Alternative Axiomatic Characterizations of the Grey Shapley Value

Document Type: Research Paper

Authors

Süleyman Demirel University, Faculty of Arts and Sciences, Department of Mathematics, 32260 Isparta, Turkey

Abstract

The Shapley value, one of the most common solution concepts of cooperative game theory is defined and axiomatically characterized in different game-theoretic models. Certainly, the Shapley value can be used in interesting sharing cost/reward problems in the Operations Research area such as connection, routing, scheduling, production and inventory situations. In this paper, we focus on the Shapley value for cooperative games, where the set of players is finite and the coalition values are interval grey numbers. The central question in this paper is how to characterize the grey Shapley value. In this context, we present two alternative axiomatic characterizations. First, we characterize the grey Shapley value using the properties of efficiency, symmetry and strong monotonicity. Second, we characterize the grey Shapley value by using the grey dividends.

Keywords


Alparslan Gök S.Z., (2012). On the interval shapley value, Optimization, pp. 1-9.

Alparslan Gök S.Z., Branzei O., Branzei R. and Tijs S., (2011). Set-valued solution concepts using interval-type payoffs for interval games, Journal of Mathematical Economics, Vol. 47, pp. 621-626.

Alparslan Gök S.Z., Branzei R. and Tijs S., (2009). Airport Interval Games and Their Shapley Value, Operations Research and Decisions, No 2, (ISSN 1230-1868)).

Alparslan Gök S.Z., Branzei R. and Tijs S., (2009). Convex interval games, Journal of Applied Mathematics and Decision Sciences, Vol. 2009, Article ID 342089, 14 pages, DOI: 10.1115/2009/342089.

Alparslan Gök S.Z., Branzei R. and Tijs S., (2010). The interval Shapley value: an axiomatization, Central European Journal of Operations Research (CEJOR), Vol. 18(2), pp. 131-140.

Aumann R.J. and Hart S., (2002). Handbook of game theory with economic applications, 3rd edn., Elsevier, Amstedam. Baker J.Jr., (1965). Airport runway cost impact study, Report submitted to the Association of Local Transport Airlines, Jackson, Mississippi.

Borm P., Hamers H. and Hendrickx R., (2001). Operations research games: a survey. TOP, Vol. 9, pp. 139-216.

Branzei R., Branzei O., Alparslan Gök S.Z. and Tijs S., (2010). Cooperative Interval Games: A Survey, Central European Journal of Operations Research (CEJOR), Vol. 18(3), pp. 397-411.

Branzei R., Dimitrov D. and Tijs S., (2003). Shapley-like values for interval bankruptcy games, Economics Bulletin, Vol. 3, pp. 1-8.

Branzei R., Dimitrov D. and Tijs S., (2008). Models in Cooperative Game Theory, Game Theory and Mathematical Methods, Springer, Berlin, Germany.

Branzei R., Mallozzi L. and Tijs S., (2010). Peer group situations and games with interval uncertainty, International Journal of Mathematics, Game Theory, and Algebra, Vol. 19(5-6), pp. 381-388.

Curiel I., Pederzoli G. and Tijs S., (1989). Sequencing games. European Journal of Operational Research, Vol. 40, pp. 344-351.

Deng J., (1982). Control problems of Grey Systems, Systems and Control Letters, Vol. 5, pp. 288-294.

Derks J. and Peters H., (1993). A Shapley Value for Games with Restricted Coalitions, International Journal of Game Theory, Vol. 21, pp. 351-360.

Dror M. and Hartman B.C., (2011). Survey of cooperative inventory games and extensions, Journal of the Operational Research Society, Vol. 62, pp. 565-580.

Fang Z., Sifeng L., Chuanmin M., (2006), Study on optimum value of Grey Matrix Game Based on Token of Grey Interval Number, Proceedings of the 6th world congress on Intelligent Control and Automation, June 21-23,

DLLn, China. Jahanshahloo G.R, Hosseinzadeh Lotfi F., Sohraiee S., (2006), Egoist’s Dilemma with interval data, Applied Mathematics and Computation, Vol. 183, pp. 94-105.

Liao Y.H., (2012). Alternative axiomatizations of the Shapley value under interval uncertainty, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 20, pp. 619-628.

Littlechild S.C. and Thompson G.F., (1977). Aircraft landing fees: A game theory approach, The Bell Journal of Economics, Vol. 8, No.1 , 186-204.

Liu S. and Lin Y., (2006). Grey Information: Theory and Practical Applications, Springer, Germany. Liu S.T., Kao C., (2009), Matrix games with interval data, Computers&Industrial Engineering, Vol. 56, pp. 1697-1700.

Meca A., (2007). A core-allocation family for generalized holding cost games, Mathematical Methods of Operations Research, Vol. 65, pp. 499-517.

Meca A., Timmer J., Garcia-Jurado I. and Borm P., (2004). Inventory games, European Journal of Operational Research, Vol. 156, pp. 127-139.

Moore R., (1979). Methods and applications of interval analysis, SIAM, Philadelphia.

Mosquera M.A., Garcia-Jurado I. and Borm P., (2008). A note on coalitional manipulation and centralized inventory management, Annals of Operations Research, Vol. 158, pp. 183-188.

Olgun M.O. and Alparslan Gök S.Z., (2013). Cooperative Grey Games and An Application on Economic Order Quantity Model, Institute of Applied Mathematics, number 2013-27, Preprint.

Palanci O., Alparslan Gök S.Z., Ergun S. and Weber G.-W., (2014). Cooperative grey games and the grey Shapley value, submitted 2014.

Peters H., (2008). Game Theory: A Multi-Leveled Approach , Springer.

Roth A., (1988). The Shapley value: essays in honor of Lloyd Shapley, Cambridge University Press, Cambridge.

Shapley L.S., (1953). A value for n -person games, Annals of Mathematics Studies, Vol. 28, pp. 307-317.

Shapley L.S., (1971). Cores of convex games, International Journal of Game Theory, vol. 1, no. 1, pp. 11-26.

Sifeng L., Forrest J. and Yang Y., (2011). Brief Introduction to Grey Systems Theory, Grey Systems and Intelligent Services (GSIS), 2011 IEEE International Conference, pp. 1-9.

Thompson G.F.,(1971). Airport costs and pricing, Unpublished PhD. Dis-sertation, University of Birmingham.

Timmer J., Borm P. and Tijs S., (2003). On three Shapley-like solutions for cooperative games with random payoffs, International Journal of Game Theory, Vol. 32, pp. 595-613.

Young H.P., (1985). Monotonic solutions of cooperative games, International Journal of Game Theory, Vol. 14, pp. 65-72.