Standardizing Representation for Equality with a Population Seat Index

TL;DR

This paper introduces a generalized apportionment framework using a population seat index (PSI) that accounts for non-linear standard functions, revealing that traditional proportional representation underestimates the contribution of less populous groups.

Contribution

It formulates a new population seat index and derives apportionment schemes that ensure absolute and relative individual equality, extending beyond traditional proportional representation assumptions.

Findings

PR assumes linear proportionality, which is often inaccurate.

The proposed scheme adjusts seat allocation based on a power function of population.

PR underrepresents less populous subgroups compared to the new method.

Abstract

Proportional representation (PR) has long been believed the ideal system for the equality of individuals in apportioning the seats of a legislature body to subgroups. We observe that PR implicitly assumes the (standard) number of representatives is proportional to the population, a situation no longer observed since 1820s. To address this issue, we suggest to formulate the apportionment problem in a broader context by explicitly specifying a standard function $f$ such that $f(p)$ is the standard, possibly fractional number of representatives for population $p$, where PR assumes $f(p)\propto p$. For this generalized apportionment problem, we give a population seat index (PSI) $\frac{f^{-1}(s)}{p}$ for quantifying the contribution of an individual in assigning $s$ seats to a population $p$, where $f^{-1}$ is the inverse of $f$. With the PSI, we derive apportioning schemes with absolute and relative individual equality. Particularly, for $s$ seats, populations $p_1, \ldots, p_k$, and a standard function $f(p) = a + b p^\gamma$ with constants $a, b, \gamma \ge 0$, the ideal, possibly fractional number of seats for subgroup $i$ is $a + \frac{(S-ka)p_i^{\gamma}}{\sum p_j^{\gamma}}$, not $\frac{Sp_i}{\sum p_j}$ calculated by PR which works only for $a=0$, $\gamma=1$. Finally, since real-world observations indicate a standard function $f \propto p^\gamma$ with $\gamma < 1$, we conclude that PR represents individuals in less populous subgroups less than individuals in more populous subgroups.