Steady Marginality: A Uniform Approach to Shapley Value for Games with Externalities

TL;DR

This paper introduces a dual marginality approach to extend the Shapley value for cooperative games with externalities, unifying two major existing extensions and providing a comprehensive framework.

Contribution

It proposes a dual marginality framework that derives McQuillin's value, complementing previous work on Pham Do and Norde's externality-free value.

Findings

Unified framework for externalities in cooperative games

Derived McQuillin's value using dual marginality approach

Closed the gap between two major externality extensions

Abstract

The Shapley value is one of the most important solution concepts in cooperative game theory. In coalitional games without externalities, it allows to compute a unique payoff division that meets certain desirable fairness axioms. However, in many realistic applications where externalities are present, Shapley's axioms fail to indicate such a unique division. Consequently, there are many extensions of Shapley value to the environment with externalities proposed in the literature built upon additional axioms. Two important such extensions are "externality-free" value by Pham Do and Norde and value that "absorbed all externalities" by McQuillin. They are good reference points in a space of potential payoff divisions for coalitional games with externalities as they limit the space at two opposite extremes. In a recent, important publication, De Clippel and Serrano presented a marginality-based axiomatization of the value by Pham Do Norde. In this paper, we propose a dual approach to marginality which allows us to derive the value of McQuillin. Thus, we close the picture outlined by De Clippel and Serrano.