The Census and the Second Law: An Entropic Approach to Optimal Apportionment for the U.S. House of Representatives

TL;DR

This paper introduces an entropic method for apportioning U.S. Congressional seats that aims to maximize fairness and proportionality by minimizing relative entropy, with broad applications in political and resource allocation systems.

Contribution

It applies principles of information theory and entropy to develop a novel, mathematically grounded apportionment method that improves fairness in legislative seat distribution.

Findings

The entropic apportionment method aligns with maximizing representational equity.

It can be used to compare different apportionment schemes and district configurations.

The approach is applicable to various multi-constituency resource allocation problems.

Abstract

The Constitutionally mandated task of assigning Congressional seats to the various U.S. States proportional to their represented populations ("according to their numbers") has engendered much contention, but rather less consensus. Using the same principles of entropic inference that underlie the foundations of information theory and statistical thermodynamics, and also enjoy fruitful application in image processing, spectral analysis, machine learning, econometrics, bioinformatics, and a growing number of other fields, we motivate and explore a method for Congressional apportionment based on minimizing relative entropy (also known as Kullback-Leibler divergence), or, equivalently, maximizing Shannon entropy. In terms of communication theory, we might say that the entropic apportionment gives each constituent as equal a voice as possible. If we view representational weight as a finite resource to be distributed amongst the represented population, the entropic measure is identical with the Theil index long employed in economics to measure inequality in the distribution of wealth or income, or in ecology to measure the distribution of biomass or reproductive fitness. Besides Congressional apportionment, the method is also directly applicable to other multi-regional or multi-constituency legislatures, to party-list proportional voting systems used in various parliamentary elections, and similar settings, where the task is to allocate a discrete number of seats or other resources, and the primary goal is one of maximal proportionality or equity. In addition, the same entropic figure-of-merit can be used in parallel to compare different choices for the total number of representatives, and then subsequently to assess different Congressional district sizes, after seats are assigned and proposed district boundaries drawn.