Math 103, Spring 2010, Solutions to Chapter 2A homework
16 points total
Problem 18 b: 4 points
The weighted voting system [17:16, 8, 4, 1] has the following winning coalitions, with the
critical player(s) in each winning coalition underlined:
{P
1
, P
2
, P
3
, P
4
}
{P
1
, P
2
, P
3
}
{ P
1
, P
2
, P
4
}
{ P
1
, P
3
, P
4
}
{ P
1
, P
2
}
{ P
1
, P
3
}
{ P
1
, P
4
}
The Banzhaf power distribution for this weighted voting system is as follows:
P
1
has 7/10 = 70% of the power
P
2
has 1/10 = 10% of the power
P
3
has 1/10 = 10% of the power
P
4
has 1/10 =
10% of the power
Remark: P
1
belongs to every winning coalition, and therefore has veto power.
P
4
has some share
in the real power, and is therefore not a dummy.
In spite of the fact that P
2
has twice the weight
of P
3
and eight times the weight of P
4
, they have all have the same fraction of the real power as
measured by Banzhaf's approach.
__________________________________________________________________
Problem 18c:
4 points
The weighted voting system [18:16, 8, 4, 1] has the following winning coalitions, with the
critical player(s) in each winning coalition underlined:
{P
1
, P
2
, P
3
, P
4
}
{P
1
, P
2
, P
3
}
{ P
1
, P
2
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 Spring '07
 Berkowitz
 Math, 0%, p1, AirTrain Newark, law firm, Banzhaf power index, Banzhaf

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