the mathematics of the electoral college (part ii)
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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”. European Economic Community of 1958. 12 votes to win. . An example of “weighted voting”. - PowerPoint PPT PresentationTRANSCRIPT
The Mathematics of the Electoral College (Part II)
E. Arthur Robinson, Jr.
Dec 3, 2010
European Economic Community of 1958. 12 votes to win.
Country VotesFrance 4Germany 4Italy 4Belgium 2Netherlands 2Luxembourg 1
An example of “weighted voting”
European Economic Community of 1958. 12 votes to win.
Country Votes Banzhaf powerFrance 4 10Germany 4 10Italy 4 10Belgium 2 6Netherlands 2 6Luxembourg 1 0
An example of “weighted voting”
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.
How does electoral college work?
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 of ways this can happen is
€
(2N)!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
Now you can choose A or B
€
(2N)!N!N!
€
2(2N)!N!N!
Probability you make a difference
Total number of ways 2N+1 voters can vote
Probability that you are the critical voter
€
22N +1
€
p =2
(2N)!N!N! ⎛ ⎝ ⎜
⎞ ⎠ ⎟
22N +1
Stirling’s formula
€
N!≈ N Ne−N 2πN
Banzhaf’s Stirling’s Formula estimate
€
N!≈ N Ne−N 2πN
€
p =2 / π( )N
=KN
Banzhaf’s Conclusion
€
p =2 / π( )N
=KN
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 2Nwhere 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
€
B = pq ≈ 2KLNN
= 2KL N
Banzhaf’s conclusion
The probability that a voter in a state with population N is critical in the Presidential Election is
Voters in the big states benefit the
most.
€
B = pq ≈ 2KLNN
= 2KL N
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
CA 54 3.344
NY 33 2.394
TX 32 2.384
FL 25 2.108
PA 23 2.018
IL 22 1.965
OH 21 1.923
MI 18 1.775
NC 14 1.629
NJ 15 1.617
VA 13 1.564
GA 13 1.529
IN 12 1.524
WA 11 1.49
TN 11 1.489
WI 11 1.486
MA 12 1.463
MO 11 1.453
MN 10 1.428
MD 10 1.366
OK 8 1.346
AL 9 1.337
WY 3 1.327
CT 8 1.317
CO 8 1.315
LA 9 1.308
MS 7 1.302
SC 8 1.278
IA 7 1.253
AZ 8 1.247
KY 8 1.243
OR 7 1.239
NM 5 1.211
AK 3 1.205
VT 3 1.192
RI 4 1.19
ID 4 1.188
NE 5 1.186
AR 6 1.167
DC 3 1.148
KS 6 1.137
UT 5 1.135
HI 4 1.132
NH 4 1.132
ND 3 1.118
WV 5 1.113
DE 3 1.095
NV 4 1.087
ME 4 1.076
SD 3 1.071
MT 3 1
State ElecVote Voter BPI
Conclusion:Voters in larger states (not smaller states) are the ones advantaged by the electoral college
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