The Mathematics of Voting and Apportionment by Sherif El-Helaly

Author:Sherif El-Helaly
Language: eng
Format: epub
ISBN: 9783030147686
Publisher: Springer International Publishing

Voters c, d, and e, who were bystanders in the deal between a and b, lost the little power they had and became dummies!

2.2.3 The Shapley–Shubik Index of Power

This power index is an application of an important game theoretic notion known as the Shapley value which is beyond the scope of this book. We shall therefore take a direct path to the Shapley–Shubik power index and refer the interested reader to [4] and [9] for information on the more general and basic notion of Shapley value.

Consider the yes-no voting system in Example 2.2.2 with four voters a, b, c, d with weights 4, 3, 2, 1, respectively, and a quota of 6. Let us examine, for example, the arrangement or permutation adcb of the voters in which a comes first, d comes second, c comes third, and b comes fourth in this specific chronological order. Voter c possesses special power in this permutation because it is the first voter to form a winning coalition together with the preceding voters. (Note that both {a} and {a, d} are losing but {a, d, c} is winning.) This leads to the following concept, which is central to the development of the Shapley–Shubik index.

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