💯Math for Non-Math Majors Unit 11 – Voting and Apportionment in Mathematics

Key Concepts and Terminology

• Voting systems determine how individual preferences are aggregated to make collective decisions
• Apportionment allocates a fixed number of representatives or resources among competing groups based on their sizes
• Fairness criteria establish standards for evaluating the equity and representativeness of voting systems
• Plurality voting selects the candidate with the most first-place votes, without requiring a majority
• Majority rule requires a candidate to receive more than half of the votes to win
  • Prevents a candidate disliked by most voters from winning due to vote splitting
• Proportional representation aims to allocate seats or influence in proportion to the share of votes received
• Quota in apportionment represents the average number of people per representative (total population divided by total representatives)
• Divisor methods in apportionment involve dividing each state's population by a common divisor to determine seat allocation

Voting Systems Overview

• Voting systems convert individual voter preferences into a collective choice or ranking of candidates
• Different voting systems can produce different outcomes for the same set of voter preferences
• Some common voting systems include plurality, runoff, instant runoff, approval, score, and Condorcet methods
• Arrow's Impossibility Theorem proves no voting system can satisfy a set of reasonable fairness criteria simultaneously
  • Highlights inherent trade-offs and limitations in designing voting systems
• Voting systems must balance competing goals such as simplicity, expressiveness, and resistance to strategic manipulation
• The choice of voting system can significantly impact election results and the incentives for candidates and voters
• Voting paradoxes, such as Condorcet's paradox and the monotonicity paradox, demonstrate counterintuitive outcomes possible under various systems

Fairness Criteria in Voting

• Majority criterion states that if a candidate is preferred by a majority of voters, they should win
• Condorcet criterion requires that if a candidate would beat every other candidate in a head-to-head matchup, they should win
  • Ensures the winner is the candidate who would beat all others in pairwise comparisons
• Independence of Irrelevant Alternatives (IIA) means adding or removing a losing candidate should not change the winner
• Monotonicity criterion requires that improving the ranking of a winning candidate should not make them lose
• Participation criterion encourages voter turnout by ensuring that voting for a preferred candidate never hurts their chances of winning
• Consistency criterion states that if an electorate is divided into separate contests, and the same candidate wins each contest, they should win the combined contest
• Neutrality means the voting system does not favor any particular candidates or alternatives
• Anonymity requires that all voters are treated equally, and swapping any two voters' ballots should not affect the outcome

Common Voting Methods

• Plurality voting is simple but vulnerable to vote splitting and the spoiler effect
• Runoff voting holds a second round between the top two candidates if no one receives a majority in the first round
  • Ensures the winner has majority support but may have lower turnout in the second round
• Instant runoff voting (IRV) simulates a series of runoffs by eliminating the last-place candidate and redistributing their votes until someone reaches a majority
  • Allows voters to express preferences without multiple rounds of voting
• Approval voting lets voters approve of multiple candidates, and the candidate with the most approvals wins
  • Encourages voters to support all acceptable candidates rather than just their favorite
• Score voting (or range voting) has voters rate each candidate on a scale, and the candidate with the highest average score wins
  • Provides more nuance than approval voting but may be more complex for voters
• Condorcet methods select the candidate who would win a head-to-head matchup against every other candidate
  • Satisfies the Condorcet criterion but can sometimes result in cyclic preferences with no clear winner

Apportionment Basics

• Apportionment is the process of distributing a fixed number of representatives or resources among groups based on their sizes
• In the United States, apportionment determines how many seats each state gets in the House of Representatives based on its population
• The apportionment problem arises because the ideal quota of representatives per person is usually a fraction, but representatives must be whole numbers
• Apportionment methods must decide how to round the ideal quotas to allocate the available seats
• The quota is calculated by dividing the total population by the total number of representatives (total population / total representatives)
  • Represents the average number of people per representative
• A state's ideal quota is its population divided by the quota (state population / quota)
  • The whole number part of the ideal quota is the state's initial allocation of seats
• Apportionment methods differ in how they distribute the remaining seats to achieve the total number of representatives required

Apportionment Methods and Paradoxes

• Hamilton's method assigns the remaining seats to the states with the largest fractional parts of their ideal quotas
  • Favors larger states but is susceptible to the Alabama paradox, where a state can lose a seat when the total number of seats increases
• Jefferson's method uses a divisor to adjust state populations until the resulting quotients sum to the desired total seats
  • Favors smaller states and avoids the Alabama paradox but can exhibit the population paradox, where a state with faster population growth loses a seat to a state with slower growth
• Webster's method is similar to Jefferson's but rounds the adjusted quotients to the nearest whole number instead of always rounding down
  • Minimizes the discrepancy between a state's share of seats and its share of the population
• The Huntington-Hill method is the current method used for U.S. congressional apportionment and uses a specific geometric mean to determine the priority of states for receiving remaining seats
• Balinski and Young's impossibility theorem proves no apportionment method can satisfy a set of desirable properties, including quota (giving each state its ideal quota rounded to the nearest whole number) and monotonicity (a state's number of seats should not decrease when the total number of seats increases)

Real-World Applications

• Voting systems are used in political elections, committee decisions, and group choice settings (clubs, awards)
• Different voting systems can alter the incentives for candidates, parties, and voters, affecting campaign strategies and voter behavior
• Some organizations and jurisdictions have adopted alternative voting systems like instant runoff voting (Australia, Ireland) or approval voting (Fargo, North Dakota) to address limitations of plurality voting
• Apportionment methods shape the balance of power in legislative bodies and can have significant impacts on policy outcomes
• Changes in apportionment, such as after a census, can shift political representation and advantage or disadvantage certain states or regions
• Gerrymandering, the manipulation of electoral district boundaries for partisan advantage, is related to apportionment and can undermine fair representation
  • Techniques like the efficiency gap measure the extent of gerrymandering
• Apportionment principles apply to other problems of proportional allocation, such as distributing resources or benefits among groups (disaster relief funds, university budgets)

Problem-Solving Strategies

• Identify the type of problem (voting system analysis, apportionment calculation) and the specific method or criteria involved
• For voting system problems, construct a preference schedule showing each voter's ranking of the candidates
  • Analyze the preference schedule according to the rules of the given voting system
  • Check for violations of fairness criteria or voting paradoxes
• In apportionment problems, calculate the quota (total population / total representatives) and each state's ideal quota (state population / quota)
  • Apply the specified apportionment method to determine the initial allocation of seats and distribute any remaining seats
  • Check for apportionment paradoxes by comparing results under different total seat numbers or population distributions
• Consider real-world constraints and implications, such as the feasibility of implementing a particular voting system or the political consequences of an apportionment outcome
• Use counterexamples to test the limitations of different methods or demonstrate paradoxes
  • Construct scenarios that highlight the differences between methods or the violation of desired properties
• Analyze the sensitivity of results to changes in the inputs, such as voter preferences, candidate fields, or population data
  • Identify conditions under which the outcome might change or paradoxes might arise
• Apply mathematical tools like rounding rules, divisor methods, and pairwise comparisons as appropriate for the specific problem type