# The Shapley Axiomatization for Values in Partition Function Games

arXiv: https://arxiv.org/abs/1308.6255 · HTML: https://mlsanity.com/paper/1308.6255/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2013.

## Built on by (most cited first, 1 of 1)

- [Explainable Graph Neural Networks via Structural Externalities](https://mlsanity.com/paper/2507.17848/) (2025)

## Abstract (arXiv)

One of the long-debated issues in coalitional game theory is how to extend the Shapley value to games with externalities (partition-function games). When externalities are present, not only can a player's marginal contribution - a central notion to the Shapley value - be defined in a variety of ways, but it is also not obvious which axiomatization should be used. Consequently, a number of authors extended the Shapley value using complex and often unintuitive axiomatizations. Furthermore, no algorithm to approximate any extension of the Shapley value to partition-function games has been proposed to date. Given this background, we prove in this paper that, for any well-defined measure of marginal contribution, Shapley's original four axioms imply a unique value for games with externalities. As an consequence of this general theorem, we show that values proposed by Macho-Stadler et al., McQuillin and Bolger can be derived from Shapley's axioms. Building upon our analysis of marginal contribution, we develop a general algorithm to approximate extensions of the Shapley value to games with externalities using a Monte Carlo simulation technique.