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 Weighted Voting Systems

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 4 Example: Explain the weighted voting system. [51 : 26, 26, 12, 12, 12, 12] Solution: The following diagram describes how to interpret this system. Weighted Voting Systems

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 5 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. Weighted Voting Systems

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 6 Example: Explain the weighted voting system. [14 : 15, 2, 3, 3, 5] Solution: Voter 1 is a dictator. Weighted Voting Systems

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 7 Example: Explain the weighted voting system. [10 : 4, 3, 2, 1] Solution: Every voter has veto power. Weighted Voting Systems

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 8 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. Weighted Voting Systems

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 9 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. Weighted Voting Systems

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 10 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. Weighted Voting Systems

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 11 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 12 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)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 13 Solution: We list all possible subsets of the set {R, D, I}. Coalitions

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 14 Coalitions In the previous example we have:

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 15 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 16 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) The Banzhaf Power Index

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 17 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 Banzhaf Power Index (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 18 The winning coalitions and critical members are: The Banzhaf Power Index (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 19 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. The Banzhaf Power Index

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 20 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) The Banzhaf Power Index

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.3, Slide 21 Solution: W e see that each board member (including A) is a critical member of exactly six The Banzhaf Power Index coalitions. All members have equal power.