The Mathematics of the Electoral College (Part II) E. Arthur Robinson, Jr. Dec 3, 2010

European Economic Community of votes to win. CountryVotes France4 Germany4 Italy4 Belgium2 Netherlands2 Luxembourg1 An example of “weighted voting”

European Economic Community of votes to win. CountryVotesBanzhaf power France410 Germany410 Italy410 Belgium26 Netherlands26 Luxembourg10 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 (23 rd 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

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

Probability you make a difference  Total number of ways 2N+1 voters can vote  Probability that you are the critical voter

Stirling’s formula

Banzhaf’s Stirling’s Formula estimate

Banzhaf’s Conclusion 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

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.

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 NY TX FL PA IL OH MI NC NJ VA GA IN WA TN WI MA MO MN MD OK AL WY CT CO LA MS SC IA AZ KY OR NM AK VT RI41.19 ID NE AR DC KS UT HI NH ND WV DE NV ME SD MT31 State ElecVote Voter BPI Conclusion: Voters in larger states (not smaller states) are the ones advantaged by the electoral college

Textbook