Share Functions for Cooperative Games with Levels Structure of Cooperation

Tinbergen Institute Discussion Paper No. 12-052/1

36 Pages Posted: 14 May 2012

See all articles by Mikel Alvarez-Mozos

Mikel Alvarez-Mozos

University of Barcelona - Department of Actuarial, Financial and Economic Mathematics

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

Oriol Tejada

ETH Zürich - CER-ETH - Center of Economic Research at ETH Zurich

Date Written: May 9, 2012

Abstract

In a standard TU-game it is assumed that every subset of the player set can form a coalition and earn its worth. One of the first models where restrictions in cooperation are considered is the one of games with coalition structure. In such games the player set is partitioned into unions and players can only cooperate within their own union. Owen introduced a value for games with coalition structure under the assumption that also the unions can cooperate among them. Winter extended this value to games with levels structure of cooperation, which consists of a game and a finite sequence of partitions defined on the player set, each of them being coarser than the previous one. A share function for TU-games is a type of solution that assigns to every game a vector whose components add up to one, and thus they can be interpreted as players' shares in the worth to be allocated. Extending the approach to games with coalition structure developed by van den Brink and van der Laan (2005), we introduce a class of share functions for games with levels structure of cooperation by defining, for each player and each level, a standard TU-game. The share given to each player is then defined as the product of her shares in the games at every level. We show several desirable properties and provide axiomatic characterizations of this class of LS-share functions.

Keywords: cooperative game, Shapley value, coalition structure, share functions, levels structure of cooperation

JEL Classification: C71

Suggested Citation

Alvarez-Mozos, Mikel and van den Brink, J.R. (René) and van der Laan, Gerard and Tejada, Oriol, Share Functions for Cooperative Games with Levels Structure of Cooperation (May 9, 2012). Tinbergen Institute Discussion Paper No. 12-052/1. Available at SSRN: https://ssrn.com/abstract=2056266 or http://dx.doi.org/10.2139/ssrn.2056266

Mikel Alvarez-Mozos (Contact Author)

University of Barcelona - Department of Actuarial, Financial and Economic Mathematics ( email )

Avda. Diagonal 690
Barcelona, 08034
Spain

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

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

Oriol Tejada

ETH Zürich - CER-ETH - Center of Economic Research at ETH Zurich ( email )

Zürichbergstrasse 18
Zurich, 8092
Switzerland
41446329693 (Phone)

HOME PAGE: http://www.cer.ethz.ch/mip/people/toriol

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