EXAMPLE 22 The Scholarship Committee

A university offers scholarships on the basis of either academic excellence or financial need. Each application for a scholarship is reviewed by two professors, who rate the student academically, and two financial aid officers, who rate the applicant’s need. If both professors or both financial-aid officers recommend the applicant for a scholarship, the dean of admissions decides whether to award a scholarship. Is it possible to assign weights to the professors, the financial-aid officers, and the dean to reflect this decisionmaking system? The answer is no.

To see why, let’s focus on the minimal winning coalitions. The participants are the two professors, and ; the financial aid officers, and ; and the dean, , who has veto power. The minimal winning coalitions (see Figure 11.2) are and .

Consider the following two winning coalitions: In , all except the financial-aid officer favors an award; while in , Professor dissents. Thus

In , we notice that is a critical voter and isn’t, while in the tables are turned because is critical while is not. If this were a weighted voting system, then in any winning coalition, the critical voters would all have greater weight than those who are not critical. Thus, would have to have both more weight than (because of the situation in ) and less weight than (because of ), which is impossible.