Finite Math A Chapter 2, Weighted Voting Systems 1 Discrete Mathematics Notes Chapter 2: Weighted Voting Systems The Power Game Academic Standards: PS.ED.2: Use election theory techniques to analyze election data. Use weighted voting techniques to decide voting power within a group. ONE PERSON ONE VOTE is an democratic idea of equality But what if the voters are not PEOPLE but are governments? countries? states? If the institutions are not equal, then the number of votes they control should not be equal. The United Nations Security Council 15 voting nations: 5 permanent members (Britain, China, France, Russia, United States), 10 nonpermanent members appointed for a 2-year rotation. Permanent members have more votes than non permanent members. Stock Holders/Shareholders: The more stock you own, the more say you have in decision making for the company. The Electoral College Each state gets a number of electors (votes) equal to the number of Senators plus the number of Representatives in Congress. California has 55 votes but North Dakota only has 3 votes. Each state is a voter but states with heavy concentration of population receive a bigger vote. This situation where each voter is not equal in the number of votes they control is called: 2.1 An Introduction to weighted voting Important terms: : A voting situation where voters are not necessarily equal in the number of votes they control. : A vote with only two choices. (usually yes/no) : The voters (symbolized by P 1, P 2, P 3, etc.) : The number of votes a player controls. : The smallest number of votes required to pass a motion.
Finite Math A Chapter 2, Weighted Voting Systems 2 Notation Example 1. [14: 8, 6, 5, 1] [q: w 1, w 2, w 3,..., w n ] q = w s = quota = total votes = Player 1 (P 1) = controls votes / has a weight of Player 2 (P 2) = controls votes Player 3 (P 3) = controls votes Player 4 (P 4) = controls vote Example 2. Given the weighted voting system [16: 8, 6, 4, 4, 3, 1], state the following: The number of players: The weight of P 5: The total number of votes: The minimum % of the quota to nearest whole %: Common Types of Quotas: Simple majority/strict majority Two-thirds majority Unanimity U.S. Senate: Simple Majority to pass an ordinary law (51 votes) 60 votes to stop a filibuster 2/3 of the votes (67) to override a presidential veto. Weighted Voting Issues Example 3: Four partners decide to start a business. P 1 buys 8 shares, P 2 buys 7 shares, P 3 buys 3 shares and P 4 buys 2 shares. One share = one vote. a. The quota is set at two-thirds of the total number of votes. Describe as a weighted voting system. b. The partnership above decides the quota is too high and changes the quota to 10 votes. c. The partnership above decides to make the quota equal to 21 votes.
Finite Math A Chapter 2, Weighted Voting Systems 3 For a weighted voting system to be legal: the quota must be at least a and no more than V Symbolically: If V w1 w2 w3... wn, then q V 2 Example 3d. What if our partnership changed the quota to 19? 4. [q: 7, 2, 1, 1, 1] What is the smallest legal quota? What is the largest legal quota? What is the value of the quota if at least two-thirds of the votes are required to pass a motion? What is the value of the quota if more than three-fourths of the votes are required to pass? 5. A committee has 4 members (P 1, P 2, P 3, P 4). P 1 has twice as many votes as P 2. P 2 has twice as many votes as P 3. P 3 and P 4 have the same number of votes. The quota is 49. Describe the weighted voting system using the notation [q: w1, w2, w3, w4] given the definitions of quota below. (Hint: write the weighted voting system as [49: 4x, 2x, x, x] and then solve for x. a) The quota is a simple majority b) The quota is more than three-fourths
Finite Math A Chapter 2, Weighted Voting Systems 4 Dictators, Dummies, and Veto Power Example 6: [11: 12, 5, 4] What do you notice about P 1? P 1 has all the power P 2 and P 3 have no power Note: If any player is a dictator, then EVERY OTHER PLAYER is a dummy. Even if there is no dictator, there may still be dummies. Example 7: [30: 10, 10, 10, 9] Example 8: [12: 9, 5, 4, 2] Is there a dictator? If P 1 chooses to vote against the motion, can the other players combine weight to meet the quota? If a player is not a dictator, but the other players cannot meet the quota without his votes, we say he has veto power. Sometimes, more than one player will have veto power. Example 9. Determine which players, if any, are: dictators, veto power, dummies a) [15: 16, 8, 4, 1] b) [18: 16, 8, 4, 1] c) [24: 16, 8, 4, 1] Example 10. Consider [q: 8, 4, 2]. Find the smallest value of q for which a) all three players have veto power b) P 2 has veto power, but P 3 does not c) P 3 is the only dummy
Finite Math A Chapter 2, Weighted Voting Systems 5 2.2/2.3 The Banzhaf Power Index Who is the most POWERFUL player? : A group of players who choose to vote together : The set of all voters. This represents a unanimous vote. Weight of the coalition: Winning coalitions Losing coalitions : Any player who MUST BE PRESENT in a winning coalition in order for it to remain a winning coalition. Note: If you subtract the critical player s votes from the coalition, the number of votes drops below the quota. Example 1: Find the critical player or critical players in each of the following coalitions. [15: 13, 9, 5, 2] a) {P 1, P 4} b) {P 2, P 3, P 4} c) {P 3, P 4} d) {P 1, P 2, P 3} [51: 30, 25, 25, 20] a) {P 1, P 3} b) {P 1, P 2, P 3} c) {P 2, P 3, P 4} d) {P 2, P 3} The Banzhaf Power Index: A player s power is proportional to the number of coalitions for which that player is critical. The more often a player is critical, the more power he holds.
Finite Math A Chapter 2, Weighted Voting Systems 6 Calculate the Banzhaf Power Index: Idea: Step 1: Make a list of all WINNING coalitions. Step 2: Determine which players are critical in each coalition. (circle, underline, highlight) Step 3: Count the total number of times each player is critical Step 4: Add the total number of times each player is critical to find the grand total number of critical players. The Banzhaf Power INDEX number for each player = step 3 step 4 The Banzhaf Power DISTRIBUTION for the weighted voting system is the % of power each player holds. Example 2: Find the Banzhaf Power index for the weighted voting system: [101: 99, 98, 3] Example 3: Find the Banzhaf Power Distribution for [4: 3, 2, 1]
Finite Math A Chapter 2, Weighted Voting Systems 7 How do you know you have all the possible coalitions written down? If n = number of players in a weighted voting system, Be systematic or use the formula! Then the number of possible coalitions is: 2 n 1 How many coalitions if 4 players? How many coalitions if 5 players? Example 4: Find the Banzhaf Power Distribution for [6: 4, 3, 2, 1] Banzhaf Coalitions: 4 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P1, P2, P4} {P3} {P1,P4} {P1, P3, P4} {P4} {P2,P3} {P2, P3, P4} {P2,P4} {P1, P2, P3, P4} {P3,P4} Example 5: Consider the weighted voting system [q: 8, 4, 2, 1]. Find the Banzhaf Power Distribution of this weighted voting system when: a) q = 8 b) q = 10 c) q = 14
Finite Math A Chapter 2, Weighted Voting Systems 8 What I expect to see for work on your homework: 1. Write down all possible coalitions and cross off losers OR just the winning coalitions. 2. Critical Players should be circled or underlined. 3. Show fraction of BPI for each player AND calculate the % for BPD. Possible Coalitions: Use these to help you: Banzhaf Coalitions: 3 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P3} {P2,P3} Banzhaf Coalitions: 4 Players {P1} {P1,P2} {P1, P2, P3} {P2} {P1,P3} {P1, P2, P4} {P3} {P1,P4} {P1, P3, P4} {P4} {P2,P3} {P2, P3, P4} {P2,P4} {P1, P2, P3, P4} {P3,P4} Where weighted voting systems/banzhaf are used: Banzhaf is used to QUANTIFY the amount of power each player holds. 1. Nassau County Board of Supervisors (see p. 55): Votes were given to districts according to population and quota was simple majority. [58: 31, 31, 28, 21, 2, 2] Banzhaf showed that two of the six counties actually had no voting power that they were actually dummy voters. Final result: 1993 court decision abolishing weighted voting in New York States. Districts were created of roughly the same population and each given one voted. 2. United Nations Security Council: Banzhaf shows that a permanent member of the council holds more than 10 times the amount of power as one of the non-permanent members. There are 5 permanent members (Britain, China, France, Russia, US) and 10 non-permanent members. This voting arrangement may change as others are being considered for permanent membership. 3. European Union Banzhaf quantifies the amount of power each nation has and shows that smaller nations such as Luxembourg and Malta still hold some power.
Finite Math A Chapter 2, Weighted Voting Systems 9 2.4/2.5 The Shapley Shubik Power Index: The Shapley-Shubik Power Index: A player s power is proportional to the number of sequential-coalitions for which that player is pivotal. The more times a player is pivotal, the more power he holds. Sequential coalition: Banzhaf: { P1, P2, P3} Shapley-Shubik: P 1, P 3, P 2 These 3 players decide to vote together. These 3 players decide to vote together. They form a coalition. P1 votes 1 st, P3 votes 2 nd, P2 votes 3 rd. Order listed in the { } doesn t matter. They form a sequential coalition. Order listed in the is important. Pivotal player: Example: Find the Pivotal Player 1. Given the weighted voting system [5: 3,2,1,1} find the pivotal player for the given sequential coalition. a) [P 1,P 4,P 3,P 2] b) [P 3,P 1,P 2,P 4] c) [P 4,P 3,P 2,P 1]
Finite Math A Chapter 2, Weighted Voting Systems 10 Counting Sequential Coalitions: List the possible sequence for 3 players. How many are there? How many sequential coalitions are there for 4 players? For 5 players? Multiplication Rule: If there are m ways to do task 1 and n ways to do task 2, then there are mxn ways to do both tasks together. Shapley-Shubik Power Distribution Factorials: If N= the number of players, then the number of sequential coalitions is N! N! = N x (N-1) x... x 3 x 2 x 1 Step 1: Make a list of all sequential coalitions Step 2: For each sequential coalition, determine the pivotal player. Step 3: For each player, count the number of times they are pivotal and divide by the number of sequential coalitions. Calculate the %. Example 2: Find the Shapely Shubik Power Distribution for [4: 3, 2, 1] Sequential Coalitions: 3 Players [P1,P2,P3] [P1,P3,P2] [P2,P1,P3] [P2,P3,P1] [P3,P1,P2] [P3,P2,P1]
Finite Math A Chapter 2, Weighted Voting Systems 11 Example 3: Find the Shapley-Shubik Power Distribution for [6: 4, 3, 2, 1] Sequential Coalitions: 4 Players [P 1,P 2,P 3,P 4] [P 2,P 1,P 3,P 4] [P 3,P 1,P 2,P 4] [P 4,P 1,P 2,P 3] [P 1,P 2,P 4,P 3] [P 2,P 1,P 4,P 3] [P 3,P 1,P 4,P 2] [P 4,P 1,P 3,P 2] [P 1,P 3,P 2,P 4] [P 2,P 3,P 1,P 4] [P 3,P 2,P 1,P 4] [P 4,P 2,P 1,P 3] [P 1,P 3,P 4,P 2] [P 2,P 3,P 4,P 1] [P 3,P 2,P 4,P 1] [P 4,P 2,P 3,P 1] [P 1,P 4,P 2,P 3] [P 2,P 4,P 1,P 3] [P 3,P 4,P 1,P 2] [P 4,P 3,P 1,P 2] [P 1,P 4,P 3,P 2] [P 2,P 4,P 3,P 1] [P 3,P 4,P 2,P 1] [P 4,P 3,P 2,P 1] Example 4: Find the Shapley-Shubik Power Distribution for [10: 8, 4, 2, 1] Sequential Coalitions: 4 Players [P 1,P 2,P 3,P 4] [P 2,P 1,P 3,P 4] [P 3,P 1,P 2,P 4] [P 4,P 1,P 2,P 3] [P 1,P 2,P 4,P 3] [P 2,P 1,P 4,P 3] [P 3,P 1,P 4,P 2] [P 4,P 1,P 3,P 2] [P 1,P 3,P 2,P 4] [P 2,P 3,P 1,P 4] [P 3,P 2,P 1,P 4] [P 4,P 2,P 1,P 3] [P 1,P 3,P 4,P 2] [P 2,P 3,P 4,P 1] [P 3,P 2,P 4,P 1] [P 4,P 2,P 3,P 1] [P 1,P 4,P 2,P 3] [P 2,P 4,P 1,P 3] [P 3,P 4,P 1,P 2] [P 4,P 3,P 1,P 2] [P 1,P 4,P 3,P 2] [P 2,P 4,P 3,P 1] [P 3,P 4,P 2,P 1] [P 4,P 3,P 2,P 1]