×

zbMATH — the first resource for mathematics

Convex interval games. (English) Zbl 1186.91024
Summary: Convex interval games are introduced and characterizations are given. Some economic situations leading to convex interval games are discussed. The Weber set and the Shapley value are defined for a suitable class of interval games and their relations with the interval core for convex interval games are established. The notion of population monotonic interval allocation scheme (pmias) in the interval setting is introduced and it is proved that each element of the Weber set of a convex interval game is extendable to such a pmias. A square operator is introduced which allows us to obtain interval solutions starting from the corresponding classical cooperative game theory solutions. It turns out that on the class of convex interval games the square Weber set coincides with the interval core.

MSC:
91A12 Cooperative games
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] L. S. Shapley, “Cores of convex games,” International Journal of Game Theory, vol. 1, no. 1, pp. 11-26, 1971. · Zbl 0222.90054 · doi:10.1007/BF01753431
[2] T. Driessen, Cooperative Games, Solutions and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988. · Zbl 0686.90043
[3] A. K. Biswas, T. Parthasarathy, J. A. M. Potters, and M. Voorneveld, “Large cores and exactness,” Games and Economic Behavior, vol. 28, no. 1, pp. 1-12, 1999. · Zbl 0957.91009 · doi:10.1006/game.1998.0686
[4] R. Brânzei, D. Dimitrov, and S. Tijs, Models in Cooperative Game Theory, Game Theory and Mathematical Methods, Springer, Berlin, Germany, 2008. · Zbl 1142.91017 · doi:10.1007/978-3-540-77954-4
[5] J. E. Martinez-Legaz, “Two remarks on totally balanced games,” Tech. Rep. TR#317, Department of Mathematics, The University of Texas, Arlington, Tex, USA, 1997.
[6] J. E. Martínez-Legaz, “Some characterizations of convex games,” in Recent Advances in Optimization, A. Seeger, Ed., vol. 563 of Lecture Notes in Economics and Mathematical Systems, pp. 293-303, Springer, Berlin, Germany, 2006. · Zbl 1152.91335 · doi:10.1007/3-540-28258-0_18
[7] H. Moulin, Axioms of Cooperative Decision Making, vol. 15 of Econometric Society Monographs, Cambridge University Press, Cambridge, UK, 1988. · Zbl 0699.90001
[8] I. Curiel, G. Pederzoli, and S. Tijs, “Sequencing games,” European Journal of Operational Research, vol. 40, no. 3, pp. 344-351, 1989. · Zbl 0674.90107 · doi:10.1016/0377-2217(89)90427-X
[9] B. O’Neill, “A problem of rights arbitration from the Talmud,” Mathematical Social Sciences, vol. 2, no. 4, pp. 345-371, 1982. · Zbl 0489.90090 · doi:10.1016/0165-4896(82)90029-4
[10] R. J. Aumann and M. Maschler, “Game theoretic analysis of a bankruptcy problem from the Talmud,” Journal of Economic Theory, vol. 36, no. 2, pp. 195-213, 1985. · Zbl 0578.90100 · doi:10.1016/0022-0531(85)90102-4
[11] I. J. Curiel, M. Maschler, and S. Tijs, “Bankruptcy games,” Zeitschrift für Operations Research. Serie A. Serie B, vol. 31, no. 5, pp. A143-A159, 1987. · Zbl 0636.90100 · doi:10.1007/BF01258644
[12] A. Charnes and D. Granot, “Prior solutions: extensions of convex nucleus solutions to chance-constrained games,” in Proceedings of the Computer Science and Statistics Seventh Symposium at Iowa State University, pp. 323-332, Ames, Iowa, USA, October 1973.
[13] J. Suijs, P. Borm, A. De Waegenaere, and S. Tijs, “Cooperative games with stochastic payoffs,” European Journal of Operational Research, vol. 113, no. 1, pp. 193-205, 1999. · Zbl 0972.91012 · doi:10.1016/S0377-2217(97)00421-9
[14] J. Timmer, P. Borm, and S. Tijs, “Convexity in stochastic cooperative situations,” International Game Theory Review, vol. 7, no. 1, pp. 25-42, 2005. · Zbl 1105.91004 · doi:10.1142/S0219198905000387
[15] S. Z. Alparslan Gök, R. Brânzei, and S. Tijs, “The interval Shapley value: an axiomatization,” no. 130, Institute of Applied Mathematics, METU, Ankara, Turkey, 2009, to appear in Central European Journal of Operations Research (CEJOR).
[16] R. Brânzei, S. Tijs, and S. Z. Alparslan Gök, “How to handle interval solutions for cooperative interval games,” preprint no. 110, Institute of Applied Mathematics, METU, 2008.
[17] S. Z. Alparslan Gök, R. Brânzei, and S. Tijs, “Cores and stable sets for interval-valued games,” Tech. Rep. 2008-17, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2008, preprint no. 90.
[18] S. Tijs, Introduction to Game Theory, vol. 23 of Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi, India, 2003. · Zbl 1018.91001
[19] S. Z. Alparslan Gök, R. Brânzei, V. Fragnelli, and S. Tijs, “Sequencing interval situations and related games,” Tech. Rep. 2008-63, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2008, preprint no. 113.
[20] R. Brânzei and S. Z. Alparslan Gök, “Bankruptcy problems with interval uncertainty,” Economics Bulletin, vol. 3, no. 56, pp. 1-10, 2008.
[21] R. J. Weber, “Probabilistic values for games,” in The Shapley Value: Essays in Honour of Lloyd S. Shapley, A. E. Roth, Ed., pp. 101-119, Cambridge University Press, Cambridge, UK, 1988. · Zbl 0707.90100
[22] L. S. Shapley, “A value for n-person games,” in Contributions to the Theory of Games, Vol. 2, Annals of Mathematics Studies, no. 28, pp. 307-317, Princeton University Press, Princeton, NJ, USA, 1953. · Zbl 0050.14404
[23] Y. Sprumont, “Population monotonic allocation schemes for cooperative games with transferable utility,” Games and Economic Behavior, vol. 2, no. 4, pp. 378-394, 1990. · Zbl 0753.90083 · doi:10.1016/0899-8256(90)90006-G
[24] T. Ichiishi, “Super-modularity: applications to convex games and to the greedy algorithm for LP,” Journal of Economic Theory, vol. 25, no. 2, pp. 283-286, 1981. · Zbl 0478.90092 · doi:10.1016/0022-0531(81)90007-7
[25] I. Dr\uagan, J. Potters, and S. Tijs, “Superadditivity for solutions of coalitional games,” Libertas Mathematica, vol. 9, pp. 101-110, 1989. · Zbl 0684.90105
[26] R. Brânzei, D. Dimitrov, and S. Tijs, “A new characterization of convex games,” Tech. Rep. 109, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2004. · Zbl 1043.91006
[27] R. Brânzei, S. Tijs, and S. Z. Alparslan Gök, “Some characterizations of convex interval games,” AUCO Czech Economic Review, vol. 2, no. 3, pp. 219-226, 2008.
[28] H. Sun, Contributions to set game theory, Ph.D. thesis, University of Twente, Twente University Press, Enschede, The Netherlands, 2003.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.