Abstract
We show that neither Peleg’s nor Tadenuma’s well-known axiomatizations of the core by non-emptiness, individual rationality, super-additivity, and max consistency or complement consistency, respectively, hold when only convex rather than balanced TU games are considered, even if anonymity is required in addition. Moreover, we show that the core and its relative interior are the only two solutions that satisfy Peleg’s axioms together with anonymity and converse max consistency on the domain of convex games.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For these two consistency axioms we use the terminology introduced by Thomson (1996) and call them max consistency and complement consistency because each name suggests how the underlying “reduced games” are defined in each case.
Voorneveld and van den Nouweland (1998) provide an axiomatization of the core which is closely related to Peleg’s result.
Although this fact is widely known, we do not know any published or unpublished paper that mentions it.
The definition of additivity is obtained by replacing \(\subseteq \) with \(=\) in the definition of super-additivity.
References
Bondareva ON (1963) Some applications of linear programming methods to the theory of cooperative games (in Russian). Probl Kibern 10:119–139
Davis M, Maschler M (1965) The kernel of a cooperative game. Nav Res Logist Q 12:223–259
Dragan I, Potters J, Tijs SH (1989) Superadditivity for solutions of coalitional games. Lib Math 9:101–110
Hwang Y-A, Sudhölter P (2001) Axiomatizations of the core on the universal domain and other natural domains. Int J of Game Theory 29:597–623
Hokari T (2005) Consistency implies equal treatment in TU games. Games and Econ Behavior 51:63–82
Maschler M, Peleg B, Shapley LS (1972) The kernel and bargaining set for convex games. Int J of Game Theory 1:73–93
Moulin H (1985) The separability axiom and equal sharing methods. J Econ Theory 36:120–148
Núñez M, Rafels C (1998) On extreme points of the core and reduced games. Ann Oper Res 84:121–133
Orshan G (1994) Non-symmetric prekernels. Discussion Paper 60, Center for Rationality and Interactive Decision Theory, Hebrew University of Jerusalem
Orshan G, Sudhölter P (2010) The positive core of a cooperative game. Int J Game Theory 39:113–136
Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15:187–200
Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17:1163–1170
Shapley LS (1967) On balanced sets and cores. Nav Res Log Q 14:453–460
Shapley LS (1971) Cores of convex games. Int J Game Theory 1:11–26
Tadenuma K (1992) Reduced games, consistency, and the core. Int J Game Theory 20:325–334
Thomson W (1996) Consistent allocation rules. RCER Working Paper 418, University of Rochester
Voorneveld M, van den Nouweland A (1998) A new axiomatization of the core of games with transferable utility. Econ Lett 60:151–155
Acknowledgements
We are grateful to an anonymous referee and the associate editor of this journal for remarks that helped to eliminate typos and to improve the writing of this paper.
Funding
Toru Hokari received support from the Japan Society for the Promotion of Science (Grant No. KAKENHI 17K03629) and Peter Sudhölter received support from the Ministerio de Economía y Competitividad (GrantNo. ECO2015-66803-313 P (MINECO/FEDER)).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hokari, T., Funaki, Y. & Sudhölter, P. Consistency, anonymity, and the core on the domain of convex games. Rev Econ Design 24, 187–197 (2020). https://doi.org/10.1007/s10058-020-00231-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10058-020-00231-6