The Mathematics of ElectionsPart I ApportionmentMark Roger.docx

The Mathematics of Elections
Part I: Apportionment
Mark Rogers
(a.k.a. The Mad Hatter)
The Mechanics of Elections
Any system of electing representatives is essentially a two-stage
process:
Apportionment: how we determine how many representatives
there should be and how those representatives are to be
distributed among various subgroups of the population as a
whole
Voting: how we choose which candidate(s) should be chosen as
those representatives
The United States Congress
Defined in Article I of the U.S. Constitution
Consists of two chambers
The House, the apportionment of which is proportional to a
state’s population
The Senate, which is not
The apportionment also affects the presidential elections.
The Electoral College weight of each state is equal to its
combined House and Senate delegation.
The Senate is comprised of two members from each state,
regardless of population.
The 108th Congress

The 110th Congress
The House of Representatives
Originally defined as 65 members for the original 13 states;
currently 435 members, plus 5 non-voting delegates for
territories
The only Constitutional requirements for apportionment are that
each state gets at least one Representative, that the general
distribution be based on population, and that each person in the
House represent at least 30,000 residents of their state.
The original proposed First Amendment would have imposed a
stepwise function for future expansions of the House’s size, but
it was never ratified.
Instead, acts of Congress have governed each increase.
“Article the First” (proposed 1789)
Proposed as the first of 12 amendments to the new Constitution
If the House began to exceed 100 seats, the distribution would
shift to one per 40,000 residents.
If the House began to exceed 200 seats, the distribution would
shift to one per 50,000 residents.
Like the Congressional-raise-limiting “Article the Second,” it
was never ratified by a sufficient number of states at the time.
The other ten amendments became the Bill of Rights.
How many Representatives is too many?
“Nothing can be more fallacious than to found our political
calculations on arithmetical principles. Sixty or seventy men
may be more properly trusted with a given degree of power than

six or seven. But it does not follow that six or seven hundred
would be proportionably a better depositary. And if we carry
on the supposition to six or seven thousand, the whole reasoning
ought to be reversed. The truth is, that in all cases a certain
number at least seems to be necessary to secure the benefits of
free consultation and discussion, and to guard against too easy a
combination for improper purposes; as, on the other hand, the
number ought at most to be kept within a certain limit, in order
to avoid the confusion and intemperance of a multitude.”
James Madison
Average Constituency
The typical number of voters an official represents
“The Constitution…must be understood, not as enjoining an
absolute relative equality, because that would be demanding an
impossibility….That which cannot be done perfectly must be
done in a manner as near perfection as can be.”
Daniel Webster, 1832
How many Representatives is too many?
Around the world, the number of legislators (and thus, the
average constituency for each) varies widely.
U.S.: 435 Representatives, for 305,532,000 people (for an
average of 702,000 constituents each)
China: 3,000, for 1,326,940,000 people (442,000 each)
India: 552, for 1,139,910,000 people (2,000,000 each)
San Marino: 60, out of 30,800 people (513 constituents per

legislator)
Nauru: 18, out of 10,000 people (556 constituents each)
A House (Re)Divided
Traditionally, new states were admitted to the Union with their
appropriate number of Representatives (typically, a small
number at the outset) added to the old total.
With each Census, the House size would be readjusted (usually
upward), and the various states’ delegations redistributed
accordingly.
1790: 65 Representatives for 3.9 million people in 13 states
1793: 105 Representatives for 4.3 million in 15 states
1813: 182 Representatives for 8.0 million in 18 states
1873: 292 Representatives for 42 million in 37 states
1893: 356 Representatives for 67 million in 44 states
Minimizing “unfairness”
Apportionment Criterion: When assigning a representative
among several parties, make the assignment so as to create the
smallest possible relative unfairness.
Minimizing “unfairness”

State legislatures could once redraw Congressional districts (as
well as their own) in any manner desired, whether “fair” or not,
most often to favor rural areas over more populous urban areas.
House Speaker Sam Rayburn (D-TX) (1882-1961) was able to
have a rural district with just 227,735 residents, while a
Houston Congressman’s had 806,701 residents.
Had the district lines been “fair,” the Houston area would have
been entitled to three to four times as many Representatives as
Rayburn’s rural area.
State-house districts often had similar disparities as great as
1000 to 1.
Vermont: 35 residents in one district, 36,000 in another
“One man, one vote”
(Well…“One person, one vote”)
In Reynolds v. Sims (1964), the U.S. Supreme Court ruled that
the Constitution’s Equal Protection Clause established a “one
man, one vote” principle, requiring each district within a state
to have the same size constituency.
Wesberry v. Sanders (1964) extended this principle to
Congressional districts as well.
Districts would thus need to be redrawn as the population
relocated over time.
“One Man, One Vote”
As a result, Congressional districts will vary quite a bit in size,
but must be reasonably equal in population.
Sparse rural areas vs. dense, multi-Representative urban areas
“Gerrymandering”
Term for redistricting designed to favor or hinder one particular

group
“packing”: concentrating the members of a group into one
district to increase their voting influence to a majority, or to
limit their voting influence to it alone
“cracking”: dividing the members of a group among several
districts, in none of which can they muster a majority, to dilute
their voting influence
Elbridge Gerry (1744-1814): governor of Massachusetts, whose
Congressional districts were redrawn in a convoluted manner to
benefit his party
The Boston Gazette lampooned the shape of one district with an
editorial cartoon likening it to a mythical creature, the
“gerrymander.”
Numerous districts of Congress have been redrawn in elaborate,
spindly shapes, such as the Texas 22nd and Illinois 4th shown
below.
Congressional districts must be contiguous in shape, but can do
so using tendrils, even as thin as a highway, to connect several
regions.
Rep. Tom DeLay (R-TX) pushed through a special re-
redistricting of the Texas Congressional districts in 2003,
following his party’s takeover of the state legislature after 140
years.

Just 2 years after the previous redistricting
The new map merged two incumbent Democrats into one
district, forcing one out of Congress.
It also divided up urban areas among the surrounding suburbs,
limiting their influence.
Rep. Frank Mascara (D-PA) was forced to run (unsuccessfully)
against colleague John Murtha after statehouse Republicans
redrew boundary lines to move him from his old district into
Murtha’s.
A tendril of Murtha’s new district extended down a street to
envelop Mascara’s house, though not his driveway.
The process can also act to increase influence.
Western states were carved out of sparsely populated territories
to maximize their presidential impact, since each state would
get at least 3 Electoral College votes (due to having one
Congressman plus two Senators) regardless of population.
One effect of the Voting Rights Act of 1965 was to create a
series of “majority-minority” districts, to redress cases of past
discrimination.
In a series of cases in the 1990s, the U.S. Supreme Court
banned gerrymandering based solely on a racial basis.
However, in 2006, the Court let the Texas redistricting stand,
ruling that gerrymandering done merely to benefit one political
party was constitutional.
The decision also upheld repeated redrawing of district lines,
not just those done after each Census.
Recent redrawings of district lines have been done by bipartisan
panels to insure that both parties enjoy “safe” districts that they
are unlikely to lose.

2002: a record-low four incumbents lost their re-election bids
The effects of “gerrymandering”
In this example, the “state” has 4 legislative districts and 64
residents, 36 “green” and 28 “purple.”
By having 44% of the population, the purple residents would
deserve 1 or 2 representatives.
In the first map, the purple residents are concentrated into one
central district, insuring they will dominate it but have little
influence in others.
The effects of “gerrymandering”
In the second map, the central area is expanded to incorporate
the other purple voters, forming an area large enough to justify
two purple-majority districts. Both they and the two “green”
districts are virtually homogenous (and thus “safe”).
In the third map, the purple residents are split up among the 4
districts, in each of which they are outnumbered 9 to 7. (The
result: no purple-majority districts.)
In the fourth map, the (minority) purple residents are split up so
as to form a 9-7 majority in three districts.
The Hamilton Method of Apportionment
A longtime method of apportionment for the House, introduced
by Alexander Hamilton (1755-1804) and adopted in 1852
A modification of the basic method of allocating delegates by
assigning each group or state an appropriate percentage of the
total number of representatives
Find the percentage of the total population contained in each
state or group.
Multiply each percentage by the number of representatives,

rounding down (to avoid potentially allocating more
representatives than are available).
Award any remaining representatives based on which group’s
“fair number” of them was rounded down the most.
Now You See Them, Now You Don’t
A study of potential expansions of the House following the 1880
Census revealed a curious paradox.
If the House were to have 299 Representatives, Alabama would
be entitled to 8 of them.
However, if the House were expanded to 300 Representatives,
Alabama would be entitled to only 7!
In other words, as the House gains an extra delegate, Alabama
would lose one, even though its percentage (and that of every
other state) had not changed.
This became known as the Alabama paradox.
Appendix: See Excel spreadsheet “Census Apportionments.”
Curiouser and Curiouser
The curious paradox was almost seen ten years earlier, in the
wake of the 1870 Census.
If the House were to have 270 Representatives, Rhode Island
would be entitled to 2 of them.
Rhode Island would be reduced to a single one!
Tiny but densely populated, Rhode Island had never had a
single Representative since the dawn of the Republic.
Appendix: See 1870 tab on “Census Apportionments.”
Who’s Got It In For The South?
The “Alabama paradox” was also seen again just ten years later,
following the 1890 Census.

If the House were to have 359 Representatives, Arkansas would
be entitled to 7 of them.
Arkansas would be entitled to just 6!
The paradox arises from attempting to reallocate previously
assigned representatives to states whose growth rates are not in
sync, rather than simply allocating any newly added ones.
Appendix: See 1890 tab on “Census Apportionments.”
Watch Closely
Some examples were extreme in their alignment-of-the-planets
timing, such as the case of Colorado following the 1900 Census.
A careful study was undertaken of every potential House size
from 350 to 400 Representatives.
In almost every case, Colorado was entitled to 3
Representatives. However…
If the House were placed at exactly 357 members, Colorado
would get only 2.
Worse than a 356-member House, and worse than a 358-member
House! (357 was the only case like this.)
Upon hearing this, one Illinois Congressman tried to have 357
specifically chosen as the number of House seats. (Jerk.)
Appendix: See 1900 Census tab on “Census Apportionments.”
Up and Down
Attempts to replace Hamilton’s method with an alternative
similarly caused Maine’s delegation to fluctuate in size.
“Now you see it and now you don’t. In Maine comes and out
Maine goes. The House increases in size and still she is out. It
increases a little more in size, and then, forsooth, in she
comes….God help the state of Maine when mathematics reach
for her and undertake to strike her down in this manner.”
Rep. Charles Edgar Littlefield (R-Maine)
Littlefield retired almost immediately afterward.

A Simpler Example
Suppose there are 47 faculty members in the sciences, 37 in the
humanities, and 16 in the professional and trade schools.
A 9-person faculty committee is to be formed.
Using Hamilton’s method, we find the “fair” number of seats
each division deserves, round down any decimals, and choose
how to allocate any remaining seats afterwards.
The 9-Person Faculty CommitteeDivisionSeats
“Deserved”Provisional Seats AwardedFinal Allocation of
SeatsSciences0.47 x 9 = 4.2344Humanities0.37 x 9 =
3.3333Professional0.16 x 9 = 1.4412Total100% x 9 = 9.0089
Now suppose that the committee is to be expanded to 10 seats.
We will use Hamilton’s method to reapportion the seats.
The 10-Person Faculty CommitteeDivisionNumber of Seats
“Deserved”Provisional Seats AwardedFinal Allocation of
SeatsSciences0.47 x 10 = 4.7045Humanities0.37 x 10 =
3.7034Professional0.16 x 10 = 1.6011Total100% x 10 =
10.00810
The professional faculty’s “fair” number of representatives has
indeed grown, but not as fast as the other two divisions, both of
which have now overtaken them in the “who’s been rounded
down the most?” category.
The Other Founding Fathers
Thomas Jefferson, John Adams, and Daniel Webster each
proposed alternatives to Hamilton’s method.
In each of their methods, the total population of the state (which
helps us find us the percentage of the total representatives the
state is entitled to) is replaced by either a smaller or larger

number.
This is done not to affect the state’s fair share, but to make the
numbers work out more easily.
Alternatives to Hamilton’s method
Jefferson’s method:
“Decrease” the total population figure (thus increasing the
expected number of representatives)
Round the “number of representatives deserved” down
Repeat until the correct number of delegates is awarded
Adams’s method:
“Increase” the total population figure (thus decreasing the
expected number of representatives)
Round the “number of representatives deserved” up
Webster’s method:
Find an alternative total population figure by trial and error
Round the “number of representatives deserved” up or down,
according to the normal rules of rounding
Representative Quotas
The “number of representatives deserved” in Hamilton’s method
is referred to as the standard quota.
Rounding down, we obtain the lower quota.
Rounding up, we obtain the upper quota.
If an apportionment allocates each state a number of
representatives between its lower and upper quotas, then it is
said to satisfy the quota rule.
In other words, a state that “deserves” 5.37 representatives
should receive either 5 or 6, not 3 or 7.

Hamilton’s method is the only one of the four “Founding
Fathers” methods that does not violate this principle, since we
added single extra representatives to some states after rounding
down their standard quotas.
The others’ altered total population figures give them an
“undeservedly” higher or lower number of “deserved”
representatives.
More Problems for Hamilton
Were the rarity of the Alabama paradox the only problem
Hamilton’s method risked, it might still be used today.
However, there are a number of other paradoxes that can occur
with it.
Population paradox: State A’s population is growing faster than
state B, yet A loses a representative to B.
A’s percentage population growth was higher than B’s, but
Hamilton’s method only takes into account the raw-number
differences (which would have been higher if B was a larger
state to begin with).
New states paradox: When a new state (and its share of new
seats) are added to the legislature, another state’s (previously
allocated) seats can end up reassigned.
Similarly, this new dilution of representation affects each state
equally on a raw-number basis, which in turn hits smaller states
harder on a percentage basis (causing their “partial
representative” numbers to fall further).
In Maine Comes, Out She Goes
In 1907, Oklahoma became the 46th state. Mindful of its rapid
oil-boom growth, the long time since the 1900 Census, and the
previous cases of the Alabama paradox, Congress chose to
simply add the 5 new Representatives it “deserved” to the
previous 386, and reallocate based on old Census data.
However, a new paradox emerged.

In a 386-member House, New York was entitled to 38 seats,
but….
In a 391-member House, New York lost one of its seats to
Maine, delaying an expected loss of Maine’s 4th seat for
another twenty years.
In the absence of a new Census, no other population figures had
been adjusted, yet New York still lost out to Maine.
It was the new states paradox; adding Oklahoma’s seats “on
top” of the others had changed the delegates for other states.
Appendix: See 1907 tab on Excel spreadsheet “Census
Apportionments.”
The Huntington-Hill
Apportionment Principle
Developed for FDR by mathematicians Edward Huntington and
Joseph Hill
Huntington: inaugural President of the Math. Assoc. of
America
Hill: Assistant Director of the U.S. Census
Their method has been used for House reapportionment since
1941.
Avoids the Alabama paradox by assigning each representative
one at a time, back from the very beginning
In essence, it calculates the unfairness of each state’s current
number of representatives, and compares it to the unfairness of
that state’s number of representatives if an extra one were
added.
The Huntington-Hill
Apportionment Principle
To find the Huntington-Hill number, calculate for each state or
group:

The formula comes from a rearranged comparison of the relative
unfairness of two competing proposed allocations.
Whichever state has the highest Huntington-Hill number should
be given the next new representative to be added in order to
minimize the relative unfairness.
Building From The Ground Up
Under the Huntington-Hill method, each group or state is given
one representative at the start.
Then, all other representatives are allotted one at a time based
on which group or state has the highest Huntington-Hill number
at that moment.
California, with a massive population (squared) figure, receives
both the 1st and 3rd bonus seats awarded, as well as the 6th,
12th, and 15th.
The “usual suspects” of large states receive the other early ones.
California’s 53rd district and North Carolina’s 13th are the last
two seats to be awarded in a 435-member House.
By a tiny margin, Utah narrowly missed out on a fourth seat.
Utah sued the Census Bureau unsuccessfully, arguing that
irregularities in Census tabulations (and undercounting of their
own Mormon missionaries) should have entitled them to the
final seat.
Appendix: See “Huntington-Hill” Excel spreadsheet.
The Faculty Committees, When Using The Huntington-Hill
MethodWhen Having 1 SeatWhen Having 2 SeatsWhen Having
3 SeatsWhen Having 4 SeatsWhen Having 5 SeatsWhen Having
6 SeatsSciencesHumanitiesProfessional

We use this table of Huntington-Hill numbers to award the 9 (or

10, or any other number) of committee seats to the various
faculty divisions, in descending order of the H-H numbers
(wherever it appears in the table).
By not stopping to reconsider old seat apportionments, we will
never take away one group’s seat to give it to another.
Coming Soon
Population projections for the 2010 Census suggest that the
trend of migration from the industrial Midwest to the South and
Southwest will continue, resulting in continued shifts in House
seats.
Utah will finally get its extra seat.
Others gaining a seat: GA, NV, NC, OR, SC
Arizona and Florida will each gain 2 seats, Texas 4.
States losing a seat: CA, IL, IA, LA, MA, MI, MN, MO, NJ,
PA
New York and Ohio will each lose 2 seats.
The Wyoming Rule
No matter the system used to divide up the House seats, all
states are guaranteed at least one, regardless of population;
thus, Wyoming with its 522,830 residents gets one
Representative, as does Montana, with its 957,861 residents.
Montana’s population is far too small to justify a second
Representative.
Wyoming is frankly too small to justify a single one, but the
Constitution mandates it.
The Wyoming Rule is a proposal to avoid this “low-end”
unfairness of large-state constituencies far exceeding the small
single-Representative constituency of small-population states.
It would increase the size of the House until the average
constituency in each state matched that of the least populous
state.
If the Wyoming Rule were enacted, the House would need to

increase to at least 585 members.
Colorado: currently 7 Representatives, would increase to 9
California: currently 53 Representatives, would increase to 70
Montana: currently 1 Representative, would increase to 2
The Ugly Conclusion
Given the many paradoxes, the question arises:
Can any method of apportionment avoid all of them?
Is there a “perfect” method of apportioning representatives?
In 1980, Michael Balinski and H. Peyton Young found the
answer.
Balinski and Young’s Impossibility Theorem: There is no
apportionment method that avoids all paradoxes and at the same
time satisfies the quota rule.
Webster was right!
“The Constitution…must be understood, not as enjoining an
absolute relative equality, because that would be demanding an
impossibility….That which cannot be done perfectly must be
done in a manner as near perfection as can be.”
Daniel Webster, 1832
Does It Make A Difference?
The 1876 presidential election was bitterly contested, as
Rutherford B. Hayes (R) trailed Samuel Tilden (D) by 19
electoral votes, with 20 electoral votes from three southern
states in dispute.
Congress voted to award the disputed electoral votes to Hayes,
giving him a 185-184 victory.
Republicans had reportedly agreed with southern states to end
Reconstruction-era troop garrisons in exchange for their

support.
Years later, Balinski and Young showed that had a different
apportionment method been used, Tilden’s lead would have
held.
What have we learned?
Dividing up a group of representatives is not easy.
Robert Burns said it best: “The best-laid plans of mice and men
often go awry.”
In a world of paradoxes and unmet quotas, no method is perfect.
Even when the seats have been assigned fairly, they may not be
divided up within a group fairly.
Small changes can have a major impact, mathematically and
historically.
All of this tells us very little about the next phase of the
election process: voting.
Now that the council’s seats have been divided up, how do we
decide who gets to fill them?
Next Friday: The Mathematics of Elections, Part II: Voting.
References
Most liberal-arts college-mathematics course (ex.: MATH 110)
textbooks
Including ours, Thomas L. Pirnot’s Mathematics All Around,
3rd edition
Alex Bogolmony’s interactive paradox explorer, at www.cut-
the-knot.org/ctk/Democracy.shtml
Census information: www.census.gov
(in particular, the stats of www.census.gov/compendia/statab/)
Complete list of projections as to which states “deserve” the
first 440 Representatives using the Huntington-Hill method:
www.census.gov/population/censusdata/apportionment/00pvalue
s.txt
Interactive electoral maps, both historic and modern:

www.270towin.com
Analysis of 2000 Presidential election given different House
sizes, at www.thirty-thousand.org/pages/Neubauer-Zeitlin.htm
And yes, of course, Google and Wikipedia.
My Mesa State homepage, at www.mesastate.edu/~mcrogers,
will have this presentation plus the spreadsheets used.
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Sheet1StatePopulation% RepresentationHamilton
NumberInteger PartFractional PartAssign Additional
Members ManuallyTotal SeatsAverage
Constituency1154752.908%2.907820.9078135158.32356446.69
8%6.697660.6976175092.039875618.557%18.5566180.5566018
5486.448834616.601%16.6005160.60051175196.853690.069%0.
069300.0693000.068566316.096%16.0964160.09640165353.974
34278.160%8.160180.1601085428.488431115.842%15.8423150.
84231165269.495473010.284%10.2840100.28400105473.01025
4674.785%4.785340.7853155093.4Total532188100.0%100.0000
955100StatePopulation% RepresentationHamilton
NumberInteger PartFractional PartAssign Additional
Constituency1154752.908%2.907820.9078135158.32356446.69
8%6.697660.6976175092.039875618.557%18.5566180.5566018
5486.448834616.601%16.6005160.60051175196.85203693.827
%3.827430.8274145092.366566312.338%12.3383120.33830125
471.97434278.160%8.160180.1601085428.488431115.842%15.8
423150.84231165269.495473010.284%10.2840100.2840010547
3.010254674.785%4.785340.7853155093.4Total532188100.0%1
00.0000946100StatePopulation% RepresentationHamilton
NumberLower QuotaUpper QuotaAssign Additional
Constituency1154752.908%2.907823135158.32356446.698%6.6
97667175092.039875618.557%18.556618190185486.448834616
.601%16.600516171175196.85203693.827%3.827434145092.36
6566312.338%12.338312130125471.97434278.160%8.16018908
5428.488431115.842%15.842315161165269.495473010.284%10
.284010110105473.010254674.785%4.785345155093.4Total532
188100.0%100.0000946100
Sheet2StatePop.% Rep.Hamilton NumberLower QuotaUpper
Quota Geom. MeanSeatsAverage
Constituency1154752.908%2.9078232.449535158.32356446.69

8%6.6976676.480775092.039875618.557%18.5566181918.4932
195197.748834616.601%16.6005161716.4924175196.853690.06
9%0.0693010.00001369.068566316.096%16.0964161716.49241
65353.97434278.160%8.1601898.485385428.488431115.842%1
5.8423151615.4919165269.495473010.284%10.2840101110.488
1105473.010254674.785%4.7853454.472155093.4Total5321881
00.0%100.000095105102
Running head: THE APPORTIONMENT PROBLEM
1
THE APPORTIONMENT PROBLEM
The Apportionment Problem
Student’s Name:
Course:
Instructor:
Date:

An ideal democratic government requires that each and every
member of the population should participate equally in the
decision making process of the country. However, in a country
having a population of more than 300 million, incorporating
each citizen in the country’s decision making process is not
practically possible as it would not only be excessively time-
consuming to gather millions of votes on each and every issue,
it would also lead to impractically lengthy discussions on every
topic brought before the legislative body.
Therefore representatives are chosen from the population and
these representatives take part in the decision making process.
The process of choosing people’s representatives is a very
important job and this should be done in a rational and fair
manner so that people of all regions within the country are
appropriately represented in the decision making body. The

most widely used means of choosing people’ representatives is
by establishing a distribution based on population, where
individual states are assigned a certain number of
representatives based on the population of that particular state.
One such method is the “Hamilton Method” of apportionment.
In this method, a fixed number of seats are distributed among
the states based on the population of each state. First the total
population of all of the states is divided by the total number of
seats to be assigned. This quotient is known as the “standard
divisor”, and represents the average number of people to be
represented by each seat. Then the population of each state is
divided by the standard divisor and the whole number portion of
that quotient is the minimum number of seats assigned to the
corresponding state. If the total number of seats assigned is less
than the desired number of seats, then the remaining seats are
distributed one at a time to the states based on the fractional
portion from dividing the state’s population by the standard
divisor by beginning with the state that had the largest
fractional portion and continuing in descending order as long as
there are still seats to distribute.
For example, given the following populations of ten states:
StatePopulation
1 15475
2 35644
3 98756
4 88346
5 369
6 85663
7 43427
8 84311
9 54730
10 25467

The Hamilton Method would produce the following distribution
of seats:
State
Pop.
% Rep.
Hamilton Number
Integer Part
Fractional
Part
Additional
Members
Total Seats
1
15475
2.908%
2.9078
2
0.9078
1
3
5158.3
2
35644
6.698%
6.6976
6
0.6976
1
7
5092.0
3
98756
18.557%

18.5566
18
0.5566
0
18
5486.4
4
88346
16.601%
16.6005
16
0.6005
1
17
5196.8
5
369
0.069%
0.0693
0
0.0693
0
0
0.0
6
85663
16.096%
16.0964
16
0.0964
0
16
5353.9
7
43427
8.160%

8.1601
8
0.1601
0
8
5428.4
8
84311
15.842%
15.8423
15
0.8423
1
16
5269.4
9
54730
10.284%
10.2840
10
0.2840
0
10
5473.0
10
25467
4.785%
4.7853
4
0.7853
1
5
5093.4
Total
532188
100.0%

100.0000
95
5
100
From the above table we see that, under the given method, State
5 has no representation at all as its population is very small as
compared to that of the other states. Thus this apportionment
scheme will appear to be extremely unfair for the people
residing in this state. On the other hand State 3 has the highest
average constituency. The actual unfairness for this
apportionment scheme can be calculated as follows:
The relative unfairness for this apportionment scheme can be
calculated as follows:
The absolute unfairness and the relative unfairness values
reveal a serious flaw in this distribution and tells us that a
different method is needed for this particular situation.
This fallacy might be resolved by redrawing the state
boundaries in order to distribute the population more uniformly.
This will alter the percentages in column 3 of the table shown
above, which would further alter the distribution of seats. This
is because any change in the population of a state changes the
percentage contribution of that state to the total population. If
the population of one or more states changes or if the state
boundaries are redrawn, these changes get reflected in the

redistribution of seats in the representative body.
For example, if the boundary between State 5 and State 6 is
adjusted (assuming, for the purposes of this example, that they
are adjacent) such that the new populations are 20369 and
65663, then the distribution would be as follows:
State
Pop.
% Rep.
Hamilton Number
Integer Part
Fractional
Part
Additional
Members
Total Seats
1
15475
2.908%
2.9078
2
0.9078
1
3
5158.3
2
35644
6.698%
6.6976
6
0.6976
1
7

5092.0
3
98756
18.557%
18.5566
18
0.5566
0
18
5486.4
4
88346
16.601%
16.6005
16
0.6005
1
17
5196.8
5
20369
3.827%
3.8274
3
0.8274
1
4
5092.3
6
65663
12.338%
12.3383
12
0.3383
0
12

5471.9
7
43427
8.160%
8.1601
8
0.1601
0
8
5428.4
8
84311
15.842%
15.8423
15
0.8423
1
16
5269.4
9
54730
10.284%
10.2840
10
0.2840
0
10
5473.0
10
25467
4.785%
4.7853
4
0.7853
1
5

5093.4
Total
532188
100.0%
100.0000
94
6
100
This redistribution of 20,000 people from State 6 to State 5 has
resulted in State 5 being assigned 4 seats, rather than the
previous 0 seats. In addition, the average constituency for each
state shows more evenness now than it was with the previous
distribution. Unfortunately, while local redistribution is often
performed for various political or socioeconomic causes,
redrawing the boundaries of states has momentous consequences
and is not a viable option once the state boundaries have been
defined.
Another issue that can arise in this kind of population based
apportionment is what is known as the “Alabama Paradox”. This
occurs when the body of legislative seats is expanded and a
state gains or loses a seat, even though its population has not
changed. This occurs because change in the number of seats
changes the standard division, and consequently changes the
Hamilton Number for each state. If a relatively large number of
seats are added to the legislative body, substantial changes in
the distribution of seats for each state would be expected.
However, when a relatively small numbers of seats are added,
the number of seats attributed to the states can be adjusted by
changing the fractional portions of the Hamilton Numbers so
that any un-apportioned seats are distributed in such a manner
that some state might gain a seat, while another state might lose
a seat. This occurred following both the 1870 and 1880 U. S.

Census. The paradox takes its name from the 1880 occurrence,
whereby it was noted that, based on the population values from
the census, if the House of Representatives contained 299 seats,
then Alabama would be entitled to 8 seats. However, if the
House contained 300 seats, then Alabama would only be entitled
to 7 seats.
One approach that can be used to help prevent the occurrence of
the Alabama Paradox is the “Hamilton-Hill” apportionment
method. This method compares the Hamilton Number with the
geometric mean of the Lower Quota and Upper Quota for a
given state. If a state’s Hamilton Number is larger than the
geometric mean, then that state is initially assigned its Upper
Quota of seats, rather than the Lower Quota. For example, with
State 2 in the original table, the Hamilton Number was 6.698.
This value is between 6 and 7. The geometric mean of 6 and 7 is
√42, or 6.481. Since 6.698 is greater than 6.481, State 2 would
initially be assigned 7 seats instead of the 6 that it was
originally assigned.
The Hamilton-Hill method minimizes the effect of small
changes in the fractional part of the Hamilton Number by
rounding up values instead of taking only the integer portion of
the Hamilton Numbers. This method might however lead to an
excessive number of seats being assigned, which have to be
adjusted consequently.
Applying the Hamilton-Hill method to the initial apportionment
gives the following results:
State
Pop.
% Rep.
Hamilton Number
Lower Quota
Upper Quota

Geom. Mean
Seats
1
15475
2.908%
2.9078
2
3
2.4495
3
5158.3
2
35644
6.698%
6.6976
6
7
6.4807
7
5092.0
3
98756
18.557%
18.5566
18
19
18.4932
19
5197.7
4
88346
16.601%
16.6005
16
17

16.4924
17
5196.8
5
369
0.069%
0.0693
0
1
0.0000
1
369.0
6
85663
16.096%
16.0964
16
17
16.4924
16
5353.9
7
43427
8.160%
8.1601
8
9
8.4853
8
5428.4
8
84311
15.842%
15.8423
15
16

15.4919
16
5269.4
9
54730
10.284%
10.2840
10
11
10.4881
10
5473.0
10
25467
4.785%
4.7853
4
5
4.4721
5
5093.4
Total
532188
100.0%
100.0000
95
105
102
Here the total number of seats allotted exceeds the total number
of available seats. As a result, it would be necessary to find out
a “modified divisor” in order to adjust the distribution such that
exactly 100 seats are distributed to the states. Once again, the
small population of State 5 results in that state being scarcely

represented.
Neither the Hamilton method nor the Hamilton-Hill methods are
perfect solutions to the apportionment problem. As we can see
from our example, an uneven population distribution can result
in some states being under-represented, while other states are
over-represented. In the extreme case, some states might not be
represented at all. Thus these methods do not always lead to an
accurate representation of the population.
One method of addressing under-representation is to assign a
minimum number of seats to each state, and then use the
apportionment techniques to distribute the remaining seats. This
is similar to what is done with the U.S. House of
Representatives where each state has at least three
representatives.
It may further be noted that a population-based distribution of
representation is essentially unfair, even when the distribution
is close to uniform and the average constituencies are
approximately equal. This might lead to a situation where the
more populous states have more power in the decision making
process of the country. It is for this reason that the United
States has two houses of Congress, with representation in the
House of Representatives being based on population, while the
Senate assigns two seats to every state regardless of population.
This two tiered approach is the most appropriate way of
addressing the limitations of strictly population-based
apportionment methods.
References:

· Methods of Apportionment. Methods of Apportionment -
History - U.S. Census Bureau. Retrieved March 18, 2016, from
http://www.census.gov/history/www/reference/apportionment/m
ethods_of_apportionment.html
· Mark Beumer, Apportionment in Theory and Practice
Retrieved March 18, 2016, from
http://www.illc.uva.nl/Research/Publications/Reports/MoL-
2010-07.text.pdf
· Karsten Schuster, Friedrich Pukelsheim, Mathias Drton &
Norman R. Draper, Seat biases of apportionment methods for
proportional representation, Electoral Studies 22 (2003) 651–
676
https://www.math.uni-
augsburg.de/emeriti/pukelsheim/2003b.pdf
Module 5 LASA Template
How to use the template:
Please do not delete anything from the template except for the
words “you fill out”. Simply place your answers where
requested below, and be sure to show and explain all of your
work. Please also attach your Excel spreadsheet (required). This
document contains a lot of information. Please read every word,
all the way to the end
***Please place your intro paragraph here (100 words min.).
Please use APA format for your intro (double spaced, size 12
font). ***
Your intro paragraph must include a thesis statement. Please
explain your paper and what you are doing here in this task.

1. Using the Hamilton method of apportionment, determine the
number of seats each state should receive.
Please use the state population values that are listed in class.
Click on the week 5 tab in the left menu, then on the LASA
link, and you will see a table of given population values to
insert in your Excel spreadsheet. Please do not use your own
population values. Thanks!
Please fill out the below table. Please also attach your Excel
spreadsheet.
State
Number of Seats (NOT population number).
1
You fill out
2
You fill out
3
You fill out
4
You fill out
5
You fill out
6
You fill out
7
You fill out
8
You fill out
9
You fill out
10
You fill out

2. Using the numbers you just calculated from applying the
Hamilton method, determine the average constituency for each
state. Explain your decision making process for allocating the
remaining seats.
Please fill out the below table. Please also attach your Excel
spreadsheet.
State
1
You fill out
2
You fill out
3
You fill out
4
You fill out
5
You fill out
6
You fill out
7
You fill out
8
You fill out
9
You fill out
10
You fill out
Please answer the following (required):
· What does Average Constituency mean?

· What does it measure?
· How is it calculated?
· What is the process for allocating surplus/remaining seats?
HINT: see pages 533-534 of the textbook.
3. Calculate the absolute and relative unfairness of this
apportionment.
Please answer the following:
· What two states will you be using for your calculations?
Please state them here.
· What is the absolute unfairness? State your final answer here.
Please show all work.
· What is the relative unfairness? State your final answer here.
Please show all work
HINT: you can find the formulas in the textbook.
4. Explain how changes in state boundaries or populations could
affect the balance of representation in this congress. Provide an
example using the results above.
Please answer the following (100 words minimum):
· Explain how changes in state boundaries or populations could

affect the balance of representation in this congress. What can
cause population increase? What can cause population decrease?
What can cause state boundaries to change? How do all of these
affect the balance of representation?
· Provide a numerical example like I showed in the example
document. Include and show all of your work. Include your
original population numbers and seat numbers, and include your
modified population numbers and seat numbers. Please explain
all of your work very clearly (with written text and math work).
5. How and why could an Alabama Paradox occur?
Please answer the question with at least 3 to 5 complete
sentences. Please discuss both HOW and WHY it occurs.
HINTS:
http://www.ctl.ua.edu/math103/apportionment/paradoxs.htm
http://en.wikipedia.org/wiki/Apportionment_paradox#Alabama_
paradox
6. Explain how applying the Huntington-Hill apportionment
method helps to avoid an Alabama Paradox.
Please answer the question with at least 3 to 5 complete
sentences.
HINT: see Section 11.2 in the textbook.
7. Based upon your experience in solving this problem, do you
feel apportionment is the best way to achieve fair
representation? Be sure to support your answer.

Answers will vary. Please provide an answer of no less than 150
words and defend your reasoning.
Hint: Note that “Apportionment” here refers to a method of
dividing seats between states to achieve fair representation.
And, Hamiltonian and Huntington methods are specific types of
Apportionment. Here is a diagram to help illustrate:
Ways of achieving
representation:
Apportionment Other
types? (you will need to research these)
Hamiltonian Huntington-Hill
Method Method
of Apportionment of Apportionment
Please evaluate the Hamiltonian method. Do you feel that it is
fair? Why or why not? Cite examples from your work above and
other examples that you find.
Please evaluate the Huntington-Hill method. Do you feel that it
is fair? Why or why not? Cite examples from your work above
and other examples that you find.
Please evaluate the Apportionment method as a whole. Do you
feel that it is fair? Why or why not? Cite examples from your

work above and other examples that you find.
8. Suggest another strategy that could be applied to achieve fair
representation either using apportionment methods or a method
of your choosing.
Answers will vary. Please provide an answer of no less than 100
words and defend your reasoning with specific examples.
Here are some examples of different methods that you might
choose and discuss:
• Jefferson’s method
• Adams’ method
• Webster’s method
---------------------------------------------------
***Please place your conclusion paragraph here (100 words
min.) Please use APA format for your intro (double spaced, size
12 font). ***
Please include your references in APA format here at the end.
Here is the grading rubric. Your goal is to achieve all in the
orange column
Unsatisfactory
Emerging
Proficient
Exemplary

Assignment Components
Application of Hamilton and Huntington-Hill formulas to
determine number of seats for each state.
Analysis of those results to determine state average
constituency.
(CO1, CO2, CO3)
Application of both the Hamilton and Huntington-Hill formulas
is inaccurate, resulting in inappropriate number of seats for
each state.
Analysis of the results from both the Hamilton and Huntington-
Hill formulas are inaccurate and results in the inappropriate
state average constituency for the correct formula; or analysis is
too underdeveloped or too unclear and interferes with
comprehension.
Application of either the Hamilton or Huntington-Hill formulas
is accurate, resulting in appropriate number of seats for each
state. However, the other formula is inaccurate.
Analysis of the results from either the Hamilton or Huntington-
Hill formulas is accurate and results in the appropriate state
average constituency for the correct formula; or analysis is
unclear and confusing.
state.

average constituency.
state, and clearly described to demonstrate mastery of all
calculations.
average constituency. Analysis is clear and detailed.
Analysis of results from above to determine if Alabama paradox
occurred.
(CO1, CO2, CO3)
Determination of whether or not an Alabama paradox occurred
in both the Hamilton and the Huntington-Hill results is
inaccurate or too underdeveloped or unclear to assess.
in both the Hamilton and the Huntington-Hill results is
somewhat accurate, but is somewhat unclear or underdeveloped.
in both the Hamilton and the Huntington-Hill results is clear
and accurate.
in both the Hamilton and the Huntington-Hill results is clear,
accurate, and uses several specific examples to support its
analysis.
Analysis of state boundaries and population effects on
representational balance.
(CO4, CO5)
Analysis of effects on representational balance is unclear,
underdeveloped, or inaccurate throughout. It lacks examples
from state boundaries or population to support its claims.
Analysis of effects on representational balance is somewhat
unclear or inaccurate in spots. It includes examples from state

boundaries or population to support its claims.
Analysis of effects on representational balance is clear and
accurate. It includes specific, accurate examples from both state
boundaries and population to support its claims.
Analysis of effects on representational balance is clear and
accurate. It includes specific, accurate examples from both state
boundaries and population to support its claims. Examples
provided are demonstrated mathematically.
Evaluation of the apportionment methods.
(CO4, CO5)
Evaluation of apportionment does not reference data found to
support the claim. It evaluates the Hamilton or the Huntington-
Hill results independently, but evaluation is underdeveloped or
inaccurate. Evaluation of the apportionment method may be
present but is inaccurate. Evaluation is clearly lacking
development and clarity.
Evaluation of apportionment vaguely references data found to
support the claim. It evaluates the Hamilton or the Huntington-
Hill results independently or evaluates the apportionment
method as a whole without using specific examples for each.
The evaluation is somewhat inaccurate or unclear.
Evaluation of apportionment references specific examples from
the data found to support the claim. It evaluates the Hamilton
and the Huntington-Hill results independently or evaluates the
apportionment method as a whole without using specific
examples for each.
Evaluation of apportionment references specific examples from
the data found to support the claim. It evaluates both the
Hamilton and the Huntington-Hill results independently, and it
evaluates the apportionment method as a whole using examples

for each.
Proposed solution to achieve fair representation.
(CO4, CO5)
Proposal is underdeveloped or inaccurate. No examples are
provided to support solution’s effectiveness.
Proposal is somewhat clear or may have elements that are
inaccurate. It provides a few to any examples, but examples do
not clearly support solution’s effectiveness.
Proposal is clear and accurate. It provides specific examples to
support the solution’s effectiveness.
Proposal is clear, accurate, and insightful. It provides a variety
of specific examples that demonstrate the solution’s
effectiveness in a variety of settings.
Writing Components
Organization
Introduction
Thesis
Transitions
Conclusion
Introduction is limited or missing entirely.
The paper lacks a thesis statement.
Transitions are infrequent, illogical, or missing entirely.
Conclusion is limited or missing entirely.
Introduction is present but incomplete or underdeveloped.
The paper is loosely organized around a thesis that may have to
be inferred.
Transitions are sporadic.

Conclusion is present, but incomplete or underdeveloped.
Introduction has a clear opening, provides background
information, and states the topic.
The paper is organized around an arguable, clearly stated thesis
statement.
Transitions are appropriate and help the flow of ideas.
Conclusion summarizes main argument and has a clear ending.
Introduction catches the reader’s attention, provides compelling
and appropriate background info, and clearly states the topic.
The paper is well organized around an arguable, focused thesis.
Thoughtful transitions clearly show how ideas relate.
Conclusion leaves the reader with a sense of closure and
provides concluding insights.
Usage and Mechanics
Grammar
Spelling
Sentence structure
Writing contains numerous errors in spelling, grammar, and/or
sentence structure that severely interferes with readability and
comprehension.
Errors in spelling and grammar exist that somewhat interfere
with readability and/or comprehension.

Writing follows conventions of spelling and grammar
throughout. Errors are infrequent and do not interfere with
readability or comprehension.
The paper is basically error free in terms of mechanics.
Grammar and mechanics help establish a clear idea and aid the
reader in following the writer’s logic.
APA Elements
Attribution
Paraphrasing
Quotations
No attempt at APA format.
APA format is attempted to paraphrase, quote, and cite, but
errors are significant.
Using APA format, accurately paraphrased, quoted, and cited in
many spots throughout when appropriate or called for. Errors
present are somewhat minor.
Using APA format, accurately paraphrased, quoted, and cited
throughout the presentation when appropriate or called for.
Only a few minor errors present.
Style
Audience
Word Choice
Writing often slips into first and/or second person.
Word choice is consistently inaccurate, unclear, or
inappropriate for the audience.
Writing sometimes slips into first and/or second person.
Word choice is sometimes inaccurate, unclear, or inappropriate
for the audience.
Writing remains in third person throughout writing.
Word choice is accurate, clear and appropriate for the audience.

Writing remains professional in third person throughout writing.
Word choice is precise, appropriate for the audience, and
memorable.

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