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.