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
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The president is elected by

1. Popular vote
2. Electoral college
3. Who has the most money
4. Who has the most popular running mate

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Sarah Palin Tina Fey?
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Electoral college

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

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2004 Kerry v Bush
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Who ran against G.W. Bush in 2000

1. Clinton
2. Hart
3. Quail
4. Gore

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270 to Win 2000 Gore V Bush
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The infamous butterfly ballot
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Florida election tallies (2000)
George W. Bush (W) 2,912,790 (50,456,002) 48.850 Republican
Al Gore 2,912,253 (50,999,897) 48.841 Democratic
Ralph Nader 97,421 (2,882,955) 1.633 Green

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The electoral college
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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)

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For McCain to win

• 103 strong GOP 60 weak GOP163
• 22 barely GOP 185
• 15 tied 200
• Barely dem 78 278

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Historical observation

• GOP almost always wins toss-ups
• This means GOP would winall weakly barely
  GOPtied 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)

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(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 A4, B3, C2, D1
• 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

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• 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?

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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)

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The Shapley-Shubik Index

• The portion of all permutations where the agent
  is pivotal
• Direct application of the Shapley value for
  simple coalitional games

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• Banzhaf calculator for electoral college

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Swing Vote 2008 Link
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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?

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Historical observation

• GOP almost always wins toss-ups
• This means GOP would winall weakly barely
  GOPtied 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)

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Banzhaf calculation

• Can NM swing the vote?
• 23 VA(13), CO(9), NV(5), NM(5), NH(4)

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VACO forms a winning coalition 23 VA(13),
CO(9), NV(5), NM(5), NH(4)

1. True
2. False

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All but VA forms a winning coalition 23
VA(13), CO(9), NV(5), NM(5), NH(4)

1. True
2. False

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If you were to vote today, who would you choose
for president

1. McCain/Palin
2. Obama/Biden
3. Cynthia McKinney/Rosa Clemente (Green)
4. Bob Barr / Wayne Allen Root (Libertarian)
5. Other or Undecided

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(MALES ONLY) Who would you choose for president
today

1. McCain/Palin
2. Obama/Biden
3. Cynthia McKinney/Rosa Clemente (Green)
4. Bob Barr / Wayne Allen Root (Libertarian)
5. Other or Undecided

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(FEMALES ONLY) Who would you choose for
president today

1. McCain/Palin
2. Obama/Biden
3. Cynthia McKinney/Rosa Clemente (Green)
4. Bob Barr / Wayne Allen Root (Libertarian)
5. Other or Undecided

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Does your vote matter?

1. Yes
2. No

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Swing votes
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Is election fraud possible in America?

• http//www.scoop.co.nz/stories/HL0310/S00211.htm

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Voting systems
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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

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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.

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Candidate Amy Brian Cathy
votes 11 16 13
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Brian Wins

• with only 40 of the vote
• Electors only vote once

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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.

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advantages/disadvantages

• OMOV
• Constituency
• Tactical voting
• Party effects (block voting)
• Wasted votes (lt majority)
• Manipulation

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Multiple step voting

• Runoffs
• Diminish tactical voting
• Majority rule (if enough steps)
• Voter burnout

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Single transferable vote a compromise

• Heres 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

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Only first two preferences shown
council members xxxx xx xxxx xxxx xxxx x x
1st preference The Shins The Kills Fiery Furnaces Fiery Furnaces Fujiya Miyagi The Bug
2nd preference The Shins Fujiya Miyagi The Bug
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Setting the quota

• Droop quota
• (votes/(seats1))1 20/416

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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

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Round 1

• Fiery furnace meet the quota. They are chosen

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Round 2

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

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Round 3

• Kills votes transferred to second choice. Shins
  reach quota no extra votes

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Round 4

• No remaining candidate meets quota. The Bug
  eliminated

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Candidate The Shins The Kills Fiery Furnaces Fujiya Miyagi The Bug
Round 1 xxxx xx xxxx xxxx xxxx x x Furnaces meet quota elected
Round 2 xxxx xx xxxx xx xxxx x xxx Furnace excess transferred to Fujiya and Bug based on second choices. No quota. Kills eliminated
Round 3 xxxx xx xxxx xx xxxx x xxx Kills votes transferred to second choice. Shins reach quota no extra votes
Round 4 xxxx xx xxxx xx xxxx x xxx No remaining candidate meets quota. Bug eliminated
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Call for nominations

• Im 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

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Recap

• Mathematics seeks optimal solution
• Voting optimally represent public opinion
• No voting system is perfect
• Outcome often depends on system employed

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Lattice models for opinion

• Renormalization in physics
• Ising/Potts model applet renormalization group
  algorithm