# A New Perspective on Impartial and Unbiased Apportionment

Ross Hyman<EMAIL_ADDRESS>Research Computing Center, University of
Chicago, Chicago IL 60637  Nicolaus Tideman<EMAIL_ADDRESS>Economics
Department, Virginia Tech, Blacksburg VA 24060

###### Abstract

How to fairly apportion congressional seats to states has been debated for
centuries. We present an alternative perspective on apportionment, centered
not on states but “families” of state, sets of states with “divisor-method”
quotas with the same integer part. We develop “impartial” and “unbiased”
apportionment methods. Impartial methods apportion the same number of seats to
families of states containing the same total population, whether a family
consists of many small-population states or a few large-population states.
Unbiased methods apportion seats so that if states are drawn repeatedly from
the same distribution, the expected number of seats apportioned to each family
equals the expected divisor-method quota for that family.

††preprint: APS/123-QED

##  I Introduction

Every ten years, the U.S. House of Representatives is reapportioned according
to the census for that decade. The Constitution specifies that
“Representatives shall be apportioned among the several States according to
their respective numbers” but prescribes no method to accomplish this. It is
not a simple matter of multiplying the total number of seats in the House by
each state’s fraction of the population, since the resulting numbers of seats
will generally not be integers. Many apportionment methods that assign each
state an integer number of seats have been proposed, including methods by
Thomas Jefferson, John Quincy Adams, Alexander Hamilton, and Daniel Webster
B&Y ; TJ ; Malk . Due to the political implications of the gain or loss of
even one seat, disputes over apportionment methods have occurred many times,
on the grounds that a proposed method is biased against states with small or
large populations. The first such dispute resulted in George Washington’s
first veto B&Y . The last such dispute, over the apportionment for the 1990
census, went to the Supreme Court Ernst .

We present an alternative way to think about apportionment, in which the
fundamental entities are not individual states but “families” of states, and
we introduce “impartial” apportionment methods and “unbiased” apportionment
methods. Impartial methods apportion seats among families of states according
to the families’ respective populations, regardless of whether a family
contains a large number of small-population states or a small number of large-
population states. For unbiased apportionment, we consider sets of states that
are repeatedly drawn from the same distribution. In each family, the number of
states and their populations, and therefore the family’s total divisor-method
quota, will, in general, differ across each sample set. Unbiased methods
apportion seats so that if states are drawn repeatedly from the same
distribution, the expected value of the total number of seats apportioned to
each family equals the expected value of the total divisor-method quota for
that family.

In sections II and III we present a history of methods used or proposed for
United States congressional apportionment, describe the Alabama paradox and
the New States paradox, motivate apportionment methods using a “slide rule,”
and show that divisor methods applied to states are not susceptible to the
Alabama paradox or the New States paradox. In sections IV \- VII we define
families of states and show that the Webster and Huntington-Hill methods
applied to families are impartial and susceptible to the New States paradox.
We find that Webster’s method applied to families of states avoids the Alabama
paradox while Huntington-Hill applied to families is susceptible to it. In
sections VIII \- XI we consider state populations that are drawn from a
specified probability distribution. We determine, for any probability
distribution for populations, the apportionment method associated with that
distribution that, when applied to states, is unbiased and avoids the Alabama
and New States paradoxes.

The link to a GitHub repository containing code for reproducing the tables in
this paper and allowing for exploration of the apportionment methods described
in this paper is in the references github .

##  II A Brief History of Apportionment.

Article I, section 2 of the U.S. Constitution specifies:

> Representatives and direct Taxes shall be apportioned among the several
> States which may be included within this Union, according to their
> respective Numbers, . . . The actual Enumeration shall be made within three
> Years after the first Meeting of the Congress of the United States, and
> within every subsequent Term of ten Years, in such Manner as they shall by
> Law direct. The Number of Representatives shall not exceed one for every
> thirty Thousand, . . .

The Constitution does not specify a method of rounding the fractions that
occur when apportioning integer numbers of representatives among states whose
populations are not integer multiples of any feasible district size.

When Congress received the results of the 1790 census, it debated several
methods of apportionment before settling on “Hamilton’s method,” which
involves deciding a size of the House, allocating seats including fractions to
all states, rounding down to integers, and then apportioning the resulting
extra seats to the states with the largest fractional remainders B&Y ; Biles .
George Washington vetoed the bill that Congress passed, the only veto of his
first term, on the ground that it gave Connecticut one representative for
every 29,605 persons counted in the Census, in violation of the Constitutional
requirement that the representation of each state not exceed one for each
30,000 B&Y ; Biles . Congress responded by adopting “Jefferson’s method,”
which involved dividing each state’s population by 33,000 and rounding down to
an integer B&Y ; Biles .

Jefferson’s method was used for apportionments for all censuses through 1830,
but Congress grew concerned over the method’s bias in favor of large-
population states. In the debate surrounding the apportionment for the 1830
census, other possibilities began to be discussed. John Quincy Adams proposed
rounding up rather than down. James Dean, a mathematics professor from
Vermont, proposed that the decision about whether a given state’s allotment
should be rounded up or down should depend on which produced a ratio of
persons per representative closer to the overall target, and Daniel Webster
proposed ordinary rounding. For the 1840 census, Congress chose Webster’s
method B&Y ; Biles .

For the apportionments from 1850 to 1900, Congress described the apportionment
it chose as Hamilton’s method, which had been their first choice in 1790,
before Washington’s veto. However, in 1850, 1870, 1880, and 1890, Congress
deliberately chose a size for the House whereby Hamilton’s method and
Webster’s method yielded the same apportionment B&Y ; Biles .

For the apportionments of 1910 and 1930, Congress went back to Webster’s
method. (Congress failed to pass a new apportionment in response to the census
of 1920.) For the apportionment in response to the 1930 census, Congress chose
to keep the House size at 435, knowing that at that size, Webster’s method
agreed with the Huntington-Hill method for the 1930 census. The Huntington-
Hill method rounds down or up depending on whether a state’s ideally assigned
number of seats, a real number that is generally not an integer, is less than
or greater than $\sqrt{S(S+1)}$, where $S$ is the integer obtained by rounding
the assigned number of seats down. Since 1940, Congress has chosen to use the
Huntington-Hill apportionment method, keeping the size of the house at 435
seats B&Y ; Biles .

In 1880, when the Census Bureau calculated the allotments under Hamilton’s
method for every House size from 275 to 350, Congress became aware that
Hamilton’s method is subject to the “Alabama paradox.” This is the phenomenon
that when the size of the House increases, it is possible for a state to lose
a seat B&Y ; Biles .

In 1907 Oklahoma became a state, and the ’New States paradox’ was discovered
when the Census Bureau calculated allotments using Hamilton’s method. The New
States paradox occurs when the size of the House increases through the
addition of a new state along with additional seats appropriate for the size
of the new state, and an existing state loses a seat B&Y ; Biles .

Michel Balinski and H. Peyton Young B&Y  identified apportionment methods that
avoid both the Alabama paradox and the New States paradox. They also showed
that if the populations of states are drawn from a uniform distribution, then
a) the Huntington-Hill method H1 ; H2 , which Congress has used since 1930, is
biased in favor of states with small populations and b) Webster’s method does
not favor states with small or large populations. In addition, they applied
bias criteria to apportionments based on the Webster and Huntington-Hill
methods and noted that the Webster apportionments showed less bias B&Y .

For the 1990 census, the state of Massachusetts, which would have gained a
seat with Webster’s method, sued the U.S government on the grounds that the
bias in favor of small-population states identified by Balinski and Young made
the use of the Huntington-Hill method for congressional apportionment
unconstitutional. The briefs in favor of the government’s position were
written by Laurence Ernst, a Census Bureau mathematician, who showed that for
every criterion and measure of bias for which Webster apportionments were the
least biased, there was a different criterion, for which the Huntington-Hill
apportionments were least biased. The court decided in favor of the federal
government, holding that, as there was no objective way to choose between the
contending bias criteria, the choice between these contending methods was
entirely subjective and within the purview of the Congress. Therefore,
continued use of the Huntington-Hill method for congressional apportionment
was constitutional Ernst .

The problem of apportioning seats in a legislature to parties under a system
of proportional representation is mathematically the same as the problem of
apportioning seats in Congress among states. For a discussion of issues of
apportionment in the context of proportional representation, see Pukelsheim
Puk .

##  III The Apportionment Slide Rule.

Figure 1: A portion of the apportionment slide rule with half-integer rounding
is shown at positions $D=100$ and $D=130$. Integer seat regions on the Seats
ruler are shown with alternating thicknesses for clarity. For $D=100$ a state
with a population of 400 is apportioned 4 seats. At $D=130$ it is apportioned
3 seats.

The integer number of seats apportioned to state $c$, $s_{c}$, would be
perfectly proportional to the population of the state, $v_{c}$, if there were
a parameter $D$ such that $s_{c}=v_{c}/D$ for every state. If the populations
of states came only in multiples of the $D$, then a state would be assigned
one seat for each $D$ persons, and there would be no apportionment problem.
The apportionment problem arises because the “quota” for state $c$,
$q_{c}=v_{c}/D$, is generally not an integer. We define the quota as the
divisor-method quota in terms of $D$, which can be any positive real number.
$D$ is the district size if seats are not rounded. It is not necessarily the
average district size $v_{T}/s_{T}$ where $v_{T}$ is the total population and
$s_{T}$ is the total number of rounded seats. The quota using $D$ and not the
quota using $v_{T}/s_{T}$ must be used for the Alabama paradox to be avoided.

Apportionment can be understood in the following visual way. Figure 1 shows
two sliding rulers, one on top of the other, with logarithmic scaling on each.
The top ruler is for state populations and the bottom ruler is for quotas and
seats. Logarithmic scaling ensures that sliding the rulers corresponds to
changing the common ratio, $D$, of population to unrounded seat quotas. For a
given value of $D$, there is a relative positioning of the two rulers, such
that the quota of seats allotted to a state with a given population is the
number of seats on the Seats Ruler directly below the population of the state
on the Populations Ruler.

In general, quotas are not integers. Most apportionment methods partition the
Seats Ruler into integer seat segments. A rounding mark, $r(f)$, is placed on
the Seats Ruler somewhere between integers $f$ and $f+1$, the precise position
depending upon the apportionment method. The segment of the Seats Ruler
between consecutive rounding marks $r(f-1)$ and $r(f)$ is the integer seat
region for $f$ seats. When the State Populations Ruler is positioned over the
Seats Ruler so that $v_{c}$ is over any part of the $f$ seat region, that is
$r(f-1)\leq q_{c}<r(f)$, state $c$ is assigned $f$ seats.

A portion of the apportionment slide rule with half-integer rounding is shown
in Figure 1 at positions $D=100$ and $D=130$. Integer seat regions on the
Seats ruler are shown with alternating thicknesses for clarity. For $D=100$,
states with 300, 400, and 500 people are apportioned 3, 4, and 5 seats
respectively. At $D=130$, states with 300, 400, and 500 people are apportioned
2, 3, and 4 seats respectively.

We refer to the set of state quotas between $f$ and $f+1$, which are the
quotas whose rounding is governed by $r(f)$, as the $f$ family of state
quotas. A vision of an apportionment slide rule with a bell-shaped
distribution of states on the Populations ruler inspired the ideas in this
paper. As $D$ is increased, the population ruler is shifted to the left, with
the logarithmic scaling preserving the shape of the distribution. Over each
family on the Seats ruler, the density of states changes from sloping up to
sloping down, as the left tail, followed by the middle, followed by the right
tail of the distribution passes by. A rounding mark that cannot move in
response to the changing shape of the density of states in its family tends to
round up more of the smaller population states at the left tail of
distribution and round down more of the large population states at the right
tail of the distribution, producing an apportionment biased in favor of small
population states. The rounding mark for each family must continuously adjust
its position, in response to the changing shape of the distribution of states
in its family, to ensure that the number of seats apportioned to the family
remains as close as possible to the unrounded quota for that family.

In general, rounding marks can be dependent on $f$, $D$, and the state
populations $\\{v\\}$, and can move if any of these change. These movements
are the cause of the various paradoxes of apportionment. For example, if $D$
is decreased (increased) the State Populations Ruler moves to the right
(left). If a rounding mark dependent on $D$ moves to the right (left) faster
than the State Populations Ruler so that it outruns a state, then that state
loses (gains) a seat, which is contrary to the motion of $D$, decreasing
(increasing) the district size if seats are not rounded. This is the Alabama
paradox. The total number of seats is adjusted by changing $D$, which
corresponds to moving the State Populations Ruler relative to the Seats Ruler.
For apportionment methods immune to the Alabama paradox, the total number of
seats will not decrease (increase) when $D$ decreases (increases).

If a state is added or removed, while $D$ and the populations $\\{v\\}$ of the
other states are held fixed, and a rounding mark dependent on state
populations consequently moves through the population of an existing state,
then that state’s allotment will change. This is the New States paradox.

The Alabama paradox will never occur if $\partial\log r/\partial\log D\geq-1$
and the New States paradox will never occur if rounding for state $c$ can
depend on $v_{c}$, $f$, and $D$ but not on the population of any other state.
Both of these conditions are satisfied if $r(f)$ depends only on $f$ and not
on $D$ or $\\{v\\}$. In other words, the rounding mark that determines the
number of seats allotted to state $c$ depends only on the integer part of
$q_{c}$ and not the quota of any other state. Such methods are called divisor
methods B&Y . A result of our paper is that there are methods that are immune
to the Alabama paradox and the New States paradox yet are not divisor methods.

##  IV Families of States.

For both impartial and unbiased apportionment, we are concerned with the
difference between the sum of the unrounded and rounded divisor-method quotas
for a group of states. For impartial apportionment, we seek to minimize this
difference while treating groups of states with the same total population
equally. For unbiased apportionment we seek to make this difference precisely
zero on average for states repeatedly drawn from the same distribution. These
two apportionment methods should agree as the number of states goes to
infinity. But there is no hope that the difference between the sum of
unrounded and rounded quotas approaches zero unless the groups include states
whose quotas are rounded up and states whose quotas are rounded down.

All the states whose quotas $q_{c}=v_{c}/D$ fall in the region $f\leq
q_{c}<f+1$, that is, all states with quotas whose integer part is $f$, we call
the $f$ family of states. Each family contains one rounding mark. All states
to the left of a rounding mark in a family are rounded down and those to the
right are rounded up. So families are permissible groups for impartial and
unbiased apportionment. In this paper we reexamine apportionment by treating
families of states, rather than individual states, as the fundamental entities
that are to be apportioned seats.

Apportioning seats to a family as a whole uniquely determines the
apportionment for the states in the family. To see this, consider the
following method that apportions seats to families. We call the sum of the
quotas of all states in the $f$ family $Q_{f}$. In general, $Q_{f}$ is not an
integer. We round $Q_{f}$ to an integer following a prescribed apportionmnet
method’s rounding rule to determine the number of seats for the $f$ family,
and we call that integer $S_{f}$. Once an integer $S_{f}$ is determined by the
chosen method, the assignment of seats to the $N_{f}$ states in the $f$ family
is uniquely determined by

$\displaystyle M_{f}$ $\displaystyle=$ $\displaystyle(f+1)N_{f}-S_{f}$
$\displaystyle M_{f+1}$ $\displaystyle=$ $\displaystyle S_{f}-fN_{f}$ (1)

where the $M_{f}$ smallest states in the family are assigned $f$ seats each
and the $M_{f+1}$ largest states in the family are assigned $f+1$ seats each.
Which states are in which group is uniquely determined by insisting that
$s_{c}\geq s_{d}$ if $v_{c}>v_{d}$.

Using this method, the rounding mark in the $f$ family is not explicitly
needed or determined. It is any number from $f$ to $f+1$ that is smaller than
the smallest state quota that is rounded up and larger than the largest state
quota that is rounded down.

Our analysis relies on two properties of families to determine impartial and
unbiased apportionments that we were not able to use for other groups of
states. One is that the boundaries of families are fixed and do not change
when rounding marks change. The other is that each family can be treated
independently of other families because each state in a family is rounded up
or down depending on the position of its family’s rounding mark and not that
of any other family’s rounding mark.

##  V Impartial Apportionment.

Table 1: Differences between Webster for families and Webster for states. State | Quota | Seats
---|---|---
|  | Webster for Families | Webster for States
North Dakota | 1.024 | 1 | 1
South Dakota | 1.166 | 1 | 1
Delaware | 1.302 | 1 | 1
Montana | 1.426 | 1 | 1
Rhode Island | 1.443 | 2 | 1
Maine | 1.791 | 2 | 2
New Hampshire | 1.812 | 2 | 2
Hawaii | 1.918 | 2 | 2
1 Family | 11.883 | 12 | 11
Louisiana | 6.124 | 6 | 6
Alabama | 6.608 | 6 | 7
South Carolina | 6.733 | 7 | 7
6 Family | 19.465 | 19 | 20
Minnesota | 7.501 | 7 | 8
Colorado | 7.596 | 8 | 8
Wisconsin | 7.748 | 8 | 8
7 Family | 22.846 | 23 | 24
Tennessee | 9.087 | 9 | 9
Massachusetts | 9.240 | 9 | 9
Arizona | 9.405 | 10 | 9
9 Family | 27.733 | 28 | 27

Apportionment methods for families can be characterized by the rounding rule
used to convert real $Q_{f}$ to integer $S_{f}$. In general, the rounding rule
can depend on $Q_{f}$, $f$, $D$, and $\\{v\\}$. If we want the rounding rule
to apportion the same numbers of seats to a family containing a large number
of small-population states as it does to another family with the same total
population, but containing a small number of large-population states, then the
rounding rule can depend only on $Q$. Furthermore, we want the number of
apportioned seats for the family to be as close as possible to $Q$ according
to a measure of closeness such as Webster rounding which minimizes $|S-Q|$ or
Huntington-Hill rounding which minimizes $|\log S-\log Q|$. We call an
apportionment method that satisfies these properties, such as Webster’s method
applied to families and the Huntington-Hill method applied to families, an
impartial method. Other divisor methods as well as Hamilton’s method, can also
be made impartial by applying them to families rather than states.

Rounding family quotas will produce the same apportionment as rounding state
quotas when there is no more than one state in a family. However, when there
is more than one state in a family, rounding $family$ quotas using any
particular rounding rule may produce a different apportionment than rounding
$state$ quotas using that same rule. This occurs because summing and then
rounding will in general produce a different result than rounding and then
summing.

Table 1 shows Webster for families (summing and then rounding) and Webster for
states (rounding and then summing) apportionments using 2020 census data
Census  for $D=v_{T}/435$. For each apportionment, the total number of
apportioned seats is 435. The apportionment is shown for families 1, 6, 7, and
9 which are the families where the two apportionments differ.

To make the total number of seats in each family as close as possible to its
family quota, Webster for Families apportions an additional seat to Rhode
Island and Arizona and one fewer seat to Alabama and Minnesota than Webster
for States. For each of the families involved, one can see that the difference
between the Webster for families and Webster for states is due to a skewed
distribution of states within those families. In the 1-Family, five states
have quotas less than 1.5 while three states have quotas above 1.5. In the
6-Family, one state has a quota less than 6.5 and two states have quotas above
6.5. In the 7-Family, all three of its states have quotas above 7.5. In the
9-Family, all three of its states have quotas less than 9.5.

##  VI The Alabama Paradox and Impartial Apportionment.

Divisor methods applied to families have many analogous properties to divisor
methods applied to states. For example, the optimization functions discussed
by Huntington H1 ; H2  and Balinsky and Young B&Y  when applied to families
instead of states, are optimized at fixed $D$ by their respective traditional
divisor methods applied to families instead of states. However, all divisor
methods applied to states are immune to the Alabama paradox but not all
divisor methods applied to families are immune. This is because unlike states,
families can change their population size as $D$ is changed. This occurs when
a state that is in one family for one value of $D$ is in a different family
for another value of $D$. There is no counterpart to $D$ dependent population
change for divisor methods applied to states, since the population of a state
remains the same as $D$ is changed.

When $D$ is decreased, and therefore the State Populations Ruler is slowly
moved to the right, a state leaves the $f$ family and enters the $f+1$ family
just as its state quota is the integer $f+1$ at the boundary between the $f$
and $f+1$ families. The family quota, $Q_{f}$, for the $f$ family decreases by
$f+1$, an integer amount, and the family quota, $Q_{f+1}$, for the $f+1$
family increases by $f+1$. This can have effects on the other states in the
two families if the rounded family quota, $S$, for a family changes by a
different amount than the unrounded family quota, $Q$, when a state enters or
leaves the family. This can happen if the rounding up or down of $Q$ depends
on the integer part of $Q$, as is the case for Huntington-Hill rounding. It
cannot happen if the rounding up or down of $Q$ does not depend on the integer
part of $Q$, as is the case for Webster rounding.

Consider two states, one with quota 0.999 and the other with quota 1.43. Each
is the only state in its family, the 0 and 1 families respectively. Using
Huntington-Hill $\sqrt{S(S+1)}$ rounding on the family quota $Q$, the first
state’s 0.999 quota is rounded up to 1 and the second state’s 1.43 quota is
rounded up to 2. As $D$ is decreased so that every state’s quota gets
magnified by 1001/999, the first state’s quota is now 1.001 and the second’s
quota is now 1.433. Now the two states are both in the 1 family with quota
2.433 which is rounded to 2. Therefore, the first state gets one seat as
before, but now the second state also gets one seat. The second state lost a
seat even though $D$ decreased. An apportionment method that can apportion
fewer seats to a state when $D$ is decreased is susceptible to the Alabama
paradox. To see this, add a third state with quota 999. Initially the total
number of seats is 1002. After decreasing $D$ the third state’s quota is 1001
and the total number of seats is 1003. The total number of seats has increased
yet the second state has lost a seat. This is the Alabama paradox.

If the rounding down or up of $Q$ depends on the integer part of $Q$, as with
Huntington-Hill rounding, then the total number of seats can go down, up, or
stay the same as $D$ decreases. In the above example, if the two states are
the only states in the country, then the total number of seats went down from
3 to 2 as $D$ decreased. For an example in which the total number of seats
remains the same, add a third state with quota initially 62.4375 which rounds
down to 62 and the total number of seats is 65. After magnification by a
factor of 1001/999, the third state’s quota is 62.5625 which rounds up to 63
and the total number of seats is also 65. In other words, if Huntington-Hill
rounding is applied to family quotas to allocate 65 seats among three states
with relative populations of 0.999, 1.43, and 62.4375, the allocation can be
achieved either with the original quotas, yielding an apportionment of 1, 2,
and 62, or by quotas scaled by 1001/999, yielding an apportionment of 1, 1,
and 63. This “multiple solution paradox” has not been mentioned previously in
the literature because it cannot happen with apportionments by state. For an
example where the total number of seats goes up when the seats apportioned to
one of the states goes down, add a fourth state with quota 999.

For conventional Webster rounding at 0.5, the first and second state are
allotted one seat each, before and after $D$ is changed, and no Alabama
paradox occurs. This is a general result. The Alabama paradox, including the
multiple solution paradox, cannot occur for Webster’s method applied to
families. This is so because if $D$ is changed so that a state enters or
leaves a family, the family quota changes by an integer amount exactly equal
to the quota of the state that entered or left the family. For Webster’s
method, the direction of the rounding of the family quota does not depend on
the integer part of the family quota so the direction of the rounding is
unchanged when a state enters or leaves a family. The rounded family quota,
like the unrounded quota, is changed by an integer amount exactly equal to the
quota of the state that entered or left the family. Therefore, the seats
apportioned to states in the family are unchanged and no Alabama paradox or
multiple solution paradox occurs for Webster’s method applied to families.

If combining states into families with the same integer part of their state
quotas is worth considering, why not also combine families together that have
the same integer part of their family quota, and so on? One disadvantage of
combining families is that such methods can violate the Alabama paradox, and
therefore can have more than one apportionment for the same total number of
seats. This happens because rounding then summing is different than summing
then rounding. To see this, consider three state quotas: 0.99999, 1.7, and
2.6. They are each in a different family. And they are rounded to 1, 2, and 3.
Decrease D so the 0.99999 state quota gets moved to 1.0 and is now in the 1
family. The quotas of the 1 family and the 2 family are now approximately 2.7
and 2.6 respectively. If we stop here, the family quotas round to 3 and 3, the
states are assigned 1, 2, and 3 seats respectively, and no Alabama paradox has
occurred. If we combine the 1 and 2 families into a family of the families
with family quotas that round down to 2, then, this family-of-families has a
quota of $2.6+2.7=5.3$, which rounds down to 5. The two families which compose
the family-of-families are assigned 3 seats and 2 seats respectively, and the
states are assigned 1, 2, and 2 seats. The Alabama paradox has occurred
because a state has lost a seat when D has decreased.

##  VII The New States paradox and Impartial Apportionment.

Webster applied to families produces the least biased apportionment for a
particular set of states, where the least biased apportionment is defined as
the apportionment that minimizes $|S_{f}-Q_{f}|$ for every family. The
method’s rounding marks are determined impartially, dependent only on $Q$ and
therefore $\\{v\\}$. On the other hand, rounding marks of apportionment
methods that are immune to the New States paradox cannot depend on $\\{v\\}$.

For example, a state with quota $2.6$ that is the only state in the 2 Family
is apportioned 3 seats and a state with quota $5.3$ that is the only state in
the 5 family is apportioned 5 seats. Add a state with quota $2.7$ without
changing $D$. The new family quota of the 2 family is $5.3$ which rounds down
to 5. Webster for families impartially apportions 5 seats to the 2 family and
5 seats to the 5 family. The $2.6$ state is apportioned $2$ seats and the
$2.7$ state is apportioned $3$ seats. The $2.6$ state has lost a seat to
maintain impartiality between one large population state with a quota of $5.3$
and two small population states with total quota $5.3$. If one instead
apportioned in a manner immune to the New States paradox, a state with quota
$5.3$ is rounded down to 5 and states with quotas $2.6$ and $2.7$ are rounded
up to 3 seats each, which is biased in favor of the small population states.
An apportionment method cannot be both impartial and immune to the New States
paradox.

##  VIII Unbiased apportionment.

If the states are independently drawn from a known or theoretical distribution
of states, then it is possible to base the apportionment method on the
distribution of states rather than a particular sample. Using a distribution
from which states are independently drawn, our task is to find the
apportionment method that is unbiased in the sense that if states are
repeatedly and independently drawn from the same distribution, then seats are
apportioned such that $\left<S_{f}\right>$, the expected value of the number
of seats apportioned to the $f$ family, is equal to $\left<Q_{f}\right>$, the
expected value of the total $f$ family quota. Such an apportionment method,
for which $\left<S_{f}\right>=\left<Q_{f}\right>$ for every $f$, we call an
unbiased apportionment.

Balinski and Young B&Y  developed implications of state populations drawn from
a uniform distribution. We generalize Balinski and Young’s work by developing
implications of a variety of population distributions. We do not adopt
Balinski and Young’s homogeneity assumption. The homogeneity assumption is
that multiplying all state populations by a scale factor will not change an
apportionment for the same total number of seats. To demonstrate that this
apparently reasonable assumption is too restrictive, consider an apportionment
method designed to be unbiased for states independently drawn from a bell-
shaped distribution of population sizes. A set of states that are all drawn
from the left end of the distribution should have rounding marks to the right
of the midpoints of integer intervals to achieve unbiased apportionments,
while a set of states with the same relative populations that are
independently drawn from the right end of the distribution should have
rounding marks to the left of the midpoints of integer intervals to achieve
unbiased apportionments, violating homogeneity. It is because of this dropping
of the homogeneity assumption that unbiased apportionment methods can be
immune to both the New States paradox and Alabama paradox and yet, in general,
not be divisor methods.

Define $p(v)$ as the density function for state population, $v$. We treat
$p(v)$ as defined for all positive real numbers, even though population must
in fact be an integer. Given a $D$, the average quota for the $f$ family is

$\left<Q_{f}\right>=\int_{fD}^{(f+1)D}p(v)\frac{v}{D}dv,$ (2)

and the expected value of the number of seats for the $f$ family is

$\left<S_{f}\right>=f\int_{fD}^{rD}p(v)dv+(f+1)\int_{rD}^{(f+1)D}p(v)dv.$ (3)

An apportionment is unbiased within the $f$ family if

$\displaystyle\left<S_{f}\right>-\left<Q_{f}\right>$ $\displaystyle=$
$\displaystyle f\int_{fD}^{rD}p(v)dv+(f+1)\int_{rD}^{(f+1)D}p(v)dv$ (4)
$\displaystyle-$ $\displaystyle\int_{fD}^{(f+1)D}p(v)\frac{v}{D}dv=0.$

In terms of the cumulative distribution function, $I(v)$, defined up to a
constant by

$\frac{dI(v)}{dv}=p(v),$ (5)

we have

$\displaystyle\left<S_{F}\right>-\left<Q_{F}\right>$ $\displaystyle=$
$\displaystyle(f+1)I((f+1)D)-fI(fD)-I(rD)$ (6) $\displaystyle-$
$\displaystyle\int_{fD}^{(f+1)D}\frac{dI(v)}{dv}\frac{v}{D}dv$
$\displaystyle=$
$\displaystyle\int_{fD}^{(f+1)D}\frac{d}{dv}\left(I(v)\frac{v}{D}\right)dv-I(rD)$
$\displaystyle-$
$\displaystyle\int_{fD}^{(f+1)D}\frac{dI(v)}{dv}\frac{v}{D}dv$
$\displaystyle=$ $\displaystyle\frac{1}{D}\int_{fD}^{(f+1)D}I(v)dv-I(rD)=0.$

The condition for an unbiased apportionment is

$I(rD)=\frac{1}{D}\int_{fD}^{(f+1)D}I(v)dv$ (7)

so that

$r(f,D)=\frac{1}{D}I^{-1}\left(\frac{1}{D}\int_{fD}^{(f+1)D}I(v)dv\right).$
(8)

The dependence of the rounding marks on $D$ means that unbiased methods are
not, in general, homogeneous divisor methods.

Since the rounding marks for unbiased apportionment are determined by an
assumed state population distribution from which the states are drawn and not
the actual state populations, it will, in general, differ from impartial
apportionment, which is determined by the actual state populations and not by
an assumed state population distribution. An apportionment from an unbiased
method will not in general minimize $|S_{F}-Q_{F}|$ for each sample drawn from
the distribution, and therefore will not be an impartial apportionment.
Likewise, Webster for families applied to sets of states repeatedly drawn from
the same distribution will not in general satisfy
$\left<S_{F}\right>=\left<Q_{F}\right>$. However, impartial and unbiased
apportionments will tend to agree if the number of states is large enough that
any particular sample of states closely resembles the distribution it is drawn
from.

##  IX Unbiased apportionments and paradoxes.

The New States paradox will not occur if the rounding marks, which depend on
the distribution from which the states are drawn, do not change when states
are added to or removed from the sample. However, if properties of the
distribution, such as its mean and standard deviation, are determined from the
sample, and they change when states are added to or removed from the sample,
then changing the sample can change the rounding marks. Determining properties
of a distribution from previous censuses while incorporating a model to update
those parameters to the current census year is immune to the New States
paradox, as long as the parameters are not further updated in response to the
inclusion of the new state, but the unbiasedness of such an apportionment is
only as accurate as the model.

The Alabama paradox will not occur if rounding marks cannot catch up with and
overtake state populations when the State Populations Ruler is moved. As the
State Populations Ruler moves to the right, $D$ decreases. Provided that
$\frac{d\log r}{d\log D}\geq-1$, rounding marks do not move faster than the
State Populations Ruler, and the Alabama paradox will not occur.

Taking the derivative of Eq.(7) with respect to $D$ we obtain

$p(rD)D\frac{drD}{dD}=\frac{1}{D}\int_{fD}^{(f+1)D}p(v)vdv.$ (9)

For this to be true

$\frac{drD}{dD}\geq 0,$ (10)

which implies

$\frac{d\log r}{d\log D}\geq-1,$ (11)

and the Alabama paradox can never occur.

We believe that unbiased apportionment methods are the first example in the
literature of apportionment methods that avoid the Alabama paradox and the New
States paradox, yet are not divisor methods because they are not homogeneous.

Power law distributions and lognormal distributions are of particular interest
as candidates for distributions from which states are drawn as there are
numerous generative models for which state populations, as well as many other
quantities, can be effectively treated as if they were drawn from such
distributions Mitz .

In the next section we show that many of the familiar divisor methods are
unbiased apportionment methods for power law distributions.

##  X Power law distributions.

Table 2: Rounding Marks for Power Law Distributions $f$ | $\beta$
---|---
| $-\infty$ | -4 | -3 | -2 | -1 | 0 | +1 | +2 | +3 | +4 | $+\infty$
0 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.37 | 0.50 | 0.58 | 0.63 | 0.67 | 1.00
1 | 1.00 | 1.36 | 1.39 | 1.41 | 1.44 | 1.47 | 1.50 | 1.53 | 1.55 | 1.58 | 2.00
2 | 2.00 | 2.42 | 2.43 | 2.45 | 2.47 | 2.48 | 2.50 | 2.52 | 2.53 | 2.55 | 3.00
3 | 3.00 | 3.44 | 3.45 | 3.46 | 3.48 | 3.49 | 3.50 | 3.51 | 3.52 | 3.54 | 4.00
4 | 4.00 | 4.45 | 4.46 | 4.47 | 4.48 | 4.49 | 4.50 | 4.51 | 4.52 | 4.53 | 5.00

Consider the power law population distribution, $p_{\beta}\propto
v^{\beta-1}$. The cumulative distribution function is $I(v)\propto v^{\beta}$.
Solving Eq. (8) for $r_{\beta}(f)$ we have

$r_{\beta}(f)=\frac{1}{D}\left(\frac{1}{D}\int_{fD}^{(f+1)D}v^{\beta}dv\right)^{1/\beta}=\left(\frac{(f+1)^{\beta+1}-f^{\beta+1}}{\beta+1}\right)^{1/\beta}.$
(12)

The rounding mark $r_{\beta}(f)$ does not depend on $D$, so power law
apportionment methods are homogeneous divisor methods. They are the only
unbiased apportionment methods with this property. This is so because power
law distribution functions are the only distributions that have the same shape
on any interval.

Examples of $r_{\beta}$ are shown in Table 2. Some values are determined by
taking the appropriate limits. Formulas for some of the values are shown
below.

$\displaystyle r_{-\infty}(f)=f,$ (13) $\displaystyle
r_{-2}(f)=\sqrt{f(f+1)},$ (14) $\displaystyle
r_{-1}(f)=\left(\log(f+1)-\log(f)\right)^{-1},$ (15) $\displaystyle
r_{0}(f)=\frac{(f+1)^{f+1}}{ef^{f}},$ (16) $\displaystyle
r_{1}(f)=f+\frac{1}{2},$ (17) $\displaystyle
r_{2}(f)=\sqrt{f(f+1)+\frac{1}{3}},$ (18) $\displaystyle r_{\infty}(f)=f+1.$
(19)

Several of these rounding rules have special names. $r_{-\infty}$ is the Adams
apportionment rounding rule. $r_{-2}$ is the Huntington-Hill rule. $r_{1}$ is
the Webster rule. $r_{\infty}$ is The Jefferson rule. There are some divisor
methods, such as Dean’s method, that are not derivable from any power law.

As Balinski and Young noted B&Y , if the states are drawn from a uniform
distribution, Webster’s method is unbiased. However, if the states are drawn
from a different distribution, then Webster’s method will be biased.

In the next section we consider unbiased apportionments for states drawn from
lognormal distributions.

##  XI Lognormal Distributions.

Figure 2 shows histograms of the distributions of the logarithms of state
populations for recent censuses Census  at 20-year intervals.

Figure 2: Histograms of the distributions of the logarithms of state
populations are shown for recent censuses at 20-year intervals.

The distributions are roughly bell-shaped. There is no single power law
distribution that can fit these distributions, since a positive exponent is
required to fit small populations and a negative exponent is required to fit
large populations. The $\beta=0$ distribution is the power law distribution
that fits census data best, in that it maximizes the likelihood of the
observed distribution for integer $\beta$, but the distribution fails at the
tails, which matter the most when considering size bias.

Table 3: Historical Statistics of Log($v$). Year | mean | standard deviation | skew | excess kurtorsis
---|---|---|---|---
2020 | 15.218 | 1.024 | -0.047 | -0.514
2010 | 15.156 | 1.019 | -0.054 | -0.537
2000 | 15.062 | 1.020 | -0.052 | -0.572
1990 | 14.939 | 1.018 | -0.015 | -0.630
1980 | 14.850 | 1.021 | -0.073 | -0.719
1970 | 14.714 | 1.063 | -0.091 | -0.719
1960 | 14.583 | 1.071 | -0.172 | -0.606

Table 3 shows moments for the logarithm of state populations for recent state
population distributions Census . The skew and excess kurtosis, both zero for
normal distributions, are sufficiently small that the lognormal distribution
is a reasonable approximation for these distributions. The mean of the
logarithm of state populations increases almost perfectly linearly with time,
with an $R^{2}$ value of $0.99$. The standard deviation of the logarithm of
state populations appears to be decreasing towards a fixed point approximately
equal to 1.

We determine the unbiased rounding marks for the lognormal distribution from
Eq. (8) by integrating and inverting its cumulative distribution function.
Expressing the cumulative distribution function in terms of $q=v/D$, we have
$I_{LN}(q,\log q_{g},\sigma)=I(qD,\log(q_{g}D),\sigma)$, where
$q_{g}=v_{g}/D$, $\log q_{g}$ is the mean of $\log q$, and $\sigma$ is the
standard deviation of $\log q$. The rounding mark for the $f$ family is
determined from

$I_{LN}(r,\log q_{g},\sigma)=\int_{f}^{f+1}I_{LN}(q,\log q_{g},\sigma)dq,$
(20)

which equals

$\displaystyle I_{LN}\left(r,\log q_{g},\sigma\right)=I_{LN}\left((f+1),\log
q_{g},\sigma\right)(f+1)$ $\displaystyle-I_{LN}\left(f,\log
q_{g},\sigma\right)f-I_{LN}\left((f+1),\sigma^{2}+\log
q_{g},\sigma\right)e^{\sigma^{2}/2}q_{g}$
$\displaystyle+I_{LN}\left(f,\sigma^{2}+\log
q_{g},\sigma\right)e^{\sigma^{2}/2}q_{g}.$ (21)

Provided that $v_{g}$ and $\sigma$ are determined a priori and not from the
actual state populations, the apportionment method associated with these
rounding marks is immune to the New States paradox and the Alabama paradox,
yet it is not a divisor method. The rounding mark between $f$ and $f+1$
depends on $f$, as it would for a divisor method, but it also depends on $D$
through $q_{g}$, which indicates that it is not homogeneous.

Table 4: Rounding Marks for Lognormal Distributions. $f$ | $q_{g}$
---|---
| 1 | 2 | 5 | 10 | 20
0 | 0.491 | 0.539 | 0.591 | 0.623 | 0.650
1 | 1.461 | 1.481 | 1.506 | 1.525 | 1.543
2 | 2.468 | 2.480 | 2.495 | 2.507 | 2.518
5 | 5.479 | 5.485 | 5.492 | 5.497 | 5.502
10 | 10.487 | 10.489 | 10.493 | 10.496 | 10.499
20 | 20.492 | 20.493 | 20.495 | 20.497 | 20.498

Table 4 shows rounding marks for lognormal distributions for various $q_{g}$
and $\sigma=1$. Looking at the $q_{g}=5$ column, one sees that the fractional
part of a family’s rounding mark starts above 0.5 and initially decreases with
increasing family number, passing through 0.5 near the mode of the
distribution, consistent with the idea that as one moves from family to
family, from the left end of the distribution to the right end of the
distribution, the fractional part of the rounding mark will be on the side of
0.5 where the distribution of states in a family is larger. As the family
number continues to increase, the fractional part of the rounding mark
continues to fall, eventually increasing again to approach 0.5, just as it
does for power law distributions. Looking at the $f=5$ row, when $q_{g}=1$,
the $5$ family is in the right tail of the distribution so its rounding mark
is at the left side of the family. As $q_{g}$ increases, the rounding marks
move to the right as the 5 family moves into the left tail of the
distribution.

The lognormal apportionment using the mean and standard deviation of the 2020
census populations Census  for which $q_{g}=5.34$, does not agree with
Webster’s method applied to families. This is because the actual distribution
of states has bumps at the tails that deviate from the lognormal distribution.
The lognormal apportionment does agree with Webster applied to states for the
2020 census.

##  XII Conclusion.

We have presented impartial and unbiased methods for Congressional
apportionment. Webster applied to families is an impartial apportionment
method that apportions the same number of seats to families of states
containing the same number of people regardless of whether a family is
composed of a large number of small-population states or a small number of
large-population states. Webster applied to families is immune to the Alabama
paradox but not the New States paradox as no method can be both impartial and
immune to the New States paradox. Unbiased methods apportion seats so that if
states are drawn repeatedly from the same distribution, the expected value of
the total number of seats apportioned to each family equals the expected value
of the total divisor-method quota for that family. Unbiased methods are immune
to the Alabama paradox and the New States paradox yet not all of them are
divisor methods. Unbiased methods for power law distributions are divisor
methods. Unbiased methods for distributions that are not homogeneous, such as
lognormal distributions, are not divisor methods.

###### Acknowledgements.

The authors thank the editor, editorial board, and reviewers for suggestions
that improved the paper, and Carlos Reyes Zgarrick for his illustration of the
apportionment slide rule.

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