The Shapley—Shubik and Banzhaf power indices as probabilities

Abstract

The Shapley — Shubik and Banzhaf indices In 1954 Lloyd Shapley and Martin Shubik published a short paper [12] in the American Political Science Review , proposing that the specialization of the Shapley value to simple games could serve as an index of voting power. That paper has been one of the most frequently cited articles in social science literature of the past thirty years, and its “Shapley—Shubik power index” has become widely known. Shapley and Shubik explained the index as follows: There is a group of individuals all willing to vote for some bill. They vote in order. As soon as a majority has voted for it, it is declared passed, and the member who voted last is given credit for having passed it. Let us choose the voting order of the members randomly. Then we may compute the frequency with which an individual … is pivotal . This latter number serves to give us our index. It measures the number of times that the action of the individual actually changes the state of affairs. … Of course, the actual balloting procedure used will in all probability be quite different from the above. The “voting” of the formal scheme might better be thought of as declarations of support for the bill and the randomly chosen order of voting as an indication of the relative degrees of support by the different members, with the most enthusiastic members “voting” first, etc.[…]

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