31. Calculate the Banzhaf index of the voting system in Exercise 30. Who is more powerful according to the Banzhaf model, a faculty member or an administrator?
31.
A faculty member will be a critical voter in a coalition that includes, in addition to , exactly two of the other three faculty members and at least two administrators. There are ways to assemble the other faculty members and ways to assemble the administrators. Thus, each faculty member has a Banzhaf power index of .
An administrator will be critical in a winning coalition that includes at least three faculty members and exactly one of the other two administrators. There are ways to assemble the faculty members and ways to get the other administrator. An administrator, with a Banzhaf power index of , is a bit less powerful on this committee than a faculty member, according to the Banzhaf model.