Voting methods are crucial for fair elections, but each has pros and cons. Plurality voting is simple but can lead to spoiler effects. Ranked-choice and pairwise comparison offer more nuanced results but can be complex.

Fairness criteria help evaluate voting systems, but Arrow's Impossibility Theorem shows no perfect system exists. Trade-offs are necessary, and the choice of system can significantly impact election outcomes and voter behavior.

Voting Methods and Fairness Criteria

Voting methods and fairness criteria

• Plurality voting involves voters choosing a single candidate, and the candidate with the most votes wins
  • Violates the majority criterion if a candidate wins without receiving more than 50% of the votes
  • Violates the Condorcet criterion if a candidate who would win in a head-to-head matchup against every other candidate loses the election (spoiler effect)
• Ranked-choice voting (instant runoff) requires voters to rank candidates in order of preference
  • If no candidate has a majority, the candidate with the fewest first-choice votes is eliminated, and their votes are redistributed to the next choice on each ballot
  • Process repeats until a candidate achieves a majority (over 50%) of the votes
  • Satisfies the majority criterion as the winner always has a majority of votes in the final round
  • Can violate the Condorcet criterion in some cases (Burlington, Vermont mayoral election in 2009)
  • Is an example of preferential voting, where voters express their preferences among candidates
• Pairwise comparison (Condorcet method) compares each candidate against every other candidate in head-to-head matchups
  • If a candidate wins all their matchups, they are the Condorcet winner and the overall winner
  • Always satisfies the Condorcet criterion by definition
  • Can fail to produce a winner if there is no Condorcet winner due to circular preferences (Condorcet paradox)
• Borda count assigns points to candidates based on their position in each voter's ranking (e.g., n-1 points for first, n-2 for second, etc., where n is the number of candidates)
  • Candidate with the most points wins the election
  • Violates the majority criterion as a candidate can win without being the first choice of a majority of voters
  • Violates the Condorcet criterion in some cases (1976 hypothetical election example with Carter, Ford, and Reagan)

Monotonicity in voting systems

• Monotonicity requires that if a voter raises a candidate in their ranking, it should not hurt that candidate's chances of winning
• Violations of monotonicity can lead to strategic voting and counterintuitive results
• Plurality voting satisfies monotonicity since raising a candidate in a voter's ranking can only help or have no effect on that candidate's chances of winning
• Ranked-choice voting can violate monotonicity in some cases where raising a candidate in a voter's ranking causes them to be eliminated earlier, changing the outcome (Burlington, Vermont mayoral election in 2009)
• Pairwise comparison satisfies monotonicity as raising a candidate in a voter's ranking can only help them in head-to-head matchups
• Borda count satisfies monotonicity since raising a candidate in a voter's ranking always increases their point total

Arrow's Impossibility Theorem implications

• Arrow's Impossibility Theorem states that no ranked voting system can simultaneously satisfy all of the following criteria:
  1. Universality: The voting system should work for any possible set of voter preferences
  2. Independence of Irrelevant Alternatives (IIA): The relative ranking of two candidates should depend only on voters' preferences between those two candidates, not on the presence or absence of other candidates
  3. Non-dictatorship: The voting system should not allow a single voter's preferences to determine the outcome regardless of other voters' preferences
  4. Pareto efficiency: If every voter prefers candidate A to candidate B, the voting system should rank A higher than B
• The theorem implies that any ranked voting system must make trade-offs between these desirable criteria
• Ranked voting systems can satisfy some of these criteria but not all simultaneously
  • Pairwise comparison satisfies Pareto efficiency and IIA but violates universality (Condorcet paradox)
  • Ranked-choice voting satisfies universality and non-dictatorship but violates IIA (spoiler effect)
  • Borda count satisfies universality, non-dictatorship, and Pareto efficiency but violates IIA (1976 hypothetical election example)
• Designers of voting systems must prioritize which criteria are most important based on the context and goals of the election
• No perfect voting system exists that satisfies all desirable fairness criteria, so trade-offs and compromises are necessary to balance competing priorities and ensure the most fair and representative outcomes possible given the limitations

Electoral Systems and Representation

• Electoral systems are the set of rules that determine how votes are translated into seats in a representative body
• Different electoral systems can lead to varying levels of proportional representation, where the proportion of seats a party wins closely matches their share of the overall vote
• Some systems, like plurality voting in single-member districts, can result in disproportionate representation
• Proportional representation systems aim to allocate seats in proportion to the votes received by each party or group
• The choice of electoral system can influence voter behavior and party strategies, potentially leading to strategic voting where voters cast ballots not for their true preferences but to achieve a desired outcome