Abstract
This paper proposes a new representation for the Public Help Index θ (briefly, PHI θ). Based on winning coalitions, the PHI θ index was introduced by Bertini et al. in (2008). The goal of this article is to reformulate the PHI θ index using null player free winning coalitions. The set of these coalitions unequivocally defines a simple game. Expressing the PHI θ index by the winning coalitions that do not contain null players allows us in a transparent way to show the parts of the power assigned to null and non-null players in a simple game. Moreover, this new representation may imply a reduction of computational cost (in the sense of space complexity) in algorithms to compute the PHI θ index if at least one of the players is a null player. We also discuss some relationships among the Holler index, the PHI θ index, and the gnp index (based on null player free winning coalitions) proposed by Álvarez-Mozos et al. in (2015).
Highlights
In the literature, a power index is defined as a measure that is meant to assess the a priori power, as the influence, or as the payoff expectation of players in a simple game.This paper concentrates on the Public Help Index θ introduced by Bertini et al (2008) and based on winning coalitions
When a simple game is defined by the set of null player free winning coalitions, the calculation of the θ index by Formula (1) is immediate
We concentrate on two power indices: the PHI θ (Bertini et al 2008), and the gnp index proposed by Álvarez-Mozos et al (2015)
Summary
A power index is defined as a measure that is meant to assess the a priori power, as the influence, or as the payoff expectation of players in a simple game. In the forthcoming paper by Stach and Bertini (2021), a representation that is based on the information contained in a set of null player free winning coalitions is given for some well-known indices like the Banzhaf (1965) index, the Rae (1969) index, Coleman’s (1971) indices to prevent action and to initiate action, Nevison’s (1979) Z index, and the König and Bräuninger (1998) index. 4, some examples of a simple game are considered to compare PHI θ with the gnp and PGI indices in order to reveal the possible application fields in which the θ index is suitable and to show how the new formula of PHI θ can be useful in its algorithmic calculation in games with at least one null player and in games that are determined by a set of null player free winning coalitions.
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