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. CountryVotes France4 Germany4 Italy4 Belgium2 Netherlands2 Luxembourg1 An example of “weighted voting”

3. European Economic Community of 1958. 12 votes to win. CountryVotesBanzhaf power France410 Germany410 Italy410 Belgium26 Netherlands26 Luxembourg10 An example of “weighted voting”

4.  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?

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 three times 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 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.

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. CA543.344 NY332.394 TX322.384 FL252.108 PA232.018 IL221.965 OH211.923 MI181.775 NC141.629 NJ151.617 VA131.564 GA131.529 IN121.524 WA111.49 TN111.489 WI111.486 MA121.463 MO111.453 MN101.428 MD101.366 OK81.346 AL91.337 WY31.327 CT81.317 CO81.315 LA91.308 MS71.302 SC81.278 IA71.253 AZ81.247 KY81.243 OR71.239 NM51.211 AK31.205 VT31.192 RI41.19 ID41.188 NE51.186 AR61.167 DC31.148 KS61.137 UT51.135 HI41.132 NH41.132 ND31.118 WV51.113 DE31.095 NV41.087 ME41.076 SD31.071 MT31 State ElecVote Voter BPI Conclusion: Voters in larger states (not smaller states) are the ones advantaged by the electoral college

32. Textbook