Mergers. Sometimes in a weighted voting system two or more players decide to mergethat

Chapter 2, Problem 72

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Mergers. Sometimes in a weighted voting system two or more players decide to mergethat is to say, to combine their votes and always vote the same way. (Note that a merger is different from a coalitioncoalitions are temporary, whereas mergers are permanent.) For example, if in the weighted voting system 37: 5, 3, 14 P2 and P3 were to merge, the weighted voting system would then become 37: 5, 44. In this exercise we explore the effects of mergers on a players power. (a) Consider the weighted voting system 34: 3, 2, 14. In Example 9 we saw that P2 and P3 each have a Banzhaf power index of 1>5. Suppose that P2 and P3 merge and become a single player P*. What is the Banzhaf power index of P*? (b) Consider the weighted voting system 35: 3, 2, 14. Find first the Banzhaf power indexes of players P2 and P3 and then the Banzhaf power index of P* (the merger of P2 and P3). Compare. (c) Rework the problem in (b) for the weighted voting system 36: 3, 2, 14. (d) What are your conclusions from (a), (b), and (c)?