§ 4.3 Hamilton’s Method§ 4.3 Hamilton’s Method

“Quipped a jubilant Hamilton, ‘The only way it could fail is if one party gained control of not just the Executive, but also the Senate and House chambers, and upon doing so, proceeded to

bring in like-minded judges!!!!’ And then the Framers all laughed and laughed and laughed.”

- AAMERICAMERICA (THE BOOK)(THE BOOK)

Hamilton’s MethodHamilton’s Method

The Idea: Give each state it’s lower quota, then assign the surplus seats.

PLANET ANDORIA

EARTH

TELLAR

VULCAN

TOTAL

POPULATIONin billions

16.2 16.1 28.3 8.9 69.5

Example: THE PLANETS OF ANDORIA, EARTH, TELLAR AND VULCAN HAVE DECIDED TO FORM A UNITED FEDERATION OF PLANETS. THE RULING BODY OF THIS GOVERNMENT WILL BE THE 139 MEMBER FEDERATION COUNCIL. APPORTION THE SEATS USING HAMILTON’S METHOD.

PLANET ANDORIA

EARTH

TELLAR

VULCAN

TOTAL

POPULATIONin billions

16.2 16.1 28.3 8.9 69.5

STANDARD QUOTA

32.4 32.2 56.6 17.8 139

Example: THE PLANETS OF ANDORIA, EARTH, TELLAR AND VULCAN HAVE DECIDED TO FORM A UNITED FEDERATION OF PLANETS. THE RULING BODY OF THIS GOVERNMENT WILL BE THE 139 MEMBER FEDERATION COUNCIL. APPORTION THE SEATS USING HAMILTON’S METHOD.

PLANET ANDORIA

EARTH

TELLAR

VULCAN

TOTAL

POPULATIONin billions

16.2 16.1 28.3 8.9 69.5

STANDARD QUOTA

32.4 32.2 56.6 17.8 139

LOWER QUOTA

32 32 56 17 137

Example: THE PLANETS OF ANDORIA, EARTH, TELLAR AND VULCAN HAVE DECIDED TO FORM A UNITED FEDERATION OF PLANETS. THE RULING BODY OF THIS GOVERNMENT WILL BE THE 139 MEMBER FEDERATION COUNCIL. APPORTION THE SEATS USING HAMILTON’S METHOD.

PLANET ANDORIA

EARTH TELLAR

VULCAN

TOTAL

POPULATIONin billions

16.2 16.1 28.3 8.9 69.5

STANDARD QUOTA

32.4 32.2 56.6 17.8 139

LOWER QUOTA

32 32 56 17 137

FRACTIONAL PART

.4 .2 .6 .8 2

Example: THE PLANETS OF ANDORIA, EARTH, TELLAR AND VULCAN HAVE DECIDED TO FORM A UNITED FEDERATION OF PLANETS. THE RULING BODY OF THIS GOVERNMENT WILL BE THE 139 MEMBER FEDERATION COUNCIL. APPORTION THE SEATS USING HAMILTON’S METHOD.

PLANET ANDORIA

EARTH TELLAR

VULCAN

TOTAL

POPULATIONin billions

16.2 16.1 28.3 8.9 69.5

STANDARD QUOTA

32.4 32.2 56.6 17.8 139

LOWER QUOTA

32 32 56 17 137

FRACTIONAL PART

.4 .2 .6 .8 2

EXTRA SEATS

1 1

Example: THE PLANETS OF ANDORIA, EARTH, TELLAR AND VULCAN HAVE DECIDED TO FORM A UNITED FEDERATION OF PLANETS. THE RULING BODY OF THIS GOVERNMENT WILL BE THE 139 MEMBER FEDERATION COUNCIL. APPORTION THE SEATS USING HAMILTON’S METHOD.

PLANET ANDORIA

EARTH TELLAR

VULCAN

TOTAL

POPULATIONin billions

16.2 16.1 28.3 8.9 69.5

STD. QUOTA

32.4 32.2 56.6 17.8 139

LOWER QUOTA

32 32 56 17 137

FRACTIONAL PART

.4 .2 .6 .8 2

EXTRA SEATS

1 1

FINAL APPORTIONMENT

32 32 57 18 139

Example: THE PLANETS OF ANDORIA, EARTH, TELLAR AND VULCAN HAVE DECIDED TO FORM A UNITED FEDERATION OF PLANETS. THE RULING BODY OF THIS GOVERNMENT WILL BE THE 139 MEMBER FEDERATION COUNCIL. APPORTION THE SEATS USING HAMILTON’S METHOD.

Hamilton’s MethodHamilton’s Method

Step 1. Calculate each state’s standard quota.

Step 2. Give to each state its lower quota.

Step 3. Give the surplus seats (one at a time) to the states with the largest fractional parts.

State Pop. (est.)

Std. Quota (S1)

Lower Quota (S2)

Connecticut

237,000 7.86

Delaware 56,000 1.86

Georgia 71,000 2.35

Kentucky 69,000 2.29

Maryland 279,000 9.25

Massachusetts

475,000 15.75

New Hampshire

142,000 4.71

New Jersey

180,000 5.97

New York 332,000 11.01

North Carolina

354,000 11.74

Pennsylvania

433,000 14.36

Rhode Island

68,000 2.25

South Carolina

206,000 6.83

Vermont 86,000 2.85

Virginia 631,000 20.92

Total 3,619,000

120

Example:Example:If the House of Representatives had been apportioned under Alexander Hamilton’s 1790s plan (with M = 120 seats) we would have the following:

State Pop. (est.)

Std. Quota (S1)

Lower Quota (S2)

Connecticut

237,000 7.86 7

Delaware 56,000 1.86 1

Georgia 71,000 2.35 2

Kentucky 69,000 2.29 2

Maryland 279,000 9.25 9

Massachusetts

475,000 15.75 15

New Hampshire

142,000 4.71 4

New Jersey

180,000 5.97 5

New York 332,000 11.01 11

North Carolina

354,000 11.74 11

Pennsylvania

433,000 14.36 14

Rhode Island

68,000 2.25 2

South Carolina

206,000 6.83 6

Vermont 86,000 2.85 2

Virginia 631,000 20.92 20

Total 3,619,000

120 111

State Pop. (est.)

Std. Quota (S1)

Lower Quota (S2)

Frac’l Part

Connecticut

237,000 7.86 7 .86

Delaware 56,000 1.86 1 .86

Georgia 71,000 2.35 2 .35

Kentucky 69,000 2.29 2 .29

Maryland 279,000 9.25 9 .25

Massachusetts

475,000 15.75 15 .75

New Hampshire

142,000 4.71 4 .71

New Jersey

180,000 5.97 5 .97

New York 332,000 11.01 11 .01

North Carolina

354,000 11.74 11 .74

Pennsylvania

433,000 14.36 14 .36

Rhode Island

68,000 2.25 2 .25

South Carolina

206,000 6.83 6 .83

Vermont 86,000 2.85 2 .85

Virginia 631,000 20.92 20 .92

Total 3,619,000

120 111 9

State Pop. (est.)

Std. Quota (S1)

L. Quota (S2)

Frac’l Part

Surplus Seats

Connecticut

237,000 7.86 7 .86 1

Delaware 56,000 1.86 1 .86 1

Georgia 71,000 2.35 2 .35

Kentucky 69,000 2.29 2 .29

Maryland 279,000 9.25 9 .25

Massachusetts

475,000 15.75 15 .75 1

New Hampshire

142,000 4.71 4 .71 1

New Jersey

180,000 5.97 5 .97 1

New York 332,000 11.01 11 .01

North Carolina

354,000 11.74 11 .74 1

Pennsylvania

433,000 14.36 14 .36

Rhode Island

68,000 2.25 2 .25

South Carolina

206,000 6.83 6 .83 1

Vermont 86,000 2.85 2 .85 1

Virginia 631,000 20.92 20 .92 1

Total 3,619,000

120 111 9 9

State Pop. S. Quota (S1)

L. Quota (S2)

Frac’l Part

Surplus

Final App. (S3)

Connecticut

237,000

7.86 7 .86 1 8

Delaware 56,000 1.86 1 .86 1 2

Georgia 71,000 2.35 2 .35 2

Kentucky 69,000 2.29 2 .29 2

Maryland 279,000

9.25 9 .25 9

Massachusetts

475,000

15.75 15 .75 1 16

New Hampshire

142,000

4.71 4 .71 1 5

New Jersey

180,000

5.97 5 .97 1 6

New York 332,000

11.01 11 .01 11

North Carolina

354,000

11.74 11 .74 1 12

Pennsylvania

433,000

14.36 14 .36 14

Rhode Island

68,000 2.25 2 .25 2

South Carolina

206,000

6.83 6 .83 1 7

Vermont 86,000 2.85 2 .85 1 3

Virginia 631,000

20.92 20 .92 1 21

Total 3,619,000

120 111 9 9 120

The Quota RuleThe Quota Rule

A state’s fair apportionment should either be its upper quota or its lower quota.

There are two ways to violate this rule:1. A state could end up with an apportionment smaller than its lower quota--a lower-quota violation.2. A state could end up with with an apportionment larger than its upper quota--an upper quota violation.

Hamilton’s MethodHamilton’s Method

Hamilton’s method satisfies the quota rule.

Hamilton’s MethodHamilton’s Method

Hamilton’s method satisfies the quota rule.

So what exactly is wrong with the method?

Hamilton’s MethodHamilton’s Method

Hamilton’s method satisfies the quota rule.

So what exactly is wrong with the method?

One problem is that it is not neutral--it consistently favors large states over smaller ones. . .

. . .it also can lead to several paradoxes, the first of which is. . .

§ 4.4 The Alabama Paradox§ 4.4 The Alabama Paradox

The Alabama ParadoxThe Alabama Paradox

The Alabama Paradox occurs when the addition of extra seats to be apportioned leads a state to lose one of its seats.

Example:Example: (example 4.5, pg 144) A small country consists of three states: A, B and C. Consider the Hamilton apportionments when M = 200 and M = 201.

State

Population

Standard Quota (M = 200)

Apportionment

A 940

B 9030

C 10,030

Total

20,000State

Population

Standard Quota (M = 201)

Apportionment

A 940

B 9030

C 10,030

Total

20,000

Example:Example: (example 4.5, pg 144) A small country consists of three states: A, B and C. Consider the Hamilton apportionments when M = 200 and M = 201.

State

Population

Standard Quota (M = 200)

Apportionment

A 940 9.4 10

B 9030 90.3 90

C 10,030 100.3 100

Total

20,000 200.00 200State

Population

Standard Quota (M = 201)

Apportionment

A 940 9.45 9

B 9030 90.75 91

C 10,030 100.80 101

Total

20,000 201.00 201