Right now, the House of Representatives apportionment allocation-rule is of timely importance, because we’re hearing in the news that the Democrats’ & Republicans’ numbers of seats are close, & that therefore a flip of one or a few seats could change everything.

We’ve currently been allocating seats by the Huntington-Hill allocation-rule. In 1991, a court in Massachusetts ruled that Huntington-Hill is unconstitutional, & that Webster’s allocation rule is the unbiased & proportional allocation-rule. But the Supreme Court overruled that Massachusetts decision, & kept the Huntington-Hill allocation-rule.

Currently, from the apportionment by the 2000 census, both New York & Ohio each have a seat less than what the unbiased proportional Webster’s method would have given them…that loss being a result of the small-state-biased Huntington-Hill (“Equal-Proportions”) method currently in use.

So, in the House, the red-states have 2 seats that should be had by the blue-states. …a disproportionate 4-seat disparity.

The Constitution says to allocate seats to parties “according to their numbers”. All agree that that means, “in proportion to their populations”.

Of course, given that seat-assignments are integers, whole numbers, genuine complete proportionality is impossible for any allocation. So we don’t make every (or any) allocation fully proportional. But it’s possible to be completely proportionally fair to states *overall*, & that’s the only way to abide by the Constitution’s proportionality-requirement. Any other allocation is in violation of the Constitution.

Which allocation-rule achieves that? Webster’s method does.

Here’s a definition & specification of Webster’s method:

X% of the national population should get X% of the seats. So multiply each state’s population % by the desired house-size (435 minus the 50 allocated as automatic 1st-seats: i.e.385). That gives each state’s “rightful-share”. They usually end in fractions, so round each state’s rightful-share to the nearest whole number of seats. Assign to it that number of seats.

But, due to the vagaries of rounding, those assignments might not sum to the desired 385 seats.

So, multiply everyone’s rightful-seats by some same “multiplier” before rounding, then then round as before.

Find the multiplier that will result in the desired number of seats being assigned.

The multiplier will be slightly greater than, or less than, 1. …depending on whether the initial allocation was too few or too many.

That completes the allocation.

(There are systematic procedures to find the right multiplier without trial-&-error, but Webster’s method is completely & adequately defined & specified by merely saying to apply the multiplier that adjusts the number of assigned seats to the desired number, 385. In fact when Daniel Webster proposed his method, & it was used, it was defined & used as I’ve described, with no mention of a systematic-procedure.)

It’s obvious that the above-described allocation is unbiased, but that can be demonstrated:

First, a few terms, & specification of the quotas-possession-space number-line:

A “quota” is an amount of population such that if a seat were assigned for every quota, seats would be assigned in proportion to their populations, & they’d total to the desired number. …or some amount of population intended to closely approximate that result.

Rounding to the nearest whole number means rounding up for states that are above the halfway-point, the .5 point of the way across their “interval” between two whole numbers of quotas, & rounding down for states that are below that halfway-point. That roundoff-point is the rounding-point for Webster. (…though other rounding-points (“R”) can be used.)

Here, the real-number-line represents the numbers of quotas, including non-integer numbers, that a state could have.

A state at the number-point x on that line has x quotas of population. Each point on the line represents an amount of population.

Any interval on that line is an interval in quotas-possession-space. …also called state-size space.

Any two consecutive whole numbers are referred to as a & b. The interval between them is (a,b).

The interval between any two numbers is expressed in that way. i.e the interval between 1 & 2 is (1,2).

I assign the number 1 to the interval size between two whole numbers on the number-line. …arbitrarily. …but not entirely arbitrary, because b – a = 1.

That a-to-b interval, called the (a,b) interval, or just (a,b), doesn’t represent a number of quotas. It represents an interval of points, each representing a possible state-size (in quotas). So (a,b) can be said to represent 1 unit of state-sizes.

Now, the demonstration that Webster is unbiased:

How many seats does Webster assign in the (a,b) interval?

Well, because the rounding-up occurs at R, then any states with a number of quotas between a & R will have a seats. Because (a,b) represents 1 unit of state-sizes, then then (a,R) a distance of R-a, represents correspondingly-less state-size-space. So, multiplying seats for a state-size by the amount of state-size space, the seats assigned in the (a,R) interval can be said to be a(R-a), because a state in that interval get a seats.

Then, for the same reason, the seats assigned in the (R,b) interval can be said to be b(b-R)

Well, for Webster, R = a + .5, the halfway between a & b. So half of the (a,b) state-size space is in (a,R), where a state gets a seats & half is in (R,b), where a state gets b seats.

Therefore, the seats assigned in (a,R) is (1/2)a , or a/2.

Likewise, the seats assigned in (R,b) is (1/2)b, or b/2

So the total seats assigned in (a,b) is:

a/2 + b/2.

Because b = a + 1, then b/2 = (a+1)/2 = a/2 + 1/2.

So, the seats assigned in (a,b) is a/2 + a/2 + 1/2.

= a + 1/2.

i.e. a + .5

I’ll use S to stand for the seats in (a,b).

S = a + .5

Now, how many quotas are in (a,b)?

Well, the state-size at a + .5 has (by definition) a + .5 quotas.

For every state-size in (a,b) above that, with more quotas, there’s a corresponding state-size in (a,b) below that, with correspondingly fewer quotas. Therefore, a + .5 is the average number of quotas for the state-sizes in (a,b)

So, multiply that a+.5 average by the amount of state-size space in (a,b).

That amount of state-size-space has been defined as 1.

So, a + .5, multiplied by `1, is a + .5

Therefore, Q, the number of quotas in (a,b) is a + .5

Q = a + .5

That can also be shown by integration.

Above it was shown that S = a + .5

So S = Q.

So S/Q = 1.

The s/q in (a,b) is 1.

That’s true in *every* interval (a,b).

Because s/q is the same in every interval, the allocation is unbiased.

Now, if you do the same determination for Huntington-Hill (HH), you find that HH over-re-represents (1,2) by a factor of about 1.057

i.e. HH over-represents (1,2) by 5.7%.

Dean over-represents (1,2) by 11%
Adams over-representss (1,2) by 33%.
Jefferson under-represents (1,2) by 33%.

…&, as shown above, Webster is unbiased, representing all (a,b) intervals equally.

So, the Constitution’s requirement for allocation proportional to population is met by Webster, & violated by HH.

Our current use of HH is in violation of the Constitution.

We should be using Webster.

Incidentally, Webster is the same as the Sainte-Lague used in Proportional-Representation.

Jefferson is the same as the d’Hondt used in proportional-representation.

So, in PR, Sainte-Lague is the name of the unbiased allocation-rule.