Chapter 2:
The Mathematics
of Power
Entry 5 – Add to Pink Toolkit
2.4 Subsets & Permutations
Subsets and Permutations
Subsets and Coalitions
Coalitions are essentially sets of players that join
forces to vote on a motion. (This is the reason
we used set notation in Section 2.2 to work with
coalitions.)
By definition a subset of a set is any
combination of elements from the set. This
includes the set with nothing in it (called the
empty set and denoted by { } ) as well as the
set with all the elements (the original set itself).
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2.4-2
Excursions in Modern Mathematics, 7e: 1.1 - 2
Subsets
Let’s look at the subsets formed by P1, P2 and P3
Now let’s look at the subsets formed by P1, P2, P3
and P4
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2.4-3
Excursions in Modern Mathematics, 7e: 1.1 - 3
Subsets of {P1, P2, P3} and {P1, P2,
P 3, P 4}
The following table shows the eight subsets of
the set {P1, P2, P3} and the 16 subsets of
the set {P1, P2, P3, P4}.
The subsets of {P1, P2, P3, P4} are organized into
two groups—the “no P4” group and the “add
P4” group. The subsets in the “no P4”
group are exactly the 8 subsets of {P1, P2, P3};
the subsets of the “add P4” group
are obtained by adding P4 to each of the 8
subsets of {P1, P2, P3}.
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2.4-4
Excursions in Modern Mathematics, 7e: 1.1 - 4
Subsets of {P1, P2, P3} and {P1, P2,
P 3, P 4}
The key observation in our example is that by
adding one more element to a set we double
the number of subsets, and this principle applies
to sets of any size. Since a set with 4 elements
has 16 subsets, a set with 5 elements must have
32 subsets and a set with 6 elements must have
64 subsets. This leads to the following key fact:
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2.4-5
Excursions in Modern Mathematics, 7e: 1.1 - 5
Subsets of {P1, P2, P3} and {P1, P2,
P 3, P 4}
Number of subsets.
A set with N elements has 2N subsets.
Now that we can count subsets, we can also
count coalitions. The only difference between
coalitions and subsets is that we don’t consider
the empty set a coalition (the purpose of a
coalition is to cast votes, so you need at least
one player to have a coalition).
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2.4-6
Excursions in Modern Mathematics, 7e: 1.1 - 6
Subsets of {P1, P2, P3} and {P1, P2,
P 3, P 4}
Number of coalitions.
A weighted voting system with N players has 2N – 1
coalitions.
Finally, when computing critical counts and Banzhaf
power, we are interested in listing just the winning
coalitions. If we assume that there are no dictators,
then winning coalitions have to have at least two
players and we can, therefore, rule out all the
single-player coalitions from our list.
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2.4-7
Excursions in Modern Mathematics, 7e: 1.1 - 7
Subsets of {P1, P2, P3} and{P1, P2, P3, P4}
Number of coalitions.
A weighted voting system with N players has 2N – 1
coalitions.
Finally, when computing critical counts and
Banzhaf power, we are interested
in listing just the winning coalitions. If we assume
that there are no dictators, then
winning coalitions have to have at least two
players and we can, therefore, rule out
all the single-player coalitions from our list.
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2.4-8
Excursions in Modern Mathematics, 7e: 1.1 - 8
Subsets of {P1, P2, P3} and
{P1, P2, P3, P4}
Number of coalitions of two or more players.
A weighted voting system with N
players has 2N - N - 1 coalitions of two or more
players.
Entry 5 – page 62 #40,44,46,48
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2.4-9
Excursions in Modern Mathematics, 7e: 1.1 - 9
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2.4-10
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2.4-11
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2.4-12
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2.4-13
Excursions in Modern Mathematics, 7e: 1.1 - 13
Entry 5 continued
do #’s 49, 51, 53
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2.4-14
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2.4-15
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2.4-16
Excursions in Modern Mathematics, 7e: 1.1 - 16
Entry 6 CW
Review Page 64
#64,65,66
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2.4-17
Excursions in Modern Mathematics, 7e: 1.1 - 17
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2.4-18
Excursions in Modern Mathematics, 7e: 1.1 - 18
(b) They are multiples
(b) They are multiples
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2.4-19
Excursions in Modern Mathematics, 7e: 1.1 - 19