Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-37/5/a/solution

Give each permanent member weight seven, each nonpermanent member weight one, and use the strict threshold . If a coalition omits a permanent member, its weight is at most , so it loses. If it includes all five permanent members and others, its weight is , which exceeds 38 exactly when . These are precisely the required winning coalitions. Thus a weighted voting game representation is
The strict-threshold convention is important: in the alternative convention requiring weight at least the quota, the quota is 39.

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