Axiomatizations of the Normalized Banzhaf Value and the Shapley Value

Posted: 4 Oct 1998

See all articles by René van den Brink

René van den Brink

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

Gerard van der Laan

Vrije Universiteit Amsterdam, School of Business and Economics; Tinbergen Institute

Abstract

A cooperative game with transferable utilities--or simply a TU-game--describes a situation in which players can obtain certain payoffs by cooperation. A solution concept for these games is a function which assigns to every such a game a distribution of payoffs over the players in the game. Famous solution concepts for TU-games are the Shapley value and the Banzhaf value. Both solution concepts have been axiomatized in various ways.

An important difference between these two solution concepts is the fact that the Shapley value always distributes the payoff that can be obtained by the 'grand coalition' consisting of all players cooperating together while the Banzhaf value does not satisfy this property, i.e., the Banzhaf value is not efficient. In this paper we consider the normalized Banzhaf value which distributes the payoff that can be obtained by the 'grand coalition' proportional to the Banzhaf values of the players. This value does not satisfy certain axioms underlying the Banzhaf value. In this paper we introduce some new axioms that characterize the normalized Banzhaf value. We also provide an axiomatization of the Shapley value using similar axioms.

JEL Classification: C71

Suggested Citation

van den Brink, J.R. (René) and van der Laan, Gerard, Axiomatizations of the Normalized Banzhaf Value and the Shapley Value. Available at SSRN: https://ssrn.com/abstract=133908

J.R. (René) Van den Brink

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

Gerard Van der Laan (Contact Author)

Vrije Universiteit Amsterdam, School of Business and Economics ( email )

De Boelelaan 1105
Department of Econometrics and Tinbergen Institute
1081 HV Amsterdam
Netherlands

Tinbergen Institute ( email )

Gustav Mahlerplein 117
Amsterdam, 1082 MS
Netherlands

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