1. Math 210G Mathematics Appreciation Dr. Joe Lakey Lecture 5: Su Voto es Su Voz

2. [The president is elected by ] A.[Popular vote] B.[Electoral college] C.[Who has the most money] D.[Who has the most popular running mate]

3. Sarah Palin = Tina Fey?

5. Electoral college Each state is allocated as many electors as it has Representatives and Senators in the United States Congress.

6. 2004: Kerry v Bush

7. [Who ran against G.W. Bush in 2000] A.[Clinton] B.[Hart] C.[Quail] D.[Gore]

8. 270 to Win270 to Win 2000: Gore V Bush

9. The infamous butterfly ballot

10. Florida election tallies (2000) George W. Bush (W) 2,912,790 (50,456,002) 48.850Republican Al Gore2,912,253 (50,999,897) 48.841Democratic Ralph Nader97,421 (2,882,955) 1.633Green

12. The electoral college

13. Battleground states NV (5, bare dem) CO (9, bare dem) NM (5, weak dem) MO (11, barely GOP) IN (11, barely GOP) OH (20, weak dem) VA (13, barely dem FL (27, barely dem) NH (4, barely dem) NC (15, tied)

14. For McCain to win… 103 strong GOP + 60 weak GOP=163 + 22 barely GOP = 185 + 15 tied =200 Barely dem: 78 = 278

15. Historical observation… GOP almost always wins “toss-ups” This means GOP would win…all weakly +barely GOP+tied +FL These would put at 227 If we add OH… 247 McCain needs 23 from… NV (5, bare dem), CO (9, bare dem), NM (5, weak dem),VA (13, barely dem),NH (4, barely dem)

17. (Penrose)-Banzhaf-(Coleman) power index Banzhaf, John F. (1965), "Weighted voting doesn't work: A mathematical analysis", Rutgers Law Review 19(2): 317-343 Example (Game Theory and Strategy P. D. Straffin ): [6; A:4, B:3, C:2, D:1] 6 votes to pass, possible majorities: AB, AC, ABC, ABD, ACD, BCD, ABCD 12 total swing votes. A = 5/12 B = 3/12 C = 3/12 D = 1/12

18. The Banzhaf Power Index: a mathematical representation of how likely a single state would be able to swing the vote Larger states have more power Is the electoral college fair? Does it reflect popular opinion?

19. The Banzhaf Power Index (Bachrach et al 08) Pivotal (critical) agent in a winning coalition is an agent that causes the coalition to lose when removed from it The Banzhaf Power Index of an agent is the portion of all coalitions where the agent is pivotal (critical)

20. The Shapley-Shubik Index The portion of all permutations where the agent is pivotal Direct application of the Shapley value for simple coalitional games

21. Banzhaf calculator for electoral college

22. Swing Vote 2008 Link

23. Daily electoral map “Conditional expectation” How does the power index change when we fix the weights for all states not considered battleground states? Can New Mexico determine the outcome of the election?

24. Historical observation… GOP almost always wins “toss-ups” This means GOP would win…all weakly +barely GOP+tied +FL These would put at 227 If we add OH… 247 McCain needs 23 from… NV (5, bare dem), CO (9, bare dem), NM (5, weak dem),VA (13, barely dem),NH (4, barely dem)

25. Banzhaf calculation Can NM swing the vote? [23; VA(13), CO(9), NV(5), NM(5), NH(4)]

26. VA+CO forms a winning coalition [23; VA(13), CO(9), NV(5), NM(5), NH(4)] A.True B.False

27. All but VA forms a winning coalition [23; VA(13), CO(9), NV(5), NM(5), NH(4)] ] A.True B.False

28. [If you were to vote today, who would you choose for president] A.McCain/Palin B.Obama/Biden C.Cynthia McKinney/Rosa Clemente (Green) D.Bob Barr / Wayne Allen Root (Libertarian) E.Other or Undecided

29. [(MALES ONLY) Who would you choose for president today] A.McCain/Palin B.Obama/Biden C.Cynthia McKinney/Rosa Clemente (Green) D.Bob Barr / Wayne Allen Root (Libertarian) E.Other or Undecided

30. [(FEMALES ONLY) Who would you choose for president today] A.McCain/Palin B.Obama/Biden C.Cynthia McKinney/Rosa Clemente (Green) D.Bob Barr / Wayne Allen Root (Libertarian) E.Other or Undecided

31. [Does your vote matter?] A.Yes B.No

32. Swing votes

34. Is election fraud possible in America? http://www.scoop.co.nz/stories/HL0310/S0 0211.htm

35. Voting systems

36. Plurality voting system Plurality voting is used in 43 of the 191 countries in the United Nations for either local or national elections. In single winner plurality voting, each voter is allowed to vote for only one candidate, and the winner of the election is whichever candidate represents a plurality of voters, that is, whoever received the largest number of votes. it is however very contentious to draw district boundary lines in this system Plurality voting is based on minimal information

37. Example: class president election ( compare to Bush, Gore, Nader ) The election for class president Each class has a president, who sits on a school council. Further assume that, in this imaginary school. Male and female students disagree on many issues; students prefer to vote for candidates of their gender. Three candidates: Amy, Brian and Cathy. Each class member gets a ballot, with these three names on it. Each voter must put an "X" by one of the names on their ballot. Votes for Amy, for Brian, and for Cathy placed in separate piles.

38. CandidateAmyBrianCathy # votes111613

39. Brian Wins with only 40% of the vote Electors only vote once

40. Plurality voting Suppose that candidates are ranked (1-3). Then Brian might be the favorite of fewer than half the voters. In some systems a runoff election among the top placing voters is called for.

41. advantages/disadvantages OMOV Constituency Tactical voting Party effects (block voting) Wasted votes (< majority) Manipulation

42. Multiple step voting Runoffs Diminish tactical voting Majority rule (if enough steps) Voter burnout

43. Single transferable vote: a compromise Here’s an example: The student council wants to organize a rock concert A list of 5 bands is considered as candidates but the council can only afford 3 bands. There are twenty council members who list their preferences

44. Only first two preferences shown # council members xxxxxxxxxx xx 1 st preference The ShinsThe KillsFiery Furnaces Fujiya & Miyagi The Bug 2 nd preference The ShinsFujiya & Miyagi The Bug

45. Setting the quota Droop quota (votes/(seats+1))+1 =20/4+1=6

46. Finding the winners Any candidate who has reached or exceeded the required quota is declared elected If not enough candidates have been elected, the count continues. If a candidate has more votes than the quota, then their surplus is transferred to other candidates according to the next preference on each voter's ballot. If no one meets the quota, the candidate with the fewest votes is eliminated and their votes are transferred. Repeat from first step until the seats are filled

47. Round 1 Fiery furnace meet the quota. They are chosen

48. Round 2 Furnace excess transferred to Fujiya and Bug based on second choices. No quota. The Kills eliminated

49. Round 3 Kills votes transferred to second choice. Shins reach quota; no extra votes

50. Round 4 No remaining candidate meets quota. The Bug eliminated

52. Candidat e The ShinsThe KillsFiery Furnaces Fujiya Miyagi The Bug Round 1xxxxxxxxxx xx Furnaces meet quota; elected Round 2xxxxxxxxxx xx xxxx xxxx Furnace excess transferred to Fujiya and Bug based on second choices. No quota. Kills eliminated Round 3xxxx xx xxxx xx xxxx xxxx Kills votes transferred to second choice. Shins reach quota; no extra votes Round 4xxxx xx xxxx xx xxxx xxxx No remaining candidate meets quota. Bug eliminated

53. Call for nominations I’m going to conduct a popularity poll I need six (6) nominations for “Favorite Bands of Math 210” Prior “American Idol” winners not allowed Your homework: figure out the “top 3” bands based on the STV method

54. Recap Mathematics: seeks optimal solution Voting: optimally represent public opinion No voting system is perfect Outcome often depends on system employed

55. Lattice models for “opinion” Renormalization in physics Ising/Potts model applet: renormalization group algorithmIsing/Potts model applet: renormalization group algorithm