11
Voting
Using Mathematics to
Make Choices.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section
Section11.3,
1.1, Slide
Slide11
11.3 Weighted Voting
Systems
• Understand the numerical
representation of a weighted
voting system.
• Find the winning coalitions in a
weighted voting system.
• Compute the Banzhaf power
index of a voter in a weighted
voting system.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 2
Weighted Voting Systems
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 3
Weighted Voting Systems
• Example: Explain the weighted voting system.
[51 : 26, 26, 12, 12, 12, 12]
• Solution: The following diagram describes how
to interpret this system.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 4
Weighted Voting Systems
• Example: Explain the weighted voting system.
[4 : 1, 1, 1, 1, 1, 1, 1]
• Solution: This is an example of a “one person
one vote” situation.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 5
Weighted Voting Systems
• Example: Explain the weighted voting system.
[14 : 15, 2, 3, 3, 5]
• Solution: Voter 1 is a dictator.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 6
Weighted Voting Systems
• Example: Explain the weighted voting system.
[10 : 4, 3, 2, 1]
• Solution: Every voter has veto power.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 7
Weighted Voting Systems
• Example: Explain the weighted voting system.
[12 : 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
• Solution: Every voter has veto power. This
describes our jury system.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 8
Weighted Voting Systems
• Example: Explain the weighted voting system.
[12 : 1, 2, 3, 1, 1, 2]
• Solution: The sum of all the possible votes is
less than the quota, so no resolutions can be
passed.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 9
Weighted Voting Systems
• Example: Explain the weighted voting system.
[39 : 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
• Solution: This system describes the voting in
the UN Security Council.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 10
Coalitions
In a weighted voting system
[4 : 1, 1, 1, 1, 1, 1, 1]
any coalition of four or more voters is a winning
coalition.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 11
Coalitions
• Example: A town has three parties:
(R)epublican, (D)emocrat, and (I)ndependent.
Membership on the town council is proportional
to the size of the parties with R having 9
members, D having 8, and I only 3. Parties vote
as a single bloc, and resolutions are passed by
a simple majority. List all possible coalitions and
their weights, and identify the winning coalitions.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 12
Coalitions
• Solution: We list all possible subsets of the
set {R, D, I}.
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Section 11.3, Slide 13
Coalitions
In the previous example we have:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 14
The Banzhaf Power Index
In the previous example, we saw that R, D, and I
each were critical voters twice. Thus, R’s Banzhaf
power index is
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 15
The Banzhaf Power Index
• Example: A law has two senior partners
(Krooks and Cheatum) and four associates (W,
X, Y, and Z). To change any major policy of the
firm, Krooks, Cheatum, and at least two
associates must vote for the change. Calculate
the Banzhaf power index for each member of
this firm.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 16
The Banzhaf Power Index
• Solution: We use {K, C, W, X, Y, Z} to
represent the firm. Since every winning coalition
includes {K, C} and any two of the other
associates, we only need to determine the
subsets of {W, X, Y, Z} with two or more
members to determine the winning coalitions.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 17
The Banzhaf Power Index
The winning coalitions and critical members are:
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 18
The Banzhaf Power Index
K and C are critical members 11 times,
whereas W, X, Y, and Z are each critical
members only 3 times. We may compute the
Banzhaf power index for each member.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 19
The Banzhaf Power Index
• Example: A 5-person air safety review board
consists of a federal administrator (A), two
senior pilots (S and T), and two flight attendants
(F and G). The intent is for the A to have
considerably less power than S, T, F, and G, so
A only votes in the case of a tie; otherwise,
cases are decided by a simple majority. How
much less power does A have than the other
members of the board?
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 20
The Banzhaf Power Index
• Solution: We see that each board member
(including A) is a critical member of exactly six
coalitions. All
members have
equal power.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 11.3, Slide 21