Abstract
Uncertainty is a daily presence in the real world. It affects our decision making and may have influence on cooperation. Often uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our actions, i.e., payoffs lie in some intervals. A suitable game theoretic model to support decision making in collaborative situations with interval data is that of cooperative interval games. Solution concepts that associate with each cooperative interval game sets of interval allocations with appealing properties provide a natural way to capture the uncertainty of coalition values into the players’ payoffs. This paper extends interval-type core solutions for cooperative interval games by discussing the set of undominated core solutions which consists of the interval nondominated core, the square interval dominance core, and the interval dominance core. The interval nondominated core is introduced and it is shown that it coincides with the interval core. A straightforward consequence of this result is the convexity of the interval nondominated core of any cooperative interval game. A necessary and sufficient condition for the convexity of the square interval dominance core of a cooperative interval game is also provided.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahuja G, Polidoro F Jr, Mitchell W (2009) Structural homophily or social asymmetry? The formation of alliances by poorly embedded firms. Strateg Manag J 30(9): 941–958
Alparslan Gök SZ (2009) Cooperative interval games. PhD. Dissertation, Middle East Technical University
Alparslan Gök SZ, Branzei R, Tijs S (2008) Cores and stable sets for interval-valued games. Preprint no. 90, Institute of Applied Mathematics, METU and Tilburg University, Center for Economic Research, The Netherlands, Center DP 17
Alparslan Gök SZ, Branzei R, Tijs S (2009) Convex interval games. J Appl Math Decis Sci, Article ID 342089, 14 p. doi:10.1115/2009/342089
Alparslan Gök SZ, Miquel S, Tijs S (2009) Cooperation under interval uncertainty. Math Methods Oper Res 69: 99–109
Alparslan Gök SZ, Weber GW (2009) Cooperative games under ellipsoidal uncertainty. Preprint no. 148, Institute of Applied Mathematics, METU (2009) (to appear in the proceedings of PCO 2010, 3rd Global conference on power control and optimization, 2–4 February 2010, Gold Coast, Queensland, Australia)
Branzei O (2005) Product innovation in heterogeneous R&D networks: pathways to exploration and exploitation. Unpublished Doctoral Dissertation, University of British Columbia
Branzei R, Dimitrov D, Tijs S (2008) Models in cooperative game theory. Springer, Berlin (CEJOR)
Branzei R, Branzei O, Alparslan Gök SZ, Tijs S (2009) Cooperative interval games: a survey. Central Eur J Oper Res (CEJOR). doi:10.1007/s10100-009-0116-0
Branzei R, Mallozzi L, Tijs S (2008) Peer group situations and games with interval uncertainty. Preprint no. 64, Dipartimento di Matematica e Applicazioni, Universita’ di Napoli (2008) (to appear in Int J Math Game Theory Algebra)
Chang C (2000) Note: remarks on the theory of the core. Naval Res Logist 47: 455–458
Charnes A, Granot D (1973) Prior solutions: extensions of convex nucleolus solutions to chance-constrained games. Proceedings of the computer science and statistics seventh symposium at Iowa State University, pp 323–332
Gillies DB (1959) Solutions to general non-zero-sum games. In: Tucker AW, Luce RD (eds) Contributions to theory of games IV, annals of mathematical studies 40. Princeton University Press, Princeton, pp 47–85
Granot D (1977) Cooperative games in stochastic characteristic function form. Manag Sci 23: 621–630
Gray B (2000) Assessing inter-organizational collaboration: multiple conceptions and multiple methods. In: Faulkner D, de Rond M (eds) Cooperative strategy: economic, business, and organizational issues. Oxford University Press, New York
Kimms A, Drechsel J (2009) Cost sharing under uncertainty: an algorithmic approach to cooperative interval-valued games. BuR—Business Research 2 (urn:nbn:de:0009-20-21721)
Lavie D, Lechner C, Singh H (2007) The performance implications of timing of entry and involvement in multipartner alliances. Acad Manag J 50(3): 578–604
Loveridge R (2000) The firm as differentiator and integrator of networks: layered communities of practice and discourse. In: Faulkner D, de Rond M (eds) Cooperative strategy: economic, business, and organizational issues. Oxford University Press, New York
Mallozzi L, Scalzo V, Tijs S (2009) Fuzzy interval cooperative games. Preprint no. 1, Dipartimento di Matematica e Applicazioni R. Caccioppoli, Università di Napoli
Mares M (2001) Fuzzy cooperative games: cooperation with vague expectations. Physica-Verlag, Heidelberg
Mares M, Vlach M (2004) Fuzzy classes of cooperative games with transferable utility. Scientiae Mathematicae Japonicae 60(2): 269–278
Moore R (1979) Methods and applications of interval analysis. SIAM Studies in Applied Mathematics, Philadelphia
Nishizaki I, Sakawa M (2001) Fuzzy and multiobjective games for conflict resolution. Physica-Verlag, Heidelberg
Puerto J, Fernández FR, Hinojosa Y (2008) Partially ordered cooperative games: extended core and Shapley value. Ann Oper Res 158: 143–159
Rafels C, Tijs S (1997) On the cores of cooperative games and the stability of the Weber set. Int J Game Theory 26: 491–499
Suijs J, Borm P, De Waegenaere A, Tijs S (1999) Cooperative games with stochastic payoffs. Eur J Oper Res 113: 193–205
Timmer J (2001) Cooperative behaviour, uncertainty and operations research. Center Dissertation Series: Tilburg
Timmer J, Borm P, Tijs S (2005) Convexity in stochastic cooperative situations. Int Game Theory Rev 7(1): 25–42
Wagner HM, Whitin TM (1958) Dynamic version of the economic lot size model. Manag Sci 5(1): 89–96
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Branzei, R., Gök, S.Z.A. & Branzei, O. Cooperative games under interval uncertainty: on the convexity of the interval undominated cores. Cent Eur J Oper Res 19, 523–532 (2011). https://doi.org/10.1007/s10100-010-0141-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10100-010-0141-z