Section 1.1, Slide 1 Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 11.3, Slide 1 11 Voting Using Mathematics to Make Choices

Section 1.1, Slide 1

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 1

11 Voting

Using Mathematics to Make Choices.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 2

Weighted Voting Systems

11.3

• 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 3



• Example: Explain the weighted voting system.[51 : 26, 26, 12, 12, 12, 12]

• Solution: The following diagram describes how to interpret this system.



• 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.



• Example: Explain the weighted voting system.[14 : 15, 2, 3, 3, 5]

• Solution: Voter 1 is a dictator.



• Example: Explain the weighted voting system.[10 : 4, 3, 2, 1]

• Solution: Every voter has veto power.



• 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.



• 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.



• 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.



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.


• 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.

Coalitions

(continued on next slide)


• Solution: We list all possible subsets of the set {R, D, I}.

Coalitions


Coalitions

In the previous example we have:


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


• 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.




• 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.




The winning coalitions and critical members are:




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.



• 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?




• Solution: We see that each board member (including A) is a critical member of exactly six


coalitions. All members have equal power.

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Section 1.1, Slide 1 Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 11.3, Slide 1 11 Voting Using Mathematics to Make Choices