49. A 5-member committee will use weighted voting. The members are assigned voting weights of 12, 7, 4, 3, and 1. They would like to set the quota so that no member has veto power and no member is a dummy voter. Help them by giving them a list of the quotas that meet their specifications. For each quota that you list, find all of the minimal winning coalitions.
49.
Call the voters , , , , and , in decreasing order by weight. The total weight of this system is 27, so the quota must be at least 14. If is not to have veto power, then the remaining voters, whose total weight is 15, must be able to pass a motion. Thus, the only possible quotas that prevent any voter from having veto power are 14 and 15. If the quota is 14, will be a dummy voter. The winning coalitions that include are , which has total weight 15, and combined with some nonempty subset of , which would have total weight at least equal to 16, which is the weight of . Voter would not be critical in any of these coalitions, since they all have at least 1 extra vote.
If the quota is 15, then has 0 extra votes, which makes it a minimal winning coalition. The weight-1 voter is not a dummy. Therefore, 15 is the only quota that meets the specifications. The other minimal winning coalitions are , , and .