EXAMPLE 16 The Revolt: Banzhaf Model

Example 9, Alice’s committee was transformed to a weighted voting system by a pact in which three weight-1 voters agreed to let Bill have their votes. Alice will be a critical voter in any winning coalition when she is joined by a losing coalition with weight 3, 4, or 5. This losing coalition could be Bill alone or Bill joined by one weight-1 voter, or it could be three or all four weight-1 voters. There is coalition consisting of Bill alone, and coalitions with Bill and one weight-1 voter. There are coalitions of three or four weight-1 voters, so Alice’s Banzhaf index is .

Bill will be a critical voter in any winning coalition when the other voters have a total weight of 2, 3, 4, or 5. These other voters could be Alice plus up to two weight-1 voters; there are of these coalitions. Without Alice, Bill would need two or more weight-1 voters; there are such coalitions, so Bill’s Banzhaf index, 22, is more than twice Alice’s.

A weight-1 voter, Zoë, must be joined by other voters with total weight 5 to form a winning coalition in which she is critical. That could be Alice and two of the three other weight-1 voters; there are of these coalitions. if Zoë is joined by Bill, exactly one of the three other weight-1 voters is needed. There are three of these coalitions too, so Zoë’s Banzhaf index is 6.