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Math 210G Mathematics Appreciation Dr. Joe Lakey Lecture 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]. Sarah Palin = Tina Fey?. Electoral college.

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the president is elected by
[The president is elected by ]
  • [Popular vote]
  • [Electoral college]
  • [Who has the most money]
  • [Who has the most popular running mate]
electoral college
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
[Who ran against G.W. Bush in 2000]
  • [Clinton]
  • [Hart]
  • [Quail]
  • [Gore]
battleground states
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
For McCain to win…
  • 103 strong GOP + 60 weak GOP=163
  • + 22 barely GOP = 185
  • + 15 tied =200
  • Barely dem: 78 = 278
historical observation
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
(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
slide18

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
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 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
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 observation24
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
Banzhaf calculation
  • Can NM swing the vote?
  • [23; VA(13), CO(9), NV(5), NM(5), NH(4)]
if you were to vote today who would you choose for president
[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
[(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
[(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
is election fraud possible in america
Is election fraud possible in America?
  • http://www.scoop.co.nz/stories/HL0310/S00211.htm
plurality voting system
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
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
Brian Wins
  • with only 40% of the vote
  • Electors only vote once
plurality voting
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
advantages/disadvantages
  • OMOV
  • Constituency
  • Tactical voting
  • Party effects (block voting)
  • Wasted votes (< majority)
  • Manipulation
multiple step voting
Multiple step voting
  • Runoffs
  • Diminish tactical voting
  • Majority rule (if enough steps)
  • Voter burnout
single transferable vote a compromise
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
Setting the quota
  • Droop quota
  • (votes/(seats+1))+1 =20/4+1=6
finding the winners
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
Round 1
  • Fiery furnace meet the quota. They are chosen
round 2
Round 2
  • Furnace excess transferred to Fujiya and Bug based on second choices. No quota. The Kills eliminated
round 3
Round 3
  • Kills votes transferred to second choice. Shins reach quota; no extra votes
round 4
Round 4
  • No remaining candidate meets quota. The Bug eliminated
call for nominations
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
recap
Recap
  • Mathematics: seeks optimal solution
  • Voting: optimally represent public opinion
  • No voting system is perfect
  • Outcome often depends on system employed
lattice models for opinion
Lattice models for “opinion”
  • Renormalization in physics
  • Ising/Potts model applet: renormalization group algorithm