Hamilton s and Jefferson s Methods

Transcription

1 Hamilton s and Jefferson s Methods Lecture 15 Sections Robb T. Koether Hampden-Sydney College Mon, Feb 23, 2015 Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

2 1 Definitions 2 Hamilton s Method 3 Jefferson s Method 4 Examples 5 Assignment Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

3 Outline 1 Definitions 2 Hamilton s Method 3 Jefferson s Method 4 Examples 5 Assignment Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

4 Apportionment Problems Definition (Apportionment Problem) The classic apportionment problem involves a representative body where each state is given a certain number of seats, according to the state s population. Let N be the number of states. Let M the number of seats. Let p 1, p 2, p 3,..., p N be the states populations. Let P = p 1 + p 2 + p p N, the total population. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

5 Definitions Definition (Standard Divisor) The standard divisor (SD) is P M. It represents the number of people that each seat represents. SD = P M. Definition (Standard Quota) The standard quota of a state is the exact fractional number of seats it should get for its fair share. It is computed as ( pi ) q i = M = p i P SD. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

6 Definitions Definition (Lower and Upper Quotas) The lower quota and the upper quota for a state are the two whole numbers nearest the standard quota for that state. If the standard quota happens to be a whole number, then the lower and upper quotas are the same. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

7 Example Example (Example) Suppose that three states A, B, and C, have populations 3 million, 6 million, and 7 million. The total population is P = 16 million. Suppose there are 50 seats to be apportioned. Find the standard divisor SD. Find each states standard quota, q 1, q 2, q 3. Find the lower and upper quotas for each state. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

8 Outline 1 Definitions 2 Hamilton s Method 3 Jefferson s Method 4 Examples 5 Assignment Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

9 Hamilton s Method Definition (Hamilton s Method) 1 Calculate the standard divisor SD. 2 Calculate each state s standard quota q i. 3 Round each one down to the lower quota L i. 4 Initially, give each state that many seats. 5 Distribute the surplus to the states with the largest fractional parts. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

10 Example Example (Example) Apply Hamilton s method to the three states A, B, and C, with populations 3 million, 6 million, and 7 million and 50 seats to be apportioned. We found SD = q 1 = 9.375, q 2 = 18.75, and q 3 = The lower quotas are 9, 18, and 21. The lower quotas add up to 48. So the surplus is = 2. Which states get the 2 extra seats? Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

11 Example VA, NC, MD, WV Example The populations of VA, NC, MD, and WV, in thousands, are 8001, 9535, 5774 and 1853 people, respectively. The total number of seats apportioned to those states is 35. Use Hamilton s method to determine how many seats each state should get. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

12 Outline 1 Definitions 2 Hamilton s Method 3 Jefferson s Method 4 Examples 5 Assignment Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

13 Jefferson s Method Definition (Jefferson s Method) By Jefferson s method, instead of the standard divisor, we use a modified divisor and recalculate until the lower quotas add up to M. As long as the surplus is positive, we try a smaller modified divisor. As long as the surplus is negative, we try a larger modified divisor. Jefferson s method was used from 1791 until 1842, when it was replaced by Webster s method. Jefferson s method involves repeated guesses until we find a number that works. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

14 Outline 1 Definitions 2 Hamilton s Method 3 Jefferson s Method 4 Examples 5 Assignment Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

15 Example Example (Example) Apply Jefferson s method to the three states A, B, and C, with populations 3 million, 6 million, and 7 million and 50 seats to be apportioned. We found SD = and q 1 = 9.375, q 2 = 18.75, and q 3 = The lower quotas are 9, 18, and 21, which add up to 48. Should the modified divisor be larger or smaller than ? Find one that works. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

16 Example VA, NC, MD, WV Example The populations of VA, NC, MD, and WV, in thousands, are 8001, 9535, 5774 and 1853 people, respectively. The total number of seats apportioned to those states is 35. Use Jefferson s method to determine how many seats each state should get. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

17 Outline 1 Definitions 2 Hamilton s Method 3 Jefferson s Method 4 Examples 5 Assignment Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18

18 Assignment Assignment Chapter 4: (Hamilton s method) Exercises 11, 12, 13, 14. Chapter 4: (Jefferson s method) Exercises 21, 23, 24, 26. Robb T. Koether (Hampden-Sydney College) Hamilton s and Jefferson s Methods Mon, Feb 23, / 18