§ Adams’ Method; Webster’s Method

Adams’ 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.

PLANETANDO RIA EARTHTELLA R VULCA N TOTAL POPULATI ON in billions STD. QUOTA MODIFIED QUOTA POP.  D FINAL APPORTIO NMENT 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.

PLANETANDO RIA EARTHTELLA R VULCA N TOTAL POPULATI ON in billions STD. QUOTA MODIFIED QUOTA POP.  FINAL APPORTIO NMENT 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’ 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 the Modified Divisor Start: Guess D ( D < SD ). End Make D larger. Make D smaller 2. Round Numbers Up. Computatio n: 1. Divide State Populations by D. 2. Round Numbers Up. 3. Add numbers. Let total = T. T < M T = M T > M

Webster’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.

PLANETANDO RIA EARTHTELLA R VULCA N TOTAL POPULATI ON in billions STD. QUOTA MODIFIED QUOTA POP.  D FINAL APPORTIO NMENT 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.

PLANETANDO RIA EARTHTELLA R VULCA N TOTAL POPULATI ON in billions STD. QUOTA MODIFIED QUOTA POP.  FINAL APPORTIO NMENT 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 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 the Modified Divisor Start: Guess D ( D < SD ). End Make D larger. Make D smaller 2. Round Numbers Convention ally. Computatio n: 1. Divide State Populations by D. 2. Round Numbers Convention ally. 3. Add numbers. Let total = T. T < M T = M T > M

A Final Comment: The Balinsky-Young Impossibility 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 Balinsky-Young Impossibility Theorem  In other words, ‘fairness’ and proportional representation are incompatible ideas.