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 Votes

France 4

Germany 4

Italy 4

Belgium 2

Netherlands 2

Luxembourg 1

An example of “weighted voting”

European Economic Community of 1958. 12 votes to win.

Country Votes Banzhaf power

France 4 10

Germany 4 10

Italy 4 10

Belgium 2 6

Netherlands 2 6

Luxembourg 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=K

N

Banzhaf’s Conclusion

€

p =2 / π( )

N=K

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 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 ≈ 2KLN

N= 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 ≈ 2KLN

N= 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

Textbook