Contemporary Mathematics

# Key Concepts

## 11.1Voting Methods

• In plurality voting, the candidate with the most votes wins.
• When a voting method does not result in a winner, runoff voting can be used to do so.
• Ranked-choice voting, also known as instant runoff voting, is one type of ranked voting system.
• The Borda count method is a type of ranked voting system in which each candidate is given a Borda score based on the number of candidates ranked lower than them on each ballot.
• When pairwise comparison is used, the winner will be the Condorcet candidate if one exists.
• Approval voting allows voters to give equally weighted votes to multiple candidates.
• When a voter finds a characteristic of a particular voting method unappealing, they may consider that characteristic a flaw in the voting method and look for an alternative method that does not have that characteristic.

## 11.2Fairness in Voting Methods

• There are several common measures of voting fairness, including the majority criterion, the head-to head criterion, the monotonicity criterion, and the irrelevant alternatives criterion.
• According to Arrow’s Impossibility Theorem, each voting method in which the only information is the order of preference of the voters will violate one of the fairness criteria.

## 11.3Standard Divisors, Standard Quotas, and the Apportionment Problem

• The apportionment problem is how to fairly divide and distribute available resources to recipients in whole, not fractional, parts.
• To distribute the seats in the U.S. House of Representatives fairly to each state, calculations are based on state population, total population, and house size, or the total number of seats to be apportioned.
• The standard divisor is the ratio of the total population to the house size, and the standard quota is the number of seats that each state should receive.

## 11.4Apportionment Methods

• Hamilton’s method of apportionment uses the standard divisor and standard lower quotas, and it distributes any remaining seats based on the size of the fractional parts of the standard lower quota. Hamilton’s method satisfies the quota rule and favors neither larger nor smaller states.
• Jefferson’s method of apportionment uses a modified divisor that is adjusted so that the modified lower quotas sum to the house size. Jefferson’s method violates the quota rule and favors larger states.
• Adams’s method of apportionment uses a modified divisor that is adjusted so that the modified upper quotas, sum to the house size. Adams’s method violates the quota rule and favors smaller states.
• Webster’s method of apportionment uses a modified divisor that is adjusted so that the modified state quotas, rounded using traditional rounding, sum to the house size. Webster’s method violates the quota rule but favors neither larger nor smaller states.

## 11.5Fairness in Apportionment Methods

• Several surprising outcomes can occur when apportioning seats that voters may find unfair: Alabama paradox, population paradox, and new-state paradox.
• Apportionment methods are susceptible to apportionment paradoxes and may violate the quota rule.
• The Balinsky-Young Impossibility Theorem indicates that no apportionment can satisfy all fairness criteria.
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