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The Mathematics of the Electoral College (Part II)

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”.

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The Mathematics of the Electoral College (Part II)

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

  2. European Economic Community of 1958. 12 votes to win. An example of “weighted voting”

  3. European Economic Community of 1958. 12 votes to win. An example of “weighted voting”

  4. 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.

  5. The Electoral Map

  6. The Election of 2008

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

  8. Is Electoral College weighted voting? • Yes --- if you think of states as voters. • But…

  9. Is Electoral College weighted voting? • Yes --- if you think of states as voters. • But… • No --- if you think of people as voters.

  10. 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

  11. 2000 Census

  12. Electoral votes 2004, 2008

  13. Electoral votes 2004, 2008 In descending order

  14. 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 threetimes that of California • Electoral College favors small states

  15. 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.

  16. 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

  17. 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

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

  19. Stirling’s formula

  20. Banzhaf’s Stirling’s Formula estimate

  21. Banzhaf’s Conclusion Voters in small states do fare better in their state elections, but by less than might be expected (!!)

  22. 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.

  23. 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.

  24. Banzhaf’s conclusion • The probability that a voter in a state with population N is critical in the Presidential Election is

  25. 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.

  26. 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.

  27. 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.

  28. 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.

  29. 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

  30. Bobby Ullman’s calculation

  31. State ElecVote Voter BPI Conclusion: Voters in larger states (not smaller states) are the ones advantaged by the electoral college

  32. Textbook

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