Chapter 11 notes (7
th
ed.)
WEIGHTED VOTING
Suppose a county commission consists of three members, one representing each of the three cities in the
county.
Voting power on the commission is proportional to the population of the cities with the commissioner
from city A getting 49 votes, the commissioner from city B getting 40 votes, and the commissioner from city C
getting 11 votes.
Since there are 100 total votes, 51 votes are required to carry any measure.
In this scenario
the representative from city C has as much power as the representative from city A since no vote can be won
without support from two commissioners.
In fact all three commissioners have equal power.
This is
counterintuitive since one might have expected A to have more than 4 times the power of C.
This is an example of a weighted voting system.
The number of votes required to win a vote, in this
case 51, is called the
quota
.
The number of votes each voter casts is called the voter’s
weight
.
The notation we
use to describe this voting system is [51: 49, 40, 11].
Let’s consider the set of all commissioners: {A, B, C}.
In the study of weighted voting, each subset is
called a
coalition
.
The weight of the coalition is defined to be the sum of the weight of its voters.
If the weight
of the coalition is greater than or equal to the quota, we call that coalition a
winning coalition
.
Otherwise, it is a
losing coalition.
Coalition
Weight
Type
{A}
49
losing
{B}
40
losing
{C}
11
losing
{A, B}
89
winning
{A, C}
60
winning
{B, C}
51
winning
{A, B, C}
100
winning
{
}
0
losing
Example:
What is wrong with the following weighted voting system? [21: 6, 5, 5, 3]
Solution:
Example:
What is wrong with the following weighted voting system? [8: 6, 5, 5, 3]
Solution:
Example:
A weighted voting system has 5 voters with weights 1, 1, 2, 2, and 3.
Find all possible values of q.
Solution:
Example:
A corporation has four partners with weights 10, 8, 7 and 1.
The bylaws of the corporation require
that twothirds of the votes are needed to pass a motion.
What is the quota?
Solution:
Example:
Consider the weighted voting system [10: 11, 4, 3].
Describe the power of each voter.
Solution:
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Chapter 11 notes (7
th
ed.)
Example:
Consider the weighted voting system [3: 2, 1, 1].
Find all coalitions with enough weight to prevent a
measure from passing.
Solution:
Such a coalition is called a
blocking coalition
.
Coalition
Weight
Type
{A}
2
losing, blocking
{B}
1
losing
{C}
1
losing
{A, B}
3
winning, blocking
{A, C}
3
winning, blocking
{B, C}
2
losing, blocking
{A, B, C}
4
winning, blocking
{
}
0
losing
Notice that all winning coalitions are blocking coalitions, but not conversely.
When a blocking coalition
consists of one voter, that voter is said to have
veto power
.
In the previous example, A has veto power.
Also,
note that the number of votes needed to block is:
(Total weight of all voters) – (quota) +1
Example:
For the weighted voting system [21: 11, 10, 9, 1], list:
a) all winning coalitions containing the first voter
b) all blocking coalitions containing the first voter
c) all voters that have veto power
d) all dummy voters
Solution:
When doing the homework, make the following two assumptions:
1) If q is not stated, then q is the smallest value needed for a majority.
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 Fall '08
 Storfer
 electoral votes, Voting system, Plurality voting system, weighted voting

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