In the second biproportional apportionment step, party and district divisors are calculated such that the row and column sums of the resulting seats matrix satisfy the constraints given by the upper apportionment.
Arguments
- votes_matrix
matrix with votes by party in rows and votes by district in columns.
- seats_cols
number of seats per column (districts/regions), predetermined or calculated with
upper_apportionment().- seats_rows
number of seats per row (parties/lists), calculated with
upper_apportionment().- method
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 likeroundwith a condition that the party that got the most votes in a district must get at least one seat ('Majorzbedingung', also called 'strongest party constrained' rule (SPC)).votes_matrixmust have row and column names to use this method. A district winner can only get a seat if they are entitled to one from the upper apportionment (seats_rows). 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) without further parameters.
It is possible to use any divisor method name listed in
proporz().
Details
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.
Note
If the maximum number of optimization iterations is reached, an error is thrown since
no solution can be found. You can overwrite the default (1000) with
options(proporz_max_iterations = ...) but it is very likely that the result is undefined
given the structure of the input parameters.
References
Oelbermann, K. F. (2016): Alternate scaling algorithm for biproportional divisor methods. Mathematical Social Sciences, 80, 25-32.
Examples
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")$districts
#> [1] 204 180 162
#>
#> attr(,"divisors")$parties
#> [1] 1.2 1.0 1.0
#>
# 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")$districts
#> I II
#> 144 31
#>
#> attr(,"divisors")$parties
#> Party A Party B
#> 0.7 2.0
#>
# 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
#>