§ 4.7 - 4.8 adams’ method; webster’s method adams’ method the idea: we will use the...
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§ 4.7 - 4.8 Adams’ Method; § 4.7 - 4.8 Adams’ Method; Webster’s MethodWebster’s Method
Adams’ MethodAdams’ Method
The Idea: We will use the Jefferson’s concept of modified divisors, but instead of rounding the modified quotas down we will round them up.
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
MODIFIED QUOTAPOP. D
FINAL APPORTIONMENT
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 ADAM’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
MODIFIED QUOTAPOP. .5060
32.02 31.82 55.93 17.59 137.35
FINAL APPORTIONMENT
33 32 56 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 ADAM’S METHOD.
Adams’ MethodAdams’ Method
Step 1. Find a modified divisor D such that when each state’s modified quota is rounded upward (this number is the upper modified quota) the total is the exact number of seats to be apportioned.
Step 2. Apportion to each state its modified upper quota.
Adams’ Method: Finding Adams’ Method: Finding the Modified Divisorthe Modified Divisor
Start: Guess D ( D < SD ).
End
Make D larger.
Make D smaller
Computation:1. Divide State Populations by D.
2. Round 2. Round Numbers Numbers Up.Up.
3. Add numbers. Let total = T.
T < M
T = M
T > M
Webster’s MethodWebster’s Method
The Idea: We will use an approach similar to both Jefferson’s and Adams’ methods, but we will round the modified quotas conventionally.
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
MODIFIED QUOTAPOP. D
FINAL APPORTIONMENT
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 WEBSTER’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
MODIFIED QUOTAPOP. 0.5
32.4 32.2 56.6 17.8 139
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 WEBSTER’S METHOD.
Webster’s MethodWebster’s Method
Step 1. Find a modified divisor D such that when each state’s modified quota is rounded conventionally (this number is the modified quota) the total is the exact number of seats to be apportioned.
Step 2. Apportion to each state its modified quota.
Webster’s Method: Finding Webster’s Method: Finding the Modified Divisorthe Modified Divisor
Start: Guess D ( D < SD ).
End
Make D larger.
Make D smaller
Computation:1. Divide State Populations by D.2. Round 2. Round Numbers Numbers ConventioConventionally.nally.
3. Add numbers. Let total = T.
T < M
T = M
T > M
A Final Comment: The A Final Comment: The Balinsky-Young Balinsky-Young
Impossibility TheoremImpossibility Theorem Like Jefferson’s Method, the methods of both Adams and Webster are free of paradox. Unfortunately, they both also imitate Jefferson’s Method in that they violate the quota rule.
In 1980, Michel Balinski and H. Peyton Young provided mathematical proof that any apportionment method that does not produce paradox violates the quota rule and that any method that satisfies the quota rule must produce a paradox.
A Final Comment: The A Final Comment: The Balinsky-Young Balinsky-Young
Impossibility TheoremImpossibility Theorem In other words, ‘fairness’ and proportional representation are incompatible ideas.