100}{2+2}}=25$

To begin the count the first preferences cast for each candidate are tallied and are as follows:

• Andrea: 65
• Carter: 15
• Brad: 20

Andrea has more than 25 votes. She therefore has reached the quota and is declared elected. She has 40 votes more than the quota so these votes are transferred to Carter, as specified on the ballots. The tallies therefore become:

• Carter: 55
• Brad: 20

Carter has now reached the quota so he is declared elected. The winners are therefore Andrea and Carter.

Imperiali method using parties

This description is based on the Imperiale method as used in the Belgian local elections.

The number of votes of every political party is divided by a series of dividers of positive numbers > 0 strating with 1 i.e 1, 2, 3, 4, etc ...). The result of these devisions are ordered in ascending order. If 2 results are the same the party with the largest initial number of votes gets the first attribution of a seat. Attribution continues until all seats have been attrubuted. Every party gets as many seats ad they have largest results.

divider	largest party	middle party	small party
VOTES = devider 1	4800	3600	1600
/2=	2400 (1)	1800 (2)	800 (9)
/3=	1600 (3)	1200 (5)	533 (15)
/4=	1200 (4)	900 (7)	400 (22)
/5=	960 (6)	720 (10)	320
/6=	800 (8)	600 (13)	…
/7=	686 (11)	514 (16)
/8=	600 (12)	450 (18)
/9=	533 (14)	400 (21)
/10=	480 (17)	360 (24)
/11=	436 (19)	327
/12=	400 (20)	…
/13=	369 (23)
/14=	343 (25)
/15=	320

In the above example 3 partys obtain respectively 4800, 3600 and 1600 votes for a council of 25 seats. Imperiali will attribute the seats as : 13 + 9 + 3 (by comparision if each seat is attributed according to the number of votes per seat by divider 400 the seats would be attributed 12 + 9 + 4). The number between brackets is the order in which seats have been attributed , the result 686 is the 11th lasters result and the aprty got the 11th seat attributed

In the Belgian system the election law imposes the following devider series : 1; 1 ½; 2; 2 ½; 3; 3 ½; 4; 4 ½; etc...