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Mathematics LibreTexts

3.7: Exercises(Concepts)

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    34189
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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 \(\left(\mathrm{V}_{1}, \mathrm{V}_{2}, \mathrm{V}_{3}\right)\). 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.

    3.7: Exercises(Concepts) is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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