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Unformatted text preview: re out the winning coalitions.
There are 4 winning coalitions:
AB ,
AC ,
BC ,
ABC
(Verify) Van Essen (U of A) Power 6 / 22 ShapleyShubik Index of Power Second, we need to compute the number of possible arrangements of
the three voters.
How many orderings of A, B, and C are there? Van Essen (U of A) Power 7 / 22 ShapleyShubik Index of Power Answer: There are 3! = 6 possible orderings ot the three people.
ABC
ACB
BAC
BCA
CAB
CBA . Van Essen (U of A) Power 8 / 22 ShapleyShubik Index of Power Third, for each ordering, we need determine which voter is pivotal.
Starting with the person of the far left we keep adding people to the
coalition until it becomes a winning coalition. We put a dot on the
pivotal person who causes the coalition to go from losing to winning.
˙
ABC
˙
AC B
˙
B AC
˙
BC A
˙
C AB
˙
C BA Van Essen (U of A) Power 9 / 22 ShapleyShubik Index of Power
˙
ABC
˙
AC B
˙
B AC
˙
BC A
˙
C AB
˙
C BA
Fourth, we count the number of times each person was pivotal:
A is pivotal 2 times, B is pivotal 2 times, C is pivotal 2 times. The ShapleyShubik Index of power for each players is:
SSI (A) = SSI (B ) = SSI (C ) = 2
1
=
6
3 They all have equal power!
Van Essen (U of A) Power 10 / 22 ShapleyShubik Index of Power: European Economic
Community Consider the EEC setup in 1958. Suppose we wanted to calculate
the SSI for each member.
France, Germany, and Italy had 4 votes.
Netherlands and Belgium had 2 votes
Luxembourg had 1 vote.
Passage required 12 or more votes. Let’ compute the SSI for France.
s Van Essen (U of A) Power 11 / 22 ShapleyShubik Index of Power: European Economic
Community The total number of orderings is 6! = 720. It would be very tedious
to work out all of the orderings and compute for which orderings
France is pivotal. Thus, we need to be smart and use some of our
counting techniques.
Note: France has 4 votes. Thus, France is pivotal if it is preceded by
8, 9, 10, or 11 votes. Why?
So we need to …gure out the di¤erent ways to get 8, 9, 10, and 11
and use our counting techniques to …gure out all of the orderings that
each case contains. Van Essen (U of A)...
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 Spring '12
 Vanessen
 The Land

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