Apportionment when seats are allocated in lots. The D’Hondt method case and political implications

Abstract

The apportionment problem involves determining how to distribute a given (non-negative) integer number among a group of individuals based on their respective sizes. In electoral systems with proportional representation, this problem arises in two situations: assigning seats to constituencies, if applicable, and distributing seats to political parties within each constituency. This paper addresses the scenario where seats are grouped into lots, extending the standard apportionment problem. We propose and analyze various apportionment methods based on the D’Hondt method for this new problem. Additionally, we examine the political implications of allocating seats not individually but in groups of varying sizes.