A quota method that satisfies the quota property and avoids the Alabama paradox, proposed by Michel Balinski and H. Peyton Young.
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
Let the exact quota for unit \(i\) be:
$$q_i = \frac{p_i \cdot H}{P}$$
where \(p_i\) is the population of unit \(i\), \(H\) is the total
number of seats (size), and \(P = \sum_i p_i\) is the total population.
The method guarantees the quota property: each unit receives either \(\lfloor q_i \rfloor\) or \(\lceil q_i \rceil\) seats, so no unit is over- or under-represented by more than one seat relative to its exact quota. Seats are awarded sequentially using the D'Hondt priority \(p_i / (1 + n_i)\), but with an upper quota cap: once a unit has received \(\lceil q_i \rceil\) seats it is excluded from further consideration. This cap prevents the runaway over-representation that the unconstrained Jefferson/D'Hondt method can produce, while preserving freedom from the "Alabama paradox," in which increasing the total house size can paradoxically cause a unit to lose a seat.
Balinski and Young proved that no apportionment method can simultaneously satisfy the quota property and avoid all paradoxes; this method is a practical compromise that prioritizes the quota property.