Abstract
This paper studies two classical solution concepts for the structure of bicooperative games. First, we define the core and the Weber set of a bicooperative game and prove that the core is always contained in the Weber set. Next, 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aubin JP (1991) Cooperative fuzzy games. Math Oper Res 6:1–13
Bilbao JM (2000) Cooperative games on combinatorial structures. Kluwer, Boston
Chua VCH, Huang HC (2003) The Shapley-Shubik index, the donation paradox and ternary games. Soc Choice Welf 20:387–403
Derks J (1992) A short proof of the inclusion of the core in the Weber set. Int J Game Theory 21:149–150
Derks J, Gilles R (1995) Hierarchical organization structures and constraints on coalition formation. Int J Game Theory 24:147–163
Felsenthal D, Machover M (1997) Ternary voting games. Int J Game Theory 26:335–351
Grabisch M, Labreuche Ch (2005a) Bi-capacities—I: definition. M öbius transform and interaction. Fuzzy Sets Syst 151:211–236
Grabisch M, Labreuche Ch (2005b) Bi-capacities—II: the Choquet integral. Fuzzy Sets Syst 151:237–259
Gillies DB (1953) Some theorems on n-person games. PhD Thesis, Princeton University Press, Princeton
Hsiao C, Raghavan T (1993) Shapley value for multi-choice cooperative games (I). Games Econ Behav 5:240–256
Ichiishi T (1981) Supermodularity: applications to convex games and the greedy algorithm for LP. J Econ Theory 25:283–286
Myerson RB (1991) Game Theory: analysis of conflict. Harvard University Press, Cambridge
Nouweland A van den, Potters J, Tijs S, Zarzuelo J (1995) Cores and related solution concepts for multi-choice games. Math Methods Oper Res 41:289–311
Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton
Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26
Schrijver A (1986) Theory of linear and integer programming. Wiley, New York
Tijs S, Branzei R, Ishihara S, Muto S (2004) On cores and stable sets for fuzzy games. Fuzzy Sets Syst 146:285–296
Weber RJ (1988) Probabilistic values for games. In: Roth AE (eds) The Shapley value: essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 101–119
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bilbao, J.M., Fernández, J.R., Jiménez, N. et al. The core and the Weber set for bicooperative games. Int J Game Theory 36, 209–222 (2007). https://doi.org/10.1007/s00182-006-0066-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-006-0066-x