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Magazine Problem 57
Consider the weighted voting system \([q: 8,4,1]\) (a) What are the possible values of \(q ?\) (b) Which values of \(q\) result in a dictator? (Who? Why?) (c) Which values of \(q\) result in exactly one player with veto power? (Who? Why?) (d) Which values of \(q\) result in more than one player with veto power? (Who? Why?) (e) Which values of \(q\) result in one or more dummies? (Who? Why?)
Problem 58
Consider the weighted voting system \([9: w, 5,2,1]\) (a) What are the possible values of \(w ?\) (b) Which values of \(w\) result in a dictator? (Who? Why?) (c) Which values of \(w\) result in a player with veto power? (Who? Why?) (d) Which values of \(w\) result in one or more dummies? (Who? Why?)
Problem 80
(a) Explain why in any weighted voting system with \(N\) players a player with veto power must have a Banzhaf power index bigger than or equal to \(\frac{1}{N}\) (b) Explain why in any weighted voting system with \(N\) players a player with veto power must have a ShapleyShubik power index bigger than or equal to \(\frac{1}{N^{*}}\)
