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Title: The Mathematics of Elections Part I: Apportionment


1
The Mathematics of ElectionsPart I
Apportionment
  • Mark Rogers
  • (a.k.a. The Mad Hatter)

2
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

3
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 states 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.

4
The 108th Congress
5
The 110th Congress
6
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 Houses size, but it was never ratified.
  • Instead, acts of Congress have governed each
    increase.

7
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.

8
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

9
Average Constituency
  • The typical number of voters an official
    represents
  • The Constitutionmust 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

10
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)

11
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

12
Minimizing unfairness
  • Apportionment Criterion When assigning a
    representative among several parties, make the
    assignment so as to create the smallest possible
    relative unfairness.

13
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 Congressmans 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 Rayburns 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

14
One man, one vote(WellOne person, one vote)
  • In Reynolds v. Sims (1964), the U.S. Supreme
    Court ruled that the Constitutions 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.

15
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

16
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

17
Gerrymandering
  • The Boston Gazette lampooned the shape of one
    district with an editorial cartoon likening it to
    a mythical creature, the gerrymander.

18
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.

19
Gerrymandering
  • Rep. Tom DeLay (R-TX) pushed through a special
    re-redistricting of the Texas Congressional
    districts in 2003, following his partys 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.

20
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
    Murthas.
  • A tendril of Murthas new district extended down
    a street to envelop Mascaras 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.

21
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

22
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.

23
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.

24
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 groups fair number of them was rounded
    down the most.

25
Now You See Them, Now You Dont
  • 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.

26
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.

27
Whos 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.

28
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.

29
Up and Down
  • Attempts to replace Hamiltons method with an
    alternative similarly caused Maines delegation
    to fluctuate in size.
  • Now you see it and now you dont. 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.

30
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 Hamiltons method, we find the fair
    number of seats each division deserves, round
    down any decimals, and choose how to allocate any
    remaining seats afterwards.

31
The 9-Person Faculty Committee
  • Now suppose that the committee is to be expanded
    to 10 seats. We will use Hamiltons method to
    reapportion the seats.

32
The 10-Person Faculty Committee
  • The professional facultys 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 whos been rounded
    down the most? category.

33
The Other Founding Fathers
  • Thomas Jefferson, John Adams, and Daniel Webster
    each proposed alternatives to Hamiltons 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 states fair
    share, but to make the numbers work out more
    easily.

34
Alternatives to Hamiltons method
  • Jeffersons 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
  • Adamss 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
  • Websters 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

35
Representative Quotas
  • The number of representatives deserved in
    Hamiltons 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.
  • Hamiltons 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.

36
More Problems for Hamilton
  • Were the rarity of the Alabama paradox the only
    problem Hamiltons method risked, it might still
    be used today. However, there are a number of
    other paradoxes that can occur with it.
  • Population paradox State As population is
    growing faster than state B, yet A loses a
    representative to B.
  • As percentage population growth was higher than
    Bs, but Hamiltons 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 states (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).

37
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
    Maines 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 Oklahomas
    seats on top of the others had changed the
    delegates for other states.
  • Appendix See 1907 tab on Excel spreadsheet
    Census Apportionments.

38
The Huntington-HillApportionment 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
    states current number of representatives, and
    compares it to the unfairness of that states
    number of representatives if an extra one were
    added.

39
The Huntington-HillApportionment 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.

40
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.
  • Californias 53rd district and North Carolinas
    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.

41
The Faculty Committees, When Using The
Huntington-Hill Method
  • 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
    groups seat to give it to another.

42
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.

43
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.
  • Montanas 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

44
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 Youngs Impossibility Theorem
    There is no apportionment method that avoids all
    paradoxes and at the same time satisfies the
    quota rule.

45
Webster was right!
  • The Constitutionmust 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

46
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,
    Tildens lead would have held.

47
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 councils seats have been divided
    up, how do we decide who gets to fill them?
  • Next Friday The Mathematics of Elections, Part
    II Voting.

48
References
  • Most liberal-arts college-mathematics course
    (ex. MATH 110) textbooks
  • Including ours, Thomas L. Pirnots Mathematics
    All Around, 3rd edition
  • Alex Bogolmonys 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/compen
    dia/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/mcro
    gers, will have this presentation plus the
    spreadsheets used.
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