Solutions for Games with General Coalitional Structure and Choice Sets

CentER Discussion Paper Series No. 2013-012

34 Pages Posted: 28 Feb 2013

See all articles by Gleb A. Koshevoy

Gleb A. Koshevoy

Russian Academy of Sciences

Takamasa Suzuki

Tilburg University - Department of Econometrics & Operations Research

Dolf Talman

Tilburg University - Department of Econometrics & Operations Research

Date Written: February 27, 2013

Abstract

In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payoff his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some specifcations of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently defined for this class. For graph games it therefore differs from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.

Keywords: Set system, nested set, rooted tree, chain, core, convexity, marginal vector

JEL Classification: C71

Suggested Citation

Koshevoy, Gleb A. and Suzuki, Takamasa and Talman, Dolf J. J., Solutions for Games with General Coalitional Structure and Choice Sets (February 27, 2013). CentER Discussion Paper Series No. 2013-012, Available at SSRN: https://ssrn.com/abstract=2225739 or http://dx.doi.org/10.2139/ssrn.2225739

Gleb A. Koshevoy (Contact Author)

Russian Academy of Sciences ( email )

Central Institute of Mathematics and Economics
117418 Moscow
Russia

Takamasa Suzuki

Tilburg University - Department of Econometrics & Operations Research ( email )

P.O. Box 90153
Tilburg, DC Noord-Brabant 5000 LE
Netherlands

Dolf J. J. Talman

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands
+31 13 466 2346 (Phone)

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