math

Mathematics Assignment

13. An executive board consists of a president (P) and three vice-presidents (V1,V2,V3)(V1,V2,V3). For a motion to pass it must have yes votes from three of the voting members, one of which must be the president's. A weighted system that could represent this situation is: [Q:P,V1,V2,V3][Q:P,V1,V2,V3] where: QQ =  PP =  V1V1 =  V2V2 =  V3V3 = 

12. Consider the weighted voting system [12: 7, 4, 1] Find the Shapley-Shubik power distribution of this weighted voting system. List the power for each player as a fraction: P1P1:    P2P2:    P3P3:   

11. Consider the weighted voting system [15: 14, 6, 4, 1] In the sequential coalition <P1,P2,P3,P4P1,P2,P3,P4 > which player is pivotal? Pivotal player =  Identify players by their number only. For example, if the answer is P2P2, enter 2

10. Consider the weighted voting system [15: 15, 8, 3, 1] Find the Banzhaf power distribution of this weighted voting system. List the power for each player as a fraction: P1P1:    P2P2:    P3P3:    P4P4:   

9. Find the Banzhaf power distribution of the weighted voting system [29: 17, 12, 11, 5] Give each player's power as a fraction or decimal value P1P1 =    P2P2 =    P3P3 =    P4P4 =   

8. Consider a weighted voting system with three players. If no one has veto power, and no one is a dummy., find the Banzhof power distribution. Player 1:    Player 2:    Player 3:    Give each value as a fraction or decimal.

7. Consider the weighted voting system [q: 8, 4, 1]. Which values of q result in a dictator (list all possible values) What is the smallest value for q that results in exactly one player with veto power who is not a dictator? What is the smallest value for q that results in exactly two players with veto power?

6. Consider the weighted voting system [12: 9, 6, 4, 1] In the coalition {P1,P2,P4P1,P2,P4} which players are critical? Critical players =  Identify players by their number only. For example, if the answer is P1P1 and P2P2, enter 1,2

5. Consider the weighted voting system [17: 9,4,3,2,2,1,1] Find the weight of the coaltion {P2,P3,P4}{P2,P3,P4} Weight = 

4. Consider the weighted voting system [15: 11, 7, 5, 2] Identify which players: Are dictators:  Have veto power, but are not dictators:  Are dummies:  Identify players by their number only. For example, if the answer is P1P1 and P2P2, enter 1,2. If there are no players that fit the description, leave the space blank.

3. Consider the weighted voting system [9: 9, 5, 2, 1] Identify which players: Are dictators:  Have veto power, but are not dictators:  Are dummies:  Identify players by their number only. For example, if the answer is P1P1 and P2P2, enter 1,2. If there are no players that fit the description, leave the space blank.

2. Consider the weighted voting system [q: 8,6,6,3,2] What is the smallest value that the quota q can take? What is the largest value that the quota q can take? What is the value of the quota if at least two-thirds of the votes are required to pass a motion?

1. Consider the weighted voting system [30: 9,8,7,6,5,3,2,2,1] Find: The total number of players:  The total number of votes:  The quota: