Transcript 4.7-4.8
§ 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.