Counting Swings in a Committee
A committee has five members with 5, 4, 3, 2 and 1 votes (15 in total). A motion passes with at least 8 votes. For a member i, a *swing* is a coalition S of the other four members that does not win by itself but wins when i joins. The (raw) Banzhaf count of i is the number of swings for i, and the *normalised* Banzhaf index of i is i's count divided by the sum of all five counts. What is the normalised Banzhaf index of the member with 4 votes? Give a reduced fraction.
Related problems
Source: Banzhaf power index (Penrose 1946, Banzhaf 1965); the weighted voting system is invented for AxiomIQ. Statement written for AxiomIQ.