2 Seat Apportionment Methods
Most democratic countries use apportionment methods to transform election results into whole numbers, which in turn indicate the number of seats that each party earns in a legislative body, administrative council, or committee. Which apportionment method does this best is not a trivial topic for mathematicians as well as for political scientists. Actually, several formulas have been proposed since the early time of representative assemblies. The following sections briefly present some of these allotment methods.
2.1 Highest Averages Methods of Allocating Seats Proportionally
Highest averages methods allocate seats proportionally to the number of votes by assigning seats in a way that assures the highest quotient by seat for each party. Other methods divide the votes by a mathematically derived quota, such as Droop and Hare quotas. In what follows are illustrative examples taken from Lijphart (1994) of the operation of some highest averages formulas in a six-member district with four parties.
# Table A.l
lijphart <- c("A"=41000, "B"=29000,"C"=17000, "D"=13000)The basic inputs for this class of functions are: 1) a list of parties, 2) a list of positive votes, and 3) a constant value for the number of seats to be returned. A numeric value (0~1) for the threshold is optional.
By default, the function prints the apportionment method being used, a snippet of the divisors used, the effective number of parties (or candidates) index for both before and after the seat allocation method takes place, and the least squares index–or Gallagher index. The effective number of parties measured the before seat allocation is a measure of the party system fragmentation, once it deals with the effective number of parties at the electoral level, while the index computed after the seat allocation, the effective number of parliamentary parties. The Gallagher’s index is an indication of the disproportionality and runs from 0 to 100. The higher the value, the higher the disproportionality of the election outcome.
2.1.1 d’Hondt
Under the D’Hondt formula, seats are allocated using divisors of 1, 2, 3, 4, etc. Thus, suggesting that the divisors are simply one more than the number of seats that party already has.
library("SciencesPo", quietly = TRUE)
# The d'Hondt will give the same results as Jefferson's method
HighestAverages(parties=names(lijphart),
votes=lijphart,
seats = 6,
method = "dh") Method: d'Hondt
Divisors: 1 2 3 4 ...
ENP: 3.36 (After): 2.57
Gallagher Index: 11.6
-------------------------
Party Seats % Seats
------- ------- ---------
A 3 50
B 2 33.3
C 1 16.7
D 0 0
-------------------------The d’Hondt is only one way of allocating seats in party list systems. Other methods include the Saint-Laguë, the modified Saint-Laguë, the Danish version, Imperiali (do not to confuse with the Imperiali quota, which is a Largest remainder method), Hungtinton-Hill, Webster, etc.
2.1.2 Saint-Laguë
Named after its founder, the pure Sainte-Laguë formula, also known as Schepers method, uses the odd-integer divisor series (1, 3, 5, 7, …) to elect candidates from political parties in approximate proportion as the proportion of votes won by the party. The outcome would be the same as the outcome produced by the Webster’s divisors method.
# The Sainte-Laguë will give the same results as the Webster's method (wb)
HighestAverages(parties=names(lijphart),
votes=lijphart,
seats = 6,
method = "sl") Method: Sainte-Laguë
Divisors: 1 3 5 7 ...
ENP: 3.36 (After): 3.6
Gallagher Index: 6.75
-------------------------
Party Seats % Seats
------- ------- ---------
A 2 33.3
B 2 33.3
C 1 16.7
D 1 16.7
-------------------------2.1.3 Modified Saint-Laguë
The modified Sainte-Laguë use divisors of 1.4, 3, 5, 7, etc. How does the d’Hondt divisor compare to the Modified Sainte-Laguë one? Which divisor method tend to favor larger parties and which one smaller parties?
HighestAverages(parties=names(lijphart),
votes=lijphart,
seats = 6,
method = "msl") Method: Modified Sainte-Laguë
Divisors: 1.4 3 5 7 ...
ENP: 3.36 (After): 3.6
Gallagher Index: 6.75
-------------------------
Party Seats % Seats
------- ------- ---------
A 2 33.3
B 2 33.3
C 1 16.7
D 1 16.7
-------------------------2.1.4 Danish
The Danish divisors (1, 4, 7, 10, …) increase so fast that large parties are quickly cut down, acting much to the benefit of smaller parties.
HighestAverages(parties=names(lijphart),
votes=lijphart,
seats = 6,
method = "danish") Method: Danish Sainte-Laguë
Divisors: 1 4 7 10 ...
ENP: 3.36 (After): 3.6
Gallagher Index: 6.75
-------------------------
Party Seats % Seats
------- ------- ---------
A 2 33.3
B 2 33.3
C 1 16.7
D 1 16.7
-------------------------2.1.5 Hungarian
The Hungarian version of Sainte-Laguë uses divisors of (1.5, 3, 5, 7, 9, …) increase so fast that large parties are quickly cut down, acting much to the benefit of smaller parties.
HighestAverages(parties=names(lijphart),
votes=lijphart,
seats = 6,
method = "hsl") Method: Hungarian Sainte-Laguë
Divisors: 1.5 3 5 7 ...
ENP: 3.36 (After): 3.6
Gallagher Index: 6.75
-------------------------
Party Seats % Seats
------- ------- ---------
A 2 33.3
B 2 33.3
C 1 16.7
D 1 16.7
-------------------------2.1.6 Webster
This approach does not use a standard divisor either. It uses instead a different divisor and modified quota such that rounding with cut-off point at algebraic mean of lower and upper quotas: wb = \(\frac{L + (L + 1)}{2}\).
HighestAverages(parties=names(lijphart),
votes=lijphart,
seats = 6,
method = "wb") Method: Webster
Divisors: 0.5 2.5 4.5 6.5 ...
ENP: 3.36 (After): 3.6
Gallagher Index: 6.75
-------------------------
Party Seats % Seats
------- ------- ---------
A 2 33.3
B 2 33.3
C 1 16.7
D 1 16.7
-------------------------2.1.7 Imperiali
The Italian Imperiali divisor system has a slowly-increasing sequence (1, 1.5, 2, 2.5, 3, …), with 0.5 difference between consecutive divisors. This is tighter than the d’Hondt divisors for which this difference is 1. Like d’Hondt, the Imperiali system is designed to encourage coalitions and secure majority governments under a PR system, but the Imperiali will be the most favorable to large parties as they will gain many seats before their quotients are reduced below those of the smaller parties.
HighestAverages(parties=names(lijphart),
votes=lijphart,
seats = 6,
method = "imperiali") Method: Imperiali
Divisors: 1 1.5 2 2.5 ...
ENP: 3.36 (After): 2.57
Gallagher Index: 11.6
-------------------------
Party Seats % Seats
------- ------- ---------
A 3 50
B 2 33.3
C 1 16.7
D 0 0
-------------------------The Belgian municipal councils were elected by the Imperiali highest averages method. The following example comes from the 2006 election in Bruges, where 47 seats were upon.
Bruges=c("CD&V/N-VA"=32092, "SP.A/Spirit"=20028,
"Flemish Interest"=13408, "Open VLD/Vivant"=9520,
"Green!"=5328, "Other"=2207)
HighestAverages(parties=names(Bruges),
votes=Bruges,
seats = 47,
method = "imperiali") Method: Imperiali
Divisors: 1 1.5 2 2.5 ...
ENP: 3.93 (After): 3.47
Gallagher Index: 3.78
----------------------------------
Party Seats % Seats
---------------- ------- ---------
CD&V/N-VA 20 42.6
SP.A/Spirit 12 25.5
Flemish Interest 8 17
Open VLD/Vivant 5 10.6
Green! 2 4.3
Other 0 0
----------------------------------2.1.8 Hungtinton-Hill
The method proposed by Joseph Hill and Edward Huntington does not use a standard divisor, but a different one and also a modified quota such that rounding with cut-off point at geometric mean: hh = \(\sqrt{L(L + 1)}\). This method makes sense only if every party is guaranteed at least one seat. It has been used for allotting seats in the US House of Representatives to the states.
HighestAverages(parties=names(lijphart),
votes=lijphart,
seats = 6, method = "hh") Method: Hungtinton-Hill
Divisors: 0 1.41 2.45 3.46 ...
ENP: 3.36 (After): 3.6
Gallagher Index: 6.75
-------------------------
Party Seats % Seats
------- ------- ---------
A 2 33.3
B 2 33.3
C 1 16.7
D 1 16.7
-------------------------2.1.9 Using Thresholds
Let’s assume that we have an election with 1,000 total voters in which five parties (A, B, C, D, and E) have gained 100 (10%), 150 (15%), 300 (30%), 400 (40%), and 50 (5%) votes, respectively. In this constituency, there are 3 seats up for election, and all votes cast are valid. What is more, this hypothetical electoral system has a 7% vote threshold, meaning that parties must get at least 7% of the total unspoiled votes cast in order to participate on the distribution of seats. Party E would then be eliminated from this contest at the outset.
If D’Hondt divisor were employed in this hypothetical election, then party C would get 1 seat (or 33% of the seats), party D 2 seats (or 67% of the seats), and parties A and B none.
const <- c("A"=100, "B"=150,"C"=300, "D"=400, "E"=50)
HighestAverages(parties=names(const),
votes=const,
seats = 3, method = "dh",
threshold = 7/100) Method: d'Hondt
Divisors: 1 2 3 NA ...
ENP: 3.51 (After): 1.8
Gallagher Index: 55.1
-------------------------
Party Seats % Seats
------- ------- ---------
D 2 66.7
C 1 33.3
A 0 0
B 0 0
E 0 0
-------------------------The following example is taken from the 2015 general elections in Spain. The region of Valencia returned 15 seats allocated using the D’Hondt method in a closed list PR system. Only lists that poll at least 3% of the total vote (which includes spoiled votes “en blanco”) can be considered.
(Valencia <- c("PP"=442005, "Podemos"=395729, "PSOE"=275680,
"C's"=221299, "IU"=68759, "PACMA"=14445, "Others"=35943)) PP Podemos PSOE C's IU PACMA Others
442005 395729 275680 221299 68759 14445 35943 blanco=8738
nulo=11891# Valencia returned 15 members
HighestAverages(parties=names(Valencia),
votes=Valencia,
seats=15, method = "dh",
threshold = 3/100)Method: d'Hondt
Divisors: 1 2 3 4 ...
ENP: 4.37 (After): 3.57
Gallagher Index: 6.33
-------------------------
Party Seats % Seats
------- ------- ---------
PP 5 33.3
Podemos 5 33.3
PSOE 3 20
C's 2 13.3
IU 0 0
Others 0 0
PACMA 0 0
-------------------------Next, the methods that divide the votes by mathematically derived quotas, such as the Droop quota, the Hare quota (or Hamilton/Vinton), and the Imperiali quota.
2.2 Largest Remainder Methods of Allocating Seats Proportionally
Highest averages methods allocate seats proportionally to the number of votes by dividing the number of votes required for each party by a quota. The quota may vary because it is the result of dividing the number of unspoiled votes (\(v\)) by some whole number close to the number of seats that would next be assigned (\(s\)). In what follows are illustrative examples of the operation of some largest remainders formulas in an eight member district with the same four parties of the previous examples.
2.2.1 Hare quota
The Hare quota is obtained by quota: \(\frac{v}{s}\).
LargestRemainders(parties=names(lijphart),
votes=lijphart,
seats = 8, method = "hare") Method: Hare
Divisors: 12500 NA NA NA ...
ENP: 3.36 (After): 4
Gallagher Index: 56.75
-------------------------
Party Seats % Seats
------- ------- ---------
NA 1 25
NA 1 25
NA 1 25
NA 1 25
NA 0 0
NA 0 0
NA 0 0
NA 0 0
NA 0 0
NA 0 0
NA 0 0
NA 0 0
-------------------------2.2.2 Droop quota
The Droop quota is obtained by quota: \(\frac{v}{(s + 1)}+1\).
LargestRemainders(parties=names(lijphart),
votes=lijphart,
seats = 8, method = "droop") Method: Droop
Divisors: 11112.11 NA NA NA ...
ENP: 3.36 (After): 4
Gallagher Index: 56.75
-------------------------
Party Seats % Seats
------- ------- ---------
NA 1 25
NA 1 25
NA 1 25
NA 1 25
NA 0 0
NA 0 0
NA 0 0
NA 0 0
NA 0 0
NA 0 0
NA 0 0
NA 0 0
-------------------------2.2.3 Hagenbach-Bischoff
The Hagenbach-Bischoff quota is obtained by \(\frac{v}{(s + 1)}\).
LargestRemainders(parties=names(lijphart),
votes=lijphart,
seats = 8, method = "hagb") 2.2.4 Imperiali quota
The Imperiali quota is obtained by \(\frac{v}{(s + 2)}\). For Italian elections during the 1950s, the quota used was a reinforced version of that quota: \(\frac{v}{(s + 3)}\).
# The 1946 Italian Constituent Assembly election results: parties and unspoiled votes
Italy = data.frame(party=c("DC", "PSIUP", "PCI", "UDN", "UQ", "PRI",
"BNL", "PdA", "MIS", "PCd'I", "CDR",
"PSd'Az", "MUI", "PCS", "PDL", "FDPR"),
votes=c(8101004, 4758129, 4356686, 1560638, 1211956,
1003007, 637328, 334748, 171201, 102393,
97690, 78554, 71021, 51088, 40633, 21853))with(Italy, LargestRemainders(parties=party,
votes=votes, seats = 556,
method = "imperiali") )with(Italy, LargestRemainders(parties=party,
votes=votes, seats = 556,
method = "imperiali.adj") )2.3 Suitable output for recycling within RMarkdown documents
The output produced by HighestAverages() and LargestRemainders() functions is of class data.frame, which R users are already familiar with, and makes the process of using the output with other applications very straightforward. Let’s take the data from 2014 Brazilian legislative elections. I will look at one electoral district–the state of Ceará–to compare how that caucus would look like if different apportionment methods were in place, keeping voters’ preference constant.
# Results for the state legislative house of Ceará (2014):
Ceara <- c("PCdoB"=187906, "PDT"=326841,"PEN"=132531, "PMDB"=981096,
"PRB"=2043217,"PSB"=15061, "PSC"=103679, "PSTU"=109830,
"PTdoB"=213988, "PTC"=67145, "PTN"=278267)Once the data is loaded, producing a table with seat apportionment results is effortless with packages like knitr, pander, gtable, xtable, etc. In the following, I demonstrate this using the kable() function from the knitr package.
mytable = HighestAverages(parties=names(Ceara),
votes=Ceara,
seats = 42, method = "dh") Method: d'Hondt
Divisors: 1 2 3 4 ...
ENP: 3.65 (After): 3.14
Gallagher Index: 47.01
library(knitr)
kable(mytable, align=c("l","c","c"), caption="Outcome under d'Hondt")| Party | Seats | % Seats |
|---|---|---|
| PRB | 21 | 50.0 |
| PMDB | 10 | 23.8 |
| PDT | 3 | 7.1 |
| PTN | 2 | 4.8 |
| PTdoB | 2 | 4.8 |
| PCdoB | 1 | 2.4 |
| PEN | 1 | 2.4 |
| PSC | 1 | 2.4 |
| PSTU | 1 | 2.4 |
| PSB | 0 | 0.0 |
| PTC | 0 | 0.0 |
If you rather prefer to visualize data using graphs, that’s fine. I’ve chosen a bar plot with colors alluding to the parties’ identification colors. The graph is made with ggplot2 library along with the SciencesPo theme theme_fte, which mimics the colors used in the http://fivethirtyeight.com.
gg <- ggplot()
gg <- gg + geom_bar( data = mytable, aes(x = reorder(Party, -Seats), y = Seats, fill = Party), stat = "identity", alpha=.9)
gg <- gg + geom_label(data = mytable,
aes(x = reorder(Party, -Seats), y= Seats + 1, label = Seats), hjust = .5)
gg <- gg + scale_y_continuous(expand = c(0, 0), limits = c(0, 25))
gg <- gg + scale_fill_party("BRA")
gg <- gg + labs(list(x = "", y = "# Seats obtained",
title = "The 2014 election in the Ceará State (seats won by party)"))
gg <- gg + theme_fte(base_size = 10, base_family = "Tahoma")
gg
Next, I use the same example above to make a plot comparing three highest-averages formulas, D’Hondt, Imperiali, and Saint-Laguë. I’ve chosen a plot resembling a Cleveland dot plot (something between a bar and dot plot), and the SciencesPo theme.
out1 = HighestAverages(
parties = names(Ceara),
votes = Ceara,
seats = 42,
method = "dh"
)Method: d'Hondt
Divisors: 1 2 3 4 ...
ENP: 3.65 (After): 3.14
Gallagher Index: 47.01
out2 = HighestAverages(
parties = names(Ceara),
votes = Ceara,
seats = 42,
method = "imperiali"
)Method: Imperiali
Divisors: 1 1.5 2 2.5 ...
ENP: 3.65 (After): 2.48
Gallagher Index: 52.15
out3 = HighestAverages(
parties = names(Ceara),
votes = Ceara,
seats = 42,
method = "sl"
)Method: Sainte-Laguë
Divisors: 1 3 5 7 ...
ENP: 3.65 (After): 3.74
Gallagher Index: 43.99
# add the method:
out1$Method = "d'Hondt"
out2$Method = "Imperiali"
out3$Method = "Saint-Laguë"
data <- rbind(out1, out2, out3)
2014 Legislative Election in Ceará (M=42)