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§ 4.7 - 4.8 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. 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 PLANET ANDOADAM’S EARTH TELLA VULCA TOTAL THE SEATS USING METHOD. RIA R N POPULATI 16.2 16.1 28.3 8.9 69.5 ON in billions STD. 32.4 QUOTA MODIFIED QUOTA POP. D FINAL APPORTIO NMENT 32.2 56.6 17.8 139 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 PLANET ANDOADAM’S EARTH TELLA VULCA TOTAL THE SEATS USING METHOD. RIA R N POPULATI 16.2 16.1 28.3 8.9 69.5 ON in billions STD. 32.4 QUOTA MODIFIED 32.02 QUOTA POP. .5060 FINAL 33 APPORTIO 32.2 56.6 17.8 139 31.82 55.93 17.59 137.35 32 56 18 139 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 Make D smaller T<M Start: Guess D ( D < SD ). Computatio n: 1. Divide State Populations by D. 2. Round Numbers Up. Make D larger. 3. Add numbers. T=M End 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. 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 PLANET ANDOWEBSTER’S EARTH TELLA VULCA TOTAL THE SEATS USING METHOD. RIA R N POPULATI 16.2 16.1 28.3 8.9 69.5 ON in billions STD. 32.4 QUOTA MODIFIED QUOTA POP. D FINAL APPORTIO NMENT 32.2 56.6 17.8 139 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 PLANET ANDOWEBSTER’S EARTH TELLA VULCA TOTAL THE SEATS USING METHOD. RIA R N POPULATI 16.2 16.1 28.3 8.9 69.5 ON in billions STD. 32.4 QUOTA MODIFIED 32.4 QUOTA POP. 0.5 FINAL 32 APPORTIO NMENT 32.2 56.6 17.8 139 32.2 56.6 17.8 139 32 57 18 139 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 Make D smaller T<M Start: Guess D ( D < SD ). Computatio n: 1. Divide State Populations by D. 2. Round Numbers Convention ally. Make D larger. 3. Add numbers. T=M End 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 A Final Comment: The Balinsky-Young Impossibility Theorem In other words, ‘fairness’ and proportional representation are incompatible ideas.