This will be a series that explores an Austrian-style election system that provides political proportionality within states, while maintaining to a large extent local representation. It is a system that is resistant to gerrymandering. Moreover, it can be implemented within the current constitution.
The first step is apportionment of representatives. While this is not a necessary component, it is useful since it provides for larger numbers of representatives, particularly for smaller states and therefore greater proportionality.
I used the cube root rule to determine the number of representatives, but did not use a priority list based on a fixed number of representatives. Rather the number of representatives for a state is based solely on a state's relative share of the total population, and not in comparison to others. This can result in the house declining or increasing slightly from the nominal size. This avoids a couple of paradoxes:
Quota violation: Where a state with a population between N/T and (N+1)/T shares of representatives is awarded N+2 or N-1 representatives. That is a state entitled to 26.5 representatives might be awarded 25 or 28 representatives.
Non-montonicity. It is possible under the current system for a state's relative share of the national population to increase, but its relative share of the national representation to decrease, or vice versa.
A quota is determined by dividing the total population by the nominal number of representatives. A state's share of representation is calculated by dividing the state's population by quota. This results in a mixed fraction of the form n+f where n is an integer and f is a fraction 0 <= f < 1. A state will receive either n or n+1 representatives. A rounding threshold of the geometric mean of (n, n+1) or sqrt( n * (n+1). If n+f >= rounding threshold it is awarded n+1 representatives.
Since each state is rounded independently, the total number of representatives may be more or less than the nominal amount.
For this example I used the 2010 apportionment population (i.e. including federally-affiliated overseas population), and a 2020 domestic population projection based on the July 1, 2019 estimated population.
For 2010, the number of representative is 676. For 2020, the number of representatives is 691. Because of the increase in representatives, 14 states gained representation, while 9 lost. While California gained a representative, its share of the total declined, just as its share of the total population declined slightly.
| State | 2010 | 2020 | Change |
| California | 82 | 83 | 1 |
| Texas | 55 | 62 | 7 |
| Florida | 41 | 46 | 5 |
| New York | 42 | 41 | -1 |
| Pennsylvania | 28 | 27 | -1 |
| Illinois | 28 | 27 | -1 |
| Ohio | 25 | 25 | |
| Georgia | 21 | 22 | 1 |
| North Carolina | 21 | 22 | 1 |
| Michigan | 22 | 21 | -1 |
| New Jersey | 19 | 19 | |
| Virginia | 18 | 18 | |
| Washington | 15 | 16 | 1 |
| Arizona | 14 | 15 | 1 |
| Massachusetts | 14 | 15 | 1 |
| Tennessee | 14 | 14 | |
| Indiana | 14 | 14 | |
| Missouri | 13 | 13 | |
| Maryland | 13 | 13 | |
| Wisconsin | 12 | 12 | |
| Colorado | 11 | 12 | 1 |
| Minnesota | 12 | 12 | |
| South Carolina | 10 | 11 | 1 |
| Alabama | 11 | 10 | -1 |
| Louisiana | 10 | 10 | |
| Kentucky | 10 | 9 | -1 |
| Oregon | 8 | 9 | 1 |
| Oklahoma | 8 | 8 | |
| Connecticut | 8 | 7 | -1 |
| Utah | 6 | 7 | 1 |
| Iowa | 7 | 7 | |
| Nevada | 6 | 7 | 1 |
| Arkansas | 6 | 6 | |
| Mississippi | 7 | 6 | -1 |
| Kansas | 6 | 6 | |
| New Mexico | 5 | 4 | -1 |
| Nebraska | 4 | 4 | |
| Idaho | 3 | 4 | 1 |
| West Virginia | 4 | 4 | |
| Hawaii | 3 | 3 | |
| New Hampshire | 3 | 3 | |
| Maine | 3 | 3 | |
| Montana | 2 | 2 | |
| Rhode Island | 2 | 2 | |
| Delaware | 2 | 2 | |
| South Dakota | 2 | 2 | |
| North Dakota | 2 | 2 | |
| Alaska | 2 | 2 | |
| Vermont | 1 | 1 | |
| Wyoming | 1 | 1 | |
