Solution
Counting Swings in a Committee
Show the problem again
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.
Worked solution
For each member, list the subsets of the other four with total t satisfying 8 - w ≤ t ≤ 7, where w is the member's weight, and count them. By enumeration the counts are 10 (weight 5), 6 (weight 4), 6 (weight 3), 2 (weight 2) and 2 (weight 1), a total of 26. For the 4-vote member the others hold 5, 3, 2 and 1, and a swing needs the others' subset total t to satisfy 4 ≤ t ≤ 7. The qualifying subsets are {5}, {3,2}, {3,1}, {5,1}, {5,2} and {3,2,1}: 6 swings, so the index is 6/26 = **3/13**. Notice the 4-vote and 3-vote members have equal power even though their weights differ.
Source: Banzhaf power index (Penrose 1946, Banzhaf 1965); the weighted voting system is invented for AxiomIQ. Statement written for AxiomIQ.