Weighted Voting Problems

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Weighted Voting Problems
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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
  qf,f,s,s,s.

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Problem 2

• 2. a. Consider the weighted voting system
  2210,10,10,10,1. Are there any dummies?
  Explain.
• b. Using your answer from part a. find the
  Banzhaf power distribution of this weighted
  voting system. (You shouldnt have to do any work
  except a little arithmetic!)
• c. Consider the weighted voting system
  q10,10,10,10,1. Find all the possible values
  of q for which the fifth participant is
  not a dummy.
• d. Consider the weighted voting system
  3410,10,10,10,w. Find all positive integers w
  where the last participant is a dummy.

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Problem 3

• 3. Consider the weighted voting systems
  9w,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?

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Problem 4

• 4. a. Verify that the weighted voting systems
  127,4,3,2 and 2414,8,6,4 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
• qw1,w2,...,wN and cqcw1,cw2,...,cwN
  always have the same Banzhaf power distribution.

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Problem 5

• 5. Consider the weighted voting system
  q5,4,3,2,1.
• a. For what values of q is there a dummy?
• b. For what values of q do all players have
  the same power?