Course Structure, Apportionment Fundamentals, and Adams' Method of Apportionment

Course Structure and Examination Breakdown

• Classroom Dynamic & Subject Matter:

  • The material combines conceptual understanding with elementary calculations (e.g., counting and basic arithmetic).
  • Conceptual aspects cover the meaning of voting systems and apportionment, while practical aspects involve applying standard algorithms.

• Test Format Details:

  • All exam questions are formatted as multiple choice.
  • First 2.5 Examinations: Purely conceptual. Even questions involving mathematical calculations are presented in a multiple-choice format.
  • Final 2.5 Examinations: Significantly more math-heavy.
  • Study Strategy: Primary focus must be on memorizing core concepts, standard definitions, and knowing how to execute mathematical procedures on those concepts.

Introduction to Apportionment

• Definition of Apportionment:

  • Apportionment is defined as the act to distribute or allocate proportionally, or to divide and assign according to some established rule of proportional distribution.

• Historical Context (The 1790 U.S. Census):

  • The 1790 Census is utilized as the primary foundational dataset for studying legislative apportionment because it features a significantly smaller total population and fewer states compared to modern U.S. demographics.
  • The goal of apportionment in this context is to determine how many seats in the House of Representatives each state should receive based on population.

• Equal vs. Proportional Representation Metaphors:

  • Classroom Analogy:
  • Consider a class of $20$ total students divided across $4$ tables/groups.
  • If Dion sits alone at one table and gets $1$ vote, while another table has $5$ students who collectively share $1$ vote, representation is unequal.
  • To allocate representatives fairly, the total population ($20$) is divided by the desired number of groups ($4$), yielding $5$ students per group/representative.
  • Districts vs. States:
  • Intra-state lines and voting districts can be redrawn within a state (a practice occurring with increasing frequency).
  • State boundaries cannot be moved. State populations vary widely (e.g., Arkansas vs. California).
  • Assigning an equal, fixed number of representatives (such as $2$ per state across $50$ states) to both low-population states (Arkansas) and high-population states (California) is inherently unfair.
  • Legislative representation must be mathematically dependent on population size.

Mathematical Foundations of Apportionment

• Target Legislative Body Parameters (1790 Census):

  • Total Available Seats: $105$
  • The seat count $105$ is fixed by law and cannot change without explicit legislation.

• Standard Divisor:

  • Definition: The average number of people represented by a single seat/representative.
  • Formula:     $\text{Standard Divisor} = \frac{\text{Total Population of the United States}}{\text{Total Number of House Seats}}$
  • 1790 Census Calculation:     $\text{Standard Divisor} = \frac{\text{Total Population}}{105} = 37\,084.51$
  • Interpretation: Under exact proportional representation, each representative serves $37\,084.51$ people. In reality, a fraction of a person cannot exist, but the exact decimal value $37\,084.51$ must be retained during intermediate calculations.

• Standard Quota:

  • Definition: The exact fractional number of representatives a state is entitled to receive based on its population and the standard divisor.
  • Formula:     $\text{Standard Quota} = \frac{\text{State Population}}{\text{Standard Divisor}}$
  • Classroom Example:
  • A group of $8$ people divided by a target size of $5$ per representative yields a quota of:       $\frac{8}{5} = 1.6$
  • State Calculation Example:
  • For a state standard quota calculated as $6.41$, exact fractional seats cannot be awarded, requiring rounding to whole numbers.

Quota Rounding Methods

• Nearest Quota (Standard Natural Rounding):

  • Applies traditional rounding rules: if the fractional part is $0.5$ or greater, round up to the next integer; if the fractional part is lower than $0.5$, round down.
  • Examples:
  • $6.41 \rightarrow 6$
  • $1.59 \rightarrow 2$

• Lower Quota (Floor Function):

  • Takes the mathematical floor of the standard quota.
  • Truncates the decimal completely, rounding down to the integer below regardless of how large the decimal is.
  • Examples:
  • $6.41 \rightarrow 6$
  • $7.999 \rightarrow 7$
  • $1.59 \rightarrow 1$

• Upper Quota (Ceiling Function):

  • Takes the mathematical ceiling of the standard quota.
  • Rounds up to the next whole integer regardless of how small the decimal is.
  • Examples:
  • $6.41 \rightarrow 7$
  • $1.59 \rightarrow 2$

Flaws of Direct Quota Summation (1790 Census Data)

• When applying rounding methods directly to all state quotas for the $105$ House seats, none of the standard rounding types sum to the required total of $105$ seats:
  • Sum of Nearest Quotas: Yields $106$ seats ($1$ seat too many; unfeasible because taking a seat away from an arbitrary state is ungrounded and unethical).
  • Sum of Lower Quotas: Yields $96$ seats ($9$ seats unassigned).
  • Sum of Upper Quotas: Yields $111$ seats ($6$ seats too many).

John Quincy Adams' Method of Apportionment

• Historical Context:

  • Proposed by John Quincy Adams, the sixth President of the United States.

• Core Philosophy:

  • Adams advocated for using the upper quota (ceiling function) to ensure smaller states were not underrepresented.

• The Modified Divisor Solution:

  • Because using the standard divisor ($37\,084.51$) alongside upper quotas produces $111$ total seats ($6$ excess seats), Adams proposed using a modified divisor ($d$).
  • To decrease the sum of upper quotas down to the target $105$ seats, the standard divisor must be increased.
  • Increasing the divisor reduces each state's quota, lowering the calculated values so their ceilings sum precisely to $105$.

• Determining the Modified Divisor:

  • Modified divisors are determined through trial and error.
  • Shifting the divisor by increments (e.g., testing $39\,000$ vs. $39\,300$) produces different quota outcomes until an exact match for total seats is found.
  • For the 1790 Census, Adams identified a modified divisor of $39\,600$

• Step-by-Step Procedure for Adams' Method:

  1. Select a modified divisor $d$ (for 1790, $d = 39\,600$).
  2. Compute each state's modified quota:      $\text{Modified Quota} = \frac{\text{State Population}}{39\,600}$
  3. Assign each state its upper quota by taking the ceiling of the modified quota:      $\text{State Seats} = \lceil \text{Modified Quota} \rceil$
  4. Sum the upper quotas across all states to verify the total equals $105$ seats.

1790 State Apportionment Calculations via Adams' Method ($d = 39\,600$)

• State 1:

  • Modified Quota calculation yields approximately $6.0041$
  • Ceiling (Upper Quota): $7$ seats.

• Delaware:

  • Modified Quota: $1.49$
  • Ceiling (Upper Quota): $2$ seats.

• Georgia:

  • Modified Quota: $2.008$
  • Ceiling (Upper Quota): $3$ seats.

• State with Population $73\,677$:

  • Calculation:     $\frac{73\,677}{39\,600} = 1.86$
  • Ceiling (Upper Quota): $2$ seats.

• Maryland:

  • Modified Quota: $8.02$
  • Ceiling (Upper Quota): $9$ seats.

• Massachusetts (MS):

  • Ceiling (Upper Quota): $12$ seats.

• New Hampshire:

  • Modified Quota: $3.38$
  • Ceiling (Upper Quota): $4$ seats.

• New Jersey:

  • Modified Quota calculation yields ceiling: $6$ seats.

• New York:

  • Modified Quota: $8.59$
  • Ceiling (Upper Quota): $9$ seats.

• North Carolina:

  • Modified Quota calculation yields ceiling: $10$ seats.

• Pennsylvania:

  • Modified Quota: $10.9$
  • Ceiling (Upper Quota): $11$ seats.

• Rhode Island:

  • Modified Quota: $0.7475$
  • Ceiling (Upper Quota): $2$ seats.

• Total Seat Allocation Summary:

  • Summing the upper quotas across all participating states under the modified divisor $39\,600$ gives exactly $105$ total seats.