41. A committee has senior members , , and , and junior members , , and . Each senior member has voting weight 3, and junior members each have voting weight 1. The quota for passing a measure is 7. Find the minimal winning coalitions of the committee and determine the Banzhaf power index for members of each class.
41.
The minimal winning coalitions would be formed by either all three senior members or two senior members and one junior member. The winning coalitions that include as a critical voter would be ; , with being either or and being one of , , (there are of these); , with as before and representing two of , , (there are 6 of these coalitions, too); or (two coalitions). Hence the Banzhaf power index of each senior member is .
A junior member can only be critical in a coalition with two senior members and no other junior members. There are 3 such coalitions—that is the Banzhaf power index of a junior member.