Power Indices, The Electoral College

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Power Indices, The Electoral College The U. S.
Presidential Election of 2004.Presented by D.
Ezra Sidran22C196001 Algorithms, Games and the
WEB.
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The Constitution Of The United StatesArticle II
Section 1. The executive Power shall be vested in
a President of the United States of America. He
shall hold his Office during the Term of four
Years, and, together with the Vice President,
chosen for the same Term, be elected, as
followsEach State shall appoint, in such Manner
as the Legislature thereof may direct, a Number
of Electors, equal to the whole Number of
Senators and Representatives to which the State
may be entitled in the Congress but no Senator
or Representative, or Person holding an Office of
Trust or Profit under the United States, shall be
appointed an Elector.
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The Election of 1800
Aaron Burr
Thomas Jefferson
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The Election of 1800
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The Constitution Of The United StatesXII
Amendment
The Electors shall meet in their respective
states, and vote by ballot for President and
Vice-President they shall name in their ballots
the person voted for as President, and in
distinct ballots the person voted for as
Vice-President, and they shall make distinct
lists of all persons voted for as President, and
of all persons voted for as Vice-President, and
of the number of votes for each, which lists they
shall sign and certify, and transmit sealed to
the seat of the government of the United States,
directed to the President of the Senate The
President of the Senate shall, in the presence of
the Senate and House of Representatives, open all
the certificates and the votes shall then be
counted--The person having the greatest number
of votes for President, shall be the President
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The Constitution Of The United StatesXXIII
Amendment
Section 1. The District constituting the seat of
Government of the United States shall appoint in
such manner as the Congress may direct A number
of electors of President and Vice President equal
to the whole number of Senators and
Representatives in Congress to which the District
would be entitled if it were a State, but in no
event more than the least populous State they
shall be in addition to those appointed by the
States, but they shall be considered, for the
purposes of the election of President and Vice
President, to be electors appointed by a State
and they shall meet in the District and perform
such duties as provided by the twelfth article of
amendment.
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Must Electors Vote For The Candidate Who Won
Their State's Popular Vote?
Not in these states ARIZONA - 8 Electoral Votes
ARKANSAS - 6 Electoral Votes DELAWARE - 3
Electoral Votes GEORGIA - 13 Electoral Votes
IDAHO - 4 Electoral Votes ILLINOIS - 22
Electoral Votes INDIANA - 12 Electoral Votes
IOWA - 7 Electoral Votes KANSAS - 6 Electoral
Votes KENTUCKY - 8 Electoral Votes LOUISIANA -
9 Electoral Votes MINNESOTA - 10 Electoral Votes
MISSOURI - 11 Electoral Votes NEW HAMPSHIRE - 4
Electoral Votes NEW JERSEY - 15 Electoral Votes
NEW YORK - 33 Electoral Votes NORTH DAKOTA - 3
Electoral Votes PENNSYLVANIA - 23 Electoral
Votes RHODE ISLAND - 4 Electoral Votes SOUTH
DAKOTA - 3 Electoral Votes TENNESSEE - 11
Electoral Votes TEXAS - 32 Electoral Votes UTAH
- 5 Electoral Votes WEST VIRGINIA - 5 Electoral
Votes
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Recent Unfaithful Electors
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The Election of 1824
John Quincy Adams
William H. Crawford
Andrew Jackson
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The Election of 1824
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The Election of 1876
Samuel J. Tilden
Rutherford B. Hayes
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The Election of 1876
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The Election of 1912
Theodore Roosevelt Bull Moose (Splinter
Republican)
Woodrow Wilson Democrat
William Howard Taft Republican
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The Election of 1912
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Power Indices Notations Definitions
I consider a weighted majority game of voting in
a legislature with n members or players
represented by a set N 1, 2, . . ., n whose
voting weights are wi, wi1, . . . , wn. The
players are ordered by their weight representing
their respective number of votes, so that wi
wi1 for all i.
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
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Power Indices Notations Definitions
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Power Indices Notations Definitions
The combined voting weight of all members of a
coalition represented by a subset T, T Í N, is
denoted by the function w(T), where w(T) å wi.
i Î T
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
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Power Indices Notations Definitions
Example T California, New York w(T) 54
33 w(T) 87
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Power Indices Notations Definitions
The decision rule is defined in terms of a quota,
q, by which a coalition of players represented
by subset T is winning if w(T) ³ q and losing if
w(T) lt q. It is customary to impose the
restriction q gt w(N)/2 to ensure a unique
decision and that the voting game is a proper
game.
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
20
Power Indices Notations Definitions
Example w(N) 538 The sum of all the states
Electoral College votes w(N) / 2 269 q
(w(N) / 2) 1 q 270 The number of Electoral
votes needed for election
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Power Indices Notations Definitions
A power index is an n-vector whose elements
denote the respective ability of each player to
determine the outcome of a general vote. The
index for each player is defined in terms of the
relative number of times that player can
influence the decision by transferring his voting
weight to a coalition which is losing without him
but wins with him. This is referred to as a
swing. Formally a swing for player i is defined
as a pair of subsets, (Ti, Ti i) such that Ti
is losing, but Ti i is winning. In terms of
voting weight, Ti is a swing if q - wi w(Ti) lt
q.
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
22
Power Indices Notations Definitions
Example Ti California, New York, Texas,
Florida, Pennsylvania, Illinois, Ohio, Michigan,
New Jersey, North Carolina w(Ti) 257 w(Ti)
Georgia 270 Ti is a swing if q - wi w(Ti)
lt q. (270 13) 257 lt 270. Therefore, Georgia
is a swing in this example.
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Power Indices Notations Definitions
    The power index for player i is defined as
the relative frequency or probability of swings
for i with respect to a coalition model where, in
some sense, each possible coalition is treated
equally if coalitions are regarded as being
formed randomly then the index is a probability.
The two indices however, employ different
probability models and are mathematically
distinct.
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
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The Shapley-Shubik Index
The Shapley-Shubik index is the probability that
i swings (or is "pivotal" in the terminology of
Shapley and Shubik) if all orderings of players
are equally likely. Thus, given a particular
swing for a member, the index is the number of
orderings of both the members of the coalition Ti
and the players not in Ti relative to the number
of orderings of the set of all players N every
reordering is counted separately. The index is
the probability of a swing for the player within
this probability model.
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
25
The Shapley-Shubik Index
For a given swing for player i, the number of
orderings of the members of the subset Ti and its
complement (apart from player i ), N- Ti -i, is
t!(n-t-1)! where t is the number of members of Ti
and n is the total number of players, members of
N. The total number of swings for i defined in
this way for this coalition model is The
index, fi , is this number as a proportion of the
number of orderings of all players in N, If all
orderings are equiprobable, it is the probability
of a swing.
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The Banzhaf Index
The Banzhaf index, on the other hand, treats all
coalitions Ti as equiprobable, players
being arranged in no particular order. A member's
power index is then the number of swings
expressed as a fraction of either the total
number of coalitions (measuring the probability
of a swing), or of the total number of swings for
all players (measuring the players relative
capacity to swing). The number of swings is then
å 1.
Ti
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
27
The Non-Normalized Banzhaf Index
The two versions of the index are defined by
expressing this number over different
denominators. The Non-Normalized Banzhaf index
(or Banzhaf Swing Probability), bi', uses the
number of coalitions which do not include i ,
2n-1, the number of subsets of Ni, as
denominator, and therefore it can be written as
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
28
The Normalized Banzhaf Index
The Normalized Banzhaf Index, bi, uses the total
number of swings for all players as the
denominator in order that it can be used to
allocate voting power among players The
normalized indices sum to unity over
players In the discussion of computation of
the Banzhaf index below it is only necessary to
consider the details of computing the swing
probability version, (2), since bi bi '/ å bi
'.
i
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
29
The Council of the European Economic Community
(1958-1972)
Between 1958 and 1972.. the ECC had six member
countries and used a system of qualified majority
voting that allocated 4 votes each to France,
West Germany and Italy, 2 votes each to Belgium
and the Netherlands and one vote to Luxembourg.
From these figures one might assume that the
smaller countries would have a disproportionately
large amount of power. For example, Luxembourg,
with 5.88 percent of the votes and less than 0.2
percent of the population, had 25 percent as many
votes as West Germany with only 0.57 of its
population Luxembourg had one vote for 310,000
people while West Germany had one vote for every
13,572,500, suggesting that Luxemburgers were
43.78 times more powerful than Germans. In fact,
however, since the number of votes required for a
decision was fixed at 12, Luxembourg's one vote
could never make any difference it was
impossible for it to add its vote to those of any
losing group of other countries with precisely 11
votes and therefore its formal voting power was
precisely zero.
COMPUTING POWER INDICES FOR LARGE VOTING GAMES by
Dennis Leech, University of Warwick http//www2.wa
rwick.ac.uk/fac/soc/economics/research/papers/twer
pleech.pdf
30
The Council of the European Economic Community
(1958-1972)
Note q set to 12 by treaty. IPGENF calculator
located here (http//www.warwick.ac.uk/ecaae/ipge
nf.htmldata_input)
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The Council of the European Economic Community
(1958-1972)
Note q set to 9 (q gt w(N)/2) IPGENF calculator
located here (http//www.warwick.ac.uk/ecaae/ipge
nf.htmldata_input)
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The Election of 2000
George W. Bush Republican
Ralph Nader Green
Al Gore Democrat
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The Election of 2000
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Conclusions Comments
Under the assumption that all votes are equally
likely (i.e., random voting), we prove that the
average probability of a vote being decisive is
maximized under a popular-vote (or simple
majority) rule and is lower under any coalition
system, such as the U.S. Electoral College
system, no matter how complicated.
How much does a vote count? Voting power,
coalitions, and the Electoral College Gelman
Katz http//jkatz.caltech.edu/papers/wp1121.pdf
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Conclusions Comments
The random voting model is, of course, a gross
oversimplification, and in fact its implications
for voting power in U.S. elections have been
extremely misleading in the political science
literature, as has been discussed by Gelman,
King, and Boscardin (1998). Under the random
voting model it is easy to see that the Electoral
College increases the voting power that is, the
probability of an individual's vote is decisive
for voters in larger states. However, this result
is not relevant in practice as we will show by
examining actual elections.
How much does a vote count? Voting power,
coalitions, and the Electoral College Gelman
Katz http//jkatz.caltech.edu/papers/wp1121.pdf
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Conclusions Comments

• All theories about Power Indices and Coalition
  Voting Systems assume random voting for
  analysis. Yet voting is anything but random.
• In Coalition Voting Systems (such as the
  Electoral College or the European Economic
  Community) players frequently have artificially
  high or low assigned weights which is inherently
  unfair.
• Artificially setting the q to be higher than q
  gt w(N)/2 can have extraordinary effects on the
  weights of players including completely negating
  the votes of some players.