A Tractable Class of Cooperative Games Defined by Directed Networks: Unanimity Decomposition and Shapley Value

TL;DR

This paper introduces a new class of cooperative games based on weighted directed graphs, providing efficient formulas for solution concepts like the Shapley value and analyzing their stability and fairness properties.

Contribution

The paper defines a tractable class of network-induced cooperative games with polynomial-time solution formulas and analyzes their core and balancedness.

Findings

Closed-form polynomial-time formulas for Shapley and Banzhaf values.

The game has a nonempty core and is totally balanced.

Provides an example where stability and fairness solutions differ.

Abstract

We introduce a class of cooperative games induced by weighted directed graphs. Specifically, the coalitional value combines an internal interaction term given by the induced subgraph game with an external component based on minimal incoming edges from outside the coalition. The resulting game has a convenient representation in terms of unanimity games. This representation enables closed-form polynomial-time formulas for the Shapley and Banzhaf values. We further establish that the game has a nonempty core and is totally balanced. The class of such games therefore provides an analytically and computationally tractable example of structured network- induced cooperative games in which stability-based allocations and fairness-based solution concepts do not coincide.