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Consider the weighted voting system \([q: 7, 3, 1]\)
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Which values of \(q\) result in a dictator (list all possible values)
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What is the smallest value for \(q\) that results in exactly one player with veto power but no dictators?
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What is the smallest value for \(q\) that results in exactly two players with veto power?
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Consider the weighted voting system \([q: 9, 4, 2]\)
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Which values of \(q\) result in a dictator (list all possible values)
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What is the smallest value for \(q\) that results in exactly one player with veto power?
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What is the smallest value for \(q\) that results in exactly two players with veto power?
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Using the Shapley-Shubik method, is it possible for a dummy to be pivotal?
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If a specific weighted voting system requires a unanimous vote for a motion to pass:
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Which player will be pivotal in any sequential coalition?
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How many winning coalitions will there be?
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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:
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Find the Banzhaf power distribution.
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Find the Shapley-Shubik power distribution
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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:
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Find the Banzhaf power distribution.
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Find the Shapley-Shubik power distribution
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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.
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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.
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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.
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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?
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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.