The aim of the current chapter is to study several solution concepts for bicooperative games. For these games introduced by Bilbao [1], we define a one-point solution called the Shapley value, as this value can be interpreted in a similar way to the classic Shapley value for cooperative games. The first result is an axiomatic characterization of this value. Next, we define the core and the Weber set of a bicooperative game and prove that the core of a bicooperative game is always contained in the Weber set. Finally, we introduce a special class of bicooperative games, the so-called bisupermodular games, and show that these games are the only ones in which the core and the Weber set coincide.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bilbao, J.M.: Cooperative Games on Combinatorial Structures. Kluwer Academic Publishers, Boston (2000)
Bilbao, J.M., Fernández, J.R., Jiménez, N., López, J.J.: Probabilistic Values for Bicooperative Games. Working paper, University of Seville (2006)
Chua, V.C.H., Huang, H.C.: The Shapley-Shubik Index, the Donation Paradox and Ternary Games. Social Choice and Welfare, 20, 387–403 (2003)
Derks, J.: A Short Proof of the Inclusion of the Core in the Weber Set. International Journal of Game Theory, 21, 149–150 (1992)
Felsenthal, D., Machover, M.: Ternary Voting Games. International Journal of Game Theory, 26, 335–351 (1997)
Freixas, J.: The Shapley-Shubik Power Index for Games with Several Levels of Approval in the Input and Output. Decision Support Systems, 39, 185–195 (2005)
Freixas, J.: Banzhaf Measures for Games with Several Levels of Approval in the Input and Output. Annals of Operations Research, 137, 45–66 (2005)
Freixas, J., Zwicker, W.S.: Weighted Voting, Abstention, and Multiple Levels of Approval. Social Choice and Welfare, 21, 399–431 (2003)
Gillies D.B.: Some Theorems on n-Person Games, Ph.D. Thesis, Princeton University Press, Princeton (1953)
Grabisch, M., Labreuche, Ch.: Bi-Capacities I: Definition, Möbius Transform and Interaction. Fuzzy Sets and Systems, 151, 211–236 (2005)
Grabisch, M., Lange, F.: Games on Lattices, Multichoice Games and the Shapley Value: A New Approach. Mathematical Methods of Operations Research, 65, 153–167 (2007)
Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, New Jersey (1970)
Shapley, L.S.: A Value for n-Person Games. In: Kuhn, H.W., Tucker, A.W. (eds) Contributions to the Theory of Games, Vol II. Princeton University Press, Princeton, New Jersey, 307–317 (1953)
Shapley, L.S.: Cores of Convex Games. International Journal of Game Theory, 1, 11–26 (1971)
Stanley, R.P.: Enumerative Combinatorics, Vol I. Wadsworth, Monterey (1986)
Weber, R.J.: Probabilistic Values for Games. In: Roth, A.E. (ed) The Shapley Value: Essays in Honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, 101–119 (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Bilbao, J.M., Fernández, J.R., Jiménez, N., López, J.J. (2008). A Survey of Bicooperative Games. In: Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds) Pareto Optimality, Game Theory And Equilibria. Springer Optimization and Its Applications, vol 17. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77247-9_8
Download citation
DOI: https://doi.org/10.1007/978-0-387-77247-9_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-77246-2
Online ISBN: 978-0-387-77247-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)