# 3.7: Exercises(Concepts)

- Page ID
- 34189

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

- Consider the weighted voting system \([q: 7, 3, 1]\)
- 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 but no dictators?
- What is the smallest value for \(q\) that results in exactly two players with veto power?

- Consider the weighted voting system \([q: 9, 4, 2]\)
- 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?
- What is the smallest value for \(q\) that results in exactly two players with veto power?

- Using the Shapley-Shubik method, is it possible for a dummy to be pivotal?

- If a specific weighted voting system requires a unanimous vote for a motion to pass:
- Which player will be pivotal in any sequential coalition?
- How many winning coalitions will there be?

- 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:
- Find the Banzhaf power distribution.
- Find the Shapley-Shubik power distribution

- 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:
- Find the Banzhaf power distribution.
- Find the Shapley-Shubik power distribution

- 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.

- 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.

- 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.

- 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?

- 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.