Discrete MathDiscrete Math4.14.1

– The Apportionment Problem: A special kind of fair The Apportionment Problem: A special kind of fair division.division.

– What if you can’t divide the indivisible objects?What if you can’t divide the indivisible objects?

– Some will get their “fair share” and some will Some will get their “fair share” and some will not…not…

– So we need some methods to solve this dilemmaSo we need some methods to solve this dilemma

What do we use Apportionment What do we use Apportionment Methods for?Methods for?

Congressional SeatsCongressional Seats

Apportion Nurses to shifts at a hospitalApportion Nurses to shifts at a hospital

Apportion Calls to a switch boardApportion Calls to a switch board

Apportion subway cars to a subway Apportion subway cars to a subway systemsystem

Apportion planes to routesApportion planes to routes

Why Are Apportionment Methods NeededWhy Are Apportionment Methods Needed??We have 50 candies and we want to apportion to 5 kids according to how We have 50 candies and we want to apportion to 5 kids according to how long they work:long they work:

You can’t get a fair share, because some of the kids will get more than their You can’t get a fair share, because some of the kids will get more than their share and some will get less.share and some will get less.

Alan Betty Connie Doug Ellie Total Minutes

150 min 78 min 173 min 204 min 295 min 900 min

150/900

16 2/3%

78/900

8 2/3%

173/900

19 2/9 %

204/900

22 2/3 %

295/900

32 7/9 % 100 %

8 1/3 pieces of candy

4 1/3pieces of candy

9 11/18pieces of candy

11 1/3pieces of candy

16 7/8pieces ofcandy

50 candies

Discrete MathDiscrete Math4.24.2

– Standard Divisor: Population (P), Number of seats Standard Divisor: Population (P), Number of seats to be apportioned (M), P/M is the standard divisor.to be apportioned (M), P/M is the standard divisor.

– Standard Quota: fraction of the total number of Standard Quota: fraction of the total number of seats that each state would be entitled to if seats that each state would be entitled to if fractional seats were possible. State X’s fractional seats were possible. State X’s population / SD.population / SD.

– Lower Quota: Standard quota rounded down.Lower Quota: Standard quota rounded down.– Upper Quota: Standard quota rounded up.Upper Quota: Standard quota rounded up.

Discrete Math

Republic of Parador: Population Data by StateState Azucar Bahia Café Diamante Esmeralda Felicidad Total

Population 1,646,0000 6,936,000 154,000 2,091,000 685,000 988,000 12,500,000

Standard Quota

32.92 138.72 3.08 41.82 13.70 19.76 250

Upper Quota

33 139 4 42 14 20 252

Lower Quota

32 138 3 41 13 19 246

In the country of Parador we will only have 250 seats in the congress.

Standard Divisor: Total population ∕ number of seats

Standard Quota is the “fair” number of seats for each state:

State population ∕ Standard Divisor

Discrete MathDiscrete Math4.34.3

– Hamilton’s method and the Quota ruleHamilton’s method and the Quota ruleStep 1: Calculate each state’s standard quota.Step 1: Calculate each state’s standard quota.

Step 2: Give each state it’s lower quota.Step 2: Give each state it’s lower quota.

Step 3: Give the surplus to the states with the largest Step 3: Give the surplus to the states with the largest fractional parts.fractional parts.

Fails Neutral criteria: Every state has the same Fails Neutral criteria: Every state has the same opportunity for favorable apportionment (Favors larger opportunity for favorable apportionment (Favors larger states).states).

Violates (Alabama paradox, Population paradox, New-Violates (Alabama paradox, Population paradox, New-State paradox)State paradox)

To be continued…

Discrete MathDiscrete Math4.3 (Continued...)4.3 (Continued...)

– Quota rule:Quota rule:Apportionment should be either its upper quota or its Apportionment should be either its upper quota or its lower quota.lower quota.

An apportionment method that guarantees that every An apportionment method that guarantees that every state will be apportioned either to its lower quota or its state will be apportioned either to its lower quota or its upper quota satisfies the rule.upper quota satisfies the rule.

Violations: Lower-quota violations and upper-quota Violations: Lower-quota violations and upper-quota violations.violations.

To be continued…

Discrete MathDiscrete Math4.44.4

– Alabama paradox: An increase in the total number Alabama paradox: An increase in the total number of seats being apportioned, in and of itself, forces of seats being apportioned, in and of itself, forces a state to lose one of its seats.a state to lose one of its seats.

Discrete MathDiscrete Math4.54.5

– Population Paradox: Occurs when a state X loses Population Paradox: Occurs when a state X loses a seat to the state Y even though X’s population a seat to the state Y even though X’s population grew at a higher rate than Y’s.grew at a higher rate than Y’s.

– New State Paradox: The addition of a new state New State Paradox: The addition of a new state with its fair share of seats affects the with its fair share of seats affects the apportionment of other states.apportionment of other states.

– * Paradoxes occur in Hamilton’s method only** Paradoxes occur in Hamilton’s method only*– * Following methods violate quota rule** Following methods violate quota rule*

Discrete MathDiscrete Math4.64.6

– Jefferson’s Method: We need to use a modified Jefferson’s Method: We need to use a modified divisor (trial and error) that will give us new divisor (trial and error) that will give us new modified quotas that, when rounded down, will modified quotas that, when rounded down, will total the exact number of seats to be apportioned.total the exact number of seats to be apportioned.

Step 1: Find the modified divisor (D, smaller than the Step 1: Find the modified divisor (D, smaller than the standard divisor) such that when each state’s modified standard divisor) such that when each state’s modified quota is rounded down, the total is the exact number of quota is rounded down, the total is the exact number of seats to be apportioned.seats to be apportioned.

Step 2: Apportion to each state it’s modified lower quota.Step 2: Apportion to each state it’s modified lower quota.

Violates (Upper Quota Rule)Violates (Upper Quota Rule)

To be continued…

Discrete MathDiscrete Math4.64.6

– Jefferson’s Method was the very first Jefferson’s Method was the very first apportionment method used by the U.S. House of apportionment method used by the U.S. House of Representatives. (Violates the quota rule).Representatives. (Violates the quota rule).

Discrete MathDiscrete Math4.74.7

– Adam’s MethodAdam’s MethodStep 1: Find the modified divisor D such that when each Step 1: Find the modified divisor D such that when each state’s modified quota is rounded upward, the total is state’s modified quota is rounded upward, the total is the exact number of seats to be apportioned.the exact number of seats to be apportioned.

Step 2: Apportion to each state its modified upper quota.Step 2: Apportion to each state its modified upper quota.

– Adam’s violations are all lower quota violations.Adam’s violations are all lower quota violations.

Discrete MathDiscrete Math4.84.8

– Webster’s Method: Round quotas to the nearest Webster’s Method: Round quotas to the nearest integer.integer.

Step 1: Find the modified divisor D such that when each Step 1: Find the modified divisor D such that when each state’s modified quota is rounded the conventional way, state’s modified quota is rounded the conventional way, the total is the exact number of seats to be apportioned.the total is the exact number of seats to be apportioned.

Step 2: Apportion to each state its modified quota Step 2: Apportion to each state its modified quota rounded the conventional way.rounded the conventional way.

May violate either quota rule, but rare.May violate either quota rule, but rare.

Comes Closest to satisfying all main requirements for Comes Closest to satisfying all main requirements for fairness.fairness.

To be continued…

Discrete MathDiscrete Math4.84.8

– Balinski and Young’s impossibility theorem: Balinski and Young’s impossibility theorem: There cannot be a perfect apportionment method. There cannot be a perfect apportionment method. Any method that does not violate the quota rule Any method that does not violate the quota rule must produce paradoxes and vice-versa.must produce paradoxes and vice-versa.

Discrete MathDiscrete MathAppendix 1Appendix 1

– The Huntington-Hill Method:The Huntington-Hill Method:Comparable to the Webster method: Find modified Comparable to the Webster method: Find modified quotas and round some down and some upward. The quotas and round some down and some upward. The difference is the cutoff point for rounding.difference is the cutoff point for rounding.

L: Lower quota L: Lower quota L+1: upper L+1: upper quota.quota.

Cutoff for Webster’s: L + (L + 1) / 2Cutoff for Webster’s: L + (L + 1) / 2– ““Arithmetic mean”Arithmetic mean”

Cutoff for Huntington-Hill: Square root of (L x (L+1))Cutoff for Huntington-Hill: Square root of (L x (L+1))– ““Geometric mean”Geometric mean”

Discrete MathDiscrete Math