EXAMPLE 20 Three-Voter Systems

We will make a list of all voting systems that have three participants, , , and . No two voting systems on the list should be equivalent.

Suppose that is a minimal winning coalition. Then can pass a motion unilaterally and is a dictator. and are dummies. Systems where or is dictator are equivalent to this one.

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If there is no dictator, every minimal winning coalition must contain either two or all three voters. Let’s consider the case in which the grand coalition is a minimal winning coalition. it is the only winning coalition, because any other winning coalition would have to be entirely contained in this coalition, which requirement 2 doesn’t allow. in this voting system, a unanimous vote is required to pass a measure. We will call this system consensus rule.

Finally, let’s suppose that there is a two-voter minimal winning coalition, . if it is the only minimal winning coalition, then is a dummy and and make all the arrangements. We will call this system the clique. Of course, the clique could be or , but these are equivalent systems.

There could be two 2-voter minimal winning coalitions, say and . Neither coalition contains the other, and there is an overlap, so all of the requirements are satisfied. in this system, has veto power. We encountered this system in Example 1 (page 463)—where Allen had veto power—and we will call it the chair veto. There are two other voting systems equivalent to this one, where or is chair.

It is possible that all three two-member coalitions are minimal winning coalitions. Because there are only three voters, any two distinct two-member coalitions will overlap, so the requirements are still satisfied. This is the majority rule system.