The apportionment problem is fundamental to the theory of fair allocation of scarce resources. It appears for example in parliamentary elections, when a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. In this talk I present a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. When designing voting rules for this new setting, an important question is to which extent proportionality guarantees can be obtained. We discuss several proportionality axioms and analyse promising voting rules with respect to these.
This is joint work with Markus Brill, Paul Gölz, Dominik Peters and Kai Wilker.