Skip to main content
Springer Nature Link
Log in

On Irreducible Min-Balanced Set Systems

  • Conference paper
  • First Online:
Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11726))

  • 565 Accesses

Abstract

Non-trivial minimal balanced systems (= collections) of sets are known to characterize through their induced linear inequalities the class of the so-called balanced (coalitional) games. In a recent paper a concept of an irreducible min-balanced (= minimal balanced) system of sets has been introduced and the irreducible systems have been shown to characterize through their induced inequalities the class of totally balanced games. In this paper we recall the relevant concepts and results, relate them to various contexts and offer a catalogue of permutational types of non-trivial min-balanced systems in which the irreducible ones are indicated. The present catalogue involves all types of such systems on sets with at most 5 elements; it has been obtained as a result of an alternative characterization of min-balanced systems.

Supported by GAČR project n. 19-04579S.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Bodjanova, S., Kalina, M.: Coarsening of fuzzy partitions. In: The 13th IEEE International Symposium on Intelligent Systems and Informatics, September 17–19, 2015, Subotica, Serbia, pp. 127–132 (2015)

    Google Scholar 

  2. Bondareva, O.: Some applications of linear programming methods to the theory of cooperative games (in Russian). Problemy Kibern. 10, 119–139 (1963)

    MATH  Google Scholar 

  3. Csóka, P., Herings, P.J.-J., Kóczy, L.Á.: Balancedness conditions for exact games. Math. Methods Oper. Res. 74(1), 41–52 (2011)

    Article  MathSciNet  Google Scholar 

  4. Ichiishi, T.: On the Knaster-Kuratowski-Mazurkiewicz-Shapley theorem. J. Math. Anal. Appl. 81, 297–299 (1981)

    Article  MathSciNet  Google Scholar 

  5. Kroupa, T., Studený, M.: Facets of the cone of totally balanced games. Math. Methods Oper. Res. (2019, to appear). https://link.springer.com/article/10.1007/s00186-019-00672-y

  6. Lohmann, E., Borm, P., Herings, P.J.-J.: Minimal exact balancedness. Math. Soc. Sci. 64, 127–135 (2012)

    Article  MathSciNet  Google Scholar 

  7. Miranda, E., Montes, I.: Games solutions, probability transformations and the core. In: JMLR Workshops and Conference Proceedings 62: ISIPTA 2017, pp. 217–228 (2017)

    Google Scholar 

  8. Peleg, B.: An inductive method for constructing minimal balanced collections of finite sets. Nav. Res. Logist. Q. 12, 155–162 (1965)

    Article  MathSciNet  Google Scholar 

  9. Peleg, B., Sudhölter, P.: Introduction to the Theory of Cooperative Games. Theory and Decision Library, series C: Game Theory, Mathematical Programming and Operations Research. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  10. Rosenmüller, J.: Game Theory: Stochastics, Information, Strategies and Cooperation. Kluwer, Boston (2000)

    Book  Google Scholar 

  11. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)

    MATH  Google Scholar 

  12. Shapley, L.S.: On balanced sets and cores. Nav. Res. Logist. Q. 14, 453–460 (1967)

    Article  Google Scholar 

  13. Shapley, L.S.: On balanced games without side payments. In: Hu, T.C., Robinson, S.M. (eds.) Mathematical Programming, pp. 261–290. Academic Press, New York (1973)

    Chapter  Google Scholar 

  14. Studený, M., Kratochvíl, V., Vomlel, J.: Catalogue of min-balanced systems, June 2019. http://gogo.utia.cas.cz/min-balanced-catalogue/

  15. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou věží 4, 18200, Prague 8, Czech Republic

    Milan Studený, Václav Kratochvíl & Jiří Vomlel

Authors
  1. Milan Studený
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Václav Kratochvíl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiří Vomlel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Milan Studený .

Editor information

Editors and Affiliations

  1. Technische Universität Dortmund, Dortmund, Germany

    Gabriele Kern-Isberner

  2. Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Zoran Ognjanović

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Studený, M., Kratochvíl, V., Vomlel, J. (2019). On Irreducible Min-Balanced Set Systems. In: Kern-Isberner, G., Ognjanović, Z. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2019. Lecture Notes in Computer Science(), vol 11726. Springer, Cham. https://doi.org/10.1007/978-3-030-29765-7_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-29765-7_37

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-29764-0

  • Online ISBN: 978-3-030-29765-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics