15. For each of the following weighted voting systems, make a list of all winning coalitions. Identify the critical voters in each coalition, and calculate the Banzhaf power index for each voter. Give the voters the names .
15.
In listing winning coalitions, the critical voters are marked with an asterisk.
(a) is dictator; the winning coalitions are and and is the only critical voter in each. Hence has a Banzhaf power index of 2, and has a Banzhaf power index of 0.
(b) This is a majority-rule system: The winning coalitions are all with two voters, and the grand coalition. The grand coalition has no critical voters, and each member of a two-voter coalition is critical. For example, is critical in and , so the Banzhaf power index of is 2. and also have Banzhaf power indices of 2.
(c) The winning coalitions are , , and . In the first two, both voters are critical, but in the grand coalition, only is critical. Hence the Banzhaf power index of is 3, while the Banzhaf power indices of and are both equal to 1.
(d) Any two of , , and can form a winning coalition. cannot form a winning coalition by joining with a single other voter, and so is a dummy. The winning coalitions are: , , , , , , and . Thus, , , and have Banzhaf power indices equal to 4, while the Banzhaf power index of is 0.
(e) can form a winning coalition with any one of the other three voters. Without it takes all three of the other voters to form a winning coalition. Thus, the winning coalitions are , , , , , , , and . The Banzhaf power indices are: , 6; , 2:, , 2; and , 2.