Abstract 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.
Kóczy et al. (Tue,) studied this question.