# The Shapley Value of Digraph Games

## TL;DR

This paper explores the calculation of the Shapley value in directed graph games, considering player dominance and cooperation constraints, and provides methods for efficient computation in cyclic digraphs.

## Contribution

It introduces a novel approach to compute the Shapley value in digraph games with dominance relations and cyclic structures, including a quick calculation method for specific characteristic functions.

## Key findings

- Shapley value for cyclic digraph games is derived and analyzed.
- A quick calculation method for certain characteristic functions is formulated.
- The paper establishes a dominance-based cooperation model among players.

## Abstract

In this paper the Shapley value of digraph (directed graph) games are considered. Digraph games are transferable utility (TU) games with limited cooperation among players, where players are represented by nodes. A restrictive relation between two adjacent players is established by a directed line segment. Directed path, connecting the initial player with the terminal player, form the coalition among players. A dominance relation is established between players and this relation determines whether or not a player wants to cooperate. To cooperate, we assume that a player joins a coalition where he/she is not dominated by any other players.The Shapley value is defined as the average of marginal contribution vectors corresponding to all permutations that do not violate the subordination of players. The Shapley value for cyclic digraph games is calculated and analyzed. For a given family of characteristic functions, a quick way to calculate Shapley values is formulated.