The Public Good index for games with several levels of approval in the input and output

TL;DR

This paper generalizes the Public Good index to $(j,k)$ simple games with multiple approval levels, providing new axiomatizations and extending its applicability beyond traditional simple games.

Contribution

It introduces a generalized Public Good index for $(j,k)$ simple games and offers corresponding axiomatizations, expanding the theoretical framework of power indices.

Findings

Generalization of the Public Good index to $(j,k)$ simple games.

Provision of axiomatizations for the new index.

Extension of power index theory to multi-level approval games.

Abstract

The Public Good index is a power index for simple games introduced by Holler and later axiomatized by Holler and Packel, so that some authors also speak of the Holler--Packel index. A generalization to the class of games with transferable utility was given by Holler and Li. Here we generalize the underlying ideas to games with several levels of approval in the input and output -- so-called $(j,k)$ simple games. Corresponding axiomatizations are also provided.