Exam 2: The Mathematics of Power: Weighted Voting

A company is run by one senior executive and three junior executives. Suppose that these four executives vote on all decisions made. In order for a decision to pass the senior executive and two junior executives must vote in favor of it. What is the Shapely-Shubik Power of the senior executive?

Consider the weighted voting system [q: 25,18,12] . What is the largest value of the quota q so that the weighted voting system would be equivalent to a weighted voting system where each member has one vote and a simple majority is needed to pass a motion?

Consider the weighted voting system [16: 12,10,4,2] . Which player(s), if any, are dummies?

Tim and his three friends Beth ,Jane, Sam pool their money to buy a pizza . Tim gives $4; Beth gives 43; while both Jane and Sam each gives $2. if there are two choices for pizza and each person gets to vote proportionally to the amont of money he or she gave , then which weighted voting system models the senario if the quota is a simple majority of votes?

Consider the weighted voting system [37: 18,13,11, w₄, 1,1] . What is the smallest possible value for the weight w₄ which makes P₄ pivotal in the sequential coalition ?

Consider the weighted voting system [q: 8,7,6,2,1,1] . What is the largest value that the quota q can be to ensure that no player has veto power?

Consider the weighted voting system [18: 8,7,6,2,1] . There is only one coalition in which P₅ is critical. Give this coalition.

Which of the weighted voting systems given below contains a dictator?

Clearly explain why it is not possible for a weighted voting system to have two dictators.

On the popular television show Survivor, each tribe usually begins with 8 members or players. Alliances between the tribe members are basically the same idea as coalitions within a voting system. Using this fact, how many different alliances are possible within a tribe?

Your college's Math Club consists of four voting members: the president, vice president, secretary, and treasurer. The president's vote counts four times as much as the secretary and treasurer's votes and two times as much as the vice-president's vote. If the secretary and treasurer each have the exactly one vote and any motion needs at least six favorable votes in order to be passed, then give the weighted voting system which models this scenario.

Consider the weighted voting system [16: 12,10,4] . What is the Shapely-Shubik Power of player one?

Your friend's family consists of two parents and three children. How many different sequential coalitions are possible?

Consider the weighted voting system[25: w1, 7, 6, 2]. What is the smallest possible value for the

many of the four share-holders have veto power?

The Party Planning Team for your local Student Union is organizing a Spring Fair. The team consists of three members: Kathy, Tim, and Sara. Each member gets to vote on one of two theme ideas; however, Kathy's vote counts twice as much as Tim's vote and three times as much as Sara's vote. If at least 75 % of the counted votes are needed in order to choose a theme, then find the Shapley-Shubik Power for each member. Give your answers as percentages rounded to the nearest whole number.

Consider the weighted voting system [16: 12,10,4,2] . In how many sequential coalitions is player P₃ pivotal?

Find the Banzhaf Power for each player in the following weighted voting system. Give your answers as fractions written in reduced form. [16: 12,10,4,2]

Consider the weighted voting system [25: 18,14, w₃] . What is the largest possible value for the weight w₃ which guarantees P₃ is the only player who is a dummy?

Consider the weighted voting system [22: 12,10,7,2] . Which player(s), if any, are dummies?