Skip to main content
Log in

On the Replication of the Pre-kernel and Related Solutions

  • Published:
Computational Economics Aims and scope Submit manuscript

Abstract

Based on the results discussed by 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, Heidelberg, 2013). which presents a dual characterization of the pre-kernel 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 consequently continuous. In addition, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game possesses 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

Notes

  1. For an overview of the most recent developments in this highly dynamic research field we refer the reader to Algaba et al. (2020, Chap. 6.  and 7.). Even there, the application of the theory of linear algebraic groups reveals to us that the Borel-groups (minimal parabolic groups) are acting on the bases of TU games, and the Shapley value remains stable whenever the change of basis is located in the same orbit. In the same vein Hernández-Lamoneda et al. (2007) were able to compute a decomposition for the space of cooperative games under the action of the symmetric group \(\text {Sym}(N)\) to identify all irreducible subspaces that are relevant to study symmetric linear solutions, this result was extended by Hernández-Lamoneda et al. (2009) for games in partition function form.

  2. In fact, they computed with their method the kernel of all weighted majority games with 5 and fewer players as well as all extreme zero-sum games with 5 players.

  3. In the sense of Aumann (1961).

  4. In the sense of Hart and Kurz (1983).

  5. The example can be reproduced while using our MATLAB toolbox MatTuGames 2022. The results can also be verified with our Mathematica package TuGames 2023a.

  6. Again, the figure has been generated with out Mathematica Package TuGames implemented within Meinhardt (2023a).

References

  • Algaba, E., Fragnelli, V., & Sánchez-Soriano, J. (Eds.). (2020). Series in Operations Research (1st ed.). New York: Chapman and Hall/CRC. https://doi.org/10.1201/9781351241410

    Book  Google Scholar 

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

    Article  Google Scholar 

  • Aumann, R. J. (1961). A survey on cooperative games without side payments. In M. Shubik (Ed.), Essays in Mathematical Economics in Honor of Oskar Morgenstern (pp. 3–27). Princeton: Princeton University Press.

    Google Scholar 

  • Aumann, R. J., Peleg, B., & Rabinowitz, P. (1965). A method for computing the kernel of \(n\)-person games. Mathematical Computation, 19, 531–551.

    Google Scholar 

  • Chang, Chih, & Hu, Cheng-Cheng. (2016). A non-cooperative interpretation of the kernel. International Journal of Game Theory. https://doi.org/10.1007/s00182-016-0529-7

    Article  Google Scholar 

  • Curiel, I. (1997). Cooperative Game Theory and Applications, Theory and Decision Library: Series C (Vol. 16). Boston: Kluwer Acad. Publ.

    Book  Google Scholar 

  • Davis, M., & Maschler, M. (1965). The kernel of a cooperative game. Naval Research Logistic Quarterly, 12, 223–259.

    Article  Google Scholar 

  • Driessen, T., & Meinhardt, H. (2005). Convexity of oligopoly games without transferable technologies. Mathematical Social Sciences, 50(1), 102–126.

    Article  Google Scholar 

  • Driessen, T. S. H., & Meinhardt, H. I. (2010). On the supermodularity of homogeneous oligopoly games. International Game Theory Review (IGTR), 12(04), 309–337. https://doi.org/10.1142/S0219198910002702

    Article  Google Scholar 

  • Funaki, Y., & Meinhardt, H. I. (2006). A note on the pre-kernel and pre-nucleolus for bankruptcy games. The Waseda Journal of Political Science and Economics, 363, 126–136.

    Google Scholar 

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

    Google Scholar 

  • Hart, S., & Kurz, M. (1983). Endogenous formation of coalitions. Econometrica, 51(4), 1047–1064.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Hernández-Lamoneda, L., Sánchez-Pérez, J., & Sánchez-Sánchez, F. (2009). The class of efficient linear symmetric values for games in partition function form. International Game Theory Review, 11(03), 369–382. https://doi.org/10.1142/S0219198909002364

    Article  Google Scholar 

  • Iñarra, E., Serrano, R., & Shimomura, K.-I. (2020). The nucleolus, the kernel, and the bargaining set: An update. Revue économique, 71, 225–266. https://doi.org/10.3917/reco.712.0225

    Article  Google Scholar 

  • Kohlberg, E. (1971). On the nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics, 20, 62–66.

    Article  Google Scholar 

  • Kohlberg, E. (1972). The nucleolus as a solution of a minimization problem. SIAM, Journal of Applied Mathematics, 23, 34–39.

    Article  Google Scholar 

  • Kopelowitz, A. (1967). Computation of the Kernels of Simple Games and the Nucleolus of \(N\)-Person Games. Technical report, RM 31, research program in game theory and mathematical economics, The Hebrew University of Jerusalem, mimeo.

  • Littlechild, S. C. (1974). A simple expression for the nucleolus in a special case. International Journal of Game Theory, 3, 21–29. https://doi.org/10.1007/BF01766216

    Article  Google Scholar 

  • Martinez-Legaz, J.-E. (1996). Dual representation of cooperative games based on Fenchel-Moreau conjugation. Optimization, 36, 291–319.

    Article  Google Scholar 

  • Maschler, M., Peleg, B., & Shapley, L. S. (1972). The kernel and bargaining set for convex games. International Journal of Game Theory, 1, 73–93.

    Article  Google Scholar 

  • Maschler, M., Peleg, B., & Shapley, L. S. (1979). Geometric properties of the kernel, nucleolus, and related solution concepts. Mathematics of Operations Research, 4, 303–338.

    Article  Google Scholar 

  • Meinhardt, H. I. (2002). Cooperative Decision Making in Common Pool Situations, Lecture Notes in Economics and Mathematical Systems (Vol. 517). Heidelberg: Springer.

    Book  Google Scholar 

  • Meinhardt, H. I. (2013). 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. Heidelberg/Berlin: Springer Publisher. https://doi.org/10.1007/978-3-642-39549-9 ISBN 978-3-642-39548-2.

    Book  Google Scholar 

  • Meinhardt, H. I. (May 2014). A note on the computation of the pre-kernel for permutation games. Technical Report MPRA-59365, Karlsruhe Institute of Technology (KIT), http://mpra.ub.uni-muenchen.de/59365/.

  • Meinhardt, H.I. (2017). Applying Lie group analysis to a dynamic resource management problem. Technical report, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany. https://doi.org/10.13140/RG.2.2.20065.15205.

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

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

    Book  Google Scholar 

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

    Book  Google Scholar 

  • Meinhardt, H.I. (Aug 2021). Deduction Theorem: The Problematic Nature of Common Practice in Game Theory. arXiv e-prints, art. arXiv:1908.00409.

  • Meinhardt, H.I. (2022). MatTuGames: A Matlab Toolbox for Cooperative Game Theory. http://www.mathworks.com/matlabcentral/fileexchange/74092-mattugames.

  • Meinhardt, H.I. (2023a). TuGames: A Mathematica Package for Cooperative Game Theory. https://github.com/himeinhardt/TuGames.

  • Meinhardt, H.I. (2023b). Analysis of cooperative games with matlab and mathematica. Technical report, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany, mimeo.

  • Meseguer-Artola, A. (1997). 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, mimeo.

  • Munkres, J. R. (1975). Topology: A First Course. Englewood Cliffs, New Jersey: Prentice-Hall.

    Google Scholar 

  • Norde, H., Pham Do, K. H., & Tijs, S. (2002). Oligopoly games with and without transferable technologies. Mathematical Social Sciences, 43, 187–207.

    Article  Google Scholar 

  • Owen, G. (1974). A note on the nucleolus. International Journal of Game Theory, 3(2), 101–103. https://doi.org/10.1007/BF01766395

    Article  Google Scholar 

  • Peleg, B., & Sudhölter, P. (2007). Introduction to the Theory of Cooperative Games, Theory and Decision Library: Series C (2nd ed., Vol. 34). Heidelberg: Springer-Verlag.

    Google Scholar 

  • Raghavan, T. E. S., & Sudhölter. (2005). The modiclus and core stability. International Journal of Game Theory, 33, 467–478. https://doi.org/10.1007/s00182-005-0207-7

    Article  Google Scholar 

  • Rockafellar, R. (1970). Convex Analysis. Princeton: Princeton University Press.

    Book  Google Scholar 

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

  • Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM, Journal of Applied Mathematics, 17, 1163–1170.

    Article  Google Scholar 

  • Serrano, R. (1997). Reinterpretation the kernel. Journal of Economic Theory, 77, 58–80.

    Article  Google Scholar 

  • Shapley, L. S. (1971). Cores of convex games. International Journal of Game Theory, 1, 11–26.

    Article  Google Scholar 

  • Sobolev, A.J. (1975). 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. Institute of Physics and Mathematics, Academy of Sciences of the Lithuanian SSR. (in Russian, English summary).

  • Stearns, R. E. (1968). Convergent transfer schemes for \(N\)-person games. Transaction of the American Mathematical Society, 134, 449–459.

    Google Scholar 

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

    Article  Google Scholar 

  • Sudhölter, P. (1997). Nonlinear self dual solutions for TU-games. In T. Parthasarathy, B. Dutta, J. A. M. Potters, T. E. S. Raghavan, D. Ray, & A. Sen (Eds.), Game Theoretical Applications to Economics and Operations Research, Theory and Decision Library: Series C (Vol. 18, pp. 33–50). Boston, MA: Springer.

    Chapter  Google Scholar 

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

    Google Scholar 

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

  • Zhao, J. (1999). A necessary and sufficient condition for the convexity in oligopoly games. Mathematical Social Sciences, 37, 189–204.

    Article  Google Scholar 

  • Zhao, J. (2018). 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.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Holger I. Meinhardt.

Ethics declarations

Conflict of interest

The author has no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author is grateful to three referees for useful comments and suggestions on an earlier draft. Moreover, the author acknowledges support by the state of Baden-Württemberg through bwHPC. In particular, the kind and excellent technical support supplied by Holger Obermaier and Peter Weisbrod is acknowledged. Of course, the usual disclaimer applies.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meinhardt, H.I. On the Replication of the Pre-kernel and Related Solutions. Comput Econ (2023). https://doi.org/10.1007/s10614-023-10428-w

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10614-023-10428-w

Keywords

Mathematics Subject Classification

Navigation