On the Replication of the Pre-Kernel and Related Solutions

Meinhardt, Holger Ingmar (2020): On the Replication of the Pre-Kernel and Related Solutions.

Abstract

Based on results discussed by Meinhardt (2013), which presents a dual characterization of the prekernel by a finite union of solution sets of a family of quadratic and convex objective functions, we could derive some results related to the single-valuedness of the pre-kernel. Rather than extending the knowledge of game classes for which the pre-kernel consists of a single point, we apply a different approach. We select a game from an arbitrary game class with a single pre-kernel element satisfying the non-empty interior condition of a payoff equivalence class, and then establish that the set of related and linear independent games which are derived from this pre-kernel point of the default game replicates this point also as its sole pre-kernel element. Hence, a bargaining outcome related to this pre-kernel element is stable. Furthermore, we establish that on the restricted subset on the game space that is constituted by the convex hull of the default and the set of related games, the pre-kernel correspondence is single-valued, and therefore continuous. In addition, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game has not a single pre-kernel point. Finally, we apply the same techniques to related solutions of the pre-kernel, namely the modiclus and anti-pre-kernel, to work out replication results for them.

• Item Type: MPRA Paper
• Language: English
• Keywords: Transferable Utility Game, Pre-Kernel, Pre-Nucleolus, Anti-Pre-Nucleolus, Modiclus, Uniqueness of the Pre-Kernel, Convex Analysis, Fenchel-Moreau Conjugation, Indirect Function, Stability Analysis.
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games

References

E. Algaba, V. Fragnelli, and J. S´anchez-Soriano, editors. Series in Operations Research. Chapman and Hall/CRC, New York, 1 edition, 2020. doi: 10.1201/9781351241410. URL https://doi.org/10.1201/9781351241410.

J. Arin and V. Feltkamp. The nucleolus and kernel of veto-rich transferable utility games. International Journal of Game Theory, 26:61–73, 1997. URL https://doi.org/10.1007/BF01262513.

R. J. Aumann. A Survey on Cooperative Games without Side Payments. In M. Shubik, editor, Essays in Mathematical Economics in Honor of Oskar Morgenstern, pages 3–27, Princeton, 1961. Princeton University Press.

Chih Chang and Cheng-Cheng Hu. A non-cooperative interpretation of the kernel. International Journal of Game Theory, pages 1–20, 2016. ISSN 1432-1270. doi: 10.1007/s00182-016-0529-7. URL http://dx.doi.org/10.1007/ s00182-016-0529-7.

I. Curiel. Cooperative game theory and applications, volume 16 of Theory and Decision Library: Series C. Kluwer Acad. Publ., Boston, 1997.

M. Davis and M. Maschler. The Kernel of a Cooperative Game. Naval Research Logistic Quarterly, 12:223–259, 1965.

T. Driessen and H. Meinhardt. Convexity of Oligopoly Games without Transferable Technologies. Mathematical Social Sciences,50(1):102–126, 2005.

T. S. H. Driessen and H. I. Meinhardt. On the Supermodularity of Homogeneous Oligopoly Games. International Game Theory Review (IGTR), 12(04):309–337, 2010. doi: 10.1142/S0219198910002702. URL https://doi.org/10.1142/ S0219198910002702.

Y. Funaki and H. I. Meinhardt. A Note on the Pre-Kernel and Pre-Nucleolus for Bankruptcy Games. The Waseda Journal of Political Science and Economics, 363:126–136, 2006.

J. Getán, J. Izquierdo, J. Montes, and C. Rafels. The Bargaining Set and the Kernel of Almost-Convex Games. Technical report, University of Barcelona, Spain, 2012.

S. Hart and M. Kurz. Endogenous Formation of Coalitions. Econometrica, 51(4):1047–1064, 1983.

L. Hernández-Lamoneda, R. Júarez, and F. Sánchez-Sánchez. Dissection of solutions in cooperative game theory using representation techniques. International Journal of Game Theory, 35(3):395–426, Feb 2007. ISSN 1432-1270. doi: 10.1007/s00182-006-0036-3. URL https://doi.org/10.1007/s00182-006-0036-3.

L. Hernández-Lamoneda, J. Sánchez-Pérez, and F. Sánchez-Sánchez. The class of efficient linear symmetric values for games in partition function form. International Game Theory Review, 11(03):369–382, 2009. doi:10.1142/S0219198909002364. URL http://www.worldscientific.com/doi/abs/10.1142/S0219198909002364.

E. Inarra, R. Serrano, and K.-I. Shimomura. The Nucleolus, the Kernel, and the Bargaining Set: An Update. Revue économique, 71:225–266, 2020. doi: 10.3917/reco.712.0225. URL https://www.cairn.info/ revue-economique-2020-2-page-225.htm.

E. Kohlberg. On the Nucleolus of a Characteristic Function Game. SIAM Journal of Applied Mathematics, 20:62–66, 1971.

J-E. Martinez-Legaz. Dual Representation of Cooperative Games based on Fenchel-Moreau Conjugation. Optimization, 36: 291–319, 1996.

M. Maschler, B. Peleg, and L.S. Shapley. The Kernel and Bargaining Set for Convex Games. International Journal of Game Theory, 1:73–93, 1972.

M. Maschler, B. Peleg, and L. S. Shapley. Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts. Mathematics of Operations Research, 4:303–338, 1979.

H. I. Meinhardt. Cooperative Decision Making in Common Pool Situations, volume 517 of Lecture Notes in Economics and Mathematical Systems. Springer, Heidelberg, 2002.

H. I. Meinhardt. The Pre-Kernel as a Tractable Solution for Cooperative Games: An Exercise in Algorithmic Game Theory, volume 45 of Theory and Decision Library: Series C. Springer Publisher, Heidelberg/Berlin, 2013. ISBN 978-3-642-39548- 2. doi: 10.1007/978-3-642-39549-9. URL https://doi.org/10.1007/978-3-642-39549-9.

H. I. Meinhardt. A Note on the Computation of the Pre-Kernel for Permutation Games. Technical ReportMPRA-59365, Karlsruhe Institute of Technology (KIT), May 2014. URL http://mpra.ub.uni-muenchen.de/59365/.

H. I. Meinhardt. Applying Lie Group Analysis to a Dynamic Resource Management Problem. Technical report, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, 2017. URL https://doi.org/10.13140/RG.2.2.20065.15205.

H. I. Meinhardt. The Pre-Kernel as a Fair Division Rule for Some Cooperative Game Models, pages 235–266. Springer International Publishing, Cham, 2018a. ISBN 978-3-319-61603-2. doi: 10.1007/978-3-319-61603-2 11. URL https://doi.org/10.1007/978-3-319-61603-2_11.

H. I. Meinhardt. The Modiclus Reconsidered. Technical report, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, 2018b. URL http://dx.doi.org/10.13140/RG.2.2.32651.75043.

H. I. Meinhardt. Reconsidering Related Solutions of the Modiclus. Technical report, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, 2018c. URL http://dx.doi.org/10.13140/RG.2.2.27739.82729.

H. I. Meinhardt. Analysis of Cooperative Games with Matlab and Mathematica. Technical report, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, 2018d. mimeo.

H. I. Meinhardt. Deduction Theorem: The Problematic Nature of Common Practice in Game Theory. arXiv e-prints, art. arXiv:1908.00409, Jul 2019. URL https://arxiv.org/abs/1908.00409.

H. I. Meinhardt. TuGames: A Mathematica Package for Cooperative Game Theory, 2020a. URL https://github.com/ himeinhardt/TuGames.

H. I. Meinhardt. MatTuGames: A Matlab Toolbox for Cooperative Game Theory, 2020b. URL http://www.mathworks. com/matlabcentral/fileexchange/35933-mattugames.

A. Meseguer-Artola. Using the Indirect Function to characterize the Kernel of a TU-Game. Technical report, Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona, Nov. 1997. mimeo.

H. Norde, K. H. Pham Do, and S. Tijs. Oligopoly Games with and without Transferable Technologies. Mathematical Social Sciences, 43:187–207, 2002.

B. Peleg and P. Sudhölter. Introduction to the Theory of Cooperative Games, volume 34 of Theory and Decision Library: Series C. Springer-Verlag, Heidelberg, 2 edition, 2007.

T.E.S. Raghavan and Sudhölter. The Modiclus and Core Stability. International Journal of Game Theory, 33:467–478, 2005. URL https://doi.org/10.1007/s00182-005-0207-7.

R. Rockafellar. Convex Analysis. Princeton University Press, Princeton, 1970.

J. Rosenmüller and P. Sudhölter. Cartels via the modiclus. Discrete Applied Mathematics, 134(1):263 – 302, 2004. ISSN 0166-218X. doi: https://doi.org/10.1016/S0166-218X(03)00227-0. URL http://www.sciencedirect.com/science/article/pii/S0166218X03002270.

D. Schmeidler. The Nucleolus of a Characteristic Function Game. SIAM, Journal of Applied Mathematics, 17:1163–1170, 1969.

R. Serrano. Reinterpretation the Kernel. Journal of Economic Theory, 77:58–80, 1997.

L. S. Shapley. Cores of Convex Games,. International Journal of Game Theory, 1:11–26, 1971.

A. J. Sobolev. The characterization of optimality principles in cooperative games by functional equations. In N. N. Vorobjev, editor, Matematicheskie Metody v Sotsial’nykh Naukakh, Proceedings of the Seminar, pages 94–151, Vilnius, 1975. Institute of Physics and Mathematics, Academy of Sciences of the Lithuanian SSR. (in Russian, English summary).

P. Sudhölter. The modified nucleolus: Properties and axiomatizations. International Journal of Game Theory, 26(2):147–182, Jun 1997a. ISSN 1432-1270. doi: 10.1007/BF01295846. URL https://doi.org/10.1007/BF01295846.

P. Sudhölter. Nonlinear Self Dual Solutions for TU-Games. In T. Parthasarathy, B. Dutta, J.A.M. Potters, T.E.S. Raghavan, D. Ray, and Sen A., editors, Game Theoretical Applications to Economics and Operations Research, volume 18 of Theory and Decision Library: Series C, pages 33–50, Boston, MA, 1997b. Springer.

P. Sudhölter. The Modified Nucleolus of a Cooperative Game. Habilitation thesis, University of Bielefeld, Bielefeld, 1993.

P. Sudhölter. The modified nucleolus as canonical representation of weighted majority games. Mathematics of Operations Research, 21(3):734–756, 1996. ISSN 0166-218X. URL http://www.jstor.org/stable/3690307.

J. Zhao. A Necessary and Sufficient Condition for the Convexity in Oligopoly Games. Mathematical Social Sciences, 37:189–204, 1999.

J. Zhao. TU Oligopoly Games and Industrial Cooperation. In Corchon and Marini, editors, Handbook of Game Theory and Industrial Organization, volume I, page 392–422. Edward Elgar Publisher, Cheltenham, 2018.