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# Week 6: Weighted Voting Systems Part 3 MATH 1014

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This 5 page Class Notes was uploaded by Amy Brogan on Thursday February 18, 2016. The Class Notes belongs to MATH 1014 at University of Cincinnati taught by Mary Koshar in Spring 2016. Since its upload, it has received 43 views. For similar materials see Mathematics of Social Choice in Mathematics (M) at University of Cincinnati.

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Date Created: 02/18/16

Koshar Amy Brogan February 15, 17, 19, 2016 Week 6: Weighted Voting Systems Part 3 Monday Review of Terms and Additions: Shapley-Shubik Power Index: quantifying the number of times a voter is pivotal relative to voting positions (permutations) o Decisive position: pivot o Formula: nPk = n!/(n-k)! Banzhaf Power Index: quantifying the number of times a voter is critical in relation to a voting combination o Decisive position: swing o n Ck = n!/k!(n-k)! Also, by looking up a BPI (Banzhaf Power Index) calculator, you can check your homework! Banzhaf Continued Example 1: [12:10, 7, 3, 2] 1. Write out all the possible combinations for the set 2. Find the winning combinations and identify the critical voters 3. Calculate the power index Example 2: [11: 8, 6, 4, 3, 1] 1. Calculate the number of possible combinations for each grouping (1 voter, 2 voters, etc). 2. Write out all the possible combinations for the set 3. Find the winning combinations and identify the critical voters 4. Calculate the power index Answers Example 1: [12:10, 7, 3, 2] Example 2: [11: 8, 6, 4, 3, 1] 1. 1 voter: {10} {7} {3} {2} 1. 5 C1 = 5 2 voters: {10, 7} {10, 3} {10, 2} {7, 3} {7, 2} {3, 2} 5 C2 = 10 5 C3 = 10 3 voters: {10, 7, 3} {10, 7, 2} {10, 3, 2} {7, 3, 2} 5 C4 = 5 4 voters: {10, 7, 3, 2} 2. (Critical voters underlined) 5 C5 = 1 2. 1 voter: {8} {6} {4} {3} {1} {10, 7} {10, 3} {10, 2} 2 voters: {8, 6} {8, 4} {8, 3} {8, 1} {10, 7, 3} {10, 7, 2} {10, 3, 2} {6, 4} {6, 3} {6, 1} {7, 3, 2} {4, 3} {4, 1} 3. Critical: {3, 1} 10: 6 times 3 voters: {8, 6, 4} {8, 6, 3} {8, 6, 1} {8, 4, 3} 7: 2 times 3: 2 times {8, 4, 1} {8, 3, 1} {6, 4, 3} {6, 4, 1} {6, 3, 1} 2: 2 times {4, 3, 1} 12 total 4 voters: {8, 6, 4, 3} {8, 6, 4, 1} {8, 6, 3, 1} {8, (6/12, 2/12, 2/12, 2/12) = all adds up to 1 4, 3, 1} {6, 4, 3, 1} 5 voters: {8, 6, 4, 3, 1} 3. (Critical voters underlined) {8, 6} {8, 4} {8, 3} {8, 6, 4} {8, 6, 3} {8, 6, 1} {8, 4, 3} {8, 4, 1} {8, 3, 1} {6, 4, 3} {6, 4, 1} {8, 6, 4, 3} {8, 6, 4, 1} {8, 6, 3, 1} {8, 4, 3, 1} {6, 4, 3, 1} {8, 6, 4, 3, 1} 4. Critical: 8: 11 times 6: 5 times 4: 5 times 3: 3 times 1: 1 time 25 total (11/25, 5/25, 5/25, 3/25, 1/25) = 1 (44%, 20%, 20%, 12%, 4%) = 100% Wednesday – Quiz Review Binary Coalitions: We have been looking at systems that are in brackets and have the weighted voters who voted yes in the situation, but there is another way to write them. In binary systems, we can also write them with a series of 1’s and 0’s. If you specify that 1 means yes, and 0 means they voted no, then systems look like: [9: 8, 5, 2, 1] {8, 2} 1010 {2} 0010 {8, 5, 2, 1} 1111 Banzhaf with voting discrepancies: Dictator: will have 100% of the power in the set because the other voters will never be critical [8: 8, 5, 2] 8 has 100% of the power Dummy: will have 0% of the power because they will never be critical [8: 8, 5, 2] both 5 and 2 are dummies because they cannot be critical Power = critical in a combination Practice: Example 1: [7: 6, 5, 2] Number of coalitions? Write out all possible coalitions. Write out all winning coalitions. Which are the critical voters? What is the Banzhaf power index for this set? Example 2: [16: 12, 10, 4, 4] Number of coalitions? Write out all possible coalitions. Write out all winning coalitions. Which are the critical voters? What is the Banzhaf power index for this set? Example 3: [26: 12, 10, 10, 7, 6, 4] 1. Are there any dictators, dummies, or veto voters in this set? 2. In the set {10b, 7, 10a, 6, 4, 12}, who is the pivotal voter? 3. In {12, 10b, 7, 4} who is critical voter? 4. How many permutations are there for this system? 5. How many coalitions are there for this system? 6. How many 3-person voting coalitions are there? 7. Find at least one example of a 4-person set that has only 1 or fewer critical voters (if possible). Answers Example 1: [7: 6, 5, 2] Example 2: [16: 12, 10, 4, 4] 3 Number of coalitions: 2 = 8 Number of coalitions: 2 = 16 = 15 + 1 voter: {6} {5} {2} empty set { } 2 voters: {6, 5} {6, 2} {5, 2} 1 voter: {12} {10} {4a} {4b} 3 voters: {6, 5, 2} 2 voters: {12, 10} {12,4a} {12, 4b} {10, Winning coalitions: {6, 5} {6, 2} {5, 4a} {10, 4b} {4a, 4b} 2} {6, 5, 2} 3 voters: {12, 10, 4a} {12, 10, 4b} {10, 4a, 4b} Critical voters: {6, 5} {6, 2} {5, 2} 4 voters: {12, 10, 4a, 4b} {6, 5, 2} I don’t have 15 coalitions. Where am I Critical count: 6 – 2 5 – 2 2 – 2 missing one? (2/6, 2/6, 2/6) (1/3, 1/3, 1/3) 4C2 = 4!/ 2!(4-2)! = 6 4C3 = 4!/ 3!(4-3)! = 4 * missing a set of 3 {12, 4a, 4b} Winning coalitions: {12, 10} {12,4a} {12, 4b} {12, 10, 4a} {12, 10, 4b} {10, 4a, 4b} {12, 4a, 4b} {12, 10, 4a, 4b} Critical voters: {12, 10} {12,4a} {12, 4b} {12, 10, 4a} {12, 10, 4b} {10, 4a, 4b} {12, 4a, 4b} {12, 10, 4a, 4b} Critical count: 12 – 6 10 – 2 4a – 2 4b – 2 (6/12, 2/12, 2/12, 2/12) (1/2, 1/6, 1/6, 1/6) Example 3: 1. They are all regular voters. No dictators (all the power), dummies (no power), or veto voters (absolutely needed for the vote). 2. Voter 10a is the pivotal voter because s/he is the last voter before the set reaches the quota. (10 + 7 = 17 + 10 = 27) 3. The critical voters are 12 and 10b because they are needed for the set to be a winning set. Either 7 or 5 could change their minds and the set would still pass 4. There are 6! Permutations = 6x5x4x3x2x1 = 720 5. There are 2 coalitions = 64 6. There are 20 coalitions for 3-person voting sets. 6! = 6! = 6 x 5 x 4 x 3 x 2 x 1 = 2x5x2 = 20 3!(6-3)! 3! 3! 3 x 2 x 1 3 x 2 x 1 1 7. One example of a 4 person set with one or fewer critical voters is {12, 10, 10, 4}

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