What�s New in US Math-Science Education

							 The Mathematics of the
 Electoral College (Part II)


E. Arthur Robinson, Jr.



                      Dec 3, 2010
 European Economic Community of
 1958. 12 votes to win.

An example of “weighted voting”

  Country       Votes
  France        4
  Germany       4
  Italy         4
  Belgium       2
  Netherlands   2
  Luxembourg    1
 European Economic Community of
 1958. 12 votes to win.

An example of “weighted voting”

  Country       Votes   Banzhaf power
  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
Electoral votes 2004, 2008
Electoral votes 2004, 2008
    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: about 4,000,000
       Wyoming: about 400,000
   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: about 4,000,000
       Wyoming: about 400,000
   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
       California: about 34,000,000
       Wyoming: about 400,000
   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


advantaged
                                            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
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