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
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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.

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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

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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)

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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.

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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.