Abstract
We generalize the concept of a cooperative non-transferable utility game by introducing a socially structured game. In a socially structured game every coalition of players can organize themselves according to one or more internal organizations to generate payoffs. Each admissible internal organization on a coalition yields a set of payoffs attainable by the members of this coalition. The strengths of the players within an internal organization depend on the structure of the internal organization and are represented by an exogenously given power vector. More powerful players have the power to take away payoffs of the less powerful players as long as those latter players are not able to guarantee their payoffs by forming a different internal organization within some coalition in which they have more power.
We introduce the socially stable core as a solution concept that contains those payoffs that are both stable in an economic sense, i.e., belong to the core of the underlying cooperative game, and stable in a social sense, i.e., payoffs are sustained by a collection of internal organizations of coalitions for which power is distributed over all players in a balanced way. The socially stable core is a subset and therefore a refinement of the core. We show by means of examples that in many cases the socially stable core is a very small subset of the core.
We will state conditions for which the socially stable core is non-empty. In order to derive this result, we formulate a new intersection theorem that generalizes the KKMS intersection theorem. We also discuss the relationship between social stability and the wellknown concept of balancedness for NTU-games, a sufficient condition for non-emptiness of the core. In particular we give an example of a socially structured game that satisfies social stability and therefore has a non-empty core, but whose induced NTU-game does not satisfy balancedness in the general sense of Billera.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aumann R.J. (1961) The core of a cooperative game without side payments. Transactions of the American Mathematical Society 98, 539–552
Aumann R.J., Peleg B. (1960) Von Neumann-Morgenstern solutions to cooperative games without side payments. Bulletin of the American Mathematical Society 66, 173–179
Behzad M., Chartrand G., Lesniak-Foster L. (1979) Graphs and Digraphs. Wadsworth, Belmont
Billera L.J. (1970) Some theorems on the core of an n-person game without side payments. SIAM Journal on Applied Mathematics 18, 567–579
van den Brink, R. (1994), Relational Power in Hierarchical Organizations, Ph.D. Dissertation, Tilburg University, The Netherlands.
van den Brink R., Gilles R.P. (2000) Measuring domination in directed graphs. Social Networks 22:1141–1157
Herings P.J.J. (1997) An extremely simple proof of the K-K-M-S theorem. Economic Theory 10, 361–367
Herings P.J.J., van der Laan G., Talman A.J.J. (2005) The positional power of nodes in digraphs. Social Choice and Welfare 24, 439–454
Ichiishi T. (1988) Alternative version of Shapley’s theorem on closed coverings of a simplex. Proceedings of the American Mathematical Society 104, 759–763
Jackson M.O. (2005) Allocation rules for network games. Games and Economic Behavior 51, 128–154
Jackson M.O., Wolinsky A. (1996) A strategic model of social and economic networks. Journal of Economic Theory 71, 44–74
van der Laan G., Talman A.J.J., Yang Z. (1998) Cooperative games in permutational structure. Economic Theory 11, 427–442
van der Laan G., Talman A.J.J., Yang Z. (1999) Intersection theorems on polytopes. Mathematical Programming 84, 25–38
Myerson R.B. (1977) Graphs and cooperation in games. Mathematics of Operations Research 2, 225–229
Nowak A.S., Radzik T. (1994) The Shapley value for n-person games in generalized characteristic function form. Games and Economic Behavior 6, 150–161
Piccione, M. and Rubinstein, A. (2003), Equilibrium in the jungle, Working Paper No. 18-2003, The Foerder Institute for Economic Research and The Sacker Institute of Economic Studies.
Predtetchinskii A., Herings P.J.J. (2004) A necessary and sufficient condition for the non-emptiness of the core of a non-transferable utility game. Journal of Economic Theory 116, 84–92
Rubinstein A. (1980) Ranking the participants in a tournament. SIAM Journal of Applied Mathematics 38, 108–111
Shapley L.S. (1973) On balanced games without side payments. In: Hu T.C., Robinson S.M. (eds) Mathematical Programming. Academic Press, New York, pp. 261–290
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Herings, P.JJ., Van Der Laan, G. & Talman, D. Socially Structured Games. Theor Decis 62, 1–29 (2007). https://doi.org/10.1007/s11238-006-9007-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11238-006-9007-1