A divisor method that rounds at the harmonic mean of consecutive integers.
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.
Value
An integer vector or matrix of the same dimensions as pop, containing the
number of seats apportioned to each unit.
Details
The Dean method finds a common divisor \(d\) and rounds each quotient at
the harmonic mean of the two surrounding integers. A quotient
\(p_i / d\) that falls between integers \(n\) and \(n + 1\) is
rounded up to \(n + 1\) if it exceeds the harmonic mean:
$$H(n,\, n+1) = \frac{2n(n+1)}{2n+1}$$
and rounded down to \(n\) otherwise. The divisor is adjusted iteratively
until the total allocation equals size.
Among the classical divisor methods, the Dean method minimizes the absolute difference in average district population (i.e., \(p_i / s_i\), where \(s_i\) is the number of seats awarded to unit \(i\)) between any two units. It falls between the Webster and Huntington-Hill methods in its treatment of small versus large units, giving slightly more seats to small units than Webster but fewer than Huntington-Hill.