A divisor method that uses standard arithmetic-mean rounding, used for US Congressional apportionment in 1842 and from 1911 to 1931. Also used in Norway and Sweden for parliamentary seat allocation (where it is known as the Sainte-Laguë method).
Arguments
- size
An integer number of seats to apportion across units, or a vector of numbers of seats, one for each column of
pop. Must be non-negative.- pop
A vector or matrix of population sizes or proportions for each unit. If a matrix is provided, the apportionment algorithm is applied columnwise: each row is a unit and each column is a replicate. For example, with congressional apportionment, the matrix would have 50 rows and as many columns as hypothetical census population scenarios.
- init
A vector or matrix of the same size as
popwith the initial number of seats allocated to each unit. Defaults to zero for all units.
Value
An integer vector or matrix of the same dimensions as pop, containing the
number of seats apportioned to each unit.
Details
The Webster method finds a common divisor \(d\) such that
standard-rounded quotients sum to the desired house size:
$$\sum_{i} \text{round}\!\left(\frac{p_i}{d}\right) = H$$
where \(p_i\) is the population of unit \(i\) and \(H\) is the total
number of seats (size). Quotients are rounded at the arithmetic mean
\(n + 0.5\) of consecutive integers \(n\) and \(n + 1\).
Among the classical divisor methods, Webster is considered the most statistically unbiased: it does not systematically favor either large or small units. It minimizes the expected absolute deviation from exact quotas when populations are drawn from a wide range of sizes.