Axiomatizations of public good indices with a priori unions

We provide axiomatizations for six variants of the Public Good Index for games with a priori unions. Two such coalitional PGIs have been introduced and alternatively axiomatized in Alonso-Meijide et al. (2008b). They assign power in two steps. In the first step, power is distributed between unions according to the PGI of the quotient game. In a second step, the Solidarity PGI splits power equally among union members while the Owen Extended PGI takes into account so-called essential parts. The other four coalitional PGIs have been introduced in Holler and Nohn (2009). The first variant elaborates the original idea of Holler (1982) that the coalitional value is a public good and only minimal winning coalitions of the quotient game are relevant. The remaining three variants also use the two-step distribution where, however, on the member stage they take into account the possibilities of players to threaten their partners through leaving their union.