A Self-dual Pseudo-divisor Quota Method for Congressional Apportionment

The topic of apportionment is a central focus of study for legislatures around the world, whether the goal is to allocate seats to political parties, or to allocate seats to the member states of a federation. The first goal is sought by parliaments employing proportional representation for parties; the second goal is sought by the United States House of Representatives and the European Parliament. Many of the leading apportionment methods were created in the late 18<sup>th</sup> century in response to requirements listed in the United States Constitution. No apportionment method perfectly satisfies all desirable properties, particularly the properties of integrality, proportionality, and quota. The Largest Remainder method satisfies quota but suffers from other paradoxes; the divisor methods like the Greatest Divisors, Major Fractions (Arithmetic Mean), Equal Proportions (Geometric Mean), Harmonic Mean, and Smallest Divisor methods satisfy proportionality but may fail quota. Some apportionment methods like Greatest Divisor unfairly favor larger parties and states, and others like Smallest Divisor unfairly favor smaller parties and states. We introduce a new method for Congressional apportionment that creates the apportionment all at once, rather than determining seats one at a time. This method always satisfies quota. It partially resembles the familiar Huntington monotone divisor methods and indeed creates a quota-capped divisor method, but can be compared as well to largest remainder methods.