Math 210G Mathematics Appreciation
Dr. Joe LakeyLecture 5: Su Voto es Su Voz
[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]
Sarah Palin = Tina Fey?
Electoral college
• Each state is allocated as many electors as it has Representatives and Senators in the United States Congress.
2004: Kerry v Bush
[Who ran against G.W. Bush in 2000]
A. [Clinton]
B. [Hart]
C. [Quail]
D. [Gore]
270 to Win 2000: Gore V Bush
The infamous butterfly ballot
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
The electoral college
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
• Banzhaf calculator for electoral college
Swing Vote 2008 Link
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)]
A. True
B. False
All but VA forms a winning coalition [23; VA(13), CO(9), NV(5), NM(5), NH(4)]
]A. True
B. False
[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
[(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
[(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
[Does your vote matter?]
A. Yes
B. No
Swing votes
Is election fraud possible in America?
• http://www.scoop.co.nz/stories/HL0310/S00211.htm
Voting systems
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.
Candidate Amy Brian Cathy
# votes 11 16 13
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
Only first two preferences shown
# council members
xxxx xx xxxxxxxx
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
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
Candidate The Shins The Kills Fiery Furnaces
Fujiya Miyagi
The Bug
Round 1 xxxx xx xxxxxxxxxxxx
x x Furnaces meet quota; elected
Round 2 xxxx xx xxxxxx
xxxxx xxx
Furnace excess transferred to Fujiya and Bug based on second choices. No quota. Kills eliminated
Round 3 xxxxxx
xxxxxx
xxxxx xxx
Kills votes transferred to second choice. Shins reach quota; no extra votes
Round 4 xxxxxx
xxxxxx
xxxxx xxx
No remaining candidate meets quota. Bug eliminated
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
• Mathematics: seeks optimal solution
• Voting: optimally represent public opinion
• No voting system is perfect
• Outcome often depends on system employed
Lattice models for “opinion”
• Renormalization in physics
• Ising/Potts model applet: renormalization group algorithm