Chapter 2, Problem 59E

Equivalent voting systems. Two weighted voting systems are equivalent if they have the same number of players and exactly the same winning coalitions.

a. Show that the weighted voting systems $\left[8:5,3,2\right]$ and $\left[2:1,1,0\right]$ are equivalent.

b. Show that the weighted voting systems $\left[7:4,3,2,1\right]$ and $\left[5:3,2,1,1\right]$ are equivalent.

c. Show that the weighted voting system discussed in Example 2.12 is equivalent to $\left[3:1,1,1,1,1\right]$.

d. Explain why equivalent weighted voting systems must have the same Banzhaf power distribution.

e. Explain why equivalent weighted voting systems must have the same Shapley Shubik power distribution.