EXAMPLE 13 A Five-Voter System
use the weighted voting system . List the winning coalitions and find their critical voters.
We will start with the grand coalition. it has a total weight of 19, so there are 7 extra votes. Since no voter has more than 7 votes, the grand coalition has no critical voters.
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Step 1 is to remove members of the grand coalition one at a time and obtain five winning coalitions: (0 extra votes, a minimal winning coalition); and (3 extra votes each; all voters except and are critical); (4 extra votes, only is critical); and (5 extra votes, only is critical).
Step 2 is to remove noncritical voters from the winning coalitions found in Step 1. We obtain , a minimal winning coalition with 0 extra votes, by removing from or by removing from . Only count this once! By removing from or from , one gets . In the same way, there are two ways to get the minimal winning coalitions and from and one other four-member coalition listed above. Finally, the minimal winning coalition can be obtained from or by removing the last voter from either one.
Table 11.5 summarizes the results and determines the Banzhaf power index of each participant. There are 11 winning coalitions in all. When compared with 120 permutations for the Shapley–Shubik index, it is manageable.
| Critical Votes | ||||||
|---|---|---|---|---|---|---|
| Coalition | Extra Votes | |||||
| 7 | 0 | 0 | 0 | 0 | 0 | |
| 0 | 0 | 1 | 1 | 1 | 1 | |
| 3 | 1 | 0 | 1 | 0 | 0 | |
| 3 | 1 | 1 | 0 | 0 | 0 | |
| 4 | 1 | 0 | 0 | 0 | 0 | |
| 6 | 1 | 0 | 0 | 0 | 0 | |
| 0 | 1 | 0 | 1 | 0 | 1 | |
| 2 | 1 | 0 | 1 | 1 | 0 | |
| 0 | 1 | 1 | 0 | 0 | 1 | |
| 2 | 1 | 1 | 0 | 1 | 0 | |
| 3 | 1 | 1 | 1 | 0 | 0 | |
| Banzhaf power index | 9 | 5 | 5 | 3 | 3 | |