A sequential priority method widely used for proportional representation elections, including in Belgium, Spain, Portugal, the Netherlands, Austria, and many other countries. Mathematically equivalent to the Jefferson method, a procedure described by Thomas Jefferson, and also known as the greatest divisors method. Compared with the Webster/Sainte-Laguë method, D'Hondt/Jefferson tends to give a slight advantage to larger units.
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 D'Hondt method allocates seats sequentially. At each step, the next seat
is awarded to the unit (party or state) with the highest quotient:
$$Q_i = \frac{p_i}{n_i + 1}$$
where \(p_i\) is the population or vote total of unit \(i\) and
\(n_i\) is the number of seats it currently holds. This process repeats
until all size seats have been awarded.
The Jefferson method finds a common divisor \(d\) such that floor-rounded
quotients sum to the desired house size:
$$\sum_{i} \left\lfloor \frac{p_i}{d} \right\rfloor = H$$
where \(p_i\) is the population of unit \(i\) and \(H\) is the total
number of seats (size). The divisor \(d\) is decreased iteratively until
this condition is met.
The Jefferson method is mathematically equivalent to the D'Hondt method: both produce the same allocation. The Jefferson divisor formulation and the sequential D'Hondt priority \(p_i / (n_i + 1)\) are two perspectives on the same apportionment rule.