VIEWS: 3 PAGES: 26 POSTED ON: 9/29/2012 Public Domain
Weighted Voting and Voting Power Section 1.5 WEIGHTED VOTING Definition: method of voting when some people have more votes than others. i.e., The votes of some voters carry more weight than the vote of others... Not all votes are considered equals! Examples: Electoral College, Corporate stockholder meetings, Mom, etc. the # of votes is based on population in each state http://www.uselectionatlas.org/USPRESIDENT/national.php?f=0 EXAMPLE: A student council has 1 representative per class, but there are 500 sophomores, 300 juniors, and 300 seniors. How could votes be weighted to represent everyone fairly? Sophomore Rep gets 5 votes, Junior & Senior Reps each get 3 votes The passage of any issue requires 6 a majority of ____ votes. Let’s pretend student council was voting to change the exam exemption policy. The proposal is to allow exemptions for everyone…(not just seniors.) COALITIONS: Could any group win Sophomores get 5 votes by themselves? NO!! Juniors get 3 votes So, join forces! Seniors get 3 votes Notation: {So;5}, {Jr;3}, {Sr;3} {So,Jr; 8} {So,Sr; 8} Remember, You need 6 votes to WIN! {Jr,Sr; 6} {So,Jr,Sr; 11} WINNING COALITIONS: Each collection is known as a Coalition. {So;5}, {Jr;3}, {So,Jr;8}, {Jr,Sr;6} … Those with enough votes to pass an issue are known as Winning Coalitions. (A winning coalition is like an alliance in “Survivor.”) A voter is essential when their vote is NECESSARY to win. (i.e. if you remove it, the winning coalition becomes a losing coalition.) So, for this example: Winning Coalitions: So,Jr;8 So,Sr;8 Jr,Sr;6 So,Jr,Sr;11 2 Sophomores are essential for _____ coalitions 2 Juniors are essential for ______ coalitions 2 Seniors are essential for _____ coalitions *Paradox: Although the votes were distributed to give more power to sophomores, the outcome is that all members have the same amount of power. Banzhaf Power Index: determines the power of a member of a voting body Step 1. Make a list of all possible coalitions Step 2. Determine which of them are winning coalitions Step 3. In each winning coalition, determine which of the players are essential players Step 4. Count the total number of times player P is essential (B) Step 5. Count the total number of times all players are essential (T) The Banzhaf Power Index of player P is given by the fraction B/T Power Index Continued: Consider the previous example: Sophomores: 2/6 Juniors: 2/6 Seniors: 2/6 The Power Index should always add up to 100% or 1. Dictators & Dummies A dictator is a member of a voting body who has all the power. A dummy is a member who has no power. #1 10 committee members vote by approval voting on 4 candidates for a new chairperson of the committee. Which candidate wins? Candidate B #2 Which candidate finishes last? Candidate D #3 If committee members # 5 and # 8 are adamantly opposed to candidates B and D and they have prior knowledge of the others’ votes, how might they have voted differently when using approval voting? Vote for A #4 If all candidates earning over 50% of the votes will be entered in a run-off, who would be in the run-off? A,B, and C #5 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes are needed to pass… Is coalition {B, C, D} a winning coalition? No #6 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes needed to pass… Which players are essential in the coalition {A, B, C; 33}? A and B #7 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes needed to pass… Which players are essential in the coalition {A, B, C, D; 36}? A #8 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes needed to pass… List 3 winning coalitions. {A,B;27}, {A,B,C;33}, {A,B,D:30}… #9 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes needed to pass… List 2 other winning coalitions. #10 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes needed to pass… Using the winning coalitions {A, B: 27} {A, B, C: 33} {A, B, D: 30} {A, C, D: 24} {A, B, C, D:36} Find the power index for voter A. 5 #11 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes needed to pass… Using the winning coalitions {A, B: 27} {A, B, C: 33} {A, B, D: 30} {A, C, D: 24} {A, B, C, D:36} Find the power index for voter B. 4 #12 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes needed to pass… Using the winning coalitions {A, B: 27} {A, B, C: 33} {A, B, D: 30} {A, C, D: 24} {A, B, C, D:36} Find the power index for voter C. 1 #13 Consider the weighted voting situation: Voter A – 15 votes; Voter B – 12 votes Voter C – 6 votes; Voter D – 3 votes 24 votes needed to pass… Using the winning coalitions {A, B: 27} {A, B, C: 33} {A, B, D: 30} {A, C, D: 24} {A, B, C, D:36} Find the power index for voter D. 1 #14 Which of Arrow’s Conditions is violated in this situation: Every student in the class voted for cake and ice cream over pizza for a class party. However, since we ranked them (along with Chick-fil-a and Brownie sundaes) and used the Borda Count Method, pizza won. Condition 3