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On Public Values and Power Indices

Bertini C., Stach I.

Decision Making in Manufacturing and Services

2015

Vol. 9, no. 1

9--25

In this paper, we analyze some values and power indices from a different point of view that are well-defined in the social context where the goods are public. In particular, we consider the Public Good index (Holler, 1982), the Public Good value (Holler and Li, 1995), the Public Help index (Bertini et al., 2008), the König and Bräuninger index (1998) also called the Zipke index (Nevison et al., 1978), and the Rae index (1969). The aims of this paper are: to propose an extension of the Public Help index to cooperative games; to introduce a new power index with its extension to a game value; and to provide some characterizations of the new index and values.

Two variations of the Public Good Index for games with a priori unions

Alonso-Meijide J. M., Casas-Mendez B., Fiestras-Janeiro G., Holler M. J.

Control and Cybernetics

2010

Vol. 39, no 3

839-855

This paper introduces two variations of the Public Good Index (Holler, 1982) for games with a priori unions. The first one stresses the public good property which suggests that all members of a winning coalition derive equal power. The second variation follows earlier work on the integration of a priori unions (Owen, 1977 and 1982) and refers to essential subsets of an a priori union when allocating power shares. Axiomatic characterizations of both indices are discussed. Numerical examples, one of them taken from a political setting, illustrate the new power indices presented in this paper.

Equity properties of the Shapley value as a power index

Malawski M.

Control and Cybernetics

2002

Vol. 31, no 1

117-127

We study power indices for simple games which have the following "uniform transfer property" : when only one losing coalition in a game becomes winning, worths of all other coalitions remaining unchanged, the index increases equally for all players in that coalition and decreases equally for all players not in that coalition. We show that both for superadditive simple games and for all simple games there is only one such index : the Shapley-Shubik index, the restriction of Shapley value to the class of simple games. Moreover, the proof of this fact does not even require the standard assumption of symmetry of power indices which can be replaced by a weaker equal treatment condition.