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Mathematics LibreTexts

3.7: Exercises(Concepts)

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  1. Consider the weighted voting system [q:7,3,1]
    1. Which values of q result in a dictator (list all possible values)
    2. What is the smallest value for q that results in exactly one player with veto power but no dictators?
    3. What is the smallest value for q that results in exactly two players with veto power?
  2. Consider the weighted voting system [q:9,4,2]
    1. Which values of q result in a dictator (list all possible values)
    2. What is the smallest value for q that results in exactly one player with veto power?
    3. What is the smallest value for q that results in exactly two players with veto power?
  3. Using the Shapley-Shubik method, is it possible for a dummy to be pivotal?
  4. If a specific weighted voting system requires a unanimous vote for a motion to pass:
    1. Which player will be pivotal in any sequential coalition?
    2. How many winning coalitions will there be?
  5. Consider a weighted voting system with three players. If Player 1 is the only player with veto power, there are no dictators, and there are no dummies:
    1. Find the Banzhaf power distribution.
    2. Find the Shapley-Shubik power distribution
  6. Consider a weighted voting system with three players. If Players 1 and 2 have veto power but are not dictators, and Player 3 is a dummy:
    1. Find the Banzhaf power distribution.
    2. Find the Shapley-Shubik power distribution
  7. An executive board consists of a president (P) and three vice-presidents (V1,V2,V3). For a motion to pass it must have three yes votes, one of which must be the president's. Find a weighted voting system to represent this situation.
  8. On a college’s basketball team, the decision of whether a student is allowed to play is made by four people: the head coach and the three assistant coaches. To be allowed to play, the student needs approval from the head coach and at least one assistant coach. Find a weighted voting system to represent this situation.
  9. In a corporation, the shareholders receive 1 vote for each share of stock they hold, which is usually based on the amount of money the invested in the company. Suppose a small corporation has two people who invested $30,000 each, two people who invested $20,000 each, and one person who invested $10,000. If they receive one share of stock for each $1000 invested, and any decisions require a majority vote, set up a weighted voting system to represent this corporation’s shareholder votes.
  10. A contract negotiations group consists of 4 workers and 3 managers. For a proposal to be accepted, a majority of workers and a majority of managers must approve of it. Calculate the Banzhaf power distribution for this situation. Who has more power: a worker or a manager?
  11. The United Nations Security Council consists of 15 members, 10 of which are elected, and 5 of which are permanent members. For a resolution to pass, 9 members must support it, which must include all 5 of the permanent members. Set up a weighted voting system to represent the UN Security Council and calculate the Banzhaf power distribution.

This page titled 3.7: Exercises(Concepts) is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.

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