Iterate and change column and row divisors such that the row and column sums of the seats matrix satisfies the constraints given by the upper apportionment.
lower_apportionment(votes_matrix, seats_cols, seats_rows, method = "round")matrix with votes by party in rows and votes by district in columns.
number of seats per column (districts/regions), predetermined or
calculated with upper_apportionment().
number of seats per row (parties/lists), calculated with
upper_apportionment().
Apportion method that defines how seats are assigned. The following methods are supported:
round: The default Sainte-Laguë/Webster method is the standard
for biproportional apportionment and the only method guaranteed to terminate.
wto: "winner take one" works like "round" with a condition that the party that
got the most votes in a district must get at least one seat ('Majorzbedingung').
The condition does not apply in a district if two or more parties have the same
number of votes and there are not enough seats for these parties. A warning is
issued in this case. Modify the votes matrix to explicitly break ties.
You can provide a custom function that rounds a matrix (i.e. the the votes_matrix divided by party and list divisors).
It is possible to use any divisor method name listed in proporz().
A seat matrix with district (columns) and party (rows) divisors stored in attributes.
The result is obtained by an iterative process ('Alternate Scaling Algorithm', see Reference). Initially, for each district a divisor is chosen using the highest averages method for the votes allocated to each regional party list in this region. For each party a party divisor is initialized with 1.
Effectively, the objective of the iterative process is to modify the regional divisors and party divisors so that the number of seats in each regional party list equals the number of their votes divided by both the regional and the party divisors.
The following two correction steps are executed until this objective is satisfied:
modify the party divisors such that the apportionment within each party is correct with the chosen rounding method,
modify the regional divisors such that the apportionment within the region is correct with the chosen rounding method.
Oelbermann, K. F. (2016): Alternate scaling algorithm for biproportional divisor methods. Mathematical Social Sciences, 80, 25-32.
votes_matrix = matrix(c(123,912,312,45,714,255,815,414,215), nrow = 3)
district_seats = c(7,5,8)
party_seats = c(5,11,4)
lower_apportionment(votes_matrix, district_seats, party_seats)
#> [,1] [,2] [,3]
#> [1,] 1 0 4
#> [2,] 4 4 3
#> [3,] 2 1 1
#> attr(,"divisors")
#> attr(,"divisors")$districts
#> [1] 207 180 161
#>
#> attr(,"divisors")$parties
#> [1] 1.15 1.00 1.00
#>
# using "winner take one"
vm = matrix(c(200,100,10,11), 2,
dimnames = list(c("Party A", "Party B"), c("I", "II")))
district_seats = setNames(c(2,1), colnames(vm))
ua = upper_apportionment(vm, district_seats)
lower_apportionment(vm, ua$district, ua$party, method = "wto")
#> I II
#> Party A 2 0
#> Party B 0 1
#> attr(,"divisors")
#> attr(,"divisors")$districts
#> I II
#> 139 22
#>
#> attr(,"divisors")$parties
#> Party A Party B
#> 0.92 2.00
#>
# compare to standard method
lower_apportionment(vm, ua$district, ua$party, method = "round")
#> I II
#> Party A 1 1
#> Party B 1 0
#> attr(,"divisors")
#> attr(,"divisors")$districts
#> I II
#> 150 21
#>
#> attr(,"divisors")$parties
#> Party A Party B
#> 0.9 1.2
#>