Introduced initially to allocate parliamentary seats among states or provinces, apportionment
methods can distribute any discrete scarce resource based on a series of claims or
budgets such as the states’ voting populations. Here, we focus on the price of representation:
how one can turn the claims, such as the number of voters, into a share of the resource,
such as parliamentary seats. The D’Hondt method naturally emerges as a competitive
equilibrium. We show that the class of parametric divisor methods is fully characterised
by a competitive equilibrium augmented by a uniform credit or debt with respect to
the original claims. Based on the insight gained from the competitive equilibrium
representation, we provide an alternative proof of majorization: divisor methods with
smaller parameters favour players with smaller budgets and vice versa . While in a
competitive equilibrium, all pay the same price, when the budget allows for the purchase
of a few items, leftovers may constitute a substantial part of the original budget.
In other words, not all are equally efficient at converting their budgets into items.
Optimisation apportionment methods, such as the Leximin, assign individual prices;
hence, each budget is fully spent, and focus on minimising price differences. We demonstrate
how the D’Hondt and Adams methods can be formulated as optimisation methods and highlight
their connection to the Leximin method.
