Comparable Axiomatizations of the Myerson Value, the Restricted Banzhaf Value, Hierarchical Outcomes and the Average Tree Solution for Cycle-Free Graph Restricted Games

Tinbergen Institute Discussion Paper 09-108/1

28 Pages Posted: 27 Nov 2009

See all articles by René van den Brink

René van den Brink

VU University Amsterdam - Department of Economics; Tinbergen Institute; Tinbergen Institute

Date Written: November 25, 2009

Abstract

We consider cooperative transferable utility games, or simply TU-games, with a limited communication structure in which players can cooperate if and only if they are connected in the communication graph. A difference between the restricted Banzhaf value and the Myerson value (i.e. the Shapley value of the restricted game) is that the restricted Banzhaf value satisfies collusion neutrality, while the Myerson value satisfies component efficiency. Requiring both efficiency and collusion neutrality for cycle-free graph games yields other solutions such as the hierarchical outcomes and the average tree solution. Since these solutions also satisfy the superfluous player property, this also `solves' an impossibility for TU-games since there is no solution for these games that satisfies efficiency, collusion neutrality and the null player property. We give axiomatizations of the restricted Banzhaf value, the hierarchical outcomes and the average tree solution that are comparable with axiomatizations of the Myerson value in case the communication graph is cycle-free. Finally, we generalize these solutions to classes of solutions for cycle-free graph games using network power measures.

Keywords: Cooperative TU-game, communication structure, Myerson value, Shapley value, Banzhaf value, hierarchical outcome, average tree solution, component efficiency, collusion neutrality

JEL Classification: C71

Suggested Citation

van den Brink, J.R. (René), Comparable Axiomatizations of the Myerson Value, the Restricted Banzhaf Value, Hierarchical Outcomes and the Average Tree Solution for Cycle-Free Graph Restricted Games (November 25, 2009). Tinbergen Institute Discussion Paper 09-108/1, Available at SSRN: https://ssrn.com/abstract=1513182 or http://dx.doi.org/10.2139/ssrn.1513182

J.R. (René) Van den Brink (Contact Author)

VU University Amsterdam - Department of Economics ( email )

De Boelelaan 1105
1081 HV Amsterdam
Netherlands

Tinbergen Institute ( email )

Burg. Oudlaan 50
Rotterdam, 3062 PA
Netherlands

Tinbergen Institute ( email )

Burg. Oudlaan 50
Rotterdam, 3062 PA
Netherlands

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