Cooperative Games in Graph Structure
By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition that is a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. For this result any information on the internal organization of the coalition is neglected.In this paper we generalize the concept of coalitional games and allow for organizational structure within coalitions. For a subset of players any arbitrarily given structural relation represented by a graph is allowed for. We then consider non-transferable utility games in which a possibly empty set of payoff vectors is assigned to any graph on every subset of players. Such a game will be called a cooperative game in graph structure or shortly graph game. A payoff vector lies in the core of the game if there is no graph on a group of players which can make all of its members better off.We define the balanced-core of a graph game as a refinement of the core. To do so, for each graph a power vector is determined that depends on the relative positions of the players within the graph. A collection of graphs will be called balanced if to any graph in the collection a positive weight can be assigned such that the weighted power vectors sum up to the vector of ones. A payoff vector lies in the balanced-core if it lies in the core and the payoff vector is an element of payoff sets of all graphs in some balanced collection of graphs. We prove that any balanced graph game has a nonempty balanced-core and therefore a nonempty core. We conclude by some examples showing the usefulness of the concepts of graph games and balanced-core. In particular these examples show a close relationship between solutions to noncooperative games and balanced-core elements of a well-defined graph game. This places the paper in the Nash research program, looking for a unifying theory in which each approach helps to justify and clarify the other.
|Date of creation:||2002|
|Date of revision:|
|Contact details of provider:|| Postal: P.O. Box 616, 6200 MD Maastricht|
Phone: +31 (0)43 38 83 830
Web page: http://www.maastrichtuniversity.nl/
More information through EDIRC
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 1998.
"Cooperative games in permutational structure,"
Other publications TiSEM
94dd61cf-8471-40af-8cc8-4, Tilburg University, School of Economics and Management.
- Kamiya, K & Talman, A.J.J., 1990.
"Variable dimension simplicial algorithm for balanced games,"
1990-25, Tilburg University, Center for Economic Research.
- Kamiya, K. & Talman, D., 1990. "Variable Dimension Simplicial Algorithm For Balanced Games," Papers 9025, Tilburg - Center for Economic Research.
- Jean Tirole, 1988. "The Theory of Industrial Organization," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262200716, March.
- Doup, T.M. & Talman, A.J.J., 1987. "A new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices," Other publications TiSEM 398740e7-fdc2-41b6-968f-4, Tilburg University, School of Economics and Management.
- Bouyssou, Denis, 1992. "Ranking methods based on valued preference relations: A characterization of the net flow method," European Journal of Operational Research, Elsevier, vol. 60(1), pages 61-67, July.
- van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 1999.
"Intersection theorems on polytypes,"
Other publications TiSEM
7cc2ec6a-3a45-4ad1-a99b-7, Tilburg University, School of Economics and Management.
- Nowak Andrzej S. & Radzik Tadeusz, 1994. "The Shapley Value for n-Person Games in Generalized Characteristic Function Form," Games and Economic Behavior, Elsevier, vol. 6(1), pages 150-161, January.
- Nash, John, 1953. "Two-Person Cooperative Games," Econometrica, Econometric Society, vol. 21(1), pages 128-140, April.
- P. Jean-Jacques Herings, 1997.
"An extremely simple proof of the K-K-M-S Theorem,"
Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(2), pages 361-367.
- Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 161-69.
When requesting a correction, please mention this item's handle: RePEc:unm:umamet:2002011. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Leonne Portz)
If references are entirely missing, you can add them using this form.