The average covering tree value for directed graph games

Anna Khmelnitskaya,Özer Selçuk,Dolf Talman

doi:10.1007/s10878-019-00471-5

Abstract

We introduce a single-valued solution concept, the so-called average covering tree value, for the class of transferable utility games with limited communication structure represented by a directed graph. The solution is the average of the marginal contribution vectors corresponding to all covering trees of the directed graph. The covering trees of a directed graph are those (rooted) trees on the set of players that preserve the dominance relations between the players prescribed by the directed graph. The average covering tree value is component efficient, and under a particular convexity-type condition it is stable. For transferable utility games with complete communication structure the average covering tree value equals to the Shapley value of the game. If the graph is the directed analog of an undirected graph the average covering tree value coincides with the gravity center solution.

Highlights

In classical cooperative game theory it is assumed that any coalition of players may form
The first possibility is when the communication between players is supposed to be feasible only along the directed paths in the digraph, for example a flow situation. This assumption is incorporated into the solution concepts of web values, in particular the tree value, and the average web value for cycle-free digraph games introduced in Khmelnitskaya and Talman (2014) and the covering values for cycle-free digraph games studied in Li and Li (2011)
In the paper we abide by the latter interpretation of a directed link and we introduce a new component efficient solution concept for digraph games

Summary

Introduction

In classical cooperative game theory it is assumed that any coalition of players may form. A spanning tree is admissible if every player has in each component in the subgraph on the set of his successors in the tree exactly one direct successor in the tree Another solution concept applicable to graph games is the gravity center solution for games with restricted communication represented by an arbitrary collection of coalitions, developed in Koshevoy and Talman (2014). The first possibility is when the communication between players is supposed to be feasible only along the directed paths in the digraph, for example a flow situation This assumption is incorporated into the solution concepts of web values, in particular the tree value, and the average web value for cycle-free digraph games introduced in Khmelnitskaya and Talman (2014) and the covering values for cycle-free digraph games studied in Li and Li (2011).

Preliminaries

The average covering tree value

Properties and stability of the average covering tree value

Undirected graph games

Discussion