The Wyoming and Cube-Root Rules are ineffectual proposals

In order to overcome the pernicious problems resulting from our undersized House of Representatives, these two formulations are frequently suggested as a basis for determining how many Representatives we should have. However, neither would enlarge the House sufficiently to mitigate the current representational inequities.

Over the last few years, several organizations have issued reports identifying various problems arising from our House of Representatives being far too small. To correct those problems, these reports generally focus on two possible algorithms for determining how many Representatives we should have: The Wyoming Rule and the Cube-Root Law. Though both of these algorithms would increase the size of the House, an analysis is needed to assess the extent to which those increases would actually mitigate those problems. But first, a brief explanation of each of these algorithms is provided.

The Wyoming Rule

The Wyoming Rule asserts that our total number of Representatives should be determined by dividing the nation’s total apportionment population by the total population of our least populous state. Applying this to the 2020 apportionment population, our total number of Representatives would be 573.¹This is calculated by dividing the total apportionment population by Wyoming’s population as follows: 331,108,434 ÷ 577,719 = 573.13  → rounds to 573 Representatives.

The Cube-Root Law

The Cube-Root “law” (or rule) asserts that the ideal size of the House of Representatives is the cube root of the total population. Applying this to the 2020 apportionment population, our total number of Representatives would be 692.²This is calculated by taking the cube root of the total apportionment population as follows: ∛331,108,434 = 691.82  → rounds to 692 Representatives.

The Analysis

For this analysis there are two scenarios: 573 Representatives (Wyoming Rule) and 692 Representatives (Cube Root). Each of these are apportioned to the states using the same algorithm currently used to apportion 435.³The Huntington-Hill method. The current apportionment along with those two alternative scenarios are provided in this table. These scenarios are evaluated from two different perspectives: Quantitatively and qualitatively.

The first section below is a quantitative evaluation of the extent to which the increases in House size restore representational parity among the states and among the citizens. The second section is a qualitative assessment of the impact of these quantitative changes. It will be shown that neither the Wyoming Rule nor the cube-root formula would increase the number of Representatives sufficiently to mitigate the corrosive problems resulting from a grossly undersized House. However, as shown in the conclusion, there is a solution which would fully mitigate these problems and restore equitable representation.

Before proceeding, it should be noted that both the Wyoming and Cube-Root concepts are conceptually flawed in such a way as to completely undermine their underlying rationales. Though it is not generally known, the Cube-Root Rule model is based on a hypothesis derived from a debunked analysis. And inherent in the Wyoming Rule is a peculiarity that renders it unsuitable as an enduring principle.

Quantitative Evaluation: Impact on representational disparities

The quantitative evaluation focuses on the how increases in House size reduce representational disparities. Ideally, there should be perfect representational parity; that is, the states’ shares of representation in the House should be exactly equal to their shares of total population (which equates to one-person-one-vote equality). For example, if a state has 0.56% of the population, perfect parity would be if it also had 0.56% of the total representation in the House. Though it is not possible to achieve perfect parity in a large multistate republic, it is possible to substantially eliminate the existing inequitable disparities simply by enlarging the House as necessary.

Baseline: 435 Representatives

The chart below is the representational disparity by state for the apportionment of 435 Representatives as of the 118^th Congress (beginning January of 2023).  For each state, the chart indicates the ratio of its share of representation to its share of population. This is explained by the two examples noted on the chart. First, Idaho has 0.46% of the representation in the US House, but 0.56% of the nation’s population. Therefore, its ratio of representation share to population share is 0.83; that is, it has 83% of the representation that it should. And Montana’s ratio of representation share to population share is 1.4, meaning that its share of representation is 40% more than it should be.

Figure 1 – 435 Representatives

Though these disparities may seem relatively small in percentage terms, they are actually quite consequential, especially with respect to the resulting interstate inequities. For example, Montanans have 82% greater political weight in the House than do Delawareans. Therefore, because the population sizes of our congressional districts vary wildly from state to state, the House of Representatives is in egregious violation of the Constitution’s one-person-one-vote equality requirement. This dispersion in the congressional district sizes can be calculated using an algorithm called “relative standard deviation”. And the relative standard deviation for the current apportionment is a massive 498%. And consider how much worse this inequity will become as the population of our country grows.

Next we will determine how much the Wyoming and cube-root formulations would reduce these inequitable interstate disparities.

The Wyoming Rule

Ironically, the Wyoming Rule would work to the detriment of the Equality State. Wyoming’s share of representation in the U.S. House would decrease from 1/435 (0.23%) to 1/573 (0.17%), equal to its share of the total population. However, that parity would only be imposed on Wyoming. All the remaining states would still be quite disparate: 23 would enjoy a greater share of representation than warranted by their share of population, and 26 would have less. That is roughly the same as the current situation relative to the apportionment of 435.

The chart below is like the previous one (Figure 1), except that the number of Representatives is increased to 573 as suggested by the Wyoming Rule.

Figure 2 – Wyoming Rule

The chart above illustrates that a small increase in the number of Representatives results in a commensurately small decrease in interstate disparities. In this scenario, the largest congressional district has 76% more residents than the smallest one, and the relative standard deviation is 404%. That is a relatively inconsequential decrease relative to the 435-Representative scenario.

The Cube-Root Rule

Wyoming would fair even worse under the cube-root model, reducing its representation from 1/435 (0.23%) to 1/692 (0.14%), which is less than its share of the total population. Of course, Wyoming is only one instance, so let’s zoom out to examine all fifty states in the same way. The next chart repeats the disparity analysis for the 692-Representative scenario .

Figure 3 – Cube-Root Rule

As before, this small additional increase in representation produces a similarly small decrease in interstate disparities. In this scenario, the largest congressional district has 75% more residents than the smallest one, and the relative standard deviation is still huge at 329%.

Inequitable Results

As shown above, neither of these proposals would lead to an apportionment that is even remotely equitable. Like the base-case scenario (435 Representatives), both alternatives would result in indefensibly egregious violations of the Constitution’s one-person-one-vote requirement.

Qualitative Evaluation: Restoring fair & democratic representation

Several organizations have published reports advocating for a larger House and identifying both the Wyoming and cube-root formulations as possible solutions. These organizations include the following:

• The Fordham University Law School’s Democracy and the Constitution Clinic ⁴Why the House of Representatives Must Be Expanded and How Today’s Congress Can Make It Happen
• FairVote.org ⁵More Members, More Voices
• The American Academy of Arts & Sciences ⁶The Case for Enlarging the House of Representatives

Though Thirty-Thousand.org takes issue with each of these reports on a few key points, all of them present compelling arguments for why the House should be substantially enlarged. These arguments, which are encapsulated below, will be used to assess the possible benefit of increasing the size of the House to that suggested by the cube-root formula. Because the cube-root formula produces a larger House size than would the Wyoming Rule, and therefore smaller congressional districts, it is more likely to produce beneficial results.

Constituents have very little access to their Representative

“By enlarging too much the number of electors [per representative], you render the representative too little acquainted with their local circumstances and lesser interests”  – James Madison (Federalist 10)

“It is a sound and important principle that the representative ought to be acquainted with the interests and circumstances of his constituents.” –  James Madison (Federalist 56)

As a result of the apportionment of 435 Representatives based on the 2020 census (Figure 1), the average congressional district size is 761,000, and the districts’ populations range from 578 thousand (Wyoming) to almost one million (Delaware). Obviously, even the smallest of these districts is too huge for a Representative to be acquainted with the interests and circumstances of his or her constituents, or for the constituents to be able to meet with or petition their Representative to the extent necessary. As a result, the vast majority of Americans are irreparably estranged from their so-called “representatives”.

American Academy:  Fundamental to the House’s status as the most purely democratic part of the government were the relatively small sizes of congressional districts. Congressmen were meant to serve in Washington while also remaining intimately familiar with the issues facing their constituents. … The bigger the district, the weaker the electoral connection.

FairVote: The ballooning number of people per representative has resulted in a Congress where members have fewer opportunities for direct contact with constituents. … Because members of the [current] Congress must represent many more people than those of the 63rd Congress in 1913, it has become much more challenging for them to protect the interests of their constituents. Individual and group voices have been diluted to the point that it is impossible for many House members to recognize (let alone appease) all of the different segments of their constituencies.

Fordham: As districts grow in size, more constituents are deprived of adequate representation. To win re-election, candidates are practically required to bargain with some interest groups over others within their districts. The interest groups and political minorities that candidates eschew are ultimately losers in the electoral process, both deprived of a functional vote and representation of their interests in the House. A larger House corrects for this issue by granting more representatives to each state and allowing those new districts to capture the preferences of a greater number of political communities. Large districts may make it harder for constituents to receive help navigating the federal bureaucracy. … Expanding the House would allow representatives to provide more hands-on help to constituents and relieve the burden on each member’s staff.

Analysis: Raising the number of Representatives to 692 would reduce the average district size to 478,480, which is still too vast for there to be satisfactory connections between the citizenry and their Representatives. To put that in perspective, the US would still have the largest average constituency size in the world, and approximately twice that of second place Mexico!⁷Pew Research: U.S. population keeps growing, but House of Representatives is same size as in Taft era. (May 31, 2018)

Gerrymandering

“The right of voting for representation is the primary right by which other rights are protected.” – Thomas Paine

“Politics have become far too important to entrust to the politicians.” – Dwight Eisenhower

With an average district size of 761,000, it is quite easy for politicians to draw districts to achieve almost any political objective. In addition to deliberate gerrymandering, having too few Representatives inevitably cause some communities of interest to effectively be disenfranchised by being subsumed into massive congressional districts.

Fordham: The more districts there are, the harder it becomes to gerrymander effectively.

The benefits of eliminating gerrymandering include increased diversity as well as allowing for additional third-party and independent candidates.

Diversity

American Academy: The size of congressional districts … has helped result in a Congress that falls far short of representing the country’s ideological and demographic diversity.

Fordham: [A] larger House would make it harder to entirely lock out a political minority of representation and cement the majority’s hold on power”, and further: “an immediate increase in House seats to accommodate for several decades of stagnation would create tremendous opportunity for new voices to enter government. Overwhelmingly, these voices would likely be younger, more female, more non-white, and more demographically representative of America than the current makeup of Congress.

Third-Party & Independent Candidates

It is indisputable that a substantial increase in the size of the House would enable additional political parties, as well as independent candidates, to be represented therein.

Fordham: [C]ongressional districts are too large for third-party candidates to reach a critical level of support, either because of high financial costs or because of the necessary compromises with interest groups. Smaller districts allow independent and third-party candidates to run issue-driven campaigns that speak specifically to the concerns of a geographic community”.

More seats in the House diminishes the significance of any single seat in determining a congressional majority. A third-party voter may therefore feel less concerned that their third-party vote will be “wasted” or will “spoil” the election of a preferred major party candidate and preferred congressional majority.

Expanding the House may help undo the bitter partisan dynamic in the chamber. A larger House would create greater opportunity for members to defect from their parties on key issues and build coalitions with other members to advance legislation and oversight prerogatives. An increase in third party representation could also reduce partisanship by denying a major party a majority or forcing some kind of coalition government.

Analysis: The ability to gerrymander would not be materially curtailed by reducing the average district population to nearly half a million. Therefore, increasing the number of Representatives to 692 could not be expected to result in a House that is adequately representative of the diversity of over 330 Americans. And given the high cost of elections in such large districts, there is no reason to believe that there would be an increase in either third-party or independent candidates.

Unequal representation among the states and the people

“Diluting the weight of votes because of place of residence impairs basic constitutional rights under the Fourteenth Amendment just as much as invidious discriminations based upon factors such as race” – Reynolds v. Sims 377 U.S. 566

The House of Representatives

Under the current apportionment of 435, Montanans have 82% greater political weight in the House than do Delawareans. And the relative standard deviation of the congressional district sizes is a massive 498% (Figure 1). Therefore, the current apportionment is clearly an egregious and unjustifiable violation of the Constitution’s one-person-one-vote requirement.

Fairvote: The current size of Congress also allows for serious disparities in the number of people per representative between the smallest districts and the largest ones.

Fordham: [T]he quality of Americans’ representation should not depend on the state they happen to live in. Increasing the number of representatives who can be allocated among the states makes the task of equal apportionment easier by increasing the pool of seats to be equally divided and produces more parity in interstate representation ratios.

Analysis: As explained above, in a House of 692 Representatives, the largest congressional district would have 75% more residents than the smallest one, and the relative standard deviation is still huge at 329% (Figure 3). This would be only marginally less egregious than the current apportionment, and therefore would not solve this problem.

The Electoral College

Fairvote: The representational disparities in the House of Representatives “also carry over to the Electoral College, where the disparity is further exacerbated by the inclusion of two electoral votes for each state”.

Analysis: Though the interstate disparity in the Electoral College is caused by both the apportionment disparity and the “small-state advantage”, the founders expected both of these disparities to be diminished as the nation grew. As with the egregious disparities in the House of Representatives, increasing the House to 692 would perpetuate the continuation of the significant representational disparities in the Electoral College.

Large districts make campaigns costlier

In addition to estranging Americans from their Representative, massive districts have two additional toxic consequences. First, the high cost of political campaigns in massive districts make it necessary for candidates to continuously prostrate themselves before various Special Interests. And second, that creates a nearly insurmountable barrier for challengers to unseat incumbents.

Incumbency Advantage

American Academy: Large districts favor incumbents as well as wealthy and well-funded candidates. Large districts also make it harder for a wide variety of challengers—including racial minorities and third-party candidates—to be elected. 

Fordham: [L]arge districts make campaigns costlier. Winning the support of a majority of voters in a 750,000-person district requires significant financial resources that candidates often need to obtain from wealthy benefactors and powerful interest groups. An expanded House would reduce the financial barrier to entry for candidates and, in effect, decrease the leverage that well-financed groups have over them.

Analysis: The huge cost of campaigning in massive congressional districts creates a nearly insurmountable advantage for the incumbent, making it virtually impossible for any challenger to unseat an incumbent.  Assuming the continuation of single-member districts, an expanded House would reduce the financial barrier to entry for candidates, but the average district size will have to be significantly smaller than 478 thousand in order for virtually any citizen to be able to seek office.

Corruption by Special Interests

Fordham: A stagnant political body is vulnerable to capture by powerful interests. The House is no exception. This concern was paramount among supporters of the proposed constitutional amendment in the Framing Era to tie the House’s membership to population growth. They feared corruption was more likely where representatives were not intimately familiar with their voters, and that the powerful would more easily commandeer a smaller body than a larger one.

Analysis: Given that these problems will inevitably result from having massive congressional districts, it could not possibly be expected that reducing the average congressional district population to nearly half a million would produce any meaningful improvements.

Congressional Workload

As explained in Thirty-Thousand.org, expanding the size of the House would better enable the Representatives to serve their constituents while, at the same time, spread their huge committee workload over a larger number of Representatives.

Fordham: [E]xpanding the House may make some administrative functions easier. The scope of Congress’s powers and work have only expanded in the modern era, and the issues that each congressperson deals with, either in representing their community or in their committee and legislative assignments, has only continued to grow. Representatives in the House arguably have too much work, and do not perform their jobs as well as they would like because of it. Smaller districts would reduce the number of salient interests representatives must consider in legislating because their districts would have fewer political communities. Additionally, a larger House allows committee work and assignments to be distributed among more members, giving each member more time to focus on their remaining tasks.

Analysis: Given that there are currently over 2,600 committee assignments relative to the House of Representatives, and given that Representatives have an average of six committee assignments each, it cannot be imagined that adding only 257 Representatives (to 435) would go far enough to improve the quality of their work.

Ordained by the Constitution: The Founders’ Rule

Though neither the Wyoming nor Cube-Root rules provide a viable solution, there is a formulation which will. The Founders’ Rule is the only solution that would, with certainty, virtually eliminate all the problems resulting from having a grossly undersized House of Representatives. In addition, this solution would bring true interstate parity to the Electoral College.

Illustrated below is a solution which conforms to the Founders’ Rule. At a House size of 6,692, near perfect equality is achieved in the House of Representatives. (To facilitate comparison with the other scenarios, the chart below is rendered on the exact same scale as the charts in Figures 1 though 3 above.)

Figure 4 – The Founders’ Rule

If the size of the House were determined by the Founders’ Rule, the disparity between the largest and smallest congressional districts would shrink to 3.7%, and the dispersion in district sizes would be only 12%.

© Thirty-Thousand.org [published 9/18/22; updated 10/29/24]