Chapter 11. Weighted Voting Systems

Goals
 Study weighted voting systems
○ Coalitions
○ Dummies and dictators
○ Veto power
 Study the Banzhaf power index
 and Shapley-Shubik
Weighted Voting Systems

In a weighted voting system, an individual voter may have
more than one vote.

The number of votes that a voter controls is called the
weight of the voter.
An example of a weighted voting system is the election of
the U.S. President by the Electoral College.
_________________________
 The voter with the largest weight is called the “first voter”,
written P1.

The weight of the first voter is represented by W1.

The remaining voters and their weights are represented
similarly, in order of decreasing weights.
Weighted Voting Systems, cont’d

The weights of the voters are usually listed as a
sequence of numbers between square
brackets.

For example, the voting system in which Angie
has a weight of 9, Roberta has a weight of 12,
Carlos has a weight of 8, and Darrell has a
weight of 11 is represented as

[12, 11, 9, 8].
Example 1

The voting system in which Angie has a weight
of 9, Roberta has a weight of 12, Carlos has a
weight of 8, and Darrell has a weight of 11 was
represented as
[12, 11, 9, 8].

In this case, P1 = Roberta, P2 = Darrell, P3 =
Angie, and P4 = Carlos.

Also, W1 = 12, W2 = 11, W3 = 9, and W4 = 8.
Weighted Voting Systems, cont’d

A simple majority requirement means that a
motion must receive more than half of the
votes to pass.

A supermajority requirement means that the
minimum number of votes required to pass a
motion is set higher than half of the total
weight.
 A common supermajority is two-thirds of the total
weight.
Weighted Voting Systems, cont’d

The weight required to pass a motion is called the
quota.

Example: A simple majority quota for the weighted
voting system [12, 11, 9, 8] would be 21.
Question:
Given the weighted voting system
[10, 9, 8, 8, 5],
find the quota for a supermajority requirement
of two-thirds of the total weight.
a. 27
b. 21
c. 26
d. 20
Weighted Voting Systems, cont’d
 The
quota for a weighted voting system
is usually added to the list of weights.
 Example:
For the weighted voting
system [12, 11, 9, 8] with a quota of 21
the complete notation is
[21 : 12, 11, 9, 8].
Example 2
 Given
the weighted voting system
[21 : 10, 8, 7, 7, 4, 4], suppose P1, P3,
and P5 vote ‘Yes’ on a motion.
 Is
the motion passed or defeated?
Example 3
 Given
the weighted voting system
[21 : 10, 8, 7, 7, 4, 4],
P1, P5, and P6 vote ‘Yes’ on a
motion. Is the motion passed or
defeated?
 suppose
P1, P4, and P6 vote ‘Yes’ on a
motion. Is the motion passed or
defeated?
 suppose
Coalitions

Any nonempty subset of the voters in a
weighted voting system is called a coalition.

If the total weight of the voters in a coalition is
greater than or equal to the quota, it is called a
winning coalition.

If the total weight of the voters in a coalition is
less than the quota, it is called a losing
coalition.
Question:
Given the weighted voting system
[27: 10, 9, 8, 8, 5]
is the coalition {P1, P4, P5} a winning
coalition or a losing coalition?
a. winning
b. losing
Example 4
 For
the weighted voting system
[8: 6, 5, 4], list all possible coalitions
and determine whether each is a
winning or losing coalition.
Example 4, cont’d

Solution: Each coalition and its status is listed
in the table below.
Coalitions, cont’d

How many coalitions are possible in a weighted
voting system with n voters?
n=1,
n=2,
n=3,
n=4
n=5
Formula for n voters
Example 5

The voting weights of EU members in a council
in 2003 are shown in the table.
Example 5, cont’d
a) If resolutions must receive 71% of the
votes to pass, what is the quota?
b) How many coalitions are possible?
Dictators and Dummies

A voter whose presence or absence in any
coalition makes no difference in the outcome
is called a dummy.

A voter whose presence or absence in any
coalition completely determines the outcome
is called a dictator.
 When a weighted voting system has a
dictator, the other voters in the system are
automatically dummies.
Weighted Voting Systems
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 isPgoing
to
4
make a difference in the outcome of the
voting.
Veto Power

In between the complete power of a dictator
and the zero power of a dummy is a level of
power called veto power.

A voter with veto power can defeat a motion
by voting ‘No’ but cannot necessarily pass a
motion by voting ‘Yes’.
 Any dictator has veto power, but a voter
with veto power is not necessarily a dictator.
Weighted Voting Systems

Veto Power
If a motion cannot pass unless player
votes in favor of the motion.
Consider [12: 9, 5, 4, 2]
has
the
power
to
obstruct
by
P
1
preventing any motion from passing.
Example 6

Consider the weighted voting system
[12: 7, 6, 4].
a) List all the coalitions and determine
whether each is a winning or losing
coalition.
b) Are there any dummies or dictators?
c) Are there any voters with veto power?
Example 6, cont’d

Solution:
a) Each coalition and its status is listed in
the table below.
Example 6, cont’d

Solution, cont’d:
b) Removing the third voter from any
coalition does not change the status of
the coalition. P3 is a dummy.
Example 6, cont’d

Solution, cont’d:
b) No voter has complete power to pass or
defeat a motion. There is no dictator.
Example 6, cont’d

Solution, cont’d:
c) If P1 is not in a coalition, then it is a losing
coalition. P1 has veto power.
Question:
In the weighted voting system
[27: 10, 9, 8, 8, 5], is P1 a:
a. dictator
b. dummy
c. voter with veto power
d. none of the above
Example 7

Consider the weighted voting system
[10: 10, 5, 4].

Are there any dummies, dictators, or
voters with veto power?
Critical Voters

If a voter’s weight is large enough so that the
voter can change a particular winning coalition
to a losing coalition by leaving the coalition,
then that voter is called a critical voter in that
winning coalition.
Question:
Given the weighted voting system
[27: 10, 9, 8, 8, 5],
is the voter P4 a critical voter in the
winning coalition {P1, P2, P4, P5}?
a. yes
b. no
Example 8

Consider the weighted voting system
[21 : 10, 8, 7, 7, 4, 4].

Which voters in the coalition {P2, P3,
P4, P5 } are critical voters in that
coalition?
The Banzhaf Power Index

The more times a voter is a critical
voter in a coalition, the more power
that voter has in the system.

The Banzhaf power of a voter is the
number of winning coalitions in which
that voter is critical.
Banzhaf Power Index, cont’d

The sum of the Banzhaf powers of all
voters is called the total Banzhaf
power in the weighted voting system.

An individual voter’s Banzhaf power
index is the ratio of the voter’s
Banzhaf power to the total Banzhaf
power in the system.
 The sum of the Banzhaf power indices of all voters
is 100%.
Banzhaf Power Index, cont’d

An individual voter’s Banzhaf power index is
calculated using the following process:
1) Find all winning coalitions for the system.
2) Determine the critical voters for each winning
coalition.
3) Calculate each voter’s Banzhaf power.
4) Find the total Banzhaf power in the system.
5) Divide each voter’s Banzhaf power by the total
Banzhaf power.
Example 9

For the weighted voting system
[18 : 12, 7, 6, 5], determine:
 The total Banzhaf power in the system.
 The Banzhaf power index of each voter.
Example 9, cont’d

Solution
Step 1: Find
all the
winning
coalitions.
Example 9, cont’d

Solution Step 2: Determine the critical
voters for each winning coalition.
 Remove each voter one at a time and
check to see whether the resulting
coalition is still a winning coalition.
 This work is shown in the next slides.
Example 9, cont’d
Example 9, cont’d
Example 9, cont’d

Solution Step 3: Count the number of
times each voter is a critical voter:
 P1: 5 times
 P2: 3 times
 P3: 3 times
 P4: 1 time

Step 4: The total Banzhaf power in
the system is 5 + 3 + 3 + 1 = 12
Example 9, cont’d

Solution Step 5:
Divide each
voter’s Banzhaf
power by the
total Banzhaf
power to find
the Banzhaf
power indices.
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
Weighted Voting Systems
Three-Player Sequential Coalitions
Weighted Voting Systems
Shapley-Shubik- Pivotal Player
The player that contributes the votes
that turn what was a losing coalition into
a winning coalition.
The Shapley-Shubik Power Index
 In each (winning) sequential coalition there is
a pivotal player--a player whose joining
causes the coalition to change from a losing
coalition to a winning coalition.
 We will use the concept of the pivotal player
to define the Shapley-Shubik Power Index.
The Shapley-Shubik
 Power Index concerns itself with sequential
coalitions--coalitions in which the order that
players join matters
The Shapley-Shubik Power Index
 The Shapley-Shubik Power Index
concerns itself with sequential coalitions-coalitions in which the order that players
join matters.
 In general, the number of sequential
coalitions with N players is:
 N ! = (N)(N - 1). . .(2)(1)
The Shapley-Shubik Power
Index
Finding the Shapley-Shubik Power Index of Player P :
Step 1. Make a list of all sequential coalitions
containing all N players.
Step 2. In each sequential coalition determine the
pivotal player.
Step 3. Count the number of times P is pivotal--call
this number S.
The Shapley-Shubik Power Index for the
player P is
the fraction S/(N !).
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.
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
Exercise 2.28
Find the Shapely-Shubik index of
following weighted voting systems
 [6:4,3,2,1]
 [7:4,3,2,1]
 [8:4,3,2,1]
 [9:4,3,2,1]
 [10:4,3,2,1]

Four player coalitions




<P1,P2,P3,P4> ;
<P1,P2,P4,P3>;
<P1,P3,P2,P4>;
<P1,P3,P4,P2>;
<P1,P4,P2,P3> ;
<P1, P4,P3,P2>;
<P2,P1,P3,P4> ;
<P2,P1,P4,P3>;
< P2,P3, P1,P4> ;
< P2,P3,P4,P1>;
< P2,P4,P3, P1> ;
< P2,P4, P1,P3 >;
<P3,P2,P1,P4> ;
<P3,P2,P4,P1>;
<P3, P1,P2,P4> ;
<P3, P1,P4,P2 >;
<P3,P4,P2,P3> ;
<P3, P4,P3,P2 >;
<P4,P2,P1,P3> ;
<P4,P2,P3,P1>;
<P4, P1,P2,P3> ;
<P4, P1,P3,P2>;
<P4,P3,P2,P1> ;
<P4, P3,P1,P2>;
The Shapley-Shubik Power
Index
 When discussing power of a coalition in
terms of the Banzhaf Index we did not
care about the order in which player’s
cast their votes.
 In other words, in Banzhaf index {P1
,P2} and {P2 ,P1} to be the same
coalition.
Example: Let us consider the coalition {P1 ,P2,P3}. How
many sequential coalitions contain these players?
 We have the following sequential coalitions:
 <P1 , P2, P3 
 P1 , P3, P2 
  P2 , P1, P3 
 P2 , P1, P3 
 P3 , P1, P2 
 P3 , P2, P1 
 We can see that there are a total of 6.
(In the first sequential coalition what we are
saying is that P1 started the coalition, then
P2 joined who in turn was followed byP3.)
Example: [10: 6, 5, 4]
 We have already seen that the 6 possible
sequential coalitions and 6 Pivotal Players
  P1 , P2, P3 
 P1 , P3, P2 
 P2 , P1, P3 
 P2 , P1, P3 
 P3 , P1, P2 
 P3 , P2, P1 
P2
P3
P1
P1
P1
P1
The Shapley-Shubik Power
Index
 The list of all of the Shapley-Shubik
Power Indices for a given election is the
Shapley-Shubik power distribution of
the weighted voting system.
Example: The European Union (revisited).
There are a total of 15! = 1,307,674,368,000
possible sequential coalitions
(and 215 - 1 = 32,767 ‘normal’ coalitions) to
consider.
Country
Votes
Banzhaf Power
Shapley-Shubik Power
France, Germany, 10
Italy, UK
1849/16,565 ≈ 11.16%
11.67%
Spain
8
1531/16,565 ≈ 9.24%
9.55%
Belgium, Greece,
Netherlands,
Portugal
5
973/16,565 ≈ 5.87%
5.52%
Austria, Sweden
4
793/16,565 ≈ 4.79%
4.54%
Denmark, Finland,
Ireland
3
595/16,565 ≈ 3.59%
3.53%
Luxembourg
2
375/16,565 ≈ 2.26%
2.07%
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
Shapley-Shubik
Homework

2, 3, 9, 11,13,16, 17,23,24,33, 34, 42,
45, 49