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