On Public Values and Power Indices

Abstract

Abstract.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.

Aggregated Power Indices for Measuring Indirect Control in Complex Corporate Networks with Float Shareholders

“…In general, power indices are used to measure the power of a player i in group decision-making bodies. However, the interpretation of this value— —assigned to player i changes and depends also on the measure (power index) itself and the context in which it is applied, see Felsenthal and Machover (2005) [ 32 ], Laruelle et al (2006) [ 33 ], Gambarelli and Stach (2009) [ 34 ], Bertini and Stach (2011) [ 35 ], Bertini et al (2013) [ 36 ], Bertini et al (2015) [ 37 ], Bertini and Stach (2015) [ 38 ], Stach (2016) [ 39 ], Bertini et al (2016) [ 40 ], Bertini et al (2017) [ 41 ], Bertini et al (2018) [ 42 ], Bertini et al (2020) [ 43 ], Staudacher et al (2021) [ 44 ], Stach and Bertini (2021) [ 28 ], Stach (2022) [ 30 ], Stach and Bertini (2022) [ 45 ], and Stach, Mercik, and Bertini (2023) [ 17 ].…”

Reformulation of Public Help Index θ Using Null Player Free Winning Coalitions

“…Bertini and Stach ( 2015 ) showed that for any simple game , for any . The minimum PHI θ index a null player i can achieve in game is shown in Theorem 5 and Corollary 3 .…”

“…Thus, for any non-null players i, j ∈ N 1 , we have: Bertini and Stach (2015) showed that for any simple game v ∈ S N ,…”

“…The PHI θ index considers all winning coalitions, not only minimal winning coalitions (as in the PGI index). Hence, as Bertini and Stach ( 2015 ) remarked, “ θ rather describes power relationships in the consumption of public goods, whereas the PGI analyzes the production of public goods. In production, one must take care to exclude free-riding; this is why the PGI considers minimal winning coalitions; in the consumption of public goods, you cannot avoid free-riding.”…”

“…5, for example, where a comparison of two indices can be found). The PHI θ index takes part of the so-called public power indices (see Bertini and Stach (2015) and Stach (2016) as θ is well-defined in the social context where goods are public. The PHI θ index considers all winning coalitions, not only minimal winning coalitions (as in the PGI index).…”