# Math 210G Mathematics Appreciation Dr. Robert Smits by KYSdIX0L

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```									           Math 210G
Mathematics Appreciation
Dr. Joe Lakey
Lecture 9: 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]
• SNL on Romney – Obama
Sarah Palin = Tina Fey?
Electoral college

• Each state is allocated as many electors
as it has Representatives and Senators in
the United States Congress.
Predicting outcomes
• 538
• Vegas oddsmaker's view
2004: Kerry v Bush
2008: Obama v McCain
[Who ran against G.W. Bush in
2000]
A.   [Clinton]
B.   [Hart]
C.   [Quail]
D.   [Gore]
Close U.S. Presidential
elections
Close U.S. Presidential
elections
• Close presidential races are nothing new
• Electoral college history: idea based on
lack of mass media
• Three times elected president lost popular
vote (Hayes, Harrison, G.W. Bush)
• Changing the system not perceived to be
in the best interest of the states.
270 to Win 2000: Gore V Bush
The infamous butterfly ballot
Florida election tallies (2000)
George W.     2,912,790      48.850   Republican
Bush (W)      (50,456,002)

Al Gore       2,912,253      48.841   Democratic
(50,999,897)

(2,882,955)
The electoral college
Historical observation…
• GOP almost always wins “toss-ups”
• Counting these puts Romney at 257 and Obama
at 281.
• If Romney were to win Ohio, this would put
Romney at 257+18=275 and Obama at 281-
18=263.
• Obama strategy should be… (1) Ohio, Ohio
Ohio and try to nail down (2) try to swing
Wisconsin (4) don’t forget CA.
(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
• 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
• Applies in situations in which one vote
might precede another.
• Banzhaf calculator for electoral college
• General Banzhaf calculator
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?
Battleground states 2012
• RCP Poll Average Electoral Votes States Obama
Romney                        Obama Romney
• Colo. 47.6% 47.8%                    09
• Fla. 47.0% 48.7%                    0 29
• Iowa 48.8% 46.8%                    60
• Nev. 49.5% 46.8%                    60
• N.H. 48.2% 47.4%                    40
• N.C. 44.7% 50.3%                    0 15
• Ohio 47.9% 45.8%                    18 0
• Va. 48.0% 48.0%                     0 13
• Wis. 49.5% 46.8%                    10 0
• Leaning/Likely State Votes          237 191
• Total Overall Votes                 281 257
Banzhaf calculation
• Can NM swing the vote?
• [37; CO(9) FL (29) IA (6) NV (6) NH (4)
NC (15) H (18) VA (13), WI (10), NM(5)]

• Other states: 233 to 191. Obama needs 37
from swing + NM
VA+OH+NM forms a winning coalition for
Obama
[37; CO(9) FL (29) IA (6) NV (6) NH (4) NC
(15) H (18) VA (13), WI (10), NM(5)]

A. True
B. False
All but FL and NC and VA forms a winning
coalition for Obama
[37; CO(9) FL (29) IA (6) NV (6) NH (4) NC
(15) H (18) VA (13), WI (10), NM(5)]
A. True
B. False
[If you were to vote today, who
would you choose for president]
A.   Gary Johnson/Jim Gray
B.   Romney/Ryan
C.   Obama/Biden
D.   Other
E.   Undecided
[Females only: If you were to vote
today, who would you choose for
president]
A.   Gary Johnson/Jim Gray
B.   Romney/Ryan
C.   Obama/Biden
D.   Other
E.   Undecided
[Males only: If you were to vote
today, who would you choose for
president]
A.   Gary Johnson/Jim Gray
B.   Romney/Ryan
C.   Obama/Biden
D.   Other
E.   Undecided