Transcript L30

L30. Sensitivity Analysis
Congressional Apportionment
Sensitivity Analysis
Quantifying Fairness
How do you distribute 435 Congressional
seats among the 50 states so that the
ratio of population to delegation size is
roughly the same from state to state?
Quite possibly one of the greatest division problems
of all time!
Quantifying Importance
How do you rank web pages for importance
given that you know the link structure
of the Web, i.e., the in-links and out-links
for each web page.
Quite possibly one of the greatest ranking problems
of all time!
Related Questions
How “close” is a state to losing a
Congressional district because of
population changes?
How to do new or deleted links that
involve a web page affect its PageRank?
Reasoning About Change
Sensitivity analysis:
How does the “answer” change if the
input data changes or if the
assumptions that underlie the
computation change?
VERY IMPORTANT IN SCIENCE & ENGINEERING
An Earlier Example
MyPi = 3.14;
R = 3961.12345;
EarthArea = 4*MyPi*R*R
Math error in MyPi, measurement error in R, model
error in spherical model, rounding error in arithmetic.
Subtext
These examples provide distinct
opportunities to review our
programming techniques.
The Apportionment
Problem
Notation
Number of states:
State populations:
Total Population:
State delegation size:
Number of seats:
n
p(1),…,p(n)
P
d(1),…,d(n)
D
Ideal: Equal Representation
Number of states:
State populations:
Total Population:
State delegation size:
Number of seats:
n
p(1),…,p(n)
P
d(1),…,d(n)
D
P
p ( n)
p(1)

 ... 
d (1)
D
d ( n)
i.e.,
p(i)
d (i ) 
D
P
And so for NY in 2000..
19004973
d ( NewYork ) 
435  29.376
281424177
But delegation size must be a whole number!!!
More Realistic…
Number of states:
State populations:
Total Population:
State delegation size:
Number of seats:
n
p(1),…,p(n)
P
d(1),…,d(n)
D
P
p ( n)
p(1)

 ... 
d (1)
D
d ( n)
Definition
An Apportionment Method determines
delegation sizes d(1),…,d(n) that are
whole numbers so that representation
is approximately equal:
p ( n)
p(1)
 ... 
d (1)
d ( n)
Jefferson Method 1790-1830
Decide on a ``common ratio’’, the ideal
number of constituents per district.
In 1790:
r = 33000
Delegation size for the i-th state is
d(i) = floor( p(i)/r )
State
Connecticut
Delaware
Georgia
Kentucky
Maryland
Massachusetts
New Hampshire
New Jersey
New York
North Carolina
Pennsylvania
Rhode Island
South Carolina
Vermont
Virginia
Pop
236841
55540
70835
68705
278514
475327
141822
179570
331589
353523
432879
68446
206236
85533
630560
Reps
7
1
2
2
8
14
4
5
10
10
13
2
6
2
19
Pop/Reps
33834
55540
35417
34352
34814
33951
35455
35914
33158
35352
33298
34223
34372
42766
33187
Jefferson Method 1790-1830
Population and the chosen common ratio
determine the size of Congress:
Year
1790
1800
1810
1820
1830
p
3615920
4889823
6584255
8969878
11931000
r
33000
33000
35000
40000
47700
D
105
141
181
213
240
Webster Method 1840
d(i) = round( p(i) / 70680 )
instead of
floor
Common
Ratio
Size of Congress Also Determined By Common Ratio
Hamilton Method (1850-1900)
This method fixes the size of
Congress.
Allocations are based on the “ideal
ratio”:
Total Population / Total Number of Seats
The 1850 Case (31 States)
D = 234
r =
21840083 / 234
= 93334
% Round 1 allocation…
for i=1:31
d(i) = floor( p(i)/r )
end
All but 14 of the 234 seats have been given out.
AL
AR
CA
CT
DE
FL
GA
IL
IN
6.798
2.047
1.768
3.973
0.971
0.768
8.073
9.123
10.590
IA
2.059
KY
LA
ME
MD
MA
MI
MS
MO
NH
NJ
NY
9.622
4.498
6.248
5.859
10.655
4.261
5.171
6.933
3.407
5.244
33.186
NC
OH
PA
RI
SC
TN
TX
VT
VI
WI
8.074
21.218
24.769
1.581
5.513
9.717
2.028
3.366
13.207
3.272
State Population / Ideal Ratio
AL
AR
CA
CT
DE
FL
GA
IL
IN
6.798
2.047
1.768
3.973
0.971
0.768
8.073
9.123
10.590
IA
2.059
KY
LA
ME
MD
MA
MI
MS
MO
NH
NJ
NY
9.622
4.498
6.248
5.859
10.655
4.261
5.171
6.933
3.407
5.244
33.186
NC
OH
PA
RI
SC
TN
TX
VT
VI
WI
8.074
21.218
24.769
1.581
5.513
9.717
2.028
3.366
13.207
3.272
floor(State Population / Ideal Ratio)
AL
AR
CA
CT
DE
FL
GA
IL
IN
6.798
2.047
1.768
3.973
0.971
0.768
8.073
9.123
10.590
IA
2.059
KY
LA
ME
MD
MA
MI
MS
MO
NH
NJ
NY
9.622
4.498
6.248
5.859
10.655
4.261
5.171
6.933
3.407
5.244
33.186
NC
OH
PA
RI
SC
TN
TX
VT
VI
WI
8.074
21.218
24.769
1.581
5.513
9.717
2.028
3.366
13.207
3.272
These 14 states most deserve an extra seat
Method of Equal Proportions
This method has been in use since 1940.
For the 2000 apportionment:
n = 50
D = 435
Determine the delegation sizes d(1:50)
Given the state populations p(1:50).
Every State Gets At Least
One District
So start with this:
d = ones(50,1)
Now “deal out” Congressional districts 51
through 435
Now Allocate the Rest…
for k = 51:435
Let i be the index of the
state that most deserves
an additional district.
d(i) = d(i) + 1;
end
Most Deserving?
The Method of Small Divisors
At this point in the “card game” deal a
district to the state having the largest
quotient p(i)/d(i).
Tends to favor big states.
Most Deserving?
The Method of Large Divisors
At this point in the “card game” deal a
district to the state having the largest
quotient p(i)/( d(i) + 1).
Tends to favor small states
Most Deserving?
The Method of Major Fractions
At this point in the “card game” deal a
district to the state having the largest
value of
( p(i)/d(i) + p(i)/(d(i)+1) )/2
Compromise via the Arithmetic Mean
Most Deserving?
The Method of Equal Proportions
At this point in the “card game” deal a
district to the state having the largest
value of
sqrt( p(i)/d(i) * p(i)/(d(i)+1) )
Compromise via the Geometric Mean
Allocation Via Equal Proportions
for k = 51:435
[z,i] = max((p./d).*(p./{1+d)))
d(i) = d(i) + 1;
end
A Sensitivity Analysis
The 435th district was awarded to North
Carolina.
Was that a “close call”? Is there another
state that “almost” won this last district?
Quantify.
Move from NC to UT
NC: 6.4593
UT: 6.4568
Equal Proportion
ranking when dealing out
the last district
North Carolina just beat out Utah for
the last congressional seat.
Can show that if 670 people move
from NC to UT, then NC loses a seat
and UT gains one
Other Questions
If Puerto Rico and/or Washington DC
become states and the total number of
representatives remains at 435, then
what states lose a congressional seat?
If the population of New York remains
fixed and all other states grow by 5%
during the2000-10 decade, then how many
seats will NY lose?
A Useful Structure Array
C = CensusData
Assigns to the structure array C the
apportionments and Census results for the
census years 1890 through 2000.
C(k) houses information pertaining to
The k-th census/apportioment.
C has these Fields
year
The year of the census. (1790,
1800,...,2000).
states
k-by-16 char array that names
existing states during the census.
pop
k-by-1 real array that
specifies the state populations.
reps
k-by-1 real array that specifies
the state apportionments.
Example
Pop = C(10).pop;
Reps = C(10).reps;
P = 0; D = 0;
for i=1:length(pop)
P = P + Pop(i);
D = D + Reps(i);
End
r = P/D
Assigns the ideal ratio for the 10th census
to r.
A Somewhat Related Problem
Gerrymandering:
The Art of
drawing
district
boundaries
So As to
favor
incumbents