Shapley’s Axiomatics for Lexicographic Cooperative Games

Full Text (PDF, 526KB), PP.1-8

Views: 0 Downloads: 0


Guram N. Beltadze 1,* Jimsher A. Giorgobiani 2

1. Departments Control Systems and Mathematics, Georgian Technical University, Georgia, Tbilisi, 0175, str. Kostava 77

2. N. Muskhelishvili Institute of Computational Mathematics, Georgian Technical University, Tbilisi, 0175, str. Kostava 77

* Corresponding author.


Received: 23 Apr. 2015 / Revised: 5 Jun. 2015 / Accepted: 1 Jul. 2015 / Published: 8 Aug. 2015

Index Terms

Game, Lexicography, Cooperative Game, Shapley’s axiomatics, Matrixs of distribution


In classical cooperative game theory one of the most important principle is defined by Shapley with three axioms common payoff fair distribution’s Shapley value (or Shapley vector). In the last decade the field of its usage has been spread widely. At this period of time Shapley value is used in network and social systems. Naturally, the question is if it is possible to use Shapley’s classical axiomatics for lexicographic cooperative games. Because of this in the article for dimensional lexicographic cooperative game is introduced Shapley’s axiomatics, as the principle of a fair distribution in the case of dimensional payoff functions, when the criteria are strictly ranking. It has been revealed that axioms discussed by Shapley for classical games are sufficient in lexicographic cooperative games corresponding with the payoffs of distribution. Besides we are having a very interesting case: according to the proved theorem, Shapley’s classical principle simultaneously transforms on the composed scalar games of a lexicographic cooperative game, nevertheless, games could not be superadditive.

Cite This Paper

Guram N. Beltadze, Jimsher A. Giorgobiani, "Shapley's Axiomatics for Lexicographic Cooperative Games", International Journal of Modern Education and Computer Science (IJMECS), vol.7, no.8, pp.1-8, 2015. DOI:10.5815/ijmecs.2015.08.01


[1]G.N. Beltadze. “A mixed extension of finite noncoalition lexicographic games”. Bulletin of the Academy of sciences of the Georgian SSR, 98, № 2 (1980), pp. 273-276 (in Russian).
[2]G.N.Beltadze, J.A.Giorgobiani. “Metastrategic extensions of Lexicographic Noncooperative Game in case of two players”. Bulletin of the Georgian National Academy of Sciences, vol 2, no 2, 2008, pp. 9-13 (in Russian).
[3]M.Salukvadze, G. Beltadze, and F. Criado. “Dyadic theoretical games models of decision - making for the lexicographic vector payoffs”. International Journal of Information Technology and Decision Making, Vol. 8, Issue 2, 2009, pp. 193-216.
[4]G.N.Beltadze. “Lexicographic noncooperative game’s mixed extension with criteria”. International Journal of Systems and Software. Asian Research Publishing Network (ARPN) Publishers. Vol 1, № 8, November 2011, pp. 247-250.
[5]G.N.Beltadze. “Lexicographic Multistage Games with Perfect Information”. Informational and Communication Technologies – Theory and Practice: Proceedings of the International Scientific Conference ICTMC-2010. Devoted to the 80th Anniversary of I.V. Prangishvili. Nova Publishers, 664 pp. USA, 2012. pp. 275-281.
[6]G.N.Beltadze. “Lexicographic Strategic Games Nonstandard Analisis”. International Journal of Intelligent Systems and Applications. Hong Kong, Volume 5, Number 7, June 2013, pp. 1-8.
[7]G.Beltadze. “Foundations of Lexicographic Cooperative Game Theory”. International Journal of Modern Education and Computer Science. Hong Kong, Volume 5, Number 3, 2013, pp. 18-25.
[8]M.Salukvadze, G. Beltadze. “The Optimal Principle of Stable Solutions in Lexicographic Cooperative Games”. International Journal of Modern Education and Computer Science. Hong Kong, Volume 6, Number 3, 2014, pp. 11-18.
[9]L. S. Shapley. “A value for n-person games”. Annals of Mathematical Studies, vol. 28, 1953, pp. 307-317.
[10]Na Xu. “Improved Weighted Shapley Value Model for the Fourth Party Logistics Supply Chain Coalition”. Hindawi Publishing Corporation, 10/2013, 5 pages. 269398
[11]O. Massol. S. Tchung-Mine. “Cooperation among liquefied natural gas suppliers: Is rationalization the sole objective?”. Energy Economics, Volume 32, Issue 4, July 2010, pp. 933-947.
[12]Ishai Menache, Asuman Ozdaglar. “Network Games Theory. Models, and Dynamics”. Synthesis Lectures on Communication Networks. Morgan and Claypool Publishers, 2011, 160 p.
[13]M. Brindha, J. K. Mendiratta. “Networked Control System – A Survey”. International Journal of .Modern Education and Computer Science, Hong Kong, Volume 5, Number 6, 2013, pp.42-48.
[14]T. P. Michalak, K.V. Aadithya, P. L. Szczepanski, B. Ravindran, R. Jennings. “Efficient Computation of the Shapley Value for Game- Theoretic Network Centrality”. Journal of Artificial Intelligence Research 46 (2013) 607-650 Submitted 9/12; published 4/13. pp. 607-650.
[15]Richard T. B. Ma, Dah Ming Chiu, John C.S. Lui, Vishal Misra, Dan Rubenstein. “Internet Economics: The Use of Shapley Value for ISP Settlement”. IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 3, June 2010, pp.775-787.
[16]P. Rathore1, G. Agnihotri, B. Khan and G. Naidu. “Transmission usage and cost allocation using Shapley value and tracing method: A comparison”. Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 3, August 2014, pp. 11-29.
[17]R. Narayanam, O. Skibski, H. Lamba and T. Michalak. “A Shapley Value-based Approach to Determine Gatekeepers in Social Networks with Applications”. ECAI, The Authors and IOS Press, T. Schaub et al. (Eds.), 2014, pp. 651-656.
[18]S. J. Brams. “Game Theory and Politics”. Dover Publications, INC, New York University, 2004, 312 p.
[19]Vinay Kumar. “Mathematics Is Science: A Topic Revisited in Context of FCS of India”. International Journal of Modern Education and Computer Science. Hong Kong, Volume 4, Number 6, 2012, pp. 17-26.
[20]R.M. Alguliyev, R.Sh. Malmudova. “Information Culture Formation as The Most Promising Direction of Individual’s General Culture”. International Journal of Modern Education and Computer Science. Hong Kong, Volume 7, Number 3, 2015, pp. 54-61.
[21]Sara Fatima, Salha Abdullah. “Improving Teaching Methodology in System Analysis and Design using Problem Based Learning for ABET”. International Journal of Modern Education and Computer Science, Hong Kong, Volume 5, Number 7, 2013, pp. 60-68.
[22]G.Gwen. “Game Theory”. Academic Press, Third Edition, 1995, 459 p.