IDEAS home Printed from https://ideas.repec.org/a/spr/jogath/v39y2010i4p585-602.html
   My bibliography  Save this article

A generalization of the Shapley–Ichiishi result

Author

Listed:
  • Jeroen Kuipers
  • Dries Vermeulen
  • Mark Voorneveld

Abstract

The Shapley-Ichiishi result states that a game is convex if and only if the convex hull of marginal vectors equals the core. In this paper we generalize this result by distinguishing equivalence classes of balanced games that share the same core structure. We then associate a system of linear inequalities with each equivalence class, and we show that the system defines the class. Application of this general theorem to the class of convex games yields an alternative proof of the Shapley-Ichiishi result. Other applications range from computation of stable sets in non-cooperative game theory to determination of classes of TU games on which the core correspondence is additive (even linear). For the case of convex games we prove that the theorem provides the minimal defining system of linear inequalities. An example shows that this is not necessarily true for other equivalence classes of balanced games.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Jeroen Kuipers & Dries Vermeulen & Mark Voorneveld, 2010. "A generalization of the Shapley–Ichiishi result," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 585-602, October.
    • Handle: RePEc:spr:jogath:v:39:y:2010:i:4:p:585-602
    DOI: 10.1007/s00182-010-0239-5
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00182-010-0239-5
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00182-010-0239-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Nimrod Megiddo, 1978. "Computational Complexity of the Game Theory Approach to Cost Allocation for a Tree," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 189-196, August.
    2. Vermeulen, A J & Potters, J A M & Jansen, M J M, 1996. "On Quasi-Stable Sets," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(1), pages 43-49.
    3. Hillas, John, 1990. "On the Definition of the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 58(6), pages 1365-1390, November.
    4. Curiel, I. & Pederzoli, G. & Tijs, S.H., 1989. "Sequencing games," Other publications TiSEM cd695be5-0f54-4548-a952-2, Tilburg University, School of Economics and Management.
    5. Aumann, Robert J. & Maschler, Michael, 1985. "Game theoretic analysis of a bankruptcy problem from the Talmud," Journal of Economic Theory, Elsevier, vol. 36(2), pages 195-213, August.
    6. S. C. Littlechild & G. Owen, 1973. "A Simple Expression for the Shapley Value in a Special Case," Management Science, INFORMS, vol. 20(3), pages 370-372, November.
    7. Granot, D & Maschler, M & Owen, G & Zhu, W.R., 1996. "The Kernel/Nucleolus of a Standard Tree Game," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(2), pages 219-244.
    8. Ichiishi, Tatsuro, 1981. "Super-modularity: Applications to convex games and to the greedy algorithm for LP," Journal of Economic Theory, Elsevier, vol. 25(2), pages 283-286, October.
    9. Daniel Granot & Jeroen Kuipers & Sunil Chopra, 2002. "Cost Allocation for a Tree Network with Heterogeneous Customers," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 647-661, November.
    10. Curiel, Imma & Pederzoli, Giorgio & Tijs, Stef, 1989. "Sequencing games," European Journal of Operational Research, Elsevier, vol. 40(3), pages 344-351, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dénes Pálvölgyi & Hans Peters & Dries Vermeulen, 2018. "Linearity of the core correspondence," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(4), pages 1159-1167, November.
    2. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 301-326, July.
    3. Michel Grabisch & Tomáš Kroupa, 2018. "The core of supermodular games on finite distributive lattices," Documents de travail du Centre d'Economie de la Sorbonne 18010, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Sergei Pechersky, 2015. "A note on external angles of the core of convex TU games, marginal worth vectors and the Weber set," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(2), pages 487-498, May.
    5. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games," Documents de travail du Centre d'Economie de la Sorbonne 16081, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    6. Michel Grabisch, 2016. "Rejoinder on: Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 335-337, July.
    7. repec:hal:pseose:hal-01372858 is not listed on IDEAS
    8. Michel Grabisch, 2016. "Rejoinder on: Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 335-337, July.
    9. Milan Studený & Václav Kratochvíl, 2022. "Facets of the cone of exact games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 35-80, February.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Csóka, Péter & Jean-Jacques Herings, P. & Kóczy, László Á. & Pintér, Miklós, 2011. "Convex and exact games with non-transferable utility," European Journal of Operational Research, Elsevier, vol. 209(1), pages 57-62, February.
    2. Kuipers, Jeroen & Mosquera, Manuel A. & Zarzuelo, José M., 2013. "Sharing costs in highways: A game theoretic approach," European Journal of Operational Research, Elsevier, vol. 228(1), pages 158-168.
    3. Peter Borm & Herbert Hamers & Ruud Hendrickx, 2001. "Operations research games: A survey," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 139-199, December.
    4. Habis, Helga, 2012. "Sztochasztikus csődjátékok - avagy hogyan osszunk szét egy bizonytalan méretű tortát? [Stochastic bankruptcy games. How can a cake of uncertain dimensions be divided?]," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(12), pages 1299-1310.
    5. Voorneveld, Mark & Grahn, Sofia, 2001. "A Minimal Test for Convex Games and the Shapley Value," Working Paper Series 2001:2, Uppsala University, Department of Economics.
    6. Hendrickx, R.L.P. & Borm, P.E.M. & Timmer, J.B., 2000. "On Convexity for NTU-Games," Discussion Paper 2000-108, Tilburg University, Center for Economic Research.
    7. Hamers, Herbert, 1997. "On the concavity of delivery games," European Journal of Operational Research, Elsevier, vol. 99(2), pages 445-458, June.
    8. Ruud Hendrickx & Judith Timmer & Peter Borm, 2002. "A note on NTU convexity," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 29-37.
    9. van Riel, A.C.R. & Semeijn, J. & Pauwels, P.F.J., 2003. "Online travel service quality: the importance of pre-transaction services," Research Memorandum 033, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    10. Brânzei, R. & Dimitrov, D.A. & Tijs, S.H., 2002. "Convex Fuzzy Games and Participation Monotonic Allocation Schemes," Discussion Paper 2002-13, Tilburg University, Center for Economic Research.
    11. Stef Tijs & Gert-Jan Otten, 1993. "Compromise values in cooperative game theory," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 1(1), pages 1-36, December.
    12. Flip Klijn & Stef Tijs & Marco Slikker, 2001. "A Dual Egalitarian Solution," Economics Bulletin, AccessEcon, vol. 3(10), pages 1-8.
    13. René van den Brink & Gerard van der Laan & Valeri Vasil'ev, 2007. "Distributing Dividends in Games with Ordered Players," Tinbergen Institute Discussion Papers 06-114/1, Tinbergen Institute.
    14. René van den Brink & Gerard van der Laan & Valeri Vasil'ev, 0000. "The Restricted Core for Totally Positive Games with Ordered Players," Tinbergen Institute Discussion Papers 09-038/1, Tinbergen Institute.
    15. Márkus, Judit & Pintér, Miklós & Radványi, Anna, 2011. "The Shapley value for airport and irrigation games," MPRA Paper 30031, University Library of Munich, Germany.
    16. Cristina Fernández & Peter Borm & Ruud Hendrickx & Stef Tijs, 2005. "Drop out monotonic rules for sequencing situations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 61(3), pages 501-504, July.
    17. Csóka, Péter & Illés, Ferenc & Solymosi, Tamás, 2022. "On the Shapley value of liability games," European Journal of Operational Research, Elsevier, vol. 300(1), pages 378-386.
    18. René Brink & Yukihiko Funaki & Yuan Ju, 2013. "Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(3), pages 693-714, March.
    19. Grahn, S. & Kuipers, J. & Vermeulen, A.J. & Voorneveld, M., 2003. "A generalization of the Shapley-Ichiishi result and its application to convex games," Research Memorandum 022, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    20. Daniel Granot & Jeroen Kuipers & Sunil Chopra, 2002. "Cost Allocation for a Tree Network with Heterogeneous Customers," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 647-661, November.

    More about this item

    Keywords

    TU games; Core; Linearity regions; Computation of Q-sets;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jogath:v:39:y:2010:i:4:p:585-602. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.