An axiomatic and non-cooperative approach to the multi-step Shapley value

Xianghui Li,Wei Zheng,Yang Li

doi:10.1051/ro/2021073

Abstract

Inspired by the two-step Shapley value, in this paper we introduce and axiomatize the multi-step Shapley value for cooperative games with levels structures. Moreover, we design a multi-step bidding mechanism, which implements the value strategically in subgame perfect Nash equilibrium for superadditve games.

Highlights

Many realistic situations can be modeled as cooperative games
A cooperative game consists of a finite player set and a characteristic function which assigns to each coalition, i.e., a subset of the player set, a real number, representing the aggregate benefit of players in it from cooperation
Considering that players may partition themselves into disjoint coalitions before cooperation, Aumann and Dreze [1] introduced a model of coalition structure

Summary

Introduction

Many realistic situations can be modeled as cooperative games. A cooperative game consists of a finite player set and a characteristic function which assigns to each coalition, i.e., a subset of the player set, a real number, representing the aggregate benefit of players in it from cooperation. We define a new solution concept, called a multi-step Shapley value, for cooperative games with levels structures. It can be interpreted as the result of bargaining of the lower-level unions on the worth of the higher-level unions they join. Perez-Castrillo and Wettstein [12] provided a non-cooperative bidding mechanism and showed that the subgame perfect Nash equilibrium (SP N E) outcomes of this mechanism always coincide with the vector of the Shapley value payoffs for zero-monotonic cooperative games. Ju and Wettstein [6] provided a non-cooperative foundation to several cooperative solution concepts, such as the Shapley value and consensus value, by using a class of bidding mechanisms that differ in the power awarded to the proposer chosen through a bidding process.

Cooperative games

Levels structures

The multi-step Shapley value and its axiomatizations

A non-cooperative approach the multi-step Shapley value

Conclusions