2. § 4.3 Hamilton’s Method A MERICA (THE BOOK) § 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.” - A MERICA (THE BOOK)

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

4. PLANETANDO RIA EART H TELLA R VULCA N TOTAL POPULA TION in billions 16.216.128.38.969.5 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.

5. PLANETANDO RIA EART H TELLA R VULCA N TOTAL POPULA TION in billions 16.216.128.38.969.5 STANDA RD QUOTA 32.432.256.617.8139 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.

6. PLANETANDO RIA EART H TELLA R VULCA N TOTAL POPULA TION in billions 16.216.128.38.969.5 STANDA RD QUOTA 32.432.256.617.8139 LOWER QUOTA 32 5617137 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.

7. PLANETANDOR IA EARTHTELLA R VULCA N TOTAL POPULAT ION in billions 16.216.128.38.969.5 STANDAR D QUOTA 32.432.256.617.8139 LOWER QUOTA 32 5617137 FRACTIO NAL PART.4.2.6.82 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.

8. PLANETANDOR IA EARTHTELLA R VULCA N TOTAL POPULAT ION in billions 16.216.128.38.969.5 STANDAR D QUOTA 32.432.256.617.8139 LOWER QUOTA 32 5617137 FRACTIO NAL PART.4.2.6.82 EXTRA SEATS 11 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.

9. PLANETANDO RIA EARTHTELLA R VULCA N TOTAL POPULATI ON in billions 16.216.128.38.969.5 STD. QUOTA 32.432.256.617.8139 LOWER QUOTA 32 5617137 FRACTION AL PART.4.2.6.82 EXTRA SEATS 11 FINAL APPORTIO NMENT 32 5718139 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.

10. Hamilton’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.

11. StatePop. (est.)Std. Quota (S1) Lower Quota (S2) Connectic ut 237,0007.86 Delaware56,0001.86 Georgia71,0002.35 Kentucky69,0002.29 Maryland279,0009.25 Massachu setts 475,00015.75 New Hampshire 142,0004.71 New Jersey 180,0005.97 New York332,00011.01 North Carolina 354,00011.74 Pennsylva nia 433,00014.36 Rhode Island 68,0002.25 South Carolina 206,0006.83 Vermont86,0002.85 Virginia631,00020.92 Total3,619,00 0 120 Example: If the House of Representatives had been apportioned under Alexander Hamilton’s 1790s plan (with M = 120 seats) we would have the following:

12. StatePop. (est.)Std. Quota (S1) Lower Quota (S2) Connectic ut 237,0007.867 Delaware56,0001.861 Georgia71,0002.352 Kentucky69,0002.292 Maryland279,0009.259 Massachu setts 475,00015.7515 New Hampshire 142,0004.714 New Jersey 180,0005.975 New York332,00011.0111 North Carolina 354,00011.7411 Pennsylva nia 433,00014.3614 Rhode Island 68,0002.252 South Carolina 206,0006.836 Vermont86,0002.852 Virginia631,00020.9220 Total3,619,00 0 120111

13. StatePop. (est.)Std. Quota (S1) Lower Quota (S2) Frac’l Part Connectic ut 237,0007.867.86 Delaware56,0001.861.86 Georgia71,0002.352.35 Kentucky69,0002.292.29 Maryland279,0009.259.25 Massachu setts 475,00015.7515.75 New Hampshire 142,0004.714.71 New Jersey 180,0005.975.97 New York332,00011.0111.01 North Carolina 354,00011.7411.74 Pennsylva nia 433,00014.3614.36 Rhode Island 68,0002.252.25 South Carolina 206,0006.836.83 Vermont86,0002.852.85 Virginia631,00020.9220.92 Total3,619,00 0 1201119

14. StatePop. (est.)Std. Quota (S1) L. Quota (S2) Frac’l Part Surplus Seats Connectic ut 237,0007.867.861 Delaware56,0001.861.861 Georgia71,0002.352.35 Kentucky69,0002.292.29 Maryland279,0009.259.25 Massachu setts 475,00015.7515.751 New Hampshire 142,0004.714.711 New Jersey 180,0005.975.971 New York332,00011.0111.01 North Carolina 354,00011.7411.741 Pennsylva nia 433,00014.3614.36 Rhode Island 68,0002.252.25 South Carolina 206,0006.836.831 Vermont86,0002.852.851 Virginia631,00020.9220.921 Total3,619,00 0 12011199

15. StatePop.S. Quota (S1) L. Quota (S2) Frac’l Part Surplu s Final App. (S3) Connecticu t 237,0007.867.8618 Delaware56,0001.861.8612 Georgia71,0002.352.352 Kentucky69,0002.292.292 Maryland279,0009.259.259 Massachus etts 475,00015.7515.75116 New Hampshire 142,0004.714.7115 New Jersey180,0005.975.9716 New York332,00011.0111.0111 North Carolina 354,00011.7411.74112 Pennsylvan ia 433,00014.3614.3614 Rhode Island 68,0002.252.252 South Carolina 206,0006.836.8317 Vermont86,0002.852.8513 Virginia631,00020.9220.92121 Total3,619,0 00 12011199120

16. The 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.

17. Hamilton’s Method  Hamilton’s method satisfies the quota rule.

18. Hamilton’s Method  Hamilton’s method satisfies the quota rule.  So what exactly is wrong with the method?

19. Hamilton’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...

20. § 4.4 The Alabama Paradox

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

22. 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. Stat e Populati on Standard Quota (M = 200) Apportionmen t A940 B9030 C10,030 Tot al 20,000 Stat e Populati on Standard Quota (M = 201) Apportionmen t A940 B9030 C10,030 Tot al 20,000

23. 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. Stat e Populati on Standard Quota (M = 200) Apportionmen t A9409.410 B903090.390 C10,030100.3100 Tot al 20,000200.00200 Stat e Populati on Standard Quota (M = 201) Apportionmen t A9409.459 B903090.7591 C10,030100.80101 Tot al 20,000201.00201