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

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

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

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**Math 210G Mathematics AppreciationDr. Joe LakeyLecture 5: Su**Voto es Su Voz**[The president is elected by ]**• [Popular vote] • [Electoral college] • [Who has the most money] • [Who has the most popular running mate]**Electoral college**• Each state is allocated as many electors as it has Representatives and Senators in the United States Congress.**[Who ran against G.W. Bush in 2000]**• [Clinton] • [Hart] • [Quail] • [Gore]**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)**For McCain to win…**• 103 strong GOP + 60 weak GOP=163 • + 22 barely GOP = 185 • + 15 tied =200 • Barely dem: 78 = 278**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)**(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**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?**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)**The Shapley-Shubik Index**• The portion of all permutations where the agent is pivotal • Direct application of the Shapley value for simple coalitional games**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?**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)**Banzhaf calculation**• Can NM swing the vote? • [23; VA(13), CO(9), NV(5), NM(5), NH(4)]**VA+CO forms a winning coalition [23; VA(13), CO(9), NV(5),**NM(5), NH(4)] • True • False**All but VA forms a winning coalition[23; VA(13), CO(9),**NV(5), NM(5), NH(4)]] • True • False**[If you were to vote today, who would you choose for**president] • McCain/Palin • Obama/Biden • Cynthia McKinney/Rosa Clemente (Green) • Bob Barr / Wayne Allen Root (Libertarian) • Other or Undecided**[(MALES ONLY) Who would you choose for president today]**• McCain/Palin • Obama/Biden • Cynthia McKinney/Rosa Clemente (Green) • Bob Barr / Wayne Allen Root (Libertarian) • Other or Undecided**[(FEMALES ONLY) Who would you choose for president today]**• McCain/Palin • Obama/Biden • Cynthia McKinney/Rosa Clemente (Green) • Bob Barr / Wayne Allen Root (Libertarian) • Other or Undecided**[Does your vote matter?]**• Yes • No**Is election fraud possible in America?**• http://www.scoop.co.nz/stories/HL0310/S00211.htm**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**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.**Brian Wins**• with only 40% of the vote • Electors only vote once**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.**advantages/disadvantages**• OMOV • Constituency • Tactical voting • Party effects (block voting) • Wasted votes (< majority) • Manipulation**Multiple step voting**• Runoffs • Diminish tactical voting • Majority rule (if enough steps) • Voter burnout**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**Setting the quota**• Droop quota • (votes/(seats+1))+1 =20/4+1=6**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**Round 1**• Fiery furnace meet the quota. They are chosen**Round 2**• Furnace excess transferred to Fujiya and Bug based on second choices. No quota. The Kills eliminated**Round 3**• Kills votes transferred to second choice. Shins reach quota; no extra votes**Round 4**• No remaining candidate meets quota. The Bug eliminated