11.5 Fairness in Apportionment Methods

Apportionment methods aim to fairly distribute seats in representative bodies like Congress. These methods face challenges like the Alabama, population, and new-states paradoxes, which can lead to counterintuitive or seemingly unfair outcomes.

Different apportionment methods, such as Hamilton, Jefferson, Adams, and Webster, have varying strengths and weaknesses. They're evaluated based on criteria like the quota rule, monotonicity, and ability to avoid paradoxes, balancing fairness and practicality in representation.

Apportionment Paradoxes

Apportionment paradoxes and fairness

Top images from around the web for Apportionment paradoxes and fairness

• 

  Aging | Introduction to Sociology View original

  Is this image relevant?

• 

  More About Correlation | Boundless Statistics View original

  Is this image relevant?

• 

  Congressional Representation | American Government View original

  Is this image relevant?

• 

  Aging | Introduction to Sociology View original

  Is this image relevant?

• 

  More About Correlation | Boundless Statistics View original

  Is this image relevant?

1 of 3

Top images from around the web for Apportionment paradoxes and fairness

• 

  Aging | Introduction to Sociology View original

  Is this image relevant?

• 

  More About Correlation | Boundless Statistics View original

  Is this image relevant?

• 

  Congressional Representation | American Government View original

  Is this image relevant?

• 

  Aging | Introduction to Sociology View original

  Is this image relevant?

• 

  More About Correlation | Boundless Statistics View original

  Is this image relevant?

1 of 3

• Alabama paradox violates the expectation that a state should not lose representation when the total number of seats in the House increases (counterintuitive)
• Population paradox violates the expectation that states with larger population growth should gain more representation compared to states with smaller population growth (seems unfair)
• New-states paradox violates the expectation that the addition of a new state to the union should not cause an existing state to lose representation (appears illogical)

Apportionment Methods

Application of Hamilton method

• Hamilton method, also known as the largest remainder method, apportions seats based on each state's quota and fractional remainders
  1. Calculate the standard divisor by dividing the total population of all states by the total number of seats in the House
  2. Determine each state's quota by dividing its population by the standard divisor
  3. Assign each state the whole number portion of its quota (lower quota)
  4. Allocate any remaining seats to the states with the largest fractional remainders until all seats are distributed
• Identifying paradoxes in the Hamilton method
  • Alabama paradox can occur when a state's fractional remainder falls just short of the cutoff for receiving an additional seat after the total number of seats increases
  • Population paradox can occur when a state with a larger population increase has a smaller fractional remainder compared to a state with a smaller population increase, causing the latter to gain a seat instead
  • New-states paradox can occur when the addition of a new state alters the standard divisor, resulting in an existing state's fractional remainder dropping below the threshold for retaining a seat

Comparison of apportionment methods

• Fairness criteria for evaluating apportionment methods
  • Quota rule requires each state to receive a number of seats within one of its upper and lower quota (based on population proportion)
  • Monotonicity ensures that no state loses a seat when the total number of seats in the House increases
  • Population monotonicity guarantees that if state A's population grows faster than state B's, state A should not lose a seat to state B
  • Avoiding the new-states paradox ensures that the addition of a new state to the union does not cause an existing state to lose a seat
• Hamilton method (largest remainder method)
  • Satisfies the quota rule by assigning seats based on each state's quota
  • Violates monotonicity, potentially leading to the Alabama paradox
  • Violates population monotonicity, potentially causing the population paradox
  • Violates the new-states paradox
• Jefferson method (greatest divisor method)
  • Violates the quota rule, as some states may receive fewer seats than their lower quota
  • Satisfies monotonicity, ensuring no state loses a seat when the total number of seats increases
  • Satisfies population monotonicity, preventing a state with faster population growth from losing a seat to a state with slower growth
  • Satisfies the new-states paradox, ensuring the addition of a new state does not cause an existing state to lose a seat
• Adams method (smallest divisor method)
  • Satisfies the quota rule by assigning seats within each state's upper and lower quota
  • Violates monotonicity, potentially causing a state to lose a seat when the total number of seats increases
  • Violates population monotonicity, potentially allowing a state with slower population growth to gain a seat from a state with faster growth
  • Violates the new-states paradox, as the addition of a new state may cause an existing state to lose a seat
• Webster method (major fractions method)
  • Satisfies the quota rule, ensuring each state receives a number of seats within its upper and lower quota
  • Satisfies monotonicity, preventing any state from losing a seat when the total number of seats increases
  • Satisfies population monotonicity, ensuring a state with faster population growth does not lose a seat to a state with slower growth
  • Violates the new-states paradox, as the addition of a new state may still cause an existing state to lose a seat

Fair Division and Proportional Representation

Apportionment and fair division

• Apportionment is a method of fair division used to allocate seats in a representative body (such as congressional apportionment)
• Proportional representation aims to ensure that the distribution of seats reflects the population distribution as closely as possible
• Divisor methods are a family of apportionment techniques that use different formulas to calculate seat allocations
• Fair division principles in apportionment seek to balance competing fairness criteria and minimize paradoxes