Chapter 2, Problem 78E

Suppose that in a weighted voting system there is a player A who hates another player P so much that he will always vote the opposite way of P, regardless of the issue. We will call A the antagonist of P.

a. Suppose that in the weighted voting system $\left[8:5,4,3,2\right]$, P is the player with two votes and his antagonist A is the player with five votes. The other two players we’ll call $P_2$ and $P_3$. What are the possible coalitions under these circumstances? What is the Banzhaf power distribution under these circumstances?

b. Suppose that in a generic weighted voting system with N players there is a player P who has an antagonist A. How many coalitions are there under these circumstances?

c. Give examples of weighted voting systems where a player A can

i. increase his Banzhaf power index by becoming an antagonist of another player.

ii. decrease his Banzhaf power index by becoming an antagonist of another player.

d. Suppose that the antagonist A has more votes than his enemy P. What is a strategy that P can use to gain power at the expense of A?