Abstract
The Shapley value, one of the most widespread concepts in operations Research applications of cooperative game theory, was defined and axiomatically characterized in different game-theoretic models. Recently much research work has been done in order to extend OR models and methods, in particular cooperative game theory, for situations with interval data. This paper focuses on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. The interval Shapley value is characterized with the aid of the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alparslan Gök SZ (2009) Cooperative interval games. PhD Dissertation Thesis, Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey
Alparslan Gök SZ, Branzei R, Fragnelli V, Tijs S (2008a) Sequencing interval situations and related games, preprint no. 113, Institute of Applied Mathematics, METU and Tilburg University, Center for Economic Research, The Netherlands, CentER DP 63
Alparslan Gök SZ, Branzei R, Tijs S (2008b) Convex interval games, preprint no. 100. Institute of Applied Mathematics, METU and Tilburg University, Center for Economic Research, The Netherlands, CentER DP 37 (J Appl Math Decis Sci, to appear)
Alparslan Gök SZ, Branzei R, Tijs S (2009a) Airport interval games and their Shapley value. Oper Res Decis Issue 1, to appear
Alparslan Gök SZ, Miquel S, Tijs S (2009) Cooperation under interval uncertainty. Math Methods Oper Res 69: 99–109
Artstein Z (1971) Values of games with denumerably many players. Int J Game Theory 1: 27–37
Aumann RJ, Hart S (eds). (2002) Handbook of game theory with economic applications, 3rd edn. Elsevier, Amsterdam
Aumann RJ, Shapley LS (1974) Values of non-atomic games. Princeton University Press, Princeton
Baker J Jr (1965) Airport runway cost impact study. Report submitted to the Association of Local Transport Airlines, Jackson, Mississippi
Branzei R, Dimitrov D, Tijs S (2003) Shapley-like values for interval bankruptcy games. Econ Bull 3: 1–8
Branzei R, Dimitrov D, Tijs S (2008a) Models in cooperative game Theory. Springer, Berlin
Branzei R, Mallozzi L, Tijs S (2008b) Peer group situations and games with interval uncertainty, preprint no. 64. Dipartimento di Matematica e Applicazioni, Universita’ di Napoli
Branzei R, Tijs S, Alparslan Gök SZ (2008c) Cooperative interval games: a survey, preprint no. 104. Institute of Applied Mathematics, METU
Drechsel J, Kimms A (2008) An algorithmic approach to cooperative interval-valued games and interpretation problems. Working paper, University of Duisburg, Essen
Harsanyi J (1959) A bargaining model for the cooperative n-person game. Ann Math Study 40: 325–355
Hart S, Mas-Colell A (1989) Potential value and consistency. Econometrica 57: 589–614
Hart S, Mas-Colell A (1996) Bargaining and value. Econometrica 64: 357–380
Janiak A, Kasperski A (2008) The minimum spanning tree problem with fuzzy costs. Fuzzy Optim Decis Mak 7: 105–118
Monderer D, Samet D, Shapley LS (1992) Weighted Shapley values and the core. Int J Game Theory 21: 27–39
Moore R (1979) Methods and applications of interval analysis. SIAM Studies in Applied Mathematics, Philadelphia
Moore R, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM Press
Moretti S, Alparslan Gök SZ, Branzei R, Tijs S (2008) Connection situations under uncertainty”, preprint no.112. Institute of Applied Mathematics, METU and Tilburg University, Center for Economic Research, The Netherlands, CentER DP 64
Owen G (1972) Multilinear extentions of games. Manage Sci 18: 64–79
Pallaschke D, Rosenmüller J (1997) The Shapley value for countably many players. Optimization 40: 351–384
Ramadan K (1996) Linear programming with interval coefficients, Master of Science Thesis, Carleton University, Ottawa
Rosenmüller J (2000) Game theory stochastics, information and cooperation. Kluwer Academic Publishers, Dordrecht
Roth A (1988) The Shapley value: essays in honor of Lloyd Shapley. Cambridge University Press, Cambridge
Shapley LS (1953) A value for n-person games. Ann Math Stud 28: 307–317
Shapley LS (1962) Values of games with infinitely many players. In: Maschler M (eds) Recent advances in game theory. Ivy Curtis Press/The Rand Co. RM-2912, Philadelphia/Santa Monica, pp 113–118
Thompson GF (1971) Airport costs and pricing. Unpublished PhD. Dissertation, University of Birmingham
Timmer JB, Borm PEM, Tijs SH (2003) On three Shapley-like solutions for cooperative games with random payoffs. Int J Game Theory 32: 595–613
Young HP (1985) Monotonic solutions of cooperative games. Int J Game Theory 14: 65–72
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors gratefully thank two anonymous referees whose detailed remarks and suggestions improved the presentation substantially, and acknowledge the support of TUBITAK (Turkish Scientific and Technical Research Council).
Rights and permissions
About this article
Cite this article
Alparslan Gök, S.Z., Branzei, R. & Tijs, S. The interval Shapley value: an axiomatization. Cent Eur J Oper Res 18, 131–140 (2010). https://doi.org/10.1007/s10100-009-0096-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10100-009-0096-0