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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
•
population of a group
average constituency =
number of representatives from that group
• “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”
•
average constituency
average constituency
absolute unfairness =
of group A
of group B
•
the absolute unfairnessof the apportionment
relative unfairness =
the smaller average constituency of the tw ogroups
• 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
“Gerrymandering”
• The Boston Gazette
lampooned the
shape of one district
with an editorial
cartoon likening it to
a mythical creature,
the “gerrymander.”
“Gerrymandering”
• 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.
“Gerrymandering”
• Rep. Tom DeLay (R-TX)
pushed through a special reredistricting 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.
“Gerrymandering”
• 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.
“Gerrymandering”
• 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 purplemajority 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 purplemajority 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
1.
2.
3.
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.
– However, if the House were expanded to 280
Representatives, 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.
– However, if the House were expanded to 360
Representatives, 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-theplanets 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 358member 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
Committee
Division
Seats
“Deserved”
Provisional
Seats Awarded
Final Allocation
of Seats
Sciences
0.47 x 9 = 4.23
4
4
Humanities
0.37 x 9 = 3.33
3
3
Professional
0.16 x 9 = 1.44
1
2
Total
100% x 9 = 9.00
8
9
• 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
Committee
Division
Number of Seats
“Deserved”
Provisional Seats Final Allocation
Awarded
of Seats
Sciences
0.47 x 10 = 4.70
4
5
Humanities
0.37 x 10 = 3.70
3
4
Professional
0.16 x 10 = 1.60
1
1
Total
100% x 10 = 10.00 8
10
• 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
– Repeat until the correct number of delegates is awarded
• 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
– Repeat until the correct number of delegates is awarded
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 oilboom 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:
(population of the state or group)2
current number of that
proposed number of that
×
group' s representatives
group' s representatives
• 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 Method
When Having
1 Seat
When Having
2 Seats
When Having
3 Seats
When Having
4 Seats
When Having
5 Seats
When Having
6 Seats
Sciences
( 47)2
= 1104.5
1× 2
( 47)2
= 368.2
2×3
( 47)2
= 184.1
3× 4
( 47)2
= 110.5
4×5
( 47)2
= 73.6
5×6
( 47)2
= 52.6
6×7
Humanities
(37)2
= 684.5
1× 2
(37)2
= 228.2
2×3
(37)2
= 114.1
3×4
(37)2
= 68.5
4×5
(37)2
= 45.6
5×6
(37)2
= 32.6
6×7
Professional
(16)2
= 128.0
1× 2
(16)2
= 42.7
2×3
(16)2
= 21.3
3×4
(16)2
= 12.8
4×5
(16)2
= 8 .5
5×6
(16)2
= 6. 1
6×7
• 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 singleRepresentative 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 185184 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-theknot.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/00pvalues.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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