A critical analysis on the notion of power

Abstract

In this paper we discuss on several ways to extend power indices defined on simple games to the context of (j, 2)-games, i.e., games with 2 possible outputs and j possible input levels. The purpose of this paper is more conceptual than technical, and shows that different power indices can be obtained in (j, 2)-games depending on how the criticality of a player is measured. Some power indices for simple games, regarded as measures of power as payment, can be obtained from the selection of two items: (1) a probabilistic model, and (2) a collection of winning coalitions. The combination of two particular probabilistic models and three collections of winning coalitions lead to six well-known power indices for simple games. The Banzhaf and the Johnston power indices are among them. The main goal of the paper is to discuss the generalizations of these six power indices from (2, 2)-games to (j, 2)-games for $j\ge 3$. The probabilistic model and the collection of winning coalitions are trivially extendable to (j, 2)-games. However, at least four possible ways of measuring the criticality of a player can be considered in this new context, and lead to different possible extensions of the six power indices to (j, 2)-games. We highlight two of them as being the most interesting under our point of view. The first one would be suitable for assessing sensitivity, because it measures the player’s ability to modify the output by a small change in his input level. The second one would be adequate to evaluate strength, because it measures the voter’s ability to modify the output by any change in his voting. The main conclusion is that different extensions of the considered power indices, as the Banzhaf and the Jonnston indices, are possible for $j\ge 3$ and that they should be taken into account when working in a multiple input context.