Chapter 2: Weighted Voting Systems

1. Chapter 2: Weighted Voting Systems

2. **Read page 51 Sec 1: Weighted Voting Systems Weighted voting system: any arrangement in which voters are not always equal in terms of the number of votes they control. Weighted voting system involve the voters which we will call players, how many votes each player controls, called weights, and how many votes are needed to win, called the quota. These weighted voting systems are usually applied when needing a yes-no vote to either pass a motion, or deny it. Think of a corporation that has an owner, a President, a CEO, etc. where each person has some power to make decisions. The owner of the company may have 8 votes while the President has 7, the CEO has 5, etc. This gives power to each person based on their position in the company. During board meetings, they might vote on certain issues within the company. These issues, or motions, will require a certain number of votes to be passed. This number is called the quota.

3. 8, 6, 4, 3 are the weights of The quota must fall between certain limits. It must be more than half the total number of votes, but less than the total number of votes. [12: 8, 6, 4, 3] is an example of a weighted voting system. In the wvs: 12 is the quota. 12 votes are needed to pass the motion. respectively. In this wvs, it would be fairly easy to pass the motion. Players one and two could vote the same way and either pass or deny the motion without any help from the other players. OR, player one can vote with player three to pass the motion. There are different ways of pairing up the players that will result in passing the motion.

4. *** TERMS YOU NEED TO KNOW!!!!! Dictator: a player whose weight is more than or equal to the quota. Ex: In [12: 13, 6, 4], player one is a dictator because its weight is 13, which is more than the quota. Dummy: a player whose weight isn't large enough to help pass a motion, or to keep a motion of being passed. If there is a dictator, the rest of the players are all dummies. So in the example above, players 2 & 3 are both dummies. However, you can have a dummy without a dictator. Ex: In [16: 12, 4, 2, 1], players 3 and 4 are dummies because their weights on influence the outcome of the vote. If P1 & P2 vote yes, P3 & P4 don't have enough votes to keep the motion from passing. Even if P3 and P4 vote with P1 & P2, their votes don't really help them at all, they can win without P3 & P4.

5. Veto Power: the power of a player who is not a dictator, but can single handedly prevent a motion from passing. Ex: In [10: 9, 6, 2, 1], P1 has veto porter because even if all the other players vote together, P1 can prevent their vote from passing. In other words, you HAVE to have P1 to pass the motion. Page 72 #1, #3, & # 7 • a) 6 players. b) 7+4+3+3+2+1=20. c) 4. d) 13/20=65% 3. You are given the relationships between the players based on their votes. If a player participates in a wvs, then he has to have at least one vote. So give the last player one vote and based the other players off that. P4 has one vote so P3 has 2, P2 would have 4, P1 has 8. That gives us this wvs: [q: 8, 4, 2, 1]. Now you have to find the quota described in a) through d). a) 2/3 of the votes: add up votes and take 2/3 of it. (2/3)*15=10

6. 7. a). [6: 4, 2, 1] in this wvs, you have veto power and a dummy. Players 1 & 2 have veto power because you have to have both in order to pass the motion. Player 3 is a dummy because it doesn't have enough power to help pass a motion or to keep a motion from passing. If P1 & P2 both vote yea, and P3 vote nay, the motion will pass. Even if P3 votes yea with P1 & P2, it doesn't matter because they can pass it without P3. b). [6: 7, 3, 1]. P1 is a dictator, and P2 & P3 are dummies. c). [10: 9, 9, 1]. Nothing. No dictator, no veto power, no dummies. Assignment: page 72. 2, 4, 6, 8-10

7. Chapter 2: Sec 2: Banzhaf Power Index Now that we know the basics of weighted voting systems, we will study how players can work together to pass a motion. Here are the terms YOU NEED TO KNOW!!!!! Coalition: a set of players that have joined forces to vote together. This can be coalitions of one player, two players, etc. Weight of a coalition: the total number of votes controlled by the players in a coalition. A winning coalition has enough weight to pass the motion. A losing coalition doesn't have enough weight to pass a motion. Critical player: any player in a coalition that is needed to make it a winning coalition. Banzhaf Power Index: a player's power is proportional to the number of times it is a critical player.

8. The total number of coalitions in a BPI is where N is the number of players. Steps to find the Banzhaf Power Index for all the players in a wvs: • Make a list of all the possible coalitions • Mark those coalitions that are winning coalitions • In each WINNING coalition, mark those players that are critical. • Count the number of times each player is critical and the number of total critical players. • The BPI is the number of times a player is critical over the number of total critical players. Ex 2.11 on page 59: [5: 3, 2, 1, 1, 1] Since this wvs has 5 players, there will be coalitions.

9. 1 player 1 2 3 4 5 2 player 12 13 14 15 23 24 25 34 35 45 3 player 123 124 125 134 135 145 234 235 245 345 4 player 1234 1235 1245 1345 2345 5 player 12345 The 31 coalitions for ANY 5 player wvs: I like to form columns based on the number of players in the coalitions. This helps to keep up with what I'm missing. 1 coalition 5 coalitions 5 coalitions **31 total coalitions 10 coalitions 10 coalitions

10. You can use the list on the previous slide for any 5 player wvs. Now that we have all the coalitions, we need to weed out any losing coalitions. [5: 3, 2, 1, 1, 1] 1 player 1 2 3 4 5 3 player 123 124 125 134 135 145 234 235 245 345 4 player 1234 1235 1245 1345 2345 5 player 12345 2 player 12 13 14 15 23 24 25 34 35 45 **all losing coalitions have been crossed out. The weights in those coalitions do not combine to give 5. Since there is no dictator, no single player coalition is a winning one. Since players 1 & 2 win together, any coalition that contains 1 & 2 will also win.

11. Here are all the winning coalitions, now we need to notate the critical players. [5: 3, 2, 1, 1, 1] 2 player 12 3 player 123 124 125 134 135 145 4 player 1234 1235 1245 1345 2345 5 player 12345 **remember the BPI is the number of times a player is critical over the number of total critical players. There are 25 total critical (underlined) players. Player 1 is critical (underlined) 11 times so his BPI is 11/25 or 44%. Player 2: 5/25 or 20%. Player 3: 3/25 or 12%. Player 4: 3/25 or 12%. Player 5: 3/25 or 12%. Hint when finding critical players: cover up the player and if the remaining players can win without the one you covered up, the covered player is NOT critical. If they can't win on their own, the covered player is critical.

12. Page 73:#11 • P1 weight: 6 P2 weight: 5. Weight of P1 & P2 = 11 • Winning coalitions: There are no single player winning coalitions because there is no dictator. 2 player winning coalitions: 3 player winning coalitions: 4 player winning coalitions: {P1, P2} {P1, P3} {P1, P2, P3} {P1, P2, P4} {P1, P3, P4} {P2, P3, P4} {P1, P2, P3, P4} c) P1 d) {P1, P2} {P1, P3} {P1, P2, P3} {P1, P2, P4} {P1, P3, P4} {P2, P3, P4} {P1, P2, P3, P4} There are 12 total CP. P1: 5/12 =