Apportion by the Huntington-Hill (Equal proportions) Method

The current method used for US Congressional apportionment, in continuous use since the Apportionment Act of 1941. Also known as the method of equal proportions.

Usage

app_huntington_hill(size, pop, init = NULL, thresh = 0)

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 pop with the initial number of seats allocated to each unit. Defaults to zero for all units.
thresh: A population threshold for assigning seats. Units with population below this threshold receive zero seats, by default. Only affects the default value of init; if init is provided, thresh is ignored.

Value

An integer vector or matrix of the same dimensions as pop, containing the number of seats apportioned to each unit.

Details

The Huntington-Hill method is a sequential priority method. Starting with one seat allocated to each unit (a constitutional minimum for Congressional apportionment), it repeatedly awards the next seat to the unit with the highest priority value: $$P_i(n) = \frac{p_i}{\sqrt{n(n + 1)}}$$ where $p_i$ is the population of unit $i$ and $n$ is the number of seats currently held by unit $i$.

This is equivalent to a divisor method that rounds each quotient at the geometric mean of the two surrounding integers: $p_i / d$ is rounded up to $n + 1$ if it exceeds $\sqrt{n(n+1)}$, and down to $n$ otherwise. Among the divisor methods, Huntington-Hill minimizes the maximum relative difference in representation between any two units.

References

Huntington, E. V. (1928). The apportionment of representatives in Congress. Transactions of the American Mathematical Society, 30(1), 85–110. doi:10.2307/1989268

Examples

app_huntington_hill(size = 435, pop = state_2020$pop)
#>  [1]  7  1  9  4 52  8  5  1 28 14  2  2 17  9  4  4  6  6  2  8  9 13  8  4  8
#> [26]  2  3  4  2 12  3 26 14  1 15  5  6 17  2  7  1  9 38  4  1 11 10  2  8  1