# What�s New in US Math-Science Education

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```							 The Mathematics of the
Electoral College (Part II)

E. Arthur Robinson, Jr.

Dec 3, 2010
European Economic Community of

An example of “weighted voting”

France        4
Germany       4
Italy         4
Belgium       2
Netherlands   2
Luxembourg    1
European Economic Community of

An example of “weighted voting”

France        4       10
Germany       4       10
Italy         4       10
Belgium       2       6
Netherlands   2       6
Luxembourg    1       0
How does electoral college work?
   Each state gets votes equal to #House
seats + 2 (=#Senate seats).
   Most states give all their electoral votes to
(plurality) winner of their popular election.
(Determined by state law)
   DC gets 3 votes (23rd Amendment, 1961).
   Electors meet in early January.
The Electoral Map
The Election of 2008
Is Electoral College weighted
voting?

   Yes --- if you think of states as
voters.
Is Electoral College weighted
voting?

   Yes --- if you think of states as
voters.

   But…
Is Electoral College weighted
voting?

   Yes --- if you think of states as
voters.

   But…

   No --- if you think of people as
voters.
Is Electoral College weighted
voting?

   Yes --- if you think of states as
voters.

   But…

   No --- if you think of people as
voters.
   Nevertheless, even in this case you
can estimate Banzhaf power of
voters
2000 Census
In descending order
Conventional wisdom
(plus 2 phenomenon)

   House seats proportional to a
state’s population
   Plus two (+2) for senate seats.
   California 53+2=55
   Wyoming 1+2=3
   Per capita representation of
Wyoming three times that of
California
   Electoral College favors small
states
Banzhaf’s question:

   How likely is a voter to affect the
popular vote in his/her state?

   Clearly, a voter in a small state is
more likely.
You as critical member of winning
coalition

   Candidates A and B.
   Suppose state has population
2N+1.
   You are the +1
   For you to be critical, N voters must
support A and N voters must
support B
    The number (2N)! this can
of ways
happen is
N!N!
You as critical member of winning
coalition

   The number of ways to have N
voters for A and N voters for B is
(2N)!
N!N!
   Now you can choose A or B

(2N)!
2
           N!N!
Probability you make a difference

   Total number of ways 2N+1 voters
can vote
2N 1
2
   Probability that you are the critical
voter
(2N)!
2       
          p
N!N! 
2N 1
2
Stirling’s formula

N! N e   N N
2N
Banzhaf’s Stirling’s Formula
estimate

N! N e  N N
2N

p
 2 /           K
N          N
Banzhaf’s Conclusion

p
 2 /          K
N          N

Voters in small states do
fare better in their state
elections, but by less than
might be expected (!!)
Example
   Alabama is 10 times the size of
Wyoming
   But voters in Wyoming have only
about 3 times the power of voters in
Alabama…
   in their state elections.
Banzhaf’s second approximation

   The probability q that a particular
state is critical in the Electoral
College vote is approximately

q = L 2N
where L is a constant
   This is very approximate at best. It
fails to take the +2 into account.
   But it is a good first step.
Banzhaf’s conclusion

   The probability that a voter in a
state with population N is critical in
the Presidential Election is
N
B  pq  2KL    2KL N
N


Banzhaf’s conclusion

   The probability that a voter in a
state with population N is critical in
the Presidential Election is
N
B  pq  2KL    2KL N
N
   Voters in the big states benefit the
most.


Example
   Alabama is 10 times the size of
Wyoming
   Voters in Wyoming have only about
1/3 the power of voters in
Alabama…
   …in the National election.
Example
   Alabama is 85 times the size of
Wyoming
   But voters in Wyoming have only
about 1/9 times the power of voters
in California…
   in the National election.
But…

   This is somewhat mitigated by the
+2 phenomenon
   Better estimates are needed.
   Exact calculations (like for the EEC
of 1958) are impossible.
   Computer simulations can be used.
Computer approximations

   John Banzhaf, Law Professor, (IBM
360), 1968
   Mark Livinston, Computer Scientist
US Naval Research Lab, (Sun
Workstation), 1990’s.
   Bobby Ullman, High School Student,
(Dell Laptop), 2010
Bobby Ullman’s calculation
State    ElecVote Voter BPI
CA         54   3.344     MS   7   1.302
NY         33   2.394     SC   8   1.278
TX         32   2.384     IA   7   1.253
FL         25   2.108     AZ   8   1.247

Conclusion:
PA         23   2.018     KY   8   1.243
IL         22   1.965     OR   7   1.239
OH         21   1.923
Voters in
NM   5   1.211
MI         18   1.775     AK   3   1.205

larger            NC
NJ
14
15
1.629
1.617
VT   3   1.192

states (not
RI   4    1.19
VA         13   1.564     ID   4   1.188

smaller           GA
IN
13
12
1.529
1.524
NE   5   1.186

states) are
AR   6   1.167
WA         11    1.49     DC   3   1.148

the ones          TN
WI
11
11
1.489
1.486
KS   6   1.137

UT   5   1.135
MA         12   1.463     HI   4   1.132

by the            MO
MN
11
10
1.453
1.428
NH   4   1.132

electoral
ND   3   1.118
MD         10   1.366     WV   5   1.113

college           OK
AL
8
9
1.346
1.337
DE   3   1.095
NV   4   1.087
WY          3   1.327     ME   4   1.076
CT          8   1.317     SD   3   1.071
CO          8   1.315     MT   3      1
LA          9   1.308
Textbook

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