Balanced contributions, consistency, and value for games with externalities

TL;DR

This paper extends the Shapley value to games with externalities by introducing balanced contributions and various forms of consistency, providing new characterizations that parallel classical results.

Contribution

It introduces balanced contributions and consistency concepts for externality games, leading to novel characterizations of a generalized Shapley value.

Findings

Characterizations of the generalized Shapley value with externalities

Introduction of balanced contributions and consistency properties

Parallel to classical Shapley value characterizations

Abstract

We consider fair and consistent extensions of the Shapley value for games with externalities. Based on the restriction identified by Casajus et al. (2024, Games Econ. Behavior 147, 88-146), we define balanced contributions, Sobolev's consistency, and Hart and Mas-Colell's consistency for games with externalities, and we show that these properties lead to characterizations of the generalization of the Shapley value introduced by Macho-Stadler et al. (2007, J. Econ. Theory 135, 339-356), that parallel important characterizations of the Shapley value.