Chapter 2, Problem 66E

Mergers. Sometimes in a weighted voting system two or more players decide to merge-that is to say, to combine their votes and always vote the same way. (Note that a merger is different from a coalition-coalitions are temporary, whereas mergers are permanent.) For example, if in the weighted voting system $\left[7:5,3,1\right]$ $P_2$ and $P_3$ were to merge, the weighted voting system would then become $\left[7:5,4\right]$ In this exercise we explore the effects of mergers on a player’s power.

a. Consider the weighted voting system $\left[4:3,2,1\right]$. In Example 2.9 we saw that $P_2$ and $P_3$ each have a Banzhaf power index of $1/5$. Suppose that $P_2$ and $P_3$ merge and become a single player $P^{\ast }$. What is the Banzhaf power index of $P^{\ast }$?

b. Consider the weighted voting system $\left[5:3,2,1\right]$. Find first the Banzhaf power indexes of players $P_2$ and $P_3$ and then the Banzhaf power index of $P^{\ast }$ (the merger of $P_2$ and $P_3$ ). Compare.

c. Rework the problem in (b) for the weighted voting system $\left[6:3,2,1\right]$.

d. What are your conclusions from (a), (b), and (c)?