91Ó°ÊÓ

Consider the weighted voting system \([q: 10,6,5,4,2]\). (a) What is the smallest value that the quota \(q\) can take? (b) What is the largest value that the quota \(q\) can take? (c) What is the value of the quota if at least two-thirds of the votes are required to pass a motion? (d) What is the value of the quota if more than two thirds of the votes are required to pass a motion?

In each of the following weighted voting systems, determine which players, if any, have veto power. (a) \([7: 4,3,3,2]\) (b) \([9: 4,3,3,2]\) (c) \([10: 4,3,3,2\) (d) \([11: 4,3,3,2]\)

A committee has four members \(\left(P_{1}, P_{2}, P_{3},\right.\) and \(\left.P_{4}\right) .\) In this committee \(P_{1}\) has twice as many votes as \(P_{2} ; P_{2}\) has twice as many votes as \(P_{3} ; P_{3}\) and \(P_{4}\) have the same number of votes. The quota is \(q=49 .\) For each of the given definitions of the quota, describe the committee using the notation \(\left[q: w_{1}, w_{2}, w_{3}, w_{4}\right] .\) (Hint: Write the weighted voting system as \([49: 4 x, 2 x, x, x],\) and then solve for \(x .)\) (a) The quota is defined as a simple majority of the votes. (b) The quota is defined as more than two-thirds of the votes. (c) The quota is defined as more than three-fourths of the votes.

A committee has six members \(\left(P_{1}, P_{2}, P_{3}, P_{4}, P_{5},\right.\) and \(\left.P_{6}\right)\). In this committee \(P_{1}\) has twice as many votes as \(P_{2} ; P_{2}\) and \(P_{3}\) each has twice as many votes as \(P_{4} ; P_{4}\) has twice as many votes as \(P_{5} ; P_{5}\) and \(P_{6}\) have the same number of votes. The quota is \(q=121 .\) For each of the given definitions of the quota, describe the committee using the notation \(\left[q: w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6}\right] .\) (Hint: Write the weighted voting system as \([121: 8 x, 4 x, 4 x, 2 x, x, x],\) and then solve for \(x .)\) (a) The quota is defined as a simple majority of the votes. (b) The quota is defined as more than two-thirds of the votes. (c) The quota is defined as more than three-fourths of the votes.

Consider the weighted voting system \([16: 9,8,7]\) (a) Write down all the sequential coalitions, and in each sequential coalition identify the pivotal player. (b) Find the Shapley-Shubik power distribution of this weighted voting system.

Find the Shapley-Shubik power distribution of each of the following weighted voting systems. (a) \([8: 8,4,2,1]\) (b) \([9: 8,4,2,1]\) (c) \([12: 8,4,2,1]\) (d) \([14: 8,4,2,1]\)

Find the Shapley-Shubik power distribution of each of the following weighted voting systems. (a) \([41: 40,10,10,10]\) (b) \([49: 40,10,10,10]\) (Hint: Compare this situation with the one in (a).) (c) \([50: 40,10,10,10]\)

Consider a weighted voting system with five players \(\left(P_{1}\right.\) through \(P_{5}\) ). (a) Find the total number of coalitions in this weighted voting system. (b) How many coalitions in this weighted voting system do not include \(P_{1} ?\) (Hint: Think of all the possible coalitions of the remaining players.) (c) How many coalitions in this weighted voting system do not include \(P_{5}\) ? [Hint: Is this really different from (b)? (d) How many coalitions in this weighted voting system do not include \(P_{1}\) or \(P_{5} ?\) (e) How many coalitions in this weighted voting system include both \(P_{1}\) and \(P_{5} ?\) [Hint: Use your answers for (a) and (d).]

(a) Given that \(10 !=3,628,800,\) find \(9 !\) (b) Find \(\frac{11 !}{10 !}\) (c) Find \(\frac{11 !}{9 !}\) (d) Find \(\frac{9 !}{6 !}\) (e) Find \(\frac{101 !}{99 !}\)

Veto power. A player \(P\) with weight \(w\) is said to have veto power if and only if \(w