Weighted Voting Problems

Problem 1 1. The disciplinary board at PU is composed of 5 members, two of whom must be faculty and three of whom must be students. To pass a motion requires at least 3 votes and at least one of these must be from a faculty member. a. Find the Banzhaf power distribution of the disciplinary board. b. Describe the board as a weighted voting system [q:f,f,s,s,s].

Problem 2 2. a. Consider the weighted voting system [22:10,10,10,10,1]. Are there any dummies? Explain. b. Using your answer from part a. find the Banzhaf power distribution of this weighted voting system. (You shouldn’t have to do any work except a little arithmetic!) c. Consider the weighted voting system [q:10,10,10,10,1]. Find all the possible values of q for which the fifth participant is not a dummy. d. Consider the weighted voting system [34:10,10,10,10,w]. Find all positive integers w where the last participant is a dummy.

Problem 3 3. Consider the weighted voting systems [9:w,5,2,1]. a. What are the possible values of w? b. Which values of w result in a dictator? Who is it and why are they a dictator? c. Which values of w result in a participant who has veto power? Who is it? d. Which values of w result in one or more dummies?

Problem 4 4. a. Verify that the weighted voting systems [12:7,4,3,2] and [24:14,8,6,4] result in the same Banzhaf power distribution. (Do the calculations side by side and look for patterns.) b. Based on your work in part a. explain why two proportional weighted voting systems [q:w1,w2,...,wN] and [cq:cw1,cw2,...,cwN] always have the same Banzhaf power distribution.

Problem 5 5. Consider the weighted voting system [q:5,4,3,2,1]. a. For what values of q is there a dummy? b. For what values of q do all players have the same power?