Competitive Equilibrium and Optimisation Approaches to Apportionment

What is it about?

Apportionment methods were originally developed to divide parliamentary seats fairly among states or regions based on population. More generally, they provide tools for allocating any limited number of indivisible resources—such as seats, positions, or funds—when different parties have competing claims. In this paper, we focus on the “price of representation”: how claims like the number of voters are converted into actual seats. We show that the widely used D’Hondt method can be understood as the outcome of a competitive market equilibrium, where all participants face the same price for seats. Building on this insight, we demonstrate that the entire family of parametric divisor methods can be characterised as competitive equilibria in which each participant receives the same adjustment to their original claim, in the form of a uniform credit or debt. This perspective provides a new and intuitive proof of a classic result: divisor methods with smaller parameters tend to favour smaller parties, while larger parameters favour larger ones. However, equal prices do not always imply equal efficiency. When participants can afford only a small number of seats, unspent “leftover” claims can be significant, meaning that some parties convert their claims into seats more efficiently than others. To address this, we examine optimisation-based apportionment methods, such as the Leximin rule, which allow different participants to face different prices. These methods ensure that all claims are fully used and explicitly aim to minimise inequalities in representation. Finally, we show that familiar rules such as the D’Hondt and Adams methods can also be expressed as optimisation problems, clarifying their relationship to Leximin and offering a unified view of competitive and optimisation-based approaches to fair allocation.

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Photo by Hansjörg Keller on Unsplash

Photo by Hansjörg Keller on Unsplash

Why is it important?

This paper is important for several reasons. First, among apportionment methods, the Adams and Jefferson (D’Hondt) rules are usually seen as the two extremes: Adams is said to favour smaller parties or regions, while Jefferson/D’Hondt benefits larger ones, with other divisor methods lying somewhere in between. Our paper shows that, from a competitive equilibrium perspective, Jefferson/D’Hondt stands out as the neutral and fair benchmark. In this interpretation, all participants face the same price for representation. Other linear divisor methods can then be understood as deliberate distortions of this equilibrium: they effectively grant free seats to some participants or require others to pay an implicit surcharge. Second, equal prices do not always translate into equal treatment. Consider two participants who enter the allocation with 3.1 and 1.9 tokens, respectively—tokens that have no value outside this specific allocation. If the price of a seat is 1, both formally face the same price, yet the first participant spends just over one token per seat, while the second spends almost two. From this perspective, an outcome where the first receives three seats and the second two may appear fairer than one strictly based on a uniform price. Third, optimisation-based apportionment methods, such as the Leximin rule, address this issue directly. They allow participants to face different implicit prices, ensuring that all available resources are fully used and that disparities in representation are minimised. These methods often produce outcomes that many would regard as more equitable than standard divisor methods. As an additional insight, we show that even the classical Adams and Jefferson/D’Hondt rules can be reformulated as optimisation problems, revealing an unexpected connection between traditional divisor methods and modern optimisation-based approaches.

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