Excursions in Modern
Mathematics
Sixth Edition
Peter Tannenbaum
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Chapter 2
Weighted Voting Systems
The Power Game
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Weighted Voting Systems
Outline/learning Objectives
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Represent a weighted voting system using a
mathematical model.
Use the Banzhaf and Shapley-Shubik indices
to calculate the distribution of power in a
weighted voting system.
Weighted Voting Systems
2.1 Weighted Voting
Systems
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Weighted Voting Systems
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The Players
The voters in a weighted voting system.
The Weights
That each player controls a certain number of
votes.
The Quota
The minimum number of votes needed to pass
a motion (yes-no votes)
Weighted Voting Systems
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Dictator
The player’s weight is bigger than or equal to
the quota.
Consider [11:12, 5, 4]
1 owns enough votes to carry a motion single
handedly.
P
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Weighted Voting Systems
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Dummy
A player with no power.
Consider [30: 10, 10, 10, 9]
P4 turns out to be a dummy! There is never
going to be a time when P4 is going to make a
difference in the outcome of the voting.
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Weighted Voting Systems
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Veto Power
If a motion cannot pass unless player votes in
favor of the motion.
Consider [12: 9, 5, 4, 2]
1 has the power to obstruct by preventing
any motion from passing.
P
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Weighted Voting Systems
2.2 The Banzhaf Power
Index
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Weighted Voting Systems
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Coalitions
Any set of players that might join forces and
vote the same way. The coalition consisting of
all the players is called a grand coalition.
Weighted Voting Systems
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Winning Coalitions
Some coalitions have enough votes to win and
some don’t. We call the former winning
coalitions and the latter losing coalitions.
Weighted Voting Systems
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Critical players
In a winning coalition, a player is said to be a
critical player for the coalition if the coalition
must have that player’s votes to win. If and
only if
W-w<q
Weighted Voting Systems
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Computing a Banzhaf Power Distribution
Step 1. Make a list of all possible winning
coalitions.
Weighted Voting Systems
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Computing a Banzhaf Power Distribution
Step 2. Within each winning coalition
determine which are the critical players. (To
determine if a given player is critical or not in a
given winning coalition, we subtract the
player’s weight from the total number of votes
in the coalition- if the difference drops below
the quota q, then that player is critical.
Otherwise, that player is not critical.
Weighted Voting Systems
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Computing a Banzhaf Power Distribution
Step 3.Count the number of times that P 1
is critical. Call this number B1 Repeat for each
of the other players to find
2,
3, ...,
N
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B B B
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Weighted Voting Systems
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Computing a Banzhaf Power Distribution
Step 4. Find the total number of times all
players are critical. This total is given by
T =B + B
1
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2
+ ... + BN
Weighted Voting Systems
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Computing a Banzhaf Power Distribution
Step 5. Find the ratio 1 =B1 T .
This gives the Banzhaf power index of
.
1
Repeat for each of the other players to find
2, 3, …, N . The complete list of ’s gives the
Banzhaf power distribution of the weighted
voting system.

P
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Weighted Voting Systems
2.3 Applications of
Banzhaf Power
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Weighted Voting Systems
Applications of Banzhaf Power
 The Nassau County Board of Supervisors
John Banzhaf first introduced the concept
 The United Nations Security Council
Classic example of a weighted voting system
 The European Union (EU)
Relative Weight vs Banzhaf Power Index
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Weighted Voting Systems
2.4 The Shapley-Shubik
Power Index
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Weighted Voting Systems
Three-Player Sequential Coalitions
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Weighted Voting Systems
Shapley-Shubik- Pivotal Player
The player that contributes the votes that turn
what was a losing coalition into a winning
coalition.
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Weighted Voting Systems
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Computing a Shapley-Shubik Power Distribution
Step 1. Make a list of all possible sequential
coalitions of the N players. Let T be the
number of such coalitions.
Weighted Voting Systems
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Computing a Shapley-Shubik Power Distribution
Step 2. In each sequential coalition determine
the pivotal player.
Weighted Voting Systems
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Computing a Shapley-Shubik Power Distribution
Step 3.Count the number of times that P 1
is pivotal. Call this number
1.
Repeat for each of the other players to find
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SS
SS2,SS3,....SSN
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Weighted Voting Systems
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Computing a Shapley-Shubik Power Distribution
Step 4. Find the ratio 1 = ss 1 T This
gives the Shapley Shubik power index of P 1 .
Repeat for each of the other players to find
 2,  3, …,  N . The complete list of  ’s gives
the Shapley-Shubik power distribution of the
weighted voting system.
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Weighted Voting Systems
The Multiplication Rule
If there are m different ways to do X, and n
different ways to do Y, then X and Y together
can be done in m x n different ways.
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Weighted Voting Systems
The Number of Sequential Coalitions
The number of sequential coalitions with N
players is N! = N x (N-1) x…x 3 x 2 x 1.
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Weighted Voting Systems
Applications of Shapley-Shubik Power
 The Electoral College
There are 51! Sequential coalitions
 The United Nations Security Council
Enormous difference between permanent and
nonpermanent members
 The European Union (EU)
Relative Weight vs Shapley-Shubik Power
Index
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Weighted Voting Systems
Conclusion
The notion of power as it applies to
weighted voting systems
 How mathematical methods allow us
to measure the power of an individual
or group by means of an index.
 We looked at two different kinds of
power indexes– Banzhaf and ShapleyShubik
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