Apportionment and Voting Notes

Introduction to Apportionment

Apportionment is dividing a whole into parts, rooted in the U.S. Constitution. Early plans were proposed by Alexander Hamilton and Thomas Jefferson.

The Hamilton Plan

  1. Standard Divisor: Total population / number of representatives. Standard Divisor = \frac{total population}{number of representatives}

  2. Standard Quota: Each state's population / standard divisor, rounded down.

  3. If the total number of representatives is less than required, assign additional representatives to states with the largest decimal remainders until the required number is reached.

The Jefferson Plan

  1. Modified Standard Divisor: A divisor chosen (by trial and error) so that the sum of the standard quotas equals the total number of representatives.

Alabama Paradox

Adding a new member can cause a state to lose a representative, which negatively affects fairness.

Fairness in Apportionment

  • Quota Rule: An apportionment plan should satisfy this rule.

  • Average Constituency: State population / number of representatives from that state, rounded to the nearest whole number.

  • Absolute Unfairness: The difference in the average constituencies.

  • Relative Unfairness: Used instead of absolute unfairness to decide which state receives another representative.

Huntington-Hill Apportionment Method

  • Based on the apportionment principle and used by the House of Representatives since 1940.

  • Implemented by calculating a Huntington-Hill number; the state with the greater number receives the next representative.

Plurality Method of Voting

The candidate with the most first-place votes wins.

Borda Count Method of Voting

Voters rank choices, assigning points to each choice.

Plurality with Elimination

Eliminate the site/candidate with the fewest first-place votes, then retake the vote.

Pairwise Comparison Voting Method

Each candidate is compared one-on-one with each other candidate to see who wins.

  • Condorcet Criterion: A voting method that elects the candidate who wins all head-to-head matchups.

Fairness of Voting Methods and Arrow's Theorem

  • Majority Criterion: Winner receives a majority of first-place votes.

  • Monotonicity Criterion: If a candidate wins, they should still win if support increases.

  • Condorcet Criterion: Candidate winning all head-to-head matchups should win.

  • Independence of Irrelevant Alternatives: Winner should remain the winner if losing candidates withdraw.

  • Arrow’s Impossibility Theorem states that no voting system can satisfy all fairness criteria.

Weighted Voting Systems

Some voters have more weight on the outcome of an election.

  • Quota: The number of votes required to pass a measure.

  • Weight: The number of votes controlled by a voter.

  • Coalition: A set of voters who vote the same way.

  • Winning Coalition: The sum of whose votes is greater than or equal to the quota.

  • Losing Coalition: The sum of whose votes is less than the quota.

  • Critical Voter: Turns a winning coalition into a losing one by leaving.

Banzhaf Power Index (BPI)

A measure of the power of a voter in a weighted voting system.