Apportionment and Voting Notes
Apportionment and Voting
Introduction to Apportionment
Apportionment is the act of dividing items between different groups according to some plan, aiming for proportionate distribution in a fair manner.
It is represented as a function that takes inputs q, n, p1, p2, …, pn, where q and n are positive integers, and the pn's are positive numbers. The output is a sequence of non-negative integers q1, q2, …, qn such that q = q1 + q2 + … + qn.
Terms:
States: Parties having a stake in the apportionment, represented as (q1 + q2 + … + q_n).
Seats: Indivisible objects divided among n states. The number of seats is more than the number of states, but this doesn't guarantee every state gets a seat.
Population: A set of n positive numbers used as the basis for apportioning seats to states.
p1, p2, …, p_n = State’s respective population
p (total population) = p1 + p2 + … + p_n.
Apportionment Problem: A method for rounding standard quotas into whole numbers, ensuring the sum equals the total number of allocated items.
Quota Rule: An apportionment method where a state's allocation with fractional parts is either the integer immediately above (upper quota) or the integer part of the state's standard quota (lower quota).
Lower Quota: Standard quota rounded down to the nearest whole number.
Upper Quota: Standard quota rounded up to the nearest whole number.
Standard Divisor: The ratio of population to seats, calculated by dividing the total population by the number of seats.
Standard\ Divisor = \frac{Total\ Population}{Number\ of\ Seats}
Standard Quota: The exact fractional number of seats a state would get if fractional seats were allowed.
Standard\ Quota (qn) = \frac{Population\ of\ a\ Particular\ Group (pn)}{Standard\ Divisor}
Types of Monotonicity
House Monotone: No state can lose a seat when the total number of allocations increases.
Population Monotone: No state can lose a seat when only the population increases.
Quota Monotone: No state can lose a seat when its quota increases.
Hamilton’s Method
Description:
First apportionment method approved in the United States of America Congress but vetoed by President George Washington in 1792.
Proposed by Alexander Hamilton.
Used intermittently between 1852 and 1901.
Tends to favor larger states.
Example:
Apportioning 250 members of the House of Representatives among regions using the 2015 National Population Census data.
Total Population = 100,981,437
Number of Seats = 250
Standard Divisor = \frac{100,981,437}{250} = 403,925.75
Standard Quota = Population of a Particular Group / Standard Divisor
Jefferson’s Method
Description:
Uses modified lower quota.
Tends to favor larger states.
If a state gets more than the integer immediately above its quota, it violates the upper quota rule.
Proposed by Thomas Jefferson in 1792 and used in US Congress from 1792 to 1840.
Example:
Using the same data, Jefferson’s Method involves adjusting the divisor to achieve a whole number allocation of seats.
Modified divisors are tested (e.g., 400,000, 395,000, 389,550) until the lower quotas sum to the total number of seats (250).
Adams’ Method
Description:
Uses modified upper quota.
Always apportions at least 1 seat to each state.
Tends to favor smaller states.
If a state gets less than the integer part of its standard quota, it violates the lower quota rule.
Proposed by John Quincy Adams but never used in the US Congress.
Webster’s Method
Description:
Based on ordinary rounding.
Proposed by Daniel Webster in 1830.
Adopted in 1842 and used in the late 19th and early 20th centuries in the US Congress.
Tends to favor smaller states.
Example 2: Apportioning Police Patrol Cars
Problem: Apportioning 30 police patrol cars to cities in Metro Manila based on the 2015 census.
Methods Used: Hamilton’s, Jefferson’s, Adams’ and Webster’s.
The results from Hamilton’s and Jefferson’s methods were the same, favoring larger cities like Quezon City.
Adams’ and Webster’s methods favored smaller cities like Makati.
Most Natural Solution for Only Two States
Theorem: Webster’s is the only apportionment method consistent with the standard two-state solution.
Paradoxes in Apportionment
Apportionment paradoxes occur when the rules for apportionment produce results that violate quota rules.
Alabama Paradox: An increase in the available number of seats causes a decrease in the number of seats for a given group.
An apportionment method that avoids the Alabama paradox is house monotone.
Population Paradox: A state’s population increases, but its allocated number of seats decreases.
The methods of apportionment that avoids the population paradox is said to be population monotone.
New States Paradox (or Oklahoma paradox): The addition of a new state, with a corresponding increase in the number of available seats, can cause a change in the apportionment of items among the other states.
Huntington-Hill’s Method
Description:
Proposed by Edward Huntington (mathematician) and Joseph Hill (Chief Statistician at the Census Bureau).
No additional transfer of a seat from one state to another will reduce the ratio between degrees of representation in any two states.
Avoids Alabama and population paradoxes, but is not consistent with the two-state standard solution.
Slightly favors small states.
Used in the US Congress since 1941.
Process
Similar to Webster’s method but minimizes percent differences in representation.
Uses geometric mean = \sqrt{n(n+1)} of each group and compares it with the quota in each group.
If lower quota > geometric mean, round up the quota.
If lower quota < geometric mean, round down the quota.
Huntington-Hill Number
When there is a choice of adding one seat to one of several states, the seat should be allocated to the state with the highest Huntington-Hill number.
The value of Huntington-Hill Number (HHN), where P_i is the population of state i and a is the number of representatives from a state:
HHN = \frac{P_i^2}{a(a+1)}
Note: Huntington-Hill method violates the lower quota and upper quota rules.
Balinski-Young Impossibility Theorem
Proved in 1980 by Michael Balinski and H. Peyton Young.
If an apportionment method does not violate the quota rule, then apportionment paradoxes are possible.
If an apportionment method does not produce apportionment paradox, then violations of the quota rule are possible.
It is impossible for a neutral apportionment method to satisfy both quota rule and population monotonicity.
Voting
Voting is applied in selecting leaders in different aspects of society.
Leaders are usually chosen based on whoever gets the most number of votes.
Preference Voting
Preference Ballot: Voters list candidates in order of preference from first to last.
Preference Schedule: A table summarizing the results of individual preference ballots for an election.
Plurality Voting
The candidate with the most first-place votes wins, but does not have to have a majority of the votes.
Fails to take into account voter preferences beyond the first choice and can lead to bad election results.
Borda Count Voting
If there are n candidates, each voter ranks candidates, with the last choice getting 1 point, the second-to-last getting 2 points, and so on, up to n points for first place.
The candidate with the highest total points is the winner.
Violates both the majority criterion and the Condorcet criterion.
Considered one of the best voting methods for elections with many candidates.
Plurality-with-Elimination Voting
Carried out in rounds; n-1 rounds are needed when there are n candidates (also called Instant-Runoff Voting or IRV).
If a candidate has a majority of first-place votes, that candidate is the winner.
If no candidate has a majority, eliminate the candidate with the fewest first-place votes.
Repeat steps 1 and 2.
When only two candidates remain in a round, the candidate with the most votes wins the election.
Pairwise Comparison Voting
Each pair of candidates is matched head-to-head; each candidate gets 1 point for a win and 1/2 point for a tie.
It needs to go through all possible pairs of candidates; the candidate with the most total points wins the election.
Fairness Criteria
Majority Criterion: If a candidate has a majority of first-place votes, that candidate is the winner.
Monotonicity Criterion: If candidate A is a winner, and in a re-election, all changes in ballots are favorable only to A, then A is the winner.
Condorcet Criterion: If a candidate is preferred by the voters over each of the other candidates in head-to-head matchups, that candidate is the winner (or Condorcet candidate).
Independence of Irrelevant Alternatives Criterion: If candidate A is a winner, and one or more of the other choices are disqualified/withdraw and the ballots are recounted, then candidate A is still the winner.
Arrow’s Impossibility Theorem
Kenneth J. Arrow (1921-2017) proved that finding an absolutely fair and decisive voting system is impossible.
Weighted Voting
Each vote has some weight attached to it.
A weighted voting system of n votes is written as \lbrace q: w1, w2, …, wn \rbrace, where q is the quota and w1, w2, …, wn is the weight of each of n voters.
Terms
Quota: Minimum weight needed for votes for a resolution to be approved.
Coalition: A group of players/voters voting the same way.
Winning Coalition: A coalition with enough weight to meet the quota.
Losing Coalition: A coalition with votes less than the quota.
Grand Coalition: A coalition consisting of all the players; the number of possible coalitions of n voters is 2^n – 1.
Critical Voter (or Critical Player): A member of a winning coalition whose votes make the difference between winning and losing.
Dummy: A player whose vote is not critical in any winning coalition.
Veto Power: When a player’s support is necessary for the quota to be met; a player has veto power if they are critical in every winning coalition.
Dictatorship: A system where a player’s weight is equal to or greater than the quota; such a player is called a dictator.
Null Voting System: When all the members of a system’s votes are less than a quota.
Banzhaf Power Index
The Banzhaf Power Index (BPI) was originally created by Lionel Penrose in 1946 and was reintroduced by John Banzhaf in 1965.
The Banzhaf power index of a voter v is given by BPI(v) = \frac{number\ of\ times\ voter\ V\ is\ critical}{total\ number\ of\ times\ all\ voters\ are\ critical}.
Steps for Computing the Banzhaf Power Index
List all winning coalitions.
In each coalition, identify the critical voters.
Count the number of times each voter is critical.
Compute the Banzhaf power index.