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AND. Chapter 15. Voting and Apportionment. WHAT YOU WILL LEARN. • Preference tables • Voting methods • Flaws of voting methods • Standard quotas and standard divisors • Apportionment methods • Flaws of apportionment methods. Section 4. Flaws of the Apportionment Methods.

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  1. AND

  2. Chapter 15 Voting and Apportionment

  3. WHAT YOU WILL LEARN • Preference tables • Voting methods • Flaws of voting methods • Standard quotas and standard divisors • Apportionment methods • Flaws of apportionment methods

  4. Section 4 Flaws of the Apportionment Methods

  5. Three Flaws of Hamilton’s Method • The three flaws of Hamilton’s method are: the Alabama paradox, the population paradox, and the new-states paradox. • These flaws apply only to Hamilton’s method and do not apply to Jefferson’s method, Webster’s method, or Adam’s method. • In 1980 the Balinski and Young’s Impossibility Theorem stated that there is no perfect apportionment method that satisfies the quota rule and avoids any paradoxes.

  6. Alabama Paradox • The Alabama paradox occurs when an increase in the total number of items to be apportioned results in a loss of an item for a group.

  7. Office 1 2 3 Total Employees 161 250 489 900 Example: Demonstrating the Alabama Paradox A large company, with branches in three cities, must distribute 30 cell phones to the three offices. The cell phones will be apportioned based on the number of employees in each office shown in the table below.

  8. Example: Demonstrating the Alabama Paradox (continued) • Apportion the cell phones using Hamilton’s method. • Does the Alabama paradox occur using Hamilton’s method if the number of new cell phones increased from 30 to 31? Explain.

  9. Office 1 2 3 Total Employees 161 250 489 900 Standard Quota 5.37 8.33 16.3 Lower Quota 5 8 16 29 Hamilton’s apportionment 6 8 16 30 Example: Demonstrating the Alabama Paradox (continued) • Based on 30 cell phones, the table is as follows: (Note: standard divisor = 900/30 = 30)

  10. Office 1 2 3 Total Employees 161 250 489 900 Standard Quota 5.55 8.61 16.84 Lower Quota 5 8 16 29 Hamilton’s apportionment 5 9 17 31 Example: Demonstrating the Alabama Paradox (continued) • Based on 31 cell phones, the table is as follows: (Note: standard divisor = 900/31 ≈ 29.03)

  11. Example: Demonstrating the Alabama Paradox (continued) • When the number of cell phones increased from 30 to 31, office one actually lost a cell phone, while the other two offices actually gained a cell phone under Hamilton’s apportionment.

  12. Population Paradox • The Population Paradox occurs when group A loses items to group B, even though group A’s population grew at a faster rate than group B’s.

  13. School A B C D E Total Population in 2003 733 1538 933 1133 1063 5400 Population in 2005 733 1539 933 1133 1112 5450 Example: Demonstrating Population Paradox A school district with five elementary schools has funds for 54 scholarships. The student population for each school is shown in the table below.

  14. Example: Demonstrating Population Paradox (continued) • Apportion the scholarships using Hamilton’s method. • If the school wishes to give the same number of scholarships two years later, does a population paradox occur?

  15. School A B C D E Total Population in 2003 733 1538 933 1133 1063 5400 Standard Quota 7.33 15.38 9.33 11.33 10.63 Lower Quota 7 15 9 11 10 52 Hamilton’s apportionment 7 16 9 11 11 54 Solution • Based on the population in 2003, the table is as follows: (Note: standard divisor = 5400/54 = 100)

  16. School A B C D E Total Population in 2005 733 1539 933 1133 1112 5450 Standard Quota 7.26 15.25 9.24 11.23 11.02 Lower Quota 7 15 9 11 11 53 Hamilton’s apportionment 8 15 9 11 11 54 Solution (continued) • Based on the population in 2005, the table is as follows: (Note: standard divisor = 5450/54 ≈ 100.93)

  17. Solution (continued) • In the school district in 2005, school B actually gives one of its scholarships to school A, even though the population in school B actually grew by 1 student and the population in School A remained the same.

  18. New-States Paradox • The new-states paradox occurs when the addition of a new group reduces the apportionment of another group.

  19. Apportionment Method Hamilton Jefferson Adams Webster May violate the quota rule No Yes Yes Yes May produce the Alabama paradox Yes No No No May produce the population paradox Yes No No No Yes No No No May produce the new-states paradox Appointment method favors Large states Large states Small states Small states Summary

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