EXAMPLE 12 The Banzhaf Model Applied to the Film Selection Committee

Let’s return to the committee of Example 4 on page 465. The members are Allen, Betty, and Cao, and they use the voting system. Let’s identify the critical voters in the grand coalition, {Allen, Betty, Cao}. if Allen leaves the coalition, what happens?

Allen is a critical voter because the remaining voters form a losing coalition when he changes his vote. This is not surprising: He has veto power.

If Betty changes her vote, the remaining voters still form a winning coalition:

Betty is not a critical voter in this coalition, and you can verify for yourself that Cao is also not a critical voter.

To determine the Banzhaf power indices of Allen, Betty, and Cao, refer to Table 11.4. The table lists all winning coalitions and the extra votes in each—that’s the total weight of the coalition minus the quota, 6. To obtain the list, the noncritical voters were removed, one by one, from the grand coalition. This yielded two winning coalitions, {Allen, Betty} and {Allen, Cao}. We cannot remove voters from these two coalitions without making losing coalitions, so we stop. The voters in each winning coalition with weight greater than the extra votes are the critical voters. We find Allen is critical in all three coalitions, while Betty and Cao are each critical in just one. Hence the Banzhaf index is (3, 1, 1).

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