Sub-coalitional egalitarian and solidarity values

This paper regards some measures for sharing (public) goods or budgets among members with different participation quotas in a binary decisionmaking process. The main characteristic of such measures is that they should have elements of solidarity with those who have a weak quota of participation in the process. These measures seem appropriate for deals that require solidarity, which contrasts with the classical power indices such as the Shapley and Shubik index or the Banzhaf index. Moreover, we provide a new representation for two
power indices—the Public Help Index ξ (proposed by Bertini and Stach in 2015) and the particularization of the solidarity value proposed by Nowak and Radzik in 1994—the ψ index—in a simple game using null player free winning coalitions.
As it is known, a set of null player free winning coalitions unequivocally determines a simple game. Finally, we compare considered power indices considering some properties in simple games.